Robust Control of an Induction Motor Drive

Transcription

1 Robust Control of an Induction Motor Drive Henrik Mosskull TRITA EE 6:8 ISSN ISBN Automatic Control School of Electrical Engineering Royal Institute of Technology (KTH) Stockholm, Sweden, 6 Submitted to the School of Electrical Engineering, Royal Institute of Technology, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

2 Copyright 6 by Henrik Mosskull Robust Control of an Induction Motor Drive Automatic Control School of Electrical Engineering Royal Institute of Technology (KTH) SE- 44 Stockholm, Sweden

3 Abstract This thesis considers robust control of an induction motor drive, consisting of an input filter, a voltage source inverter and one or several induction motors in parallel. The motor torque is here controlled by using the method Indirect Self Control (ISC), and power oscillations between the inverter and the input filter are damped by means of a stabilization controller in an outer feedback loop. Closed-loop performance with ISC is analyzed under the assumption of a stiff inverter input voltage. It is shown how parameter errors influence the torque loop and the conclusion is that the motor leakage inductance should not be overestimated, especially not with a large desired control bandwidth. It is also shown that model errors enhance cross coupling, but that the performance is quite insensitive even to large parametric errors. Based on the closed-loop model, expressions for the controller parameters are derived to obtain required stability margins. For design of the stabilization controller to suppress oscillations between the inverter and the input filter, it is shown that the effects of time delays and limited torque (or current) control bandwidth are important and cannot be neglected. From models including these non-ideal properties of the control system, explicit expressions for stabilization controllers to use with ISC as well as field-oriented control (FOC) are derived. This is valuable as stabilization often is designed through costly and time-consuming manual tuning. In the controller design, the trade-off between tight torque control and stability of the DC-link is also explicitly considered. In this way reasonable stability margins are obtained, while minimizing the negative effects on torque control. Stability of the closed-loop drive with the proposed stabilization is validated through realistic hardware-in-the-loop simulations using real control HW and SW. Using models obtained from frequency domain system identification, stability of the non-linear closed-loop drive is verified by combining stability results for linear systems with the small gain theorem for the non-linear model errors. This thesis considers the input filter dynamics in connection with torque control of an induction motor. The key result is a model-based framework for simultaneous treatment of DC-link stability and efficient torque control.

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5 Acknowledgements This work is the result of a collaboration between Bombardier Transportation in Västerås and KTH in Stockholm. I would first of all like to express my gratitude to the people who made this work possible in the first place. These include my supervisor at KTH, Prof. Bo Wahlberg and Peter Oom and Dr. Kent Öhrn at Bombardier Transportation. My technical supervisor at Bombardier Transportation was Johann Galic. Without his experience and deep knowledge of drives, power electronics and control, this work would not have been completed. Thanks for sharing your time and your wisdom, Johann! I would also like to thank Prof. Stefan Östlund for support regarding general issues related to induction motor control and the people introducing me to the subject, Dr. Steffen Richter, Dr. Peter Krafka and Michael Rampe, whom I have had the pleasure to work with. I want to thank Prof. Anders Helmersson for valuable help with numerically solving the LMIs in Chapter 9 and Prof. Elling Jacobsen for teaching me robust multivariable feedback control. Eva Sandberg was kind to help me with the layout in Microsoft Word and Karin Karlsson-Eklund guided me through the administrative routines at KTH. This work was partly supported by The Swedish Science Foundation, which is gratefully acknowledged. Finally, I would like to thank my family for support and encouragement.

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7 Contents Introduction.... Background.... Induction Motor Drive Torque Control of an Induction Motor Field-Oriented Control (FOC) Direct Torque Control (DTC) DC-Link Stability Constant Power Operation Stabilization with Perfect Torque Control Improved Stabilization Structure Improved Modeling for Stabilization Synthesis Input Filter Oscillations with DC-DC Converters Notation and Nomenclature Outline and Contributions... Voltage Source Inverter...5. Three-Phase Voltage Source Inverter Modulation Pulse Width Modulation Six-Step Operation Induction Motor Physical Description Equivalent Circuit Diagrams Inverse Gamma Model Stator Coordinates Synchronous and Rotor Flux Coordinates Current Control Gamma Model Stator Coordinates... 43

8 viii Contents 3.4. Slip Frequency Control Controllability Analysis Input Constraints Polar Representation Poles and Zeros Relative Gain Array (RGA) Scaling Singular Values Flux Control Robustness Time Delays Limitations Imposed by Input Constraints Disturbance Rejection Reference Tracking Singular Values Revisited Summary Field-Oriented Control Current Control with Active Damping Rotor Flux Estimation Closed-Loop Torque Dynamics Indirect Self Control Discrete-Time ISC Control Law Prediction Average Torque Continuous-Time ISC Control Law Prediction Average Torque Closed-Loop System Model without Time Delays... 86

9 Contents ix 6.3. Model with Time Delays and Disturbance Rejection Controller Tuning Zero - Pole Cancellation Loop Shaping Gain Scheduling Simulations Summary ISC with Observer Torque and Stator Flux Estimation Full-Order Observer Observer Equations Parameter Variations Steady State Stator Flux Estimation Steady-State Torque Estimation Model of Closed-Loop System with Observer Model with Zero Observer Gain Model with Non-Zero Observer Gain Summary Sensitivity and Robustness Analysis Performance Requirements Sensitivity Requirements Input Requirements Perfect Field Orientation Flux Estimation with Open-Loop Observer Rotor Parameter Errors Additional Parameter Variations Estimated Currents for Torque Estimation Flux Estimation with Closed-Loop Observer Non-Parametric Uncertainty Structured Singular Value Uncertainty Description Model for Controller Evaluation Controller Evaluation... 4

10 x Contents Simulations Summary Linear Controller Design µ-synthesis LFT Controller Linear Process Model Extracting the Stator Frequency Model for Controller Synthesis Results Summary DC-Link Stability Linear Model of the Drive Indirect Self Control (ISC) Field-Oriented Control (FOC) Feedback Representation of the Drive Stability Analysis Unstable Examples Revisited Normalization of Coupling Vector Stability and Performance Requirements Alternative Stabilization Methods Current Reference Modification Stator Voltage Modification Stator Frequency Modification Modification of Average Filter Stabilization Controller Design Design for Maximum Stability Margin Improved Disturbance Rejection Approximation of G c Practical Stabilization Controllers Summary Verification of Stability Margins from Measurements.85. Stability and Robustness Analysis Input-Output Stability Stability Analysis... 87

11 Contents xi..3 Robustness Implementation Pulse Patterns Stabilization and Torque Controllers Linear Stability Analysis Identification of Inverter Input Admittance Coasting Driving Braking....4 Linearization Errors Linearization Errors Experimental Result Power Iterations Summary... 8 Summary and Suggestions for Future Work.... Summary.... Future Work... 4 A Drive Data...7 B Drive Data Set... 7 Drive Data Set... 7 Motor Parameters... 8 Input Filter Data... 8 Tractive Effort... 8 Closed-Loop Equations... Real-Valued Space Vector Representation... Closed-Loop System without Time Delays... Non-Linear Equations of Decoupling... Steady State Relations... 3 Linearization of Decoupling... 4 Transfer Function Representation... 6 Linearization of Feedback Controllers... 9 Closed-Loop Equations... 9 Closed-Loop System with Time Delays... 33

12 xii Contents Linear Model of Prediction Closed-Loop Equations C Steady State Analysis...37 Stator Flux Estimation Error Torque Estimation Error D Non-Zero Observer Gain...45 E Space Vectors...49 F Singular Values with Space Vector Representation...53 Bibliography...57

13 Chapter Introduction This thesis treats robust control of an induction motor drive used for DC traction applications. An induction motor drive consists of an input filter, a voltage source inverter and one or several induction motors in parallel. The input filter, which is directly connected to the line, is needed as an energy reservoir and to suppress switching harmonics generated by the inverter. During control design, the input filter is often neglected. To compensate for a varying filter voltage in a practical application, the inverter modulation ratio is modified to ensure that the desired stator voltages always are applied to the motors. However, with a resonant input filter, this kind of suppression of voltage variations may lead to power oscillations between the inverter and the input filter. There is hence a trade-off between efficient disturbance rejection and stability of the drive. In this thesis, the problem of ensuring robust torque control of a drive including input filter is studied in detail. In a traction application, the operating conditions, such as speed, load and temperature, vary over a wide range. Furthermore, the electrical environment of the drive depends on the number of surrounding trains and feeder stations, and the distances between them. In practice all process signals needed for parameter adaptation are not feasibly measurable and all parameter variations are not practical to implement. To always guarantee satisfying performance, robust control algorithms are therefore required. Section. motivates the need of further research in the field and Section. presents induction motor drives in more detail. Commonly used torque control methods are briefly discussed in Section.3, whereas a fairly detailed introduction to the problem of power oscillations between the inverter and input filter is given in Section.4. Section.5 then introduces notation and nomenclature, while Section.6 gives the outline of the thesis and highlights the major contributions.

14 Introduction. Background The behavior of an induction motor is accurately described by well known mathematical models. Still, for practical use, the motor parameters must be estimated and correctly adapted to the operating conditions. As highperformance control methods are model based, the closed-loop performance is influenced by the accuracy of the estimated motor parameter. Here it is valuable to know which motor parameters are most critical and how model errors affect performance. These issues have been explored for the so-called classical field-oriented control or vector control. However, for the control scheme Indirect Self Control (ISC), on which this thesis focuses, few results have been published in the literature. This thesis aims at filling this gap. Traditionally, torque control of an induction motor drive is designed while neglecting the dynamics of the input filter. The effects of a varying DC-link voltage, i.e., the input voltage to the inverter, are then counteracted by adapting the modulation ratio of the inverter. This suppression of DC-link voltage variations gives the inverter a negative-resistance behavior as seen from the DC link. In connection with a poorly damped input filter, this property of the controlled inverter may destabilize the drive. Such stability problems are not unique for induction motor drives but have been treated also for other kinds of switching converters, e.g. DC-DC converters. For these applications, the stability problem has been solved by adding resistance to the input filter. However, for high power drives, this solution is not feasible due to the large additional power losses. Here the problem has instead been solved by adding fictitious resistance. The behavior of the inverter is then changed through control to emulate an increased resistance. In a torque controlled drive, this may be achieved by modifying the torque reference as a function of variations in the DC-link voltage. Explicit schemes to do this have been derived using simple models of the controlled inverter. However, the approximations made during design of stabilization have turned out to be too crude to always stabilize a drive in practice. To handle these cases, more general stabilization structures have been introduced. Tuning of the stabilization functionality is, however, usually left to the user. Practical experience shows that stabilization needs to be adapted to the operating point, which in practice often leads to a non-negligible amount of manual tuning. The goal of this thesis is to find explicit ways of optimizing the design of DClink stabilization. It is also the intention to examine different approaches for stabilization regarding performance and implementation issues. This area of research is relatively little explored. Much effort in recent research has been spent on speed sensorless and nonlinear control of induction motors. For the

15 Introduction 3 induction motor drives studied in this thesis, we assume that speed sensors are present, which usually is the case in a traction application.. Induction Motor Drive The three-phase induction motor was invented already in the 9 th century. Compared to DC motors, induction motors have higher power densities and are mechanically more robust, which make them the ideal motor in many applications. On the other hand, they require AC power supplies and more sophisticated control. First with the invention of field-oriented control (or vector control) in the late 96 s, [7], [8], control of an induction motor could be compared to that of a separately excited DC motor. The need of advanced power electronics further delayed the widespread use of induction motors in industrial and traction applications till the 98 s. Today, however, squirrel-cage induction motors fed voltage source inverters (VSI) is standard in traction applications [75]. A typical DC propulsion system in an electric train may therefore be depicted as in Figure -, where the induction motor drive is connected to the DC supply voltage E(t) either via an overhead line or a third rail. The torque generated at the motor shaft is denoted T(t) in Figure - and the motor speed is represented by ω m (t). i d DC + E - + U d - Voltage Source Inverter Motor ω m, T Figure -: DC propulsion system in a traction application. Each inverter feeds one or several induction motors in parallel and power can flow in both directions, i.e., either from the electric power supply to the motors or vice versa. This way energy generated by braking trains can be used to feed trains that consume power. Especially in a metro system with frequent accelerations and decelerations, the possibility of regeneration generates large power savings. If the line is not receptive, the braking energy is dissipated in a braking resistance, which not is shown in Figure -. An example of a metro with overhead DC voltage supply is shown in Figure -.

16 4 Introduction Figure -: Guangzhou metro. The three-phase AC voltage for the induction motors is generated from a DC voltage through the voltage source inverter in Figure -. The DC-link voltage U d (t) is fed by the supply voltage E(t) via an input filter, composed of a capacitor with capacitance C and an inductor with inductance L and resistance R. The capacitor is an energy reservoir for the VSI, whereas the inductor is needed to give appropriate suppression of high frequency harmonics generated by the inverter. By explicitly showing the resistance, the induction motor drive in Figure - can be represented as in Figure -3. Actually, the values of R and L in Figure -3 should also include resistance and inductance of the overhead line. This will, however, be neglected in this thesis as the inductance of the overhead line normally is small compared to the filter inductance. Figure -3: Model of an induction motor drive.

17 Introduction 5 In traction applications, space and weight constraints tend to keep the capacitance C of the input filter relatively small. Moreover, to meet regulations on harmonics suppression, or to assure a certain required input impedance of the drive, the inductance L has to be relatively large. In combination with a small resistance R to keep the power losses acceptable, the resulting input filters get poorly damped with high resonance peaks at the frequency ω = / (LC). This is illustrated in Figure -4, showing the Bode plot of the transfer function from DC-link current i d (t) to line current i(t), which is given by the following equation i () t = i () t. (.) pcl+ pcr+ d Here p is the differential operator and the filter data are taken from Appendix A (input filter with 35.6 Hz resonance frequency). The large resonance peak is hence a consequence of a desired small gain of the filter at high frequencies to suppress switching frequency harmonics. To quantify the damping properties of the input filter, we use the damping factor ζ, which is defined by C ζ = R. (.) L The damping factor for the filter used to generate Figure -4 is only.. Magnitude [abs] Phase [deg] Bode Diagram Frequency [Hz] 3 Figure -4: Bode plot of input filter transfer function.

18 6 Introduction.3 Torque Control of an Induction Motor The schematically shown controller in Figure -3 controls the operation of the inverter through the coupling vector k(t). Based on measured motor and DClink quantities, the coupling vector is determined to make the motor torque follow a reference, which is denoted by T ref in Figure -3. Given that the induction motor is a non-linear multi-input multi-output (MIMO) system with varying parameters, control of these systems is a challenging task that has received rather much attention in the literature over the last decades, see e.g. [], [68], [83]. In most of these publications, the influence of the power supply is neglected. The inverter is consequently assumed to be connected to a stiff DC-link voltage, which is assumed to be independent of the consumed power. The controlled drive may then be represented by the block diagram in Figure -5, where the torque is influenced by the reference signal via the system G c and through a disturbance in the DC-link voltage through a transfer function G d. The influence of the motor speed is not explicitly shown in the figure as we often will consider the speed to be constant in this thesis. In traction applications the speed is normally slowly varying due the large mass of a train. (Practical experience also shows no indications of increased stability problems during slippery conditions). Actually, also the flux of the motor will be regulated although this is not explicitly shown in Figure -5. Figure -5: Model of a controlled induction motor. Here the dynamics of the input filter are not considered..3. Field-Oriented Control (FOC) A major breakthrough in the area of induction motor control was taken with the invention of field-oriented control (FOC) or vector control in the late 96s. Until then, induction motors had been controlled using so called scalar control methods, like the volt-hertz control. Here the magnitude and frequency of the stator voltage are determined from steady-state properties of the motor, which leads to poor dynamic performance. FOC, on the other hand, uses a vector model of the drive that is valid also during transients, which facilitates

19 Introduction 7 faster control. The idea with field orientation is to mimic control of DC motors also for induction motors. To accomplish this, the motor equations are described in a coordinate system oriented to the (rotor) flux in the motor. The torque and flux can then be controlled through different components of the stator current. In the rotating coordinate system, all motor quantities are constant at steady state and PI controllers can hence be used to ensure zero steady-state errors. It has later been shown that the classical field-oriented controller can be interpreted as asymptotic exact linearization [5]. Fieldoriented control was developed by a research group lead by Prof. Werner Leonard in Brauschweig, Germany and the concept was initially published in [7] and [8]. See also the well-known book on control of electrical drives by Leonard [4]. Closed-loop stability and performance achived with fieldoriented control have rigorously been treated in a number of papers, see for example [5] and [64]. Selection of controller parameters has been discussed in for example [4], and design of flux estimators in [5]. Recently, much research effort has been spent on using FOC without speed sensors. Basics of field-oriented control will briefly be described in Chapter 5. As explained above, the effect of the power supply (including input filter) is often neglected in connection with induction motor control. In cases where the input filter actually is considered, such as in [3], [6] and [7], variations in the DClink voltage are assumed to be perfectly compensated for. Due to inevitable time delays, perfect cancellation of the influence of a varying DC-link voltage is never possible in practice. In Chapter 5 time delays are considered and an expression for the transfer function G d in Figure -5 is derived. The influence of disturbances in the DC-link voltage on the torque is important when studying DC-link stability, which is done in Chapter and Chapter..3. Direct Torque Control (DTC) In parallel to the development of FOC, a different class of controllers for induction motors, the so called Direct Torque Control (DTC) methods, emerged in the 98s. The two classical DTC schemes are the methods invented by Takahashi [8] and Depenbrock [8]. In principle, these methods keep the stator flux space vector on a pre-defined track and control the torque by varying the speed of the flux along the track. Hence, the behavior of the motor is controlled via the stator flux and not the stator current as with FOC. With the method by Takahashi, the stator flux track is circular, whereas a hexagonal track is used with Direct Self Control (DSC), i.e., the method suggested in [8]. The speed of the flux vector along the track is affected by using so called zero vectors (which result in zero stator voltage) and the

20 8 Introduction insertion of these vectors is controlled though a torque hysteresis controller. If the speed of the stator flux space vector along the track is larger than the speed of the rotor flux, then the load angle (the angle between the fluxes) increases, which implies that also the torque increases. When the torque reaches an upper hysteresis threshold, a zero vector is applied and the stator flux comes to a halt. The load angle then decreases and the torque eventually reaches a lower hysteresis threshold, where the stator flux is released along the track again. The DSC is illustrated in Figure -6, where the zero voltages are marked with dots. Figure -6: Illustration of the operation of the DTC control method DSC, where Ψ s represents the stator flux and Ψ r the rotor flux of the induction motor. The torque of the induction motor depends on the load angle δ. The dots on the hexagon represent the locations of so-called zero vectors. Here the stator flux space vector rests and as the rotor flux still rotates, the load angle and hence also the torque decreases. When the torque reaches a lower threshold, the zero vector is removed and the stator flux starts moving and the load angle and torque increases. As the torque reaches an upper threshold, a zero voltage in inserted and the stator flux comes to a halt. Advantages with these types of hysteresis algorithms are simple implementation and excellent dynamic performance also with low switching frequencies. The DSC therefore has successfully been used in high power traction applications, using fairly slowly switching Gate Turn-Off Thyristors (GTOs) [74], [77]. At low speeds, small stator voltages are required, which means that short voltage pulses need to be generated. With DSC, the voltage pulses between two zero vectors (along one hexagon side) are generated by switchings in one phase only. The duration of such pulses are therefore lower

21 Introduction 9 limited by the so-called minimum on time, i.e., the shortest time between two switchings of a GTO, which is in the range of 5 µs [3]. This lower stator voltage limit is a restriction of DSC, which stimulated the development of the method Indirect Self Control (ISC) as a low-speed support for DSC [3], [33], [34], [49]. With ISC, short voltage pulses are generated by switchings in different phases and the duration of the pulses are no longer lower limited by the minimum on times. Whereas the DSC directly generates the switching commands for the power semiconductors, the ISC generates the switching commands indirectly via a stator voltage reference. This reference then corresponds to an average stator voltage to be applied during a pulse period and is separately converted to switching commands by a modulator. To use a modulator separate from the controller has the benefit that requirements on for example the switching frequency and the generated harmonics can be more easily and directly addressed. In traction applications, the harmonics generated by the drive must be predictable as they may disturb the signaling system and hence in the end cause accidents. This is a further reason why ISC with a PWM modulator may be preferred over DSC. Although the term DTC often is associated with hysteresis methods, we will here classify also ISC as a DTC method. This route is followed also in the survey on DTC schemes given in [4], where control methods operating with closed torque loops without current controllers are referred to as DTC methods. ISC hence differs from the classical FOC methods due to the absence of explicit current controllers, but also in that the ISC is oriented to the stator flux, rather than to the rotor flux. As stator flux estimation is less parameter dependent compared to rotor flux estimation [9], stator flux orientation is often regarded as less sensitive to parameter errors compared to rotor flux orientation. However, relatively few results on performance of ISC with model errors have been published in the literature. This thesis aims at filling this gap by performing a detailed analysis of the closed-loop performance with ISC. In Chapter 6 the closed-loop equations are derived when applying the ISC to an induction motor. These equations are then used to derive expressions for the controller parameters to achieve certain performance requirements. In Chapter 7 and Chapter 8 stator flux estimation and the effects of model errors on steady-state as well as dynamic performance are considered. Remark: In [66] a version of DSC is presented where the switchings are determined through hybrid control techniques. The focus here is to minimize the torque ripple, while keeping the switching frequency as low as possible. No emphasis is put on generating predictable switching frequency harmonics.

22 Introduction.4 DC-Link Stability In a traction application, where a small capacitor of the input filter is preferred due to space and weight constraints, the assumption of an ideal power supply for the inverter is no longer valid. Here the DC-link voltage is not independent of the generated torque and power may start to oscillate between the inverter and the poorly damped input filter. Note that this holds even if the internal torque feedback loop, represented by the blocks G c and G d in Figure -5, is stable. Such problems have been analyzed for open-loop control of induction motors in [] and [43]. The practically more interesting case with closed-loop torque control is discussed in e.g. [3], [6], [7], [35], [8] and [86]. The analysis is however constrained by the assumption of perfect rejection of DClink voltage variations, which implies constant power operation. This is never achieved in practice, but the assumption of constant power operation makes it possible to simply understand the stability problems from an electric-circuit point of view. The simple circuit model also naturally leads to the ideas for stabilization of the drive suggested in the literature. In Subsection.4. we will review the results achieved with the constant power assumption. A similar treatment is found in [3]. We also present an example of stabilization in Subsection.4., which, under the assumption of perfect torque control (disturbance rejection and reference tracking), would stabilize the drive at all operating points. However, as shown by two examples in Subsection.4.3, the simplifying assumptions of perfect control may be too optimistic in a practice. These examples then motivate the use of a more flexible stabilization structure. For proper synthesis of this stabilization scheme, an improved model of the drive is suggested in Subsection Constant Power Operation Often the switching commands to the inverter are modified to counteract for variations in the DC-link voltage. Ideally, this compensation is perfect and the stator voltages become completely independent of the DC-link voltage. This will be referred to as perfect disturbance rejection and means that the DC-link voltage has no effect on the power, i.e., the product P(t) = U d (t)i d (t) is assumed to be constant. By differentiating the expression for the constant power we get id P id () t = Ud () t = U d () t, (.3) Ud U d Y

23 Introduction where subscripts denote steady-state operating point values. Under the assumption of constant power operation, Equation (.3) hence forms a linear model of the inverter in Figure -3 as seen from the DC link, where Y is the input admittance of the inverter. By defining the corresponding equivalent inverter resistance R inv as the inverse of Y in (.3), the drive in Figure -3 may be modeled by the simple electric circuit in Figure R L i d + E C U d R inv - - Figure -7: Linear model of the induction motor drive under the assumption of perfect disturbance rejection (constant power operation). Here the voltage source inverter is modeled by an equivalent resistance R inv. As a consequence of the constant power assumption, the equivalent resistance is negative at positive power, see (.3). This follows as for example a decrease of DC-link voltage must result in an increase of DC-link current to keep the product of DC-link voltage and DC-link current constant. An increased current means that the voltage is further decreased and we intuitively realize that a negative resistance may be bad for stability. That this is indeed the case can be established more rigorously by examining the damping factor ζ drive of the circuit in Figure -7, i.e., ( ) L L ζ drive = + YR ζ + Y ζ + Y, (.4) C C where ζ is the damping factor of the input filter, see (.). From the definition of the admittance in (.3), it follows that Y is proportional to P. The expression for ζ drive in (.4) then gives that positive operating point power leads to decreased damping of the drive, whereas negative power increases damping. For the drive to be stable, the damping factor ζ drive must be positive, which through (.3) and (.4) can be stated in terms of the operating point power as RC P Ud. (.5) L For the drive with input filter with 35.6 Hz resonance frequency defined in Appendix A, the stability condition (.5) is only satisfied for powers less than

24 Introduction % of nominal power. Hence, these kinds of stability problems need to be considered for practical operation of the drive. Remark: From the characteristic polynomial of the circuit in Figure -7, it follows that besides condition (.5), also the following relation must be satisfied for stability U d P. (.6) R The additional condition (.6) is, however, usually fulfilled in a practical application and (.5) is the critical limitation. Remark: In trains with AC voltage supply, the DC-link voltage is controlled by a line converter. DC-link instability is therefore less of a problem in AC traction applications compared to DC applications..4. Stabilization with Perfect Torque Control Under the assumption of constant power operation, we showed that the inverter in Figure -3 simply can be modeled as a resistance. At positive power, this equivalent resistance is negative and for powers larger than the limit in (.5) the drive becomes unstable. To solve these kinds of problems, the DC-link voltage could be stabilized by additional power electronics, as proposed in [7] or by adding resistance to the input filter as discussed for DC-DC converters in []. An alternative, which neither requires extra hardware nor increases power losses, is to modify the input impedance of the inverter. From the expression for the damping factor ζ drive in (.4), it seems natural to improve damping by making the inverter act like a positive resistance (positive admittance), instead of a negative, as discussed in [3], [6] and [58]. This means that e.g. a decrease in DC-link voltage should result also in a decrease of DC-link current, not an increase as with constant power operation. One way to force this behavior is to modify the torque reference of the drive as suggested in [35] and [8], i.e., T ref () t () t ρ Ud T ref () t, P Ud = ρ Ud T ref () t, P <, Ud () t (.7)

25 Introduction 3 where T ref is an external torque reference and P = T ω m. The parameter ρ in (.7) is set to either one or two, referred to as linear and quadratic stabilization, respectively. In braking ρ could also be set to zero, which turns stabilization off. Around an operating point with positive power, the DC-link current is increased by an increase in torque at positive speed and by a decrease of torque at negative speed. The opposite holds at negative operating point power. We then realize that with torque reference modification according to (.7), an increase in DC-link voltage should also result in an increase of DC-link current (and vice versa), which hence should make the inverter behave like a positive resistance. To analytically evaluate the effect of the torque reference modification (.7), we use the following power balance approximation () () () ω () i tu t = T t t, (.8) d d m which is valid if power losses of the inverter and motor are neglected. (In case of more than one motor fed in parallel by the inverter, the torque in (.8) corresponds to the total torque of the drive.) Apart from perfect disturbance rejection, we also assume perfect reference tracking, i.e., T(t) = T ref (t), and insert the stabilization expression (.7) into the power balance equation (.8). The relation between the DC-link current and DC-link voltage can then be linearized and the inverter can be modeled by (cf. Figure -7) R U d P ( ρ ) P inv = U d P < ( ρ + ) P,,. (.9) For ρ = the equivalent resistance R inv with positive power is infinite and consequently the DC-link current is unaffected by variations in the DC-link voltage. With ρ =, the inverter behaves like a positive resistance, which improves the damping of the drive (compared to the input filter). In braking the equivalent resistance is positive for both choices of ρ (actually the equivalent resistance is positive also without stabilization, i.e., with ρ = ). Note that with the torque reference modification (.7), the power consumed (or generated) by the inverter is no longer constant but is affected by variations in the DC-link voltage. This means that the stabilization scheme to some extent worsen disturbance rejection. Also the effective torque dynamics are effected (slowed down) by the stabilization as a change in the external torque reference results in a disturbance in the DC-link, which is counteracted

26 4 Introduction by modifying the internal torque reference through (.7). In general, stabilization therefore is a trade-off between damping and control performance..4.3 Improved Stabilization Structure In the previous subsection we showed that under the assumption of perfect control (reference tracking as well as disturbance rejection), the stabilization scheme (.7) would stabilize the drive at all operating points (with all input filters), see (.9). However, the assumption of perfect torque control never holds in practice, which means that the stability analysis performed above may not be valid. This is also illustrated through the two examples shown in Figure -8, where stabilization according to (.7) with ρ = is applied. In Figure -8.A the induction motor drive is simulated at zero torque with the input filter in Appendix A with resonance frequency 8.8 Hz. From the expression for the equivalent resistance (.9), (or the stability condition (.5)) this should give a stable system, which however is contradicted by the simulation. In Figure -8.B the drive is simulated close to the maximum (pullout) torque with the input filter with resonance frequency 53 Hz in Appendix A. We see that the drive is unstable, although the corresponding equivalent resistance is positive according to (.9). The simulations are performed at 6% and 5% of nominal speed, where an ISC controller is used giving a control bandwidth of around 4 Hz with a pulse period of.9 ms. Voltage [V] DC-link voltage Voltage [V] DC-link voltage Time [s] Time [s] Figure -8: (A) Simulation showing stability problems of an induction motor drive at zero torque. The simulation is performed at 6% of nominal speed. (B) Unstable example of a drive at maximum (pull-out) torque at 5% of nominal speed. Hence, we conclude that the simplified analysis failed in predicting stability and that stabilization according to (.7) does not guarantee a stable drive. The

27 Introduction 5 stabilization therefore has to be modified. A general drawback with stabilization according to (.7) is that it does not affect the drive at zero torque. This for example means that this stabilization method cannot be used to stabilize the unstable example at zero torque presented in Figure -8.A. Further, the stabilization method (.7) offers little flexibility as stabilization only can be adjusted via the exponent ρ. We therefore propose to use a more flexible stabilization structure as () () () T t = T t + KU t, (.) ref ref d where K is a stabilization controller. Note that the DC-link voltage in (.) represents deviations from a nominal value and that the stabilization scheme (.) allows for stabilization also at zero torque. However, by directly modifying the torque reference as in (.), a constant offset in the DC-link voltage would give a constant nonzero torque contribution. This is of course not acceptable and the low frequency contribution must be removed in practice. It may also be advantageous to avoid exiting the torque reference with too high frequencies due to problems with limited control bandwidth (or to avoid aliasing in a digital implementation). The DC-link voltage used in (.) is therefore first filtered by a band-pass filter before added to the torque reference. We already here point out that the band-pass filter will be neglected during synthesis of the stabilization controller K, but will be included during controller evaluation. By linearizing the expression in (.7), this stabilization method can be represented as in (.), where the stabilization controller K is given by ρ T sgn ( ωm ) K =. (.) U Note that the stabilization controller (.) is zero at zero torque and that the effect of the parameter ρ is seen to be a scaling factor for the gain K. Here we may also note that the stabilization method proposed in [86] can be rewritten as T ptc Tref () t = T ref () t + k Ud () t, (.) U d ptc + K where k and T C are constants. Stabilization in [86] is only suggested at positive power, which means that K in (.) is equivalent to (.) with a low-pass filter with time constant T C instead of band-pass filter (that would be d

28 6 Introduction used together with (.) and (.7) in practice). An advantage with the implementation (.), however, is that it is easier to vary the gain of K through the parameter k compared to the exponent ρ in (.7). Still, the method proposed in [86] offers no stabilization at zero torque as the stabilization controller then is zero. Remark: Stabilization methods similar to (.), where variations in the DClink voltage are added to some other reference signal, have been proposed in connection with field-oriented control in [3], [6] and [7]. Here the reference signals are the quadrature stator current reference and the reference stator voltage. In Section.3 it will be shown that these schemes are equivalent to stabilization according to (.). This means that a specific behavior obtained with one stabilization scheme can be obtained also with the others. It will be shown how to convert the stabilization controllers between the different methods. The detailed treatment of stabilization will therefore be concentrated to the method (.)..4.4 Improved Modeling for Stabilization Synthesis The task is now to synthesize the stabilization controller K in (.) to stabilize the drives at all operating points with a large variety of input filters (including the two cases simulated above). From the results of the previous subsections we realize that for adequate synthesis of the stabilization controller, we need better models of the drive, where the assumption of perfect control is relaxed. We therefore model the controlled induction motor as in Figure -5, i.e., () ( ) () ( ) () T t = G p T t + G p U t, (.3) c ref d d where G c and G d are transfer functions (we neglect variations in the motor speed, which is considered to be constant). In order to link the AC and DC sides of the inverter in Figure -3, we use the power balance equation (.8). If we consider the speed ω m to be fixed, this equation can be linearized to give ωm Tωm id () t = T() t Ud () t, (.4) U U d where the quantities in (.4) hence represent deviations from an operating point. d

29 Introduction 7 The DC-link voltage and the DC-link current are also linked through the input filter in Figure -3. As a matter of fact, the DC-link voltage depends on the line voltage and the DC-link current in the following way () ( ) () ( ) () U t = G p E t Z p i t, (.5) d E DC d where the transfer functions G E (s) and Z DC (s) are given by GE ( s) = scl+ scr+, ( ) sl + R ZDC s = scl+ scr+. (.6) Equations (.), (.3), (.4) and (.5) now give the linear model of the drive including stabilization controller which is shown in Figure -9. Note that the band-pass filter for the stabilization controller is not explicitly shown but will be considered during stability analysis in Chapter. Figure -9: The shaded block represents a linear feedback model of an induction motor drive. To stabilize the drive, an outer feedback controller K is introduced. In practice also a band-pass filter is used in front of the stabilization controller but is not explicitly shown in this figure. With explicit expressions for the linear models G c and G d in (.3) (which are derived in Chapter 5 and Chapter 6), the model in Figure -9 will be used to synthesize stabilization controllers K in Chapter. Stability of the system in Figure -9 will be treated with both FOC and ISC.

30 8 Introduction.4.5 Input Filter Oscillations with DC-DC Converters Stability problems with input filter oscillations are not restricted to induction motor drives. The problem is also well known in connection with DC-DC converters, which may be modeled by the general diagram in Figure -. i d + DC Input Filter Z o U d Z i Converter Z - Figure -: DC-DC converter with input filter. Here Z o denotes the output impedance of the input filter and Z i the input impedance of the converter. Stability problems of DC-DC converters with duty-cycle control have been treated in [5] and [5] and problems with current-programmed control in [36] and []. More recent publications on the topic are given in [4] and [8]. The stability problem has also here been attributed to the negative resistance behavior of the converter with efficient suppression of DC-link voltage variations. In DC-DC converter applications, the stability problem has been solved by increasing the damping factor of the input filter by adding resistance, see for example []. For large power applications, like electric trains, additional resistance of the input filter is not feasible due to the large power losses. As described above, the approach here is instead to modify the behavior of the inverter. Resistive behavior is then achieved through active damping and hence not by adding physical resistance. In [4] and [53] it is recognized that stability of the circuit in Figure - can be determined through the poles of the transfer function +Z o /Z i, where Z o is the output impedance of the input filter and Z i is the input impedance of the converter. Through Nyquist analysis it is then concluded that if the converter shows a negative resistance behavior at the resonance peak of the input filter, the drive is likely to be unstable (here dynamic models of the input impedance are used). In Chapter we will use the same approach by rewriting the drive in Figure -3 as a feedback system containing the loop +Z o /Z i (or +Z o Y by using the inverter input admittance Y). This representation will be used to state performance requirements of the controlled drive in terms of the inverter input impedance. We then use the stabilization controller K to shape the admittance to obtain these desired properties.

31 Introduction 9.5 Notation and Nomenclature As this contribution originates from problems in traction applications, the used vocabulary often associates with traction drives. For example driving and braking are used to indicate the direction of the power flow of the drive. Driving means power flowing from the DC link to the motors and consequently the word braking is used for power flow in the other direction. For operation with zero torque we introduce the term coasting. Much of the analysis and synthesis is based on linear models of the components of the drive. With zero initial conditions, a linear system with input u(t) and output y(t) can be described with Laplace transforms as ( ) G( s) U( s) Y s =. Here U(s) and Y(s) are the Laplace transforms of the input and output signals, respectively and G(s) is the transfer function of the system. To represent the relation between the time domain signals we will use the differential operator p = d/dt. The output y(t) of the system with transfer function G(s) and input u(t) can then be written as () G( p) u() t y t =. Now, p is conventionally used to represent the number of pole pairs of an induction motor. In order to distinguish between the differential operator and the number of pole pairs, the latter will be denoted by n p. The mathematical treatment in this thesis will to a large extent be based on so called space vectors, which will be typed in bold face. Below some used notation is defined: Subscripts A, B, C Three-phase components α, β Real and imaginary parts of a space vector in stator coordinates d, q Real and imaginary parts of a space vector in synchronous coordinates ref Reference value Steady state value

32 Introduction Superscripts s Abbreviations CM DSC DTC ECD FOC IGBT ISC LFT LMI LPV NP RGA RP RS PWM VM VSI Space vector in stator coordinates (only used with FOC) Current Model Direct Self Control Direct Torque Control Equivalent Circuit Diagram Field-Oriented Control Insulated Gate Bipolar Transistor Indirect Self Control Linear Fractional Transformation Linear Matrix Inequality Linear Parameter Varying Nominal Performance Relative Gain Array Robust Performance Robust Stability Pulse Width Modulation Voltage Model Voltage Source Inverter.6 Outline and Contributions The contents of this thesis can be grouped into three main parts, where the first part gives a general introduction to induction motor drives as shown in Figure -3. The second part treats robust torque and flux control of an induction motor, corresponding to Figure -5, where the inverter is connected to an ideal power supply. The restriction of an ideal power supply is finally removed in the last part of the thesis, where hence stabilization of the feedback system in Figure -9 is discussed. It should be noted that different

33 Introduction parameter sets are used in the different parts. For the investigations of internal torque control, the first data set in Appendix A is used, whereas the second data set is used during the treatment of DC-link stability. Part I: Introduction to an Induction Motor Drive This part of the thesis describes the different components of the drive shown in Figure -3, i.e., the voltage source inverter and the induction motor. Relevant properties of the inverter fed induction motor from a control point of view are then investigated through a controllability analysis. Chapter Voltage Source Inverter This chapter gives details of the three-phase voltage source inverter relevant for the analysis in this thesis. Also the generation of switching commands (modulation) for the power semi conductors is briefly discussed and the concepts of Pulse Width Modulation (PWM) and Six-Step Operation are presented. Chapter 3 Induction Motor This chapter gives a brief physical description of an induction motor but also presents mathematical models of it. The mathematical models are expressed using so-called complex-valued space vectors. Two different models are considered to prepare for the treatment of the two control methods FOC and ISC. The ideas behind these two control methods are also presented here. Chapter 4: Controllability Analysis In this chapter a controllability analysis of the inverter fed induction motor is performed. This means that dynamic properties of the system are investigated, but also effects of input limitations on reference tracking and disturbance rejection are considered. The intention is to reveal fundamental properties and limitations of the plant from a control point of view. The results of this chapter have been published in the following conference paper H. Mosskull. Controllability Analysis of an inverter fed induction machine. In: American Control Conference, Boston, Ma, 4. Part II: Robust Control of an Induction Motor with Ideal Power Supply This part of the thesis describes torque (and flux) control of the induction motor as shown in Figure -5. That is, the feedback connection via the input filter is not considered, but the inverter is assumed to be connected to an ideal power supply. The closed-loop system G c in Figure -5 is examined with two types of torque control, Field-Oriented Control (FOC) and Indirect Self

34 Introduction Control (ISC). The focus is on ISC, where tuning issues as well as sensitivity to model errors are treated. Finally, linear controllers using modern control theory are designed and compared to the ISC. Chapter 5 Field-Oriented Control Here the classical field-oriented control (FOC) of induction motors is presented. New is the consideration of non-perfect suppression of variations in the DC-link voltage. Chapter 6 Indirect Self Control This chapter presents the control method Indirect Self Control (ISC) and derives the equations for the corresponding closed-loop system, where time delays and disturbances are carefully treated. Based on these equations also suggestions on how to set the controller parameters are given. Parts of the results of this chapter have been published in H. Mosskull. Tuning of field-oriented controller for induction machines. In Conference on Power Electronics, Machines and Drives, Bath, UK,. Chapter 7 ISC with Observer In order to implement the control method ISC certain process quantities are needed. As these may not be measured in practice, they have to be estimated by an observer. In this chapter it is shown how an observer affects the closedloop system. Also the effects of model errors on steady state estimates of the control variables torque and stator flux magnitude are examined. New is also the representation of the ISC with observer as a de-coupling controller in series with feed-forward and feedback controllers. Chapter 8 Sensitivity and Robustness Analysis Based on the linear models derived in Chapter 7, this chapter examines the consequences of model errors on closed-loop performance with ISC. Parametric as well as non-parametric model errors are treated. The results with non-parametric uncertainty have been presented in H. Mosskull. µ-analysis of indirect self control of an induction machine. In 6 th IFAC World Congress, Prague, Czech Republic, 5. Chapter 9 Linear Controller Design This chapter investigates the possibilities to design linear controllers for the induction motor in presence of non-parametric uncertainty. The design is

35 Introduction 3 based on µ-synthesis and is first done for fixed operating points. To handle varying operating points, the design is extended to Linear Parameter Varying (LPV) controllers. The work on LPV controllers has been presented in the following conference paper H. Mosskull. Linear parameter varying controller for an induction machine. In 4 nd Conference on Decision and Control, Maui, Hawaii, 3. Part III: DC-Link Stability This part of the thesis analyzes stability of the induction motor drive due to the interaction between the voltage source inverter and the input filter, i.e., the feedback system shown in Figure -9. The analysis is followed by synthesis of stabilization controllers and evaluation of stability using measurements. Chapter DC-Link Stability This chapter treats problems with power oscillations between the inverter and the input filter of the induction motor drive in Figure -3. The stability problem is analyzed using the feedback model in Figure -9 and stabilization controllers are designed to stabilize the drive. The problem is treated with ISC as well as FOC. The work on DC-link stability has been published in the following conference papers, where the first two treat stabilization with ISC and the third treats stabilization with FOC H. Mosskull. Stabilization of an induction machine drive. In th European Conference on Power Electronics and Applications, Toulouse, France, 3. H. Mosskull. Some issues on stabilization of an induction machine drive. In 43 rd Conference on Decision and Control, the Bahamas, 4. H. Mosskull, Stabilization of an induction motor drive with resonant input filter, in th European Conference on Power Electronics and Applications, Dresden, Germany, 5. Chapter Verification of Stability Margins from Measurements This chapter presents a method to verify stability of an induction motor drive from measurements. Here linear models of the drive are identified from experiments but also non-linear effects are considered using the small gain theorem. The work on validation from measurements has been presented in

36 4 Introduction H. Mosskull, B. Wahlberg and J. Galic. Validation of stability for an induction machine drive using measurements. In 3 th IFAC Symposium on System Identification, Rotterdam, The Netherlands, 3. M. Barenthin, H. Mosskull, H. Hjalmarsson, B. Wahlberg. Validation of stability for an induction machine drive using power iterations. In 6 th IFAC World Congress, Prague, Czech Republic, 5. Chapter Summary and Suggestions for Future Work This chapter summarizes the results of the thesis and presents ideas for future work. The results of Chapter and Chapter have also been presented in Licentiate Thesis H. Mosskull. Stabilization of an induction machine drive, Licentiate thesis, Automatic Control, Dept. of Signals, Sensors and Systems, Royal Institute of Technol., Stockholm, Sweden, 3. Journal Papers in preperation H. Mosskull, J. Galic and B. Wahlberg. Stabilization of an induction motor drive part I: modeling and analysis. H. Mosskull, J. Galic and B. Wahlberg. Stabilization of an induction motor drive part II: synthesis and experiments.

37 Chapter Voltage Source Inverter This chapter gives a brief introduction to three-phase voltage (two-level) source inverters and to pulse-width modulation (PWM). The interested reader is referred to [54] for further details on power electronics, and to [3] for a rigorous treatment of pulse width modulation for power inverters.. Three-Phase Voltage Source Inverter The control input to the induction motor is the three-phase stator voltage, which is generated through the voltage source inverter in Figure -3. A threephase voltage source inverter is depicted in Figure -, consisting of three legs, one for each phase. The legs contain IGBTs (Insulated Gate Bipolar Transistor) in parallel with diodes. i d + U d i A i B A B C i C - Figure -: Three-phase voltage source inverter. The IGBT is a transistor, which can either conduct or block current in the direction of the arrow in the figure (in the other direction it always blocks). In each inverter leg, only one of the IGBTs is turned on at a time to prevent a short circuit of the DC link. If for example the upper IGBT in phase A is turned on and the lower is turned off, a positive current i A (t) must pass through the upper IGBT, whereas a negative current must go through the upper diode. If we neglect voltage drops of the IGBTs and diodes, the potential at point A then in both cases equals the higher potential of the capacitor in Figure -.

38 6 Voltage Source Inverter For the other case, with the upper IGBT turned off and the lower turned on, the potential at point A equals the lower potential of the capacitor. Hence, simplified we may consider the legs as switches as shown in Figure -. Shown in this figure are also the three stator windings of the induction motor and the three stator voltages (or phase voltages) denoted u A (t), u B (t) and u C (t). Note that stator voltages refer to voltages across the stator windings and not to the inverter output voltages (i.e., the differences in potentials between the terminals and the zero point of the inverter). i d + U d - i A i B B + u B - u A u C + A C i C Figure -: Simplified model of a three-phase voltage source inverter connected to the stator windings of an induction motor. Shown here are also the three stator voltages u A, u B and u C. Through the switches in Figure -, the potentials at the points A, B and C can each be set to two different values. If we assign potential zero to the point in the figure, these two values are ±U d (t)/. We will denote the potentials v A (t), v B (t) and v C (t), and use the following representation () () () v t = k t U t, (.) A A d where k A (t) only can take the two values ±.5. By also introducing the notation v star (t) for the potential at the star point of the motor, the stator voltages in Figure - may be expressed as () () () () () () () () () () () () () ua t va t vstar t ka t ub t = vb t vstar t = kb t Ud () t vstar () t. (.) uc t vc t vstar t kc t k t

39 Voltage Source Inverter 7 The vector k(t) with elements k A (t), k B (t) and k C (t) in (.) is the coupling vector, which we have used previously without a formal definition. Through (.), the stator voltages have been expressed using three equations (three-dimensional vectors), one per phase. If the motor is Y-coupled and the star point is not connected, as in Figure -, it follows from Kirchoff s current law that the three stator currents always add to zero. This implies that also the sum of the stator voltages vanishes, i.e., () () () u t + u t + u t. (.3) A B C The constraint (.3) implies that the three stator voltages (as well as the currents) are not independent and all information about the stator voltages may be captured by a two-dimensional quantity. For this purpose, the socalled space vector representation has been introduced. Space vectors are complex numbers and the transformation from three-dimensional quantities, like the ones in (.), to the complex-valued space vectors is described in Appendix E. Due to the constraint (.3), all information about the three-phase stator voltage in (.) is contained in the space vector of the stator voltage, which is given by s () t = () tu () t u k, (.4) where space vectors are denoted in bold face. By comparing the space vector equation (.4) to the corresponding three-phase expression (.), we note that the term containing the potential at the star point has disappeared in (.4). This is because the space vector transformation maps equal offsets to all three phase quantities, so called zero sequences, to zero. To motivate why zero sequences have no influence, we note that the voltages across the stator impedances in Figure - (the stator voltages) are not affected by adding or subtracting the same amount to all potentials at the points A, B and C. This also means that the zero sequence of the three-phase coupling vector does not affect the three-phase stator voltage (note that k(t) does not in general satisfy the constraint (.3)). As shown in Appendix E, the space vector transformation may be interpreted as a projection of a three-phase quantity onto the plane defined by the constraint (.3) (plus a scaling). With two possible positions of each of the three switches, the inverter in Figure - may be put in eight different states. Two of these inverter states result in zero stator voltage. These are the combinations with all switches in the upper position or all switches in the lower position. The corresponding (zero) space vectors are sometimes referred to as the zero voltages. The remaining six non-zero stator voltage space vectors are shown in Figure -3, d

40 8 Voltage Source Inverter where the length of each vector is /3U d (with the specific scaling of the space vector transformation, see Appendix E). Figure -3: The vectors u s u s6 represent the six non-zero stator voltage space vectors that can be generated by the three-phase voltage source inverter. The three (physical) stator voltages u A (t), u B (t) and u C (t) can be obtained from a space vector by projecting it onto unit vectors in the directions of u s, u s3 and u s5 in Figure -3. For example this means that the real part of a space vector corresponds to the A-component of the three-phase quantity.. Modulation At each point in time, only the two zero voltages or the six non-zero space vectors in Figure -3 can be applied to the motor. Based on how these actual stator voltage vectors are chosen, we may differentiate between two groups of control methods for induction motors. With methods like the DSC (see Subsection.3.), the controller basically works in continuous time and directly determines the switching instants (selects between the eight possible vectors). However, with ISC, the controller generates a voltage reference to be applied in average over a certain time interval, called the pulse period. This reference voltage is then separately converted into switching commands by a modulator and we will refer to this mapping of the average voltage reference to switching times as modulation. With an average based controller, we naturally consider average values of linear combinations of the u si s. By

41 Voltage Source Inverter 9 forming linear combinations of the eight possible space vectors, the entire hexagon in Figure -3 can be reached (in average over a pulse period). With an average based control method, we could hence divide the controller block in Figure -3 into two blocks. That is, one controller block that generates a reference stator voltage and one modulation block that converts the voltage into the coupling vector k(t). In the following two subsections, two kinds of modulation principles are presented, namely pulse width modulation (PWM) and six-step operation... Pulse Width Modulation With Pulse Width Modulation (PWM), the coupling vector is determined to make the average stator voltage (space vector) over a time interval T p, called the pulse period, equal to a reference value. From Expression (.4) we see that the DC-link voltage directly affects the stator voltage. In order to compensate for variations in the DC-link voltage, we therefore assume that U d is measured and used to normalize the coupling vector. We hence assume that the coupling vector is generated to satisfy t k + T p ( tk ) ( ) usref ( ) d T k τ τ =. (.5) U t p tk d k The bar over the DC-link voltage in (.5) represents an average value over a pulse period, which is used to suppress switching frequency harmonics. Now, Expression (.5) only specifies the average of the coupling vector over a pulse period. For the actual behavior in time, we assume that the switching instants are chosen such that the voltage pulses are applied in the middle of the pulse periods, as schematically illustrated in Figure -4. T sw T p t k t k+ Figure -4: Schematic illustration of generation of voltage pulses. Shown here is that the switching times T sw are calculated to put the voltage pulses in the middle of the pulse periods. This has the benefit that sampled values of motor quantities at the times t k, t k+ and so on correspond to fundamental values. t

42 3 Voltage Source Inverter This has the advantage that sampled values of process signals at t k, t k+ and so on correspond to fundamental values and no additional filtering is required to remove the high frequency switching harmonics [34]. The controller then never observes the harmonics, and these will therefore be neglected during the analysis. We hence approximate k(t) in the expression for the stator voltage space vector (.4) by its low frequency component k (t). If we also replace the discrete-time stator voltage reference in (.5) by a continuous-time signal, we can model the coupling vector as usref ( t Td ) k() t k () t, (.6) U t T d ( ) where T d is a time delay. The time delay arises as the pulses are applied in the middle of the pulse periods, which in average corresponds to a time delay of half a pulse period. Additional time delays may also be introduced due to precalculation of the switching times (see Subsection 6..). By introducing the notation D(p) for a time delay of T d seconds, and A(p) for the moving average over a pulse period, Equation (.6) can be linearized and represented as sref sref d Ud k() t = D( p) u ( t) u D( p) A( p) Ud ( t). (.7) U Note that the quantities in (.7) represent deviations from an operating point. A delta notation could have been used to stress this, but is omitted for notational convenience. To obtain expression (.7), we also assumed that the time delay in the stator voltage reference is compensated for in steady state, i.e., we use u sref (t) instead of u sref (t-t d ). If we now linearize the equation for the stator voltage (.4) and use the linearized expression for the coupling vector (.7), it follows that deviations in the stator voltage around an operating point can be written u u u. (.8) sref () t = D( p) () t + ( D( p) A( p) ) U () t s sref d U d Equation (.8) is illustrated in Figure -5, where the normalization of the coupling vector due to DC-link voltage variations is clearly interpreted as a feed-forward compensation. Due to the average and the time delay, we see that the compensation will not be perfect though. Note that for a pure time delay of T d s, it follows that normalization of the coupling vector is beneficial, i.e., the magnitude of -D(ω) is less than one, only if ω T d < π /3. d

43 Voltage Source Inverter 3 Figure -5: Linear feedforward model of the normalization of the coupling vector to suppress variations in the DC-link voltage. An example of modulation is shown in Figure -6, using so called space vector modulation [87]. Figure -6.A shows a stator voltage (solid) generated from a sinusoidal reference voltage (dashed). The frequency of the sinusoidal reference voltage in the example is Hz and the used switching frequency is 55 Hz. The frequency content of the actual stator voltage is shown in Figure -6.B. It shows that the signal basically consists of a component at the stator frequency ( Hz) and high frequency components being multiples of the switching frequency. The main switching frequency component in the example occurs around Hz, which is twice the switching frequency. Due to the low-pass character of the induction motor, the high frequency components have little effect and the motor is mainly affected by the fundamental component. Actual stator voltage and reference stator voltage Fourier transform of stator voltage 5 8 Voltage [V] Time [s] Frequency [Hz] Figure -6: (A) Phase voltage generated by an inverter (solid) from a Hz sinusoidal reference voltage (dashed). (B) Fourier transform of the stator voltage... Six-Step Operation Ideally, the inverter generates sinusoidal stator voltages, which produce a torque without harmonics (if we neglect switching frequency harmonics, which are very high). Sinusoidal voltages imply a stator voltage space vector

44 3 Voltage Source Inverter with constant magnitude that rotates with constant frequency in the complex plane in Figure -3. From this figure we realize that the amplitude of sinusoidal voltages is upper limited by the distance to the middle of one of the sides of the hexagon, which is given by / 3U d. If we increase the reference voltage beyond this limit, harmonics are introduced as the length of the space vector cannot be kept constant during a revolution. This is sometimes referred to as overmodulation and makes it possible to increase the fundamental of the stator voltage beyond / 3U d (by also introducing harmonics). Maximum fundamental voltage is generated in so called six-step operation (alternatively block mode or hex mode), which is illustrated by a phase voltage in Figure -7 with a stator frequency of 65 Hz. Here no zero voltages are used and the different non-zero voltage vectors are stepped through in sequence. By comparing Figure -7 with Figure -6 we see that the extra switchings in Figure -6 make it possible to reduce the fundamental but also to reduce low frequency harmonics. With six-step operation the voltage amplitudes are not adjustable (cannot normalize the coupling vector as no zero voltages are used) but directly depend on the DC-link voltage. It also follows that the inverter switching frequency equals the stator frequency. The stator voltage in Figure -7 can be expanded into a Fourier series and the first term (the fundamental) is given by /π U d. The maximum magnitude of the fundamental stator voltage possible to generate with the inverter is therefore limited by [54] us Ud. (.9) π Note that the limit in (.9) is given for peak values of the phase voltages. Actual stator voltage and reference stator voltage Fourier transform of stator voltage 5 8 Voltage [V] Frequency [Hz] Frequency [Hz] Figure -7: (A) Maximum stator voltage generated by an inverter. The solid curve is the actual stator voltage and the dashed curve is the reference voltage. The stator frequency is 65 Hz. (B) Fourier transform of the stator voltage.

45 Voltage Source Inverter 33 Remark: In six-step operation the switching frequency equals the stator frequency, which means that the switching frequency in the example shown Figure -7.A is much lower than the switching frequency used to generate Figure -6.A. The time delay between the reference and the output voltages, which is related to the pulse period and hence to the switching frequency, is then larger in Figure -7.A compared to in Figure -6.A. The time delay limits the achievable control bandwidth and the effectiveness of disturbance rejection as shown in Figure -5.

46

47 Chapter 3 Induction Motor The intention with this chapter is to give some insight into the operation of an induction motor but also to show how it can be represented mathematically. The mathematical description will be based on complex-valued space vectors, which gives a very compact representation of the induction motor in the form of equivalent circuit diagrams (ECDs). Two such diagrams, the inverse Γ- model and the Γ-model, are presented in Section 3. and will be used when describing Field-Oriented Control (FOC) in Chapter 5 and Indirect Self Control (ISC) in Chapter 6. To prepare for these chapters, we derive some relevant motor equations in Sections 3.3 and 3.4. This chapter does not give a comprehensive description of the modeling of an induction motor. Only aspects needed for the forthcoming chapters are treated. For a detailed description of induction motors, see e.g. [39] and [4]. 3. Physical Description The induction motor consists of a stationary part, the stator, and a rotating part, the rotor. A stator with a three-phase distributed stator winding is shown in Figure 3-. Figure 3-: Illustration of the stator of an induction motor.

48 36 Induction Motor The rotor often is of the squirrel cage type, which simply consists of a number of rotor bars connected through two end rings as shown in Figure 3-. Just as with the stator, the electrical effect of the rotor may be modeled as an equivalent three-phase winding, see for example [63]. Figure 3-: Squirrel cage rotor with rotor bars connection by two end rings. Applying a sinusoidal three-phase voltage to the terminals of an induction motor results in a rotating flux in the stator. This is schematically illustrated with the large rotating magnet in Figure 3-3. Most of the flux propagates to the rotor, where the rotor bars hence see a magnetic flux passing by. Consequently, voltages across the rotor bars are induced, which in turn give rise to rotor currents. These rotor currents generate flux and the rotor is therefore represented as a small magnet in Figure 3-3. Together with the original flux, the additional flux produced by the rotor winding generates a torque, which strives to align the two fluxes. The torque hence forces the rotor to follow the applied stator flux. Note, however, that a non-zero torque requires that the stator flux rotates asynchronously to the rotor (as opposed to the case with synchronous motors). Otherwise no currents are induced, as the flux seen by the rotor bars then is constant. This fact also motivates the name asynchronous motor, which also is used for this type of motor. The frequency of the induced electrical rotor quantities equals the frequency of the flux seen by the rotor bars, which is called the slip frequency and is denoted by ω. The slip frequency hence corresponds to the difference between the frequency of the rotating flux and the electrical rotational frequency of the rotor. By electrical rotor frequency we mean n p times the mechanical rotor speed ω m, where n p is the number of pole pairs of the motor. The phase windings of the stator are usually arranged to give multiple poles, see [39]. With n p pole pairs, one physical rotation of the rotor corresponds to n p rotations for the electrical quantities of the rotor. The relevant rotor frequency in this case therefore is n p ω m.

49 Induction Motor 37 Figure 3-3: Schematic picture of an induction motor. The magnetic field produced by the stator is represented by the large rotating magnet. The varying flux induces currents and hence also flux in the rotor, which is illustrated by the small magnet in the figure. Torque is generated to align the stator and rotor fluxes. Although torque is the main control variable, i.e., the main quantity we want to control, usually also the flux in the motor is controlled. The induction motor is designed to operate at a certain nominal flux and running with for example larger fluxes drives the motor into saturation, which produces large currents in the stator windings. Using a too small flux leads to a reduced maximum torque (reduced pull-out torque, see Subsection 3.4.). In practical applications, the flux may be varied as a function of load as to minimize the total stator current. 3. Equivalent Circuit Diagrams Mathematically, the induction motor can be compactly described by using complex-valued space vectors. Magnetically linear motors are assumed with inductances varying sinusoidally with the rotor position. It should be noted that different models exist that correctly describe the motor as seen from the stator terminals (at least if iron losses are neglected). Depending on the purpose, a certain model, or equivalent circuit diagram (ECD), may be preferred in a certain situation. For example, with a control scheme oriented to the rotor flux, such as the field-oriented control (FOC) presented in Chapter 5, an ECD with simple rotor equations may be preferred. Such an ECD is the so called inverse Γ-model, which is shown in Figure 3-4. The ECD consists of a stator mesh and a rotor mesh, corresponding to the two three-phase windings of the motor. The two meshes are connected by the magnetizing inductance L m

50 38 Induction Motor and the rotor mesh also contains the so called rotor EMF as a voltage source. Each winding contains resistance, where R s denotes stator resistance and R r denotes rotor resistance. In practice, each winding also contains leakage inductance as not all flux produced in one winding passes through to the other winding. However, in the inverse Γ-model all leakage inductance has been put in the stator mesh, where the total leakage inductance is denoted L σ. This is not really physically correct, but simplifies the rotor equations. Seen from the stator terminal, the representation is correct, although for example the internal rotor current i r is scaled compared to the current that would be measured on a real motor. Finally, i s s (t), ψ s s (t) and ψ r s (t) in the ECD represent the stator current, the stator flux and the rotor flux. i s s R s L R r i m s i r s u s s. s s L m. s r jn p m r s Figure 3-4: Inverse Γ-model ECD of the induction motor. Remark: For convenience we use the term flux although we actually mean flux linkage. For the control method Indirect Self Control (ISC), which is introduced in Chapter 6, the stator flux is the central quantity. In this case the Γ-model shown in Figure 3-5 is suitable, where all leakage inductance is put in the rotor mesh to simplify the expressions for the stator equations. The term Γmodel refers to the location of the inductances in the ECD in Figure 3-5 (which also motivates the name inverse Γ-model for the ECD in Figure 3-4). Although rotor quantities and parameters in Figure 3-4 are not equal to those in Figure 3-5, the same notation is still used for simplicity. It should be clear from the context which model is used in a certain chapter. In connection with ISC, all quantities refer to the Γ-model and in connection to FOC all quantities refer to the inverse Γ-model. The model in Figure 3-4, as well as the model in Figure 3-5, is given in stator fixed coordinates, where space vectors are expressed in a coordinate system attached to the stator. However, when working with FOC in Chapter 5, we will use a description in rotor flux

51 Induction Motor 39 coordinates (see Section 3.3). In order to distinguish between quantities in the two coordinate systems, quantities of the inverse Γ-model in stator fixed coordinates are written with superscripts s. Figure 3-5: Γ-model ECD of the induction motor. In the following two sections we derive the equations needed when discussing FOC in Chapter 5 and ISC in Chapter Inverse Gamma Model In this section we derive the equations needed in connection with the treatment of field-oriented control (FOC) in Chapter 5. With FOC, the motor torque and flux are controlled via different components of the stator current, similarly to controlling a separately excited DC motor. In Subsection 3.3. the motor equations in so-called stator coordinates are presented. However, in order to work with constant quantities at steady state (as opposed to sinusoidal), the motor equations are often written in synchronous coordinates in connection with current control of induction motors. These coordinates are introduced in Subsection 3.3. and the ideas behind field-oriented current control are presented in Subsection Stator Coordinates From the ECD in Figure 3-4 it follows that the induction motor is described by the following equations in stator coordinates

52 4 Induction Motor d s s s R r s Lσ is () t = us () t ( Rs + Rr) is () t jnpωm ψ r () t, (3.) dt Lm s E d s s R r s ψr () t = Rris () t jnpωm ψ r () t, (3.) dt Lm where E s is the so called back EMF. The torque is not directly given in Figure 3-4 but can be calculated from rotor flux and stator current as [39] 3 s * s 3 T () t = np Im{ ( ψ r () t ) i s () t } = np( ψr α () t isβ () t ψrβ () t is α () t ),(3.3) where the asterisk denotes complex conjugate and α and β denote the real and imaginary parts of the (stator-fixed) space vectors. () t 3.3. Synchronous and Rotor Flux Coordinates The equations in Subsection 3.3. are said to be in stator coordinates. This means that the space vectors are represented in a coordinate system that is attached to the stator (does not rotate). Also the rotor quantities are expressed in this fixed coordinate system. The real and imaginary parts of a general space vector Q s (t) in stator coordinates are denoted Q α and Q β, respectively. β q Q(t) d θ (t) Figure 3-6: Illustration of synchronous coordinates. The synchronous coordinate system with axes denoted by d and q is displaced the angle θ relative to the stator fixed coordinate system with axes denoted α and β. We now introduce a change of coordinates as α j ( t) s Q() t = Q () t + jq () t Q () t e θ, (3.4) d q

53 Induction Motor 4 where θ (t) is the angle of a rotating coordinate system, see the illustration of the coordinate transformation in Figure 3-6. The new quantity Q(t) is said to be in synchronous coordinates and the d- component is referred to as the direct component and the q-components is referred to as the quadrature component of the space vector. In order to express the motor equations from the previous subsection in synchronous coordinates, we need to know how derivates are affected by the change of coordinates. For that purpose we note that the derivative of a space vector in stator coordinates can be expressed as That is, jθ () t ( Q t e ) s d Q () t = () = dt = Q() t + j Q() t e = p+ j () t Q() t e. ( θ () ω ) ( ω ) j t jθ() t ( ω ) j θ () t (3.5) s Q () t e = p+ j () t Q () t, (3.6) where d ω() t = θ() t. (3.7) dt From the definition of a space vector in synchronous coordinates in (3.4) and the expression for the derivatives in (3.5), it follows that the two motor equations (3.) and (3.) are transformed to d () () ( () R r Lσ is t = us t Rs + Rr + jlσω t ) is() t jnpωm ψ r () t,(3.8) dt Lm d R r ψr () t = Rris() t + j( ω () t npωm() t ) ψ r () t. (3.9) dt Lm The torque equation (3.3), finally, may now be represented as 3 3 T () t = np Im{ ψ r () t is () t } = np ( ψrd () t isq () t ψrq () t isd () t ). (3.) The synchronous coordinates are often chosen as rotor flux coordinates, i.e., θ is chosen as the angle of the rotor flux space vector. In rotor flux coordinates, the q-component of the flux therefore is zero by definition. The equation for the stator current space vector (3.) then becomes

54 4 Induction Motor d Rr L s() t s() t ( Rs Rr jl () t ) s() t jnp m r () t dt σω ω i = u + + i σ L ψ, (3.) m where we used the notation Ψ r = Ψ rd. The corresponding equation for the rotor flux space vector can be separated into one equation for the rotor flux magnitude and one equation for the stator frequency as d ψ dt ω R =, (3.) r () t Ri () t ψ () t r r sd r Lm () t r () t () t Rrisq = npωm +, (3.3) ψ where ω is the frequency of the rotor flux space vector, see (3.7). Finally, the torque can be calculated from rotor flux and stator current as 3np T () t = ψ r () t isq() t, (3.4) which follows from (3.) Current Control For a separately excited DC motor, the torque is proportional to the product of flux and armature current. By keeping the flux constant (through a constant magnetization current), the torque may be controlled by varying the armature current. For an induction motor, the torque equation (3.3) is a little bit more complicated. However, in rotor flux coordinates, the situation resembles that of the DC motor, see the torque equation (3.4). Here the torque is given as the product of the rotor flux magnitude Ψ r and the quadrature current component i sq. From the rotor flux equation (3.), we see that the rotor flux Ψ r is only influenced by the direct current component i sd. The control strategy with FOC therefore is to use the direct current component to regulate the rotor flux to a constant value and then to use the quadrature component to control the torque (the current components can be modified independently). By working in rotor flux coordinates, we hence can reuse ideas from DC-motor control.

55 Induction Motor Gamma Model With ISC, the stator flux is a central quantity and for that purpose the Γ-model, with its simple stator equations is used when modeling the induction motor. The motor equations of the Γ-model in stator coordinates are given in Subsection 3.4., where also the idea behind ISC (and other DTC techniques) is presented. In Subsection 3.4., steady-state relations between the slip frequency and the torque are derived, which will be used in Chapter Stator Coordinates By using the ECD in Figure 3-5, the fluxes of the induction motor can be described in stator coordinates by [76] ψ () t = R i () t + u () t, (3.5) s s s s R r R r ψ r() t = ψs() t + jnpωm() t ψ r() t. (3.6) Lσ Lσ Note that the equations are in stator coordinates, although no superscripts s are used. Further, the stator current is given by () i () s t = + ψs t ψ r () t, (3.7) Lσ Lm Lσ and the torque may be calculated as 3np 3n * p T () t = Im{ ψ s () t i s() t } = ( ψs α () t isβ () t ψsβ () t is α () t ), (3.8) where α and β denote the real and imaginary components of the space vectors. By inserting the expression for the stator current (3.7) into the torque equation (3.8), we may also compute the torque as 3 np * T () t = Im{ ψs () t ψ r () t }. (3.9) By using a polar representation of the space vectors as () ψ () L σ () () j s() t j r() t j u() t s t = s t e χ, r t = ψr t e χ, s t = use χ ψ ψ u, (3.) the torque in (3.9) can be written as ()

56 44 Induction Motor 3 n T t t t t t, (3.) p () = ψs() ψr () sin ( χs() χr ()) Lσ δ () t where the load angle δ(t) was defined as the difference between the angles of the stator and rotor flux space vectors. The load angle is illustrated in Figure 3-7. β Ψ s (t) Figure 3-7: Definition of load angle δ(t), which is the angle between the stator and rotor fluxes. With DTC methods, torque is controlled by increasing or decreasing the load angle. The idea with DTC methods is to keep the stator flux magnitude constant and affect the torque via the load angle δ. As the rotor flux varies relatively slowly compared to the stator flux, an increase in load angle transiently gives an increase in torque. The load angle is affected by varying the frequency of the stator flux space vector. In the following subsection, steady-state relations between the load angle, the torque and the slip frequency are derived. These will be needed during the treatment of ISC in Chapter 6. δ(t) Ψ r (t) α 3.4. Slip Frequency Control In this subsection we derive a steady-state relation between the torque and the slip frequency (with constant stator flux), which will be used in connection with ISC treated in Chapter 6. We also derive an expression for the so called pull-out torque, which is the maximum torque that can be generated with a constant stator flux magnitude. With the polar representation (3.), the rotor equation (3.6) can be represented as

57 Induction Motor 45 d jχ () ( () ) () () () r t jχr() t ψr t e = r t j r t r t e dt ψ + χ ψ ω () t Rr jδ() t R r jχr () t = ψs() t e + jnpωm() t ψr () t e. Lσ L σ (3.) By separating Equation (3.) into real and imaginary parts, it follows that ψ r () t = ψs() t cosδ () t ψr () t, (3.3) T T σ () t () t σ ψ s ω () t npωm() t = sinδ () t, (3.4) Tσ ψ r ω () t where the rotor leakage time constant T σ = L σ /R r was introduced. At steady state, the rotor flux magnitude is constant, which means that (3.3) reduces to ψ = ψ cosδ, (3.5) r s where subscripts denote steady-state values. By using (3.5), Equation (3.4) gives the following expression for the steady state slip frequency ω = tan δ. (3.6) T σ We may now insert the two steady state relations (3.5) and (3.6) into the torque equation (3.), to write the steady-state torque as 3 n T =. (3.7) p r R ψ ω r Equation (3.7) shows that if the rotor flux is constant, the torque is proportional to the slip frequency. If instead of regulating the rotor flux, the stator flux is kept constant, the rotor flux magnitude varies with the load and expression (3.7) gets a little more complicated. By again using the steady state relation (3.5) and the following trigonometric identity cos δ =, (3.8) + tan δ

58 46 Induction Motor the torque equation (3.7) can be written as 3np ωtσ T = L + ω T ψ s σ ( σ ). (3.9) The steady-state torque according to (3.9) is plotted in Figure 3-8 as a function of slip frequency with constant stator flux magnitude. Here we see that around the origin, the steady state torque depends almost linearly on slip frequency but eventually reaches a maximum for positive slip frequencies and a minimum for negative slip frequencies. The maximum (or minimum) steady state torque is called the pull-out torque or break-down torque and occurs at ω =. (3.3) The pull-out torque is given by T T σ 3np = ψ s. (3.3) 4L σ pull out Increasing the slip frequency above the value (3.3) results in a decreasing torque. The increased slip frequency, however, increases losses in the rotor, which may damage the motor. The steady state slip frequency should therefore always be upper limited by (3.3). Torque [Nm] Torque as a function of slip frequency Slip frequency [Hz] Figure 3-8: Steady state torque as a function of steady state slip frequency. Here it is seen that the torque approximately varies linearly with the slip frequency for small slip frequencies. Above the so called pull-out slip frequency (for positive slip frequencies) the torque decreases with increasing slip frequency, while losses increase in the motor. A control algorithm therefore should limit the steady state slip to values below the pull-out slip frequency.

59 Chapter 4 Controllability Analysis In this chapter we perform an input-output controllability analysis of the inverter fed induction motor to reveal its relevant properties from a control point of view. This analysis includes dynamic properties, but also consequences of constraints on the inputs are considered. The controllability analysis is performed at zero torque and nominal flux at three different stator frequencies, namely % (OP), 5% (OP) and 9% (OP3) of base speed ω base. It will be shown that the induction motor has a sharp resonance peak at the operating point stator frequency, where the height of the resonance peak increases with the motor speed. It is also shown that coupling between the inputs and outputs may give large disturbances in torque at pure flux control at higher speeds with input uncertainty. Further, the upper limitation of the DC-link voltage magnitude reduces the possibilities to suppress DC-link voltage disturbances at higher speeds. As the DC-link voltage excites a high gain direction of the plant, such disturbances have a large effect on the outputs. Physical constraints on the input, i.e., the three-phase stator voltage, are discussed in Section 4.. To analyze the effects of these input constraints, but to also consider torque and the flux magnitude as control variables, a linear model of the induction motor on polar form is derived in Section 4.. Poles and zero of the induction motor model are examined in Section 4.3 and Section 4.4 calculates the relative gain array (RGA). Section 4.5 introduces a scaled representation and Section 4.6 shows the singular values of the motor. Section 4.7 shows that perfect flux control may be sensitive to errors at the plant input. Limitations due to time delays are discussed in Section 4.8 and Section 4.9 treats limitations imposed by input constraints. Physical motivation to some of the results is given in Section 4. and Section 4. summarizes the results obtained in the chapter. Remark: The controllability analysis is only carried out for rotor speeds below base speed. Above base speed, the stator voltage magnitude saturates and only the frequency of the stator voltage can be modified. Here the induction motor hence represents a system with one input and two outputs. However, DC-link disturbances still affect the magnitude of the stator voltage.

60 48 Controllability Analysis 4. Input Constraints In this thesis we consider induction motors fed by a voltage source inverter. The fundamental stator voltage magnitude therefore is limited by us () t u s () t Ud, (4.) π where U d is the DC-link voltage, see Subsection... This is a constraint on the stator voltage amplitude but we also should put restrictions on the stator voltage frequency to prevent the steady state slip frequency, ω, from exceeding the pull-out slip frequency (3.3), i.e., ω = ω n ω, (4.) p m where ω is the steady state stator frequency. Note that with constant rotor speed, the constraint on the deviation of the (steady-state) stator frequency around an operating point with zero slip frequency becomes ω = ω. (4.3) In the sequel we will use linearized models where deviations of signals around stationary operating points are modeled. A delta notation as in (4.3) could then have been used. However, to simplify notation, the deltas will be omitted. T σ T σ 4. Polar Representation The outputs of the motor are here considered to be the torque and the stator flux magnitude. Further, from the input constraints given by (4.) and (4.3), it makes sense to consider the stator voltage magnitude and frequency as inputs. We therefore derive a linear model with the mentioned inputs and outputs. To facilitate this, we use the polar notation of the space vectors introduced by (3.). With the polar representation, the torque can be expressed by (3.), i.e., 3 np T = ψs() t ψr () t sinδ () t. (4.4) L σ To obtain a linear model of the process, the motor equations (3.5) and (3.6) are rewritten in the following way

61 Controllability Analysis 49 Rs ψs R = s + ψs + ψr cosδ + us cosδus (4.5) Lm Lσ Lσ ψ r = ψs cosδ ψ r (4.6) T T σ σ Rs ψ r us δ = us sinδ sinδus ωu L σ ψ ψ + (4.7) s s Rs ψ r ψ s us δ = + sinδ + sinδus npωm, (4.8) Lσ ψs Tσ ψr ψs where δ is the load angle, see (3.), and d ωu () t = χu () t, δus() t = χu () t χs() t. (4.9) dt Equations (4.4) and (4.5)-(4.8) now form a (real-valued) non-linear state space model of the induction motor. Inputs are the stator voltage magnitude and frequency and outputs are the stator flux magnitude and the torque (via (4.4)). Acting on the system is also the rotor speed, which is considered as a disturbance. The non-linear model can be linearized around stationary operating points. From (4.5)-(4.8) it follows that at an operating point with torque, stator flux and mechanical rotor speed specified by T, Ψ s and ω m, the stationary states are given by and the stationary inputs by ψ s ψs cosδ R ψ s s ω sinδ cosδ + ψ L r σ arctan = δ, (4.) us sin Rs δ + δ L m L σ 4 L σ arcsin T 3 npψ s

62 5 Controllability Analysis ψ sr s sin + δ us cosδus Lm Lσ =. (4.) ω npωm + tan δ T σ By linearizing the equations (4.4) - (4.8) we now obtain a linear model of the plant, which we represent as () () () () t ψ s t us t = G + Gv ωωm t T t ωu yt () ut () (), (4.) where G is a x transfer matrix representing the dynamics from inputs to outputs and G vω is a x matrix representing the dynamics from the rotor speed to the outputs. Further, the notation y(t) = (Ψ s (t) T(t)) T, u(t) = (u s (t) ω u (t)) T was introduced. In addition to the rotor speed, also the DC-link voltage is considered a disturbance. The influence of the DC-link voltage on the three-phase stator voltage was discussed in connection with PWM in Subsection.., where also a method to suppress the influence of a varying DC-link voltage was introduced. Here we neglect this compensation and thus model the stator voltage as () t () t U () t d us u sref. (4.3) U d Equation (4.3) can be linearized and represented in polar form as usref u() t = uref () t + U d Ud () t. (4.4) G Note that the DC-link voltage only affects the magnitude of the stator voltage space vector in (4.4). We also see that the DC-link voltage adds to the input of the plant. If we move the disturbance to the output instead, the linearized model of the induction motor will be represented by Ud

63 Controllability Analysis 5 () t () t Ud y () t = Gu() t + ( GGUd Gv ω ) Gu() t + Gvv() t, (4.5) ωm where the disturbance vector v(t) = (U d (t) ω m (t)) T and the disturbance transfer function G v were introduced. The model (4.5) is visualized in Figure 4-. v G v u G y Figure 4-: Linear induction motor model. The input to the motor, which here is represented by the stator voltage magnitude and stator voltage frequency, affects the outputs, i.e., the stator flux magnitude and torque via the transfer function G. The outputs are also affected by the disturbances DC-link voltage and motor speed via the disturbance transfer function G v. 4.3 Poles and Zeros The poles and zeros of the linear process model G derived in the previous section are shown in Figure 4- at zero torque and varying stator frequency. Pole-zero map Real Axis Figure 4-: Pole-zero map of the induction motor. x s and o s denote poles and zeros at the three operating points OP, OP and OP3. Here it is seen that the poles are in the left half plane, which means that the motor model is stable. One poorly damped pole pair moves along the imaginary axis with approximately the operating point stator frequency. The poles here are shown at zero operating point torque. At non-zero torque the locations of the poles are only slightly changed. Imag Axis

64 5 Controllability Analysis The locations of the poles and zeros at the three operating points defined above are marked with x s and o s, respectively. The system is stable at all operating points with one LHP zero and four LHP poles. As the stator frequency increases, two poles approach the zero, whereas the remaining two poles move along the imaginary axis with the stator frequency. Consequently, the system gets less damped with increasing rotor speed. The smaller the stator resistance, the closer the poles get to the imaginary axis. 4.4 Relative Gain Array (RGA) The relative gain array (RGA) was introduced by Bristol in 966 as a measure of interactions for decentralized control []. The RGA of a non-singular square matrix G, denoted RGA(G) or Λ(G) is a square matrix defined by RGA( G) = G ( G ) T, (4.6) where the operation x denotes element by element multiplication (Hadamard or Schur product). However, the RGA also is an indicator of sensitivity to uncertainty. Large RGA elements mean poor robustness for inverse based controllers in presence of independent input uncertainty, [7]. By large is here meant values above, [7]. On the other hand, large RGA elements also mean that there is strong coupling and even nominal performance with a diagonal controller may not be satisfactory. Hence, in general, large RGA elements indicate a plant that is inherently difficult to control. The maximum RGA elements for the induction motor at the three operating points are plotted in Figure 4-3 as solid, dashed and dotted lines. As seen, there are peaks in the RGA elements corresponding to the operating point stator frequency and the heights increase with the stator frequency. Most problematic are large RGA elements around the desired bandwidth. One possibility to avoid potential problems is then to set the bandwidth well above the operating point stator frequency, where the RGA elements are small. Due to bandwidth constraints, for example due to time delays, this may not be possible at higher stator frequencies (where low switching frequencies give long time delays) and the RGA indicator limits the achievable bandwidth at those operating points. With a bandwidth independent of the operating point, it follows that it probably has to be less than the stator frequency at OP3. Remark: Many control algorithms aim at decoupling the control of torque and flux, for example the ISC described in Chapter 6. Such controllers are inverse based and are consequently sensitive to diagonal input uncertainty.

65 Controllability Analysis 53 Maximum element of RGA(G) Angular Frequency [rad/s] 4 Figure 4-3: Maximum RGA elements at the three considered operating points. The solid curve shows the RGA elements at OP, the dashed curve shows the RGA elements at OP and the dotted curve shows the RGA elements at OP3. We note that the peaks of the RGA elements appear at the operating point stator frequencies and increase with higher speed. Large RGA elements for example indicate sensitivity to independent input uncertainty. 4.5 Scaling To easily examine performance requirements and limitations caused by input constraints, the system should be scaled as proposed in [7]. Performance requirements are here given as constraints on the control errors e(t), i.e., the difference between a reference r(t) and the output y(t). We let the unscaled signals and system in (4.5) be represented with hats, i.e., yˆ t = Gu ˆˆ t + Gˆ vˆ t. (4.7) () () () We now introduce scaling to make all signals less than one in magnitude. The scaled input u(t) is therefore generated as () D u() t u t u ˆ v =, (4.8) where D u is a diagonal matrix with the maximum expected values of the inputs along the diagonal. By similarly defining the scaling matrices D e for the maximum accepted control errors and D v for the maximum expected disturbances, the scaled system is represented by where () () () y t = Gu t + G v t, (4.9) v

66 54 Controllability Analysis G = D GD ˆ, G = D Gˆ D. (4.) e u v e v v The limit on the (deviations in the) stator voltage magnitude follows from (4.), and although (4.) only was set as a steady state restriction on the stator voltage frequency, it will used to scale this quantity. Hence, by choosing the smallest deviation of the asymmetric limits we get umax max min Ud us, us,, u max =. (4.) π T σ We require that the control errors are less than 5% of rated torque and flux (see Appendix A) and assume that the maximum disturbances on the inputs are % of nominal DC-link voltage and 5% of the pull-out slip frequency, respectively. Reference values are scaled as r = Rr, where R=D - e D r and the new input is upper limited by one. The matrix D r defines the maximum expected reference changes, which we set to % of rated torque and % of rated flux. 4.6 Singular Values The largest and smallest singular values show the highest and lowest gains of the system over all directions, i.e., G( jω) d( ω) σ ( G( jω) ) σ ( G( jω) ), d( ω), (4.) d ω ( ) where d(ω) is a -dimensional vector with complex elements. Small singular values mean that large inputs are needed in the corresponding input direction to affect the output and this might of course be a problem since there are constraints on the inputs. Especially problematic are cases where disturbances or references act in the low gain output direction. In Figure 4-4 the maximum and minimum singular values at OP are shown as solid lines. The maximum singular value shows a resonance peak at the operating point stator frequency, which could be expected from the pole-zero map in Figure 4-. The minimum singular value on the other hand is well damped but decreases with the operating point stator frequency. The dashed lines in Figure 4-4 show the gains in the output flux and torque directions, i.e.,

67 Controllability Analysis 55 ( ω) G j ( ω) y ( ω) ( ω) G j y G j y G j y =, (4.3) where y is set to ( ) T and ( ) T, respectively. Finally, the dotted lines show the gains in the disturbance directions, i.e., plotting (4.3) with the columns of G v instead of y. We see that the gain in the output flux direction follows the minimum singular values. This means that flux control requires large inputs. The gain in the output torque direction is higher, although not the maximum gain. It also turns out that the gain in the direction of the rotor speed disturbance coincides with the gain in the output torque direction and, at least for frequencies around and above the resonance, the gain in the direction of the DC-link voltage disturbance is very high. As the DC-link disturbance actually enters at the plant input, the dotted curve also is proportional to the gain from the DC-link disturbance to the outputs, see (4.5). Hence, disturbances in the DC link have large impact of the outputs at the resonance frequency. 6 4 Singular Values Singular Values [db] Angular Frequency [rad/s] Figure 4-4: Singular values of the induction motor at OP. The maximum and minimum singular values are shown as solid lines, the gains in the output torque and flux directions are shown as the dashed lines and the gains in the disturbance directions are shown as the dotted lines. Although the condition number, i.e., the ratio between the largest and smallest singular values, is large at many frequencies, the RGA, which is independent of scaling, indicated robustness problems only around the resonance frequencies, see Section 4.4.

68 56 Controllability Analysis 4.7 Flux Control Robustness As an example to illustrate the robustness issues discussed in connection with the RGA analysis in Section 4.4, we consider pure flux control at OP. With scaling according to Section 4.5, the input direction for pure flux control at the resonance frequency is given by (.8.997e -j. ) T, which is the input direction giving minimum gain. Now, the gain of the system in this direction is 4dB compared to the gain in the input direction ( ) T, which is 33dB. Hence, a very small error in the input direction of pure flux control at the resonance causes large errors. The critical component is the error in amplitude, which corresponds to only 9 V. The flux control issue is also visible in Figure 4-5, where the gains in the input directions ( ) T (dotted line) and ( ) T (solid line) are shown together with the maximum and minimum singular values (dashed lines). At the resonance frequency, the gain in the input direction ( ) T is seen to be much larger than the minimum gain, corresponding to pure flux control. We also see that the stator voltage magnitude acts in the direction with minimum gain and the stator voltage frequency in the direction giving maximum gain for low frequencies. For high frequencies the situation is the opposite. 8 Singular Values of G*[ ] T and G*[ ] T 6 Singular Values [db] Angular Frequency [rad/s] Figure 4-5: Gains in the stator voltage magnitude and frequency directions. The gain in the input direction ( ) T is plotted as the dotted line and the gain in the input direction ( ) T is plotted as the solid line. Figure 4-6 shows the responses in torque (solid lines) and flux (dashed lines) at steps in stator voltage magnitude and frequency (% of U d and /T σ ), respectively (the results are generated by simulating the non-linear motor

69 Controllability Analysis 57 equations). A step in the stator voltage magnitude results in oscillations in torque as well as flux. The stationary effect of the step is a change of flux. The responses in torque and flux to a step in the frequency of the stator voltage are much smoother and the stationary effect is a change of torque. Torque and flux responses to step in stator voltage magnitude Time [s] Torque and flux responses to step in stator voltage frequency Time [s] Figure 4-6: Responses to a step in (A) stator voltage magnitude and (B) in stator voltage frequency. Torques are represented with solid lines and fluxes with dashed lines. We see that a step in the stator voltage magnitude gives oscillating responses, whereas the responses to a step in the stator voltage frequency are smooth. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. 4.8 Time Delays To represent a realistic controlled drive, a time delay should be added at the system input, cf. Figure -5. Assuming that the time delay, T d, is equal in all directions, the control bandwidth ω B is limited by ω B </T d, [7]. The time delays in a realistic application vary as a function of motor speed. Largest time delays would appear around base speed with six-step operation. With for example a time delay of ms would give a bandwidth limitation of 8 Hz. 4.9 Limitations Imposed by Input Constraints Consider perfect control, i.e., that u(t) is chosen as () = () () u t G Rr t G G v t, (4.4) which gives e(t) =. This follows from the model of the plant in (4.9). The question is if selecting u(t) this way still satisfies the input constraint u. As the constraint on the stator voltage frequency (4.) actually is a restriction v

70 58 Controllability Analysis only at steady state, we will also consider the modified input constraint u. In the following two subsections, limitations caused by the input constraints are separately examined for disturbances and references Disturbance Rejection If only disturbance rejection is considered, it follows from the ideal control law (4.4) that the available inputs are sufficient to perfectly cancel the effects of the disturbances if the following condition is met ( HG ( j ) Gv ( j )) σ ω ω <, (4.5) where H is the identity matrix in case of the input constraint u and the vector ( ) in case of u. Examining one disturbance at the time, we replace G v in (4.5) by its columns. As the DC-link disturbance actually adds to the stator voltage magnitude, which follows from Equation (4.4), it turns out that (4.5) is satisfied as long as (if we use u Ψ s ω which is valid apart from very low speeds) U d ω. (4.6) π.ψ The relation (4.6) is satisfied with equality for ω = 44 rad/s, which means that the DC-link voltage disturbance can not be completely rejected at OP3 (475 rad/s). For rotor speed disturbances it turns out that even with the maybe too conservative constraint u, disturbances can be rejected for frequencies up to 5 rad/s, which is sufficient, as rotor speed disturbances usually are low frequency disturbances. s 4.9. Reference Tracking To perfectly track reference changes without violating the input constraints, we need the following condition to hold ( HG ( j ) R ( j )) σ ω ω <, ω ωr, (4.7) where ω r is the frequency up to which reference tracking is required and the constant matrix H is defined as in the previous subsection. Now, if bandwidth is actually only required for torque control, one should study the inputs

71 Controllability Analysis 59 required by the torque reference alone and similarly for pure flux control. This means replacing the matrix R in (4.7) by its columns. From Figure 4-4 it is clear that flux control requires larger inputs than torque control. It follows that with the constraint u and with reference step heights of % of rated torque and % of rated flux, the frequencies where the inputs reach their limits for pure torque control become 6 Hz, 8 Hz and 7 Hz at the three operating points. The corresponding values for flux control are 7 Hz, 38 Hz and 36 Hz. If we instead consider the constraint u, the maximum torque bandwidths increase to 43 Hz, 6 Hz and 44 Hz, whereas the bandwidth for flux control becomes unconstrained. This result can partly be understood from Figure 4-5. If the restriction on the stator voltage frequency is made weaker, the solid line in Figure 4-5 is simply shifted upwards (together with the maximum and minimum gains). The gain in the output torque direction turns out to be upper bounded by the gain in the input direction ( ) T and the torque control bandwidth is consequently upper bounded by the stator voltage magnitude constraint. The flux control bandwidth, however, increases with the limit on the stator voltage frequency. 4. Singular Values Revisited In this section we try to physically motivate the resonance peak of the maximum singular value at the operating point stator frequency seen in Section 4.6. We will do this by examining the gain of the motor from stator voltages to stator currents. This gain has been studied in for example [3]. At constant speed we can form the Laplace transforms of the motor equations (3.) and (3.) to obtain the following transfer function representation where ( s) () t = ( p) () t i G u, (4.8) s s s s s s R r + pωm s Lm G =. (4.9) r r Lσs + Rs + Rr + Lσ jnpωm s + Rs jnpωm L m Lm jn R R Note that the transfer function in (4.9) has complex-valued coefficients and is therefore typed in bold face (in equivalence to space vectors). For systems that can be represented on this form, it is shown in Appendix F that the

72 6 Controllability Analysis singular values of the corresponding real-valued representation of the system can be calculated as s s G ( jω ) and ( jω ) G. (4.3) The inputs giving maximum gains are hence given by the so called positive and negative sequences, i.e., jωt () and () t = ae t = ae jωt us u s. (4.3) The gains (4.3) are shown at the speeds ω m =, and 5 Hz in Figure 4-7. We see that the gain is largest at low frequencies. This may be physically motivated as only the winding resistances limit the current at low frequencies, whereas the current is also limited by the inductances at higher frequencies. Singular values of G Singular values of G - - Frequency [Hz] 3 - Frequency [Hz] 3 Singular values of G Frequency [Hz] Figure 4-7: Maximum and minimum gains of the induction motor in stator coordinates at (A) Hz, (B) Hz and (C) 5 Hz mechanical speed. Maximum gains appear at low frequencies. Stationary currents are only limited by the winding resistances, whereas inductances also limit sinusoidal currents.

73 Controllability Analysis 6 With constant synchronous speed ω (and constant ω m ) it follows from the expression for the derivatives (3.6) that the transfer function in synchronous coordinates, which we here denote G, relates to the corresponding transfer function in stator coordinates (4.8) as ( ) = s ( + ) G s G s jω. (4.3) Compared to the representation in stator coordinates, the argument is hence shifted from s to s+jω. The corresponding gains to (4.3) for the system in synchronous coordinates, i.e., G, are shown at ω m =, and 5 Hz in Figure 4-8. The largest gain is no longer at zero frequency but at the synchronous frequency ω (note that the physical currents in stator coordinates still have the large gain a zero frequency). Singular values of G Singular values of G - - Frequency [Hz] 3 - Frequency [Hz] 3 Singular values of G Frequency [Hz] Figure 4-8: Maximum and minimum gains of the induction motor in synchronous coordinates at (A) Hz, (B) Hz and (C) 5 Hz mechanical speed. Due to the coordinate transformation, the highest gain now appears at the synchronous frequency. In Figure 4-4 the singular values of the polar representation were shown with the stator voltage magnitude and frequency as inputs. This figure shows a

74 6 Controllability Analysis large gain from the stator flux magnitude to the outputs stator flux magnitude and torque around the operating point stator frequency. We will now show that this resonance follows from the resonance of the transfer function G. In Appendix F it is shown that deviations in power around an operating point can be approximated by (here with signals in stator coordinates) 3 () Re s {( ) () ( ()) } * s s s P t = i u t + u i t *. (4.33) s s s s With a sinusoidal variation in the stator voltage magnitude we have u + e e. (4.34) s jωt jωt s s ( ) () ˆ j ωt χu u sin ˆ s t = U ωt = e U s u j s Here we see that if the frequency ω of the varying stator voltage magnitude equals the operating point stator frequency ω, then a constant term is generated. From above we know that a DC-voltage gives a large DC-current (in stator coordinates). However, due to the multiplication by the steady-state stator voltage in (4.33), the effect of the large DC-current on the power is a disturbance oscillating with frequency ω. In Appendix F the transfer function from stator voltage magnitude to power is derived, see Equation (F.4). By neglecting power losses of the induction motor, the power may be approximated by the product T(t)ω m (t). By dividing the transfer function expression (F.4) by ω m, we may then approximate the transfer function from stator voltage magnitude to torque as 3 us ω m ( Gd ( ) + Gd ( jω )). (4.35) Here we see that the resonance in the transfer function G at the operating point stator frequency ω (which also is present in G d ) appears in the transfer function (4.35) as well, cf. Figure 4-4. (Note that the torque does not depend on the chosen coordinate system). 4. Summary In this chapter we have examined properties of an inverter fed induction motor. Usually, the induction motor equations are expressed using space vectors in stator or synchronous coordinates. In this chapter we have used a linear model of the motor on polar form with the stator voltage magnitude and frequency as

75 Controllability Analysis 63 inputs. This way the input constraint of the stator voltage magnitude is easily analyzed. (Note that with this representation, signals are constant at steady state just as with a representation in synchronous coordinates). Outputs of the linear model are the torque and the stator flux magnitude. It has been shown that the linear motor model is stable at all examined operating points. The dynamics of the motor strongly vary with the motor speed with a poorly damped pole pair that moves along the imaginary axis (with small negative real part). These poles give the maximum singular value of the motor a sharp resonance peak at the operating point stator frequency. The height of the peak increases with the speed of the motor. These properties have been examined in for example [3]. Here we have extended the analysis of the linear motor model to also consider the relative gain array (RGA) and consequences of input limitations. It was shown that also the RGA elements show a resonance peak at the operating point stator frequency, where the height increases with the speed. Large RGA elements mean that the cross coupling of the plant is significant. For good nominal performance, a decoupling controller is required. However, a decoupling controller can lead to poor robust performance in presence of independent input uncertainty. We then conclude that it may be difficult to achieve both good nominal and robust performance of the induction motor at higher speeds. It was shown through an example that perfect flux control may be very sensitive to model errors. Inputs with perfect flux control act in a low gain direction of the induction motor, which means that large inputs are needed for fast flux control. The coupling of the motor (indicated by the RGA elements) then means that model errors may cause pure flux control to accidentally excite high gain directions instead of the desired ideal (low-gain) flux direction. It was shown that disturbances in the DC-link voltage excite the resonance of the motor and hence cause large disturbances in torque and flux. Variations in the rotor speed, however, give a well damped response in torque and flux. In Subsection.. it was shown that rejection of DC-link voltage disturbances can be interpreted as a feedforward compensation at the plant input. The DClink voltage affects the magnitude of the stator voltage and hence compensation of such disturbances means modifying the reference stator voltage, or more precisely, the inverter coupling vector. At higher speeds, modification of the magnitude of the coupling vector space vector is limited due to an upper magnitude constraint and disturbances in the DC-link voltage therefore have a large effect on the motor outputs.

76

77 Chapter 5 Field-Oriented Control This chapter briefly describes the classical so called field-oriented control (FOC) of induction motors. As pointed out in Section 3.3, the idea with FOC is to work in rotor flux coordinates, where the motor torque can be controlled via the quadrature component of the stator current, while keeping the rotor flux constant through the corresponding d-component of the current. In rotor flux coordinates all electrical quantities are constant at steady state as opposed to sinusoidal as in a coordinate system fixed to the stator. This means that ordinary PI controllers are sufficient to completely remove steady-state errors. We will assume that the reference values of the current components are calculated from torque and rotor flux references as (cf. (3.) and (3.4)) () t () Tref isdref () t = ψ rref () t, isqref () t =. (5.) L 3n ψ t m p rref Section 5. presents the FOC control law expressed in rotor flux coordinates. To apply this control in practice, the orientation of the rotor flux in needed and for this purpose the rotor flux is estimated by an observer. The problem of rotor flux estimation is therefore very briefly touched upon in Section 5.. In order to treat DC-link stability in Chapter, we need the transfer functions G c and G d in the torque model (.3). These transfer functions are therefore derived in Section Current Control with Active Damping Here we will not directly use the original FOC control law, but FOC with active damping as proposed in [3] and [6]. This means that the control law in rotor flux coordinates is given by ( ) () t = ( jω () t Lσ R ) () t + F( p) () t () t u i i i. (5.) sref a s sref s

78 66 Field-Oriented Control In (5.) F is a PI controller, whereas the first term is used for decoupling and so called active damping. The control law is illustrated in Figure 5-, where the induction motor is represented by the block G e and the back EMF E. This representation follows from the stator current equation (3.), which can be written as R r is() t = s() t jnpωm ψr () t Lσ p Rs Rr jω L u σ L. (5.3) m Ge ( p) () t E In Figure 5- also the influence of the DC-link voltage on the stator voltage is shown. Disturbance rejection as described in Subsection.. is not included here. The stator voltage is therefore only modeled as a multiplication of the reference by the DC-link voltage, normalized by its nominal value. In Figure 5- this relation is linearized and represented as s us () t = u sref () t + u Ud () t. (5.4) U d G Ud Figure 5-: Block diagram of the closed-loop system with field-oriented control including active damping. Apart from the PI feedback controller F, the FOC control law consists of the term jω L σ i s, which removes the cross coupling in the system G e, and the active damping part R a i s, which is used to improve rejection of the back EMF E. During controller design, the back EMF E in Figure 5- is namely treated as a disturbance. Note, however, that the rotor flux magnitude actually depends on the d-component of the stator current through (3.).

79 Field-Oriented Control 67 To motivate the introduction of active damping, we study the transfer functions from reference and back EMF to the stator current in Figure 5-. Without active damping we have i GF G s () t = sref () t () t + GF i + GF E, (5.5) where the system G is given by G ( s) =, (5.6) Rs + R r Lσ s+ Lσ which is obtained after decoupling, see Figure 5-. A convenient choice of controller parameters is to set F = (α c /s)[g ] - where α c is a design parameter. This choice is suggested in [6] and gives a loop gain of α c /s, which means that the transfer functions in (5.5) can be evaluated as αc p is () t = isref () t G E () t. (5.7) α p+ α p+ c The transfer function from reference to output is here a simple first order system with time constant α c. The performance of the closed-loop system, however, also depends on the ability to suppress the disturbance E. With active damping, the transfer function G in (5.5) is replaced by P i, see Figure 5-. To reach the same behavior from references to output as in (5.7), we now choose the controller as F = (α c /s)p i -. Again, the result is given by (5.7) with G replaced by P i. As long as the gain of P i is smaller than the gain of G, we have improved disturbance rejection and hence the closed-loop performance. In [6] it is suggested to set the active damping parameter R a as It then follows that a c σ s r c R = α L R R. (5.8) i ( ) P s = L σ ( s+ α ) c. (5.9) From (5.6) and (5.9) we see that the high frequency asymptotes are the same for the two systems G and P i. The low frequency properties of P i may, however, be influenced through the parameter α c, which determines the location of the pole. If α c is chosen larger than (R s +R r )/L σ, then the low frequency gain of P i is smaller than the low frequency gain of G. This is

80 68 Field-Oriented Control illustrated in Figure 5-, where the dashed curve is the Bode plot of the system G and the solid curve is the Bode plot of P i. With the approach for selecting the controller parameters discussed above, i.e., F = (α c /s)p - i, it follows with P i according to (5.9) that the PI controller parameters become k = α L, k = α L. (5.) p c σ i c σ Magnitude [abs] Phase [deg] Bode plot of G' and P i (R +R )/L s r σ α c Frequency [rad/s] 3 Figure 5-: Bode plot of the two systems G (dashed) and P i (solid). We note that the gain of P has been decreased at low frequencies compared to G. This is a result of the active damping. i ref F F+R a E G F+R a u s G e i s x jl Figure 5-3: Representation of FOC with active damping as a two-degree of freedom controller. The effect of the active damping term can also be understood by rewriting the system in Figure 5- as in Figure 5-3. Here we have incorporated the active damping term in a new feedback controller F =F+R a, where hence the original proportional gain has been increased by R a. Additionally, we here have a prefilter F/(F+R a ) on the references. This is a special case of a general

81 Field-Oriented Control 69 so called two-degree of freedom controller. The gain in the feedback loop can now be increased to improve disturbance rejection, while keeping the transfer function from references to output the same by using the prefilter. The prefilter F/(F+R a ) is a lag filter with high frequency gain k p /(k p +R a ), which hence slows down the response from references to the outputs. Now, as already mentioned, the rotor flux magnitude in (5.9) is actually not independent of the stator currents, which follows from the rotor flux equation (3.). To also incorporate this dependence into the closed-loop model, we cannot use a complex-valued space vector representation, as the rotor flux magnitude depends only on the direct component of the stator current. We therefore introduce a real-valued description, with -dimensional vectors (denoted in plain face) with the real and imaginary components of the complex-valued space vectors as elements. With this representation, the closed-loop system can be drawn as in Figure 5-4. U d G Ud i sref F + - P i s Figure 5-4: Real-valued linear model of closed-loop system when a field-oriented controller is applied to an induction motor. Here also the influence of the DC-link voltage is shown as a disturbance. Here the PI controller F and the system P are given by where k p k + P i + s F =, P = Ln m pωm Lσ ( s αc ) P,(5.) kp k + + i R r s P i P =. (5.) LL m R σ r ( s+ αc ) s+ Rr Lm

82 7 Field-Oriented Control The non-zero factor P is a result of the interaction due to the back EMF. A modification of the d-component of the input affects the the d-component of the stator current. This in turn means that the rotor flux is affected, which via the back EMF also affects the q-component of the stator current. For large values of α c we may use that +P, which follows as the gain of P decreases with frequency and P () R r /(α c L σ ) <<. The diagonal terms in P are then both approximately equal to P i in (5.9). Finally, in the real-valued representation used in Figure 5-4, the block G Ud can be evaluated as G Ud Rs Lσ ωω Re{ s} L r m R u ψ r = =. (5.3) Ud Im{ s} U d Lσ R u s + ω + ω Lm R r The expression for u s used in (5.3) follows from the stator current equation (3.), using steady state values of the stator current components obtained from (3.) and (3.3). Figure 5-5: Linear model of the closed-loop system including time delay when FOC is applied to an induction motor. Here it is assumed that the stator currents used for cross-coupling are predicted to compensate for the time delay. So far we have not considered time delays, which for example are introduced due to modulation, see Section.. Due to the rotating space vectors, time delays lead to phase shifts, which destroy field orientation. The cross coupling between the two current components then increase and performance degrades. If we for simplicity assume that the phase shift is compensated for by predicting the stator currents, we may add time delays to the block diagram in Figure 5-4 as shown in Figure 5-5. Actually, the disturbance U d also influences the prediction. For small time delays, this effect is small however.

83 Field-Oriented Control 7 Time delays will be more carefully treated in connection with ISC in Chapter Rotor Flux Estimation The stator current needed to implement the feedback loops in Figure 5- is measured. However, the controller works in rotor flux coordinates, and the orientation of the rotor flux is needed for the transformations between stator coordinates and rotor flux coordinates. The angle of the rotor flux space vector is not measured but must be estimated. Good performance of FOC therefore requires accurate estimation of the rotor flux angle, from which also the frequency ω needed in the control law (5.) can be calculated. Imperfect field-orientation introduces cross coupling as e.g. a voltage applied to affect only the current in the d-direction in reality affects both current components. In Figure 5-6 the transformations between the coordinate systems are explicitly shown as well as the fact that a three-phase VSI is used to generate the stator voltages. Figure 5-6: Illustration of current control where coordinate transformations are explicitly shown. The observer block in Figure 5-6 is hence used to estimate the angle of the rotor flux space vector χ r. With so-called Direct Field-Orientation (DFO), χ r is obtained by directly estimating the rotor flux, which via Equation (3.) can be done as t ˆ s() ˆ ˆs R r s ψˆ ( ) ˆ r t = Rri s τ jnpωm r ( τ) dτ ˆ ψ. (5.4) L m

84 7 Field-Oriented Control This estimator corresponds to a simulation of the rotor flux equation with the stator current as input. This model is therefore refered to as the current model (CM). The current model may also be implemented by using the rotor flux equation in estimated rotor flux coordinates, i.e., (3.9). By separating this equation into real and imaginary parts, we arrive at the corresponding equations (3.) and (3.3). Through the slip relation (3.3), we may hence estimate the derivative of the rotor flux angle χ r as ( τ) ( ) ( τ) ( ) d Ri ˆ ˆ Ri ˆ ˆ χ () = ˆ ω ( τ) = + ω + ω, (5.5) dt r sq r sqref r t np m np m ψˆ r τ ψrref τ where we approximated the rotor flux and the quadrature stator current component by their reference values (the estimated flux reference converges to its constant reference value and the current controllers make the current reference follow their reference values). By estimating the angle of the rotor flux space vector indirectly via an estimation of the frequency ω is called Indirect Field-Orientation (IFO). It has been shown in [9] that the CM with IFO is quite robust to model errors with regards to stability. Incorrect parameter estimates, however, lead to deteriorated performance. The current model in (5.4) is sensitive to R r and L m. Note that also the estimate of L m affects the performance with the IFO implementation of CM as the d- component of the stator current reference is set according to (5.). From the ECD in Figure 3-4 we may also express the rotor flux in terms of the stator voltage. We then obtain the so-called voltage model (VM), which is given by ( τ ˆ τ ) τ ˆ σ () t s s s s r = s s s s ψˆ ( t) u ( ) R i ( ) d L i t. (5.6) Equation (5.6) is expressed in stator coordinates, i.e., this is the DFO version of VM, which also can be implemented in synchronous coordinates giving an IFO scheme. At high speeds, the voltage model gives an accurate stator flux estimate, since the back EMF dominates the stator voltage. However, at low and zero speed, the voltage model is well known to give non-satisfactory performance. This is due to sensitivity to the voltage drop across the stator resistance and to inherent signal integration problems at low excitation frequencies, see [38] and [9]. A further problem with stator flux estimation according to (7.), is that this equation cannot be directly used in practice as it contains a pure integration. This gives a marginally stable system due to the pole at the origin (this pole corresponds to a pole pair at ±jω in a real-valued representation in synchronous or polar coordinates). Also, pure integration

85 Field-Oriented Control 73 may give integrator drift, see e.g. [7]. To solve the integration problem, the pure integration can be replaced by a low-pass filter. In this case also the static effect of the filter should be compensated for as is done with the Statically Compensated Voltage Model (SCVM), see [7]. Due to the problems with VM at low speeds, the rotor flux is here preferably estimated using the current model, while the voltage model is suitable at higher speeds, see e.g. [], [5] and [37]. The CM and VM may both be seen as reduced order observers, as they do not require the full model of the motor to be simulated. Using a full-order observer with a correction term, a seemless transition between the two models can be achieved, see [5]. This gives a good rotor flux angle estimate over the entire speed range. 5.3 Closed-Loop Torque Dynamics In this section we consider the torque dynamics to obtain the expressions for the transfer functions G c and G d in the torque model (.3). These will be needed in connection with DC-link stability treated in Chapter. If we linearize the torque equation (3.4), it follows that deviations in the torque around an operating point obey Lm ω 3np 3npψ r R r T () t = ψ i () t + i ψ () t = i t Lm p + R r ( ) r sq sq r s (), (5.7) where the rotor flux was expressed as a function of the direct component of the stator current through (3.). The stator currents in (5.7) are related to the reference currents and the DC-link voltage disturbance as shown by the block diagram in Figure 5-5. We now introduce the following notation for this closed-loop system () () () i t = G i t + G U t, (5.8) s ic sref id d where it follows from Figure 5-5 that the transfer functions G ic and G id are given by ( ), ( ) ic id Ud G = I + PDF PDF G = I + PDF PG. (5.9) If we assume that i sqref is chosen according to (5.), it follows from the torque equation (5.7) and the current control equation (5.8) that

86 74 Field-Oriented Control Lm Lm ω 3 R ω n ψ R T t Gic Tref t Gid Ud t L m Lm p + p R + r R r G G r p r r () = () + () c d.(5.) We may hence write the transfer function G c from torque reference to torque as Lm ω R r PDF Gc = G ic = Lm + P DF s + R r. (5.) We note that G c only depends on the quadrature current control, since the transfer function P is lower triangular and F is diagonal. Further, from (5.) we have that the disturbance transfer function G d is given by G d 3n ψ = Lm ω R Lm s + Rr p r r G id. (5.) If we use that L m >> L σ, it follows from the expression for G Ud in (5.3) that, at higher speeds, G Ud L σ ψrω ω R r U. (5.3) d Using this approximation we may write the transfer function G id in (5.9) as G id ψrω U d L σ ( s+ α ) c S Lσ ω S Rr S LL σ ω nω + Rr m p m P S, (5.4)

87 Field-Oriented Control 75 where the sensitivity functions for the two current loops S and S are given by Lσ ( s+ αc ) Sii =. (5.5) L s+ α + DF σ ( ) The second term in the second component in (5.4) is very small at low frequencies due to the sensitivity function, and goes to zero at high frequencies due to P. In between it reaches values of around ω n p ω m /α c with controller parameters according to (5.) (in both current loops), which are less than one for large values of α c. By neglecting this term, the expression for G id can then be approximated by G id S P G Ud c L Rr ii ω σ. (5.6) We can now insert the approximation for G id in (5.6) into the expression for G d in (5.) to get 3n pψ r Gd SPGUd. (5.7) A U d Ψ r ω U d Ψ r ω U d T ref /Ψ 3n r p i sqref F + D + + P i sq 3n p Ψ r T Figure 5-7: Block diagram of the induction motor controlled with FOC. Here also normalization of the coupling vector to compensate for DC-link voltage variations has been added. Just as with the transfer function for G c in (5.), the expression for G d in (5.7) only depends on the quadrature current control. The transfer functions in (.3) with FOC can then be illustrated by Figure 5-7, where suppression of DC-link disturbances according to Subsection.. has been added.

88

89 Chapter 6 Indirect Self Control This chapter describes Indirect Self Control (ISC), which is a control scheme for induction motors. Following the classification in [4], ISC is a so called Direct Torque Control (DTC) method, where torque is controlled directly and not indirectly via the stator currents. With ISC the torque is controlled by appropriately adjusting the stator flux in a coordinate system aligned to the stator flux space vector. In this chapter we derive a continuous-time linear model of the closed-loop system with ISC. Time delays and compensation for time delays are treated as well as rejection of DC-link voltage and rotor speed disturbances. It is shown that the orientation to the stator flux leads to a partial decoupling of flux and torque control. At zero torque, the two loops are completely decoupled, whereas a small coupling from flux to torque exists at non-zero torques. The closed-loop model is also used to derive expressions for the controller parameters to achive certain performance requirements. The discrete-time ISC control law as stated in [49] is presented in Section 6.. In this thesis, analysis of the closed-loop system is done in continuous time. A continuous-time version of the ISC control law is therefore derived in Section 6.. The closed-loop equations when applying ISC are presented in Section 6.3 and Section 6.4 discusses tuning of the controllers. Finally, some simulation results are shown in Section 6.5. Remark: ISC is sometimes also referred to as Indirect Stator-Quantity Control or Indirect Stator-Flux-Oriented Control. 6. Discrete-Time ISC This section introduces the discrete-time ISC control law given in [33] and [49]. The control law is derived in Subsection 6... Due to time delays, the control law has to be calculated in advance, which means that certain motor quantities have to be predicted. This issue is discussed in Subsection 6...

90 78 Indirect Self Control Finally, a simple method to suppress switching frequency harmonics in the measured torque is presented in Subsection Control Law Just as with the control method DSC described in Subsection.3., ISC controls the motor torque by appropriately adjusting the stator flux space vector. With a discrete-time implementation of ISC, the controller then determines an appropriate update of the stator flux vector over a pulse period, as illustrated in Figure 6-. The magnitude of the stator flux is updated to stay at a desired (usually constant) value and the angle increment is determined to influence the load angle and hence also the torque, cf. the torque equation (3.). Figure 6-: Stator flux increment over a pulse period, composed of a magnitude update Ψ s and a flux increment χ s. From Equation (3.7) we know the relation between slip frequency and torque at steady state. This knowledge can then be used to form the desired steadystate stator frequency ω ref as

91 Indirect Self Control 79 ω R = n ω + T, (6.) ω r ref p m ref 3 npψ r where ω ref is the desired steady-state slip frequency. Over a pulse period, the desired steady-state update of the stator flux angle then corresponds to χ s = ω ref T p, where ω ref is given by (6.). To improve the dynamic properties, as well as ensuring zero steady state errors, a PI controller is added to the angle increment. The angle increment with ISC is therefore calculated as ref R r χs( tk) = npωm( tk) + T ref ( tk) Tp( tk) + 3 npψ r ( tk) ω ref ( t ) k p R + Knr + TpTnr q T, ref tk T tk p= 3 npψ r k PI controller t k r ( ( ) ( )) ( t ) ω ( ) k (6.) where q is the shift operator, i.e., for example q - ω (t k ) = ω (t k- ). Equation (6.) gives the desired angle update χ s of the stator flux space vector in Figure 6-. We now also consider the magnitude update Ψ s to keep the stator flux magnitude at its reference value. To do that, we first express the stator flux space vector at time t k+ in Figure 6- as j( χ ( ) ) ( ) s tk + χ ψ s s t k j χs ( tk ) s( tk+ ) = ( ψs( tk) + ψs( tk) ) e = + s( tk) e ψ s( tk) ψ ψ.(6.3) + K ψ ( t ) By using the representation in (6.3), we can write the increment of the stator flux space vector in Figure 6- as j χs( tk) ( ψ ) ( ) ( ) ( ) k ( ) ψ s = ψ s tk+ ψ s tk = + K tk e s t ψ. (6.4) k With ISC, the scaled magnitude update K Ψ, defined in (6.3), is generated by a P controller, i.e., ( ) ( ) ( ) ( ) Kψ tk = Pψ ψsref tk ψs tk, (6.5)

92 8 Indirect Self Control where P Ψ is a proportional gain. We have now determined the desired stator flux increment Ψ s in (6.4) through the components K Ψ and χ s given by (6.5) and (6.). The next step is to generate this flux increment through the available control input, i.e., the stator voltage. From the stator equation (3.5), it follows that the increment of the stator flux space vector over a pulse period is generated as t k+ ( ( τ) R ( τ) ) dτ T ( R ) ψ = u i = u i, (6.6) s s s s p s s s tk where the bars over the signals indicate average values over the pulse period. Hence, in order to generate the desired stator flux increment (6.4), the average stator voltage over the pulse period should satisfy ( ) ψs ψs u = + R i + R iˆ t + T t T T s s s s s k p k s s. (6.7) In (6.7) we approximated the average stator current by a predicted value in the middle of the pulse period, where the argument of the estimated stator current means that the estimation corresponds to time t k +½T p, given information available up to time t k. By inserting the stator flux update (6.4) into the expression for the average voltage (6.7), the stator voltage reference can be written as u ( ) j χs( tk ) ( ψ ( )) ( ) + K tk e ψ s tk t = + R iˆ t + T t T ( ) sref k s s k p k s. (6.8) Equation (6.8), together with the expressions for K Ψ and χ s in (6.5) and (6.), then forms the time-discrete ISC control law. Remark: In [3], the discrete-time ISC is calculated as u ( ) j χs( tk ) ( ψ ( k )) + K t e s t t ψ = ψ t + R t + T t T ψ ( k) ( t ) ( ) iˆ ( ) sref k sref k s s k p k s s k i.e., the reference flux magnitude is used instead of the actual magnitude. We will show in Subsection that changes in the flux reference give large disturbances in the torque with the implementation (6.9). When talking about the ISC control law, we will refer to (6.8) and not (6.9) unless explicitly mentioned.,(6.9)

93 Indirect Self Control Prediction The ISC control law (6.8) determines an average stator voltage u sref (t k ) based on reference values and on information of the state of the motor at time t k. By using some modulation scheme, the reference voltage is then mapped to switching times for the inverter phases as illustrated in Figure -4 in Subsection... If long voltage pulses are needed, the time from the beginning of the pulse period to the first switch may be very short, even shorter than the actual calculation time in a real-time implementation. To avoid such problems, the switching times are calculated in advance, using predicted information about the state of the motor. This is schematically illustrated in Figure 6-, where the prediction time is denoted by T s. Hence, the switching times to be applied within the pulse period between t k and t k+ are calculated already at time t k -T s, using predicted motor quantities valid at time t k. T sw T p T s t k t k+ t Figure 6-: Schematic illustration of precalculations of switching times by the time T s. As calculation of the switching times T sw requires the state of the motor at time t k, the state at t k has to be predicted using information available at time t k-t s. By including the extra time delay due to the prediction, the ISC control law (6.8) is replaced by u ( t T ) sref k s ( ) + R iˆ t + T t T, s s k p k s j χs( tk Ts) ( + K ( )) ˆ ψ tk Ts e ψ s( tk tk Ts) = + T p (6.) where hence the stator flux and stator current are predicted. Note that we do not consider predictions of K Ψ and χ s in (6.) as a time delay of these quantities has less effect on the performance compared to a time delay in the stator flux.

94 8 Indirect Self Control 6..3 Average Torque Due to the inverter switchings, the actual torque contains switching frequency harmonics, cf. Section.. These high frequency harmonics are unavoidable and should not be reacted on by the control. This means that the feedback and feed-forward terms of the ISC control law in (6.) and (6.5) only should involve the low frequency components of the measured signals. By using symmetrical pulse patterns, where pulses are applied in the middle of the pulse periods, the low frequency part of the signals may simply be extracted by sampling at the pulse period instants [34]. If the inverter switching times are calculated at time t k -T s, which does not coincide with a pulse period instant, we could predict signals at time t k using measurements at time t k-. An alternative way to estimate the fundamental of a signal is to form the average over a pulse period, using intermediate measurements. We will assume that this is done for the torque, which means that the control error in (6.) is replaced by where (assume T p /T s is an integer) ( ) ( ) ( ) T t T = T t T T t T, (6.) error k s ref k s k s Tp / Ts Ts T ( t T ) = T( t kt ). (6.) k s k s T p k = 6. Continuous-Time ISC This section derives a continuous-time model of the discrete-time ISC control law presented in Section 6.. We first consider the case without predictions in Subsection 6.., and include the predictions in Subsection Control Law In order to derive a continuous-time ISC control law, we introduce the following new controller parameters P Knr Tnr K ψ ψ p =, Kp =, Ki =, (6.3) T T T p p p

95 Indirect Self Control 83 where P Ψ is the P gain of the discrete-time flux controller and K nr and T nr are the parameters for the discrete-time torque controller, see (6.) and (6.5). By using the gains in (6.3), a continuous-time model of the discrete-time control law can be achieved as the limit of (6.8) as the pulse period goes to zero. In order to do this, we start by expressing the exponential in (6.8) as where j χ ( t ) ( ) ( ) ( χ ( )) s k e = + j χ t +Ο t, (6.4) s k s k ( χs( tk) ) lim Ο =. (6.5) Tp It then follows that the first term in (6.8) can be written where ( t ) k ( ) ( ) ψ χ s s = Kψ + j + ψ s tk, (6.6) T p T p ( ) ( ( )) ( ( ) ( )) ( ( )) = T K ψ ψ t ψ t +Ο χ t +Ο χ t. (6.7) p p sref k s k s k s k We now want to evaluate the limit of (6.6) as the pulse period T p goes to zero. With the new controller gains (6.3), it follows from the expression for χ s in (6.) that χ s Rr lim = npωm() t + T ref () t + Tp 3 npψ r () t Rr + K p + Ki ( Tref () t T() t ). p 3 npψ r () t Tp (6.8) As the pulse period goes to zero, the limit in (6.8) is consequently bounded and the limit of in (6.7) is zero. It then follows from (6.6) that ( t ) ψ lim = K ( ψ ( t ) ψ ( t χ )) + lim j ψ ( t ), (6.9) s ψ s k p sref k s k s k Tp T Tp p T p where the limit of χ s /T p as T p is given by (6.8). The resulting continuous-time control law hence becomes

96 84 Indirect Self Control u ( ψ ) () () () () t () i () t = u t + ju t ψ t + R t, (6.) u sref T s s s where the quantities u T (t) and u Ψ (t) are given by c ( ) ψ () () () uψ t = Kp ψsref t ψs t, (6.) R u t = n t + T t + () ω () r T p m ref 3 npψ r () t () Rr K p + Ki T ref t T t p 3 npψ r t ( () ()) (). (6.) From the control law (6.) and the illustration of u c in Figure 6-3, we see that the controller outputs are aligned to the stator flux space vector. We may consequently interpret ISC as a stator-flux oriented scheme (direct fieldorientation). The flux controller modifies the magnitude of the stator flux via a voltage contribution in the direction of the stator flux through u Ψ. The torque controller, on the other hand, affects the frequency of the stator flux space vector (and hence the torque via the load angle) through the component u T. Figure 6-3: Orientation of the ISC feedback controller outputs to the stator flux space vector. Note that the signals u Ψ and u T are not voltages measured in V. The notation u reflects that these signals are seen as process inputs in stator flux coordinates. 6.. Prediction In Subsection 6.. it was shown how time delays were handled by predicting the stator flux and the stator current space vectors. In the discrete-time control law (6.), the stator flux is only predicted due to the time delay of a sampling interval T s, which is due to precalculation. The stator current, on the other

97 Indirect Self Control 85 hand, is additionally predicted.5t p to estimate the fundamental current in the middle of the next pulse period. However, the approximation of the stator flux derivative in (6.7) can also be considered as an approximation in the middle of the pulse period, i.e., p ( t + T ) ψ ( t ) ψ ψ s s k p s k = ψ s ( tk + T p). (6.3) T T p One may then argue that the effect of the prediction should be modeled by predicting the entire continuous-time control law in (6.) to the middle of the next pulse period. However, the terms calculated from the reference values and control errors were not considered to be predicted. We will therefore only predict the stator flux and the stator current, and hence use the following continuous-time model of ISC u t = u t + ju t ψ ˆ t+ T t + R iˆ t+ T t, (6.4) () ( ψ () ()) ( ) ( ) sref T s d s s d where T d represents the total time delay, i.e., Td = Tp + Ts. (6.5) 6..3 Average Torque If we choose to use the average of the torque when computing the torque error in (6.), then the torque in (6.) is replaced by t T () t = T( τ) dτ AT( t ) T. (6.6) p t Tp In (6.6) we also introduced the formal operator A for the average value over a pulse period. In the Laplace domain, A is therefore given by ( ) ( st p A s = e ). (6.7) st p 6.3 Closed-Loop System In this section we derive a linear model of the closed-loop system when ISC is applied to an induction motor. The closed-loop model is first derived without

98 86 Indirect Self Control time delays in Subsection 6.3., and the effects of time delays as well as disturbance rejection are added in Subsection Model without Time Delays The continuous-time ISC control law (6.) is illustrated in Figure 6-4, where the signal y contains the control variables stator flux magnitude and torque, i.e., y=(ψ s T) T and hence the reference r=(ψ sref T ref ) T. Note that rejection of DC-link voltage disturbances as discussed in Subsection.. is not considered here but will be added first in Subsection Figure 6-4: Continuous-time model of the closed-loop system when ISC is applied to an induction motor (neglecting time delays). The block G in Figure 6-4 represents the induction motor and F fwnl and F NL are the feed-forward and feedback controllers. The signal u c was defined in Equation (6.) and may be represented as u c () t u () t ju () t = + = ψ T () t s () t () () ψ R ψsref ψ r Kp j Kpi + Ki + p 3 npψ r () t Tref t T t F r y NL R ψ sref () t r + j + jn pωm() t, 3 npψ r () t Tref () t F r fwnl (6.8)

99 Indirect Self Control 87 which follows from the expressions for u Ψ (t) and u T (t) in (6.) and (6.). In (6.8) we also formally defined the complex-valued controller blocks F NL and F fwnl shown in Figure 6-4. Note also that the average of the torque as discussed in Subsection 6..3 is not explicitly shown in Figure 6-5 or Equation (6.8). It will, however, be considered when examining torque control later. By using a real-valued representation as defined in connection with FOC in Section 5., it is shown in Appendix B, that the model in Figure 6-4 can be linearized to give the model in Figure 6-5. Without time delays, variations in the motor speed are perfectly compensated for and ω m therefore does not affect the closed-loop system at all. Figure 6-5: Linear model of the closed-loop system with ISC (neglecting time delays). In Appendix B it is also shown that, except for torques close to the pull-out torque, the system P appearing in Figure 6-5 equals ψ s s P( s) = stσ +. (6.9) TT σ st σ + 3 n Pψ r stσ + Rr ( stσ + ) To motivate why the system P is lower triangular, we consider the orientation of the controller outputs shown in Figure 6-3. Here we see that the first component of the input, u Ψ, affects the stator flux magnitude. From the torque equation (3.), we realize that a change of the stator flux magnitude also affects the torque, as long as the steady-state load angle δ is non-zero (which is equivalent to T ). Hence, with nonzero torque, the first component of the input affects both components of the output. A modification of u T, on the other hand, only affects the frequency of the stator flux space vector. This results in a change of the load angle and hence the torque, but not in a change of the stator flux magnitude. Note that P in (6.9) has the same structure as the

100 88 Indirect Self Control system P achieved with FOC in (5.). Note, however, that the systems are not equal though, although we use the same notation. The controllers F fw and F in Figure 6-5 are the linear transfer functions from the control errors to the control inputs and are given by ψ K p Ffw ( s) = Rr, F( s) Rr = K p K. (6.3) i 3 npψ + r 3 npψ r s Finally, the disturbance transfer function G Ud in Figure 6-5 can be evaluated as Ri s scosϕ ψ sω ω DUd =. (6.3) U d Ri s ssinϕ + ψsω In the expression for D Ud in (6.3), the constant ϕ denotes the steady state angle between the stator current and the stator flux space vectors. This angle is shown in Figure 6-6, where some steady state space vectors are drawn at 7.5 Hz mechanical rotor speed (8 % of base speed) and a slip frequency of Hz (8 % of pull-out slip frequency). For illustration purposes, the flux vectors have been normalized such that the length of the stator flux is half the length of the stator voltage vector. The fluxes would have been very small otherwise. Figure 6-6: Space vectors at 7.5 Hz mechanical rotor speed and Hz slip frequency.

101 Indirect Self Control 89 As seen in Figure 6-4, the ISC control law consists of several feedback loops. From the linear model in Figure 6-5, it follows that apart from the feedback though the PI controllers, these feedback loops realize a transformation of the motor transfer function G into the system P. From (6.9) we see that the system P is diagonal at zero torque. This means that control of torque and flux control is completely decoupled (as the feedback and feedforward controllers are also diagonal). In general, the system P is lower triangular and the torque control does not affect the flux loop, whereas the flux control affects the torque. As the flux reference normally is kept constant, the coupling from the flux reference to torque is not a practical problem. The idea of ISC can hence be interpreted as (partial) decoupling followed by PI control. This is explicitly shown in Figure 6-7, where the ISC controller is represented by the feedback and feedforward blocks F and F fw, but also through the decoupling block G - P. From the control law (6.) we realize that the decoupling block corresponds to the orientation to the stator flux space vector. Actually, also the influence of a varying rotor flux magnitude has been included in the system P, see Appendix B. The effect is small though. Figure 6-7: Linear model of the closed-loop system with ISC expressed as an inverse-based controller Model with Time Delays and Disturbance Rejection The models shown in the previous subsection did not consider time delays or compensation of DC-link voltage disturbances. Apart from the outputs of the feedback and feedforward controllers, the signals of the control law (needed for field-orientation) are predicted to compensate for the time delay, see Subsection 6... In Appendix B it is shown that if the time delays are short, we may model the system as shown in Figure 6-8. Here the decoupling block is hence not affected by the time delay. In Figure 6-8 we see that rotor speed variations are not perfectly cancelled in presence of time delays. We have also added normalization of the coupling

102 9 Indirect Self Control vector to compensate for variations in the DC-link voltage as discussed in Subsection... In the figure D ω = ( -n p ) T, see Equation (B.3) in Appendix B, and D Ud is given by expression (6.3). Figure 6-8: Linear model of closed-loop system with ISC and disturbance rejection. During treatment of DC-link stability in Chapter, we will need explicit expressions for the transfer functions G c and G d in the torque equation (.3). From Figure 6-8 and the shape of the transfer functions P, F and F fw, it follows that and ( ) ( fw) ( Ffw ) PD F + PDF Gc = I + PDF PD F + F = + ( ) Ud ( ) ( ) P Gd = I + PDF PD DA = PDUd DA D Ud = + + PDF P + PDF DUd (6.3) (6.33) From (6.3) it follows that D Ud /D Ud at higher speeds, which means that the transfer function G d in (6.33) then can be approximated as PDUd ( DA) Gd. (6.34) + P DF Note that the transfer functions G c and G d in (6.3) and (6.34) are not affected by the flux controller. This also follows from the representation of the closedloop system in Figure 6-9, which is valid at higher speeds.

103 Indirect Self Control 9 Figure 6-9: Representation of the closed-loop system valid at higher speed. Here it is clear that the flux control affect the torque control loop but not the other way around. We also see that the disturbances only affect the torque loop. From Figure 6-9 it is clear that the two disturbances U d and ω m only add to the torque loop (at higher speeds). This is because the two disturbances (approximately) add to the stator voltage in the same direction as u T, i.e., orthogonal to the stator flux space vector, see (6.4) and Figure 6-3. For the rotor speed this is clear as the compensation, which completely cancels the disturbance without time delays, is part of u T, see (6.). To see that also U d adds in the direction orthogonal to the stator flux space vector, we note from (.8) that U d adds in the direction of the stator voltage space vector u s. From Figure 6-6 we see that u s is approximately orthogonal to Ψ s at higher speeds as then R s i s << u s. 6.4 Controller Tuning In this section we discuss how to set the torque and flux controller parameters to achieve certain performance requirements (here we do not consider effects of imperfect field-orientation when selecting the parameters, see further Subsection 7.3.). We will treat the two loops in Figure 6-9 separately and neglect the interaction. We also note that we may interpret the torque controller as a feedback controller and a prefilter as shown in Figure 6-. We only design F to reach desired properties of the feedback loop.

104 9 Indirect Self Control Figure 6-: Torque control loop where the ISC is interpreted as a feedback controller and a prefilter. The prefilter F pre in Figure 6- is given by F pre ( K p + ) Ffw + F K = = F K K i p i s +, (6.35) s + which is a lead filter with high frequency gain +/K p. The Bode plot of the prefilter (6.35) is shown for an example in Figure 6-. In Section 5. it was shown that the effect of using active damping with FOC is equivalent to using a feedback controller and a pre-filter, just like the structure in Figure 6- (cf. Figure 5-3). With FOC and active damping, the prefilter was selected as a lag filter with high frequency gain k p /(k p +R a ) to slow down the response from the reference. Magnitude [abs] Phase [deg].5..5 Bode plot of prefilter F pre 3 Frequency [Hz] Figure 6-: Bode plot of prefilter with ISC torque control. The effect of the prefilter is an increased gain at higher frequencies. In the following two subsections we discuss two possibilities of controller tuning. For simplicity, we approximate the effect of the average operator A in

105 Indirect Self Control 93 the torque loop by an additional time delay of half a pulse period. The time delay in the torque loop is then modified from T d s to T da s, where Tda = Td + Tp = Tp + Ts. (6.36) Remark: Note that we here give expressions for the continuous-time controller parameters. In a real application, the discrete-time implementation of ISC must be used, where the corresponding discrete-time controller parameters can be determined from the continuous-time parameters from (6.3) Zero - Pole Cancellation From the expressions for the system P and the feedback controller F in (6.9) and (6.3), we see that by choosing the controller parameters as Tσ ψ Kp =, Ki =, Kp =, (6.37) T T ψ T ISC ISC s ISC where T ISC is a design parameter, the loop gain becomes L = / st ISC in both control loops. Due to the average operator, the time delays in the two loops however differ. In the flux control loop, the time delay is T d s and the phase margin φ m then becomes π φm = arg{ L ( jωc) } + π = Tdωc, (6.38) where ω c the crossover frequency. If we for example require a phase margin of 6, it follows that the crossover frequency has to be restricted to π 6 ωc = or TISC Td. (6.39) T 6T π ISC d The phase margin in the torque control loop is similarly given by π φm = arg{ L( jωc) } + π = Tdaωc, (6.4) and a requirement of a phase margin of 6 then leads to π 6 ωc = or TISC Tda. (6.4) T 6T π ISC da

106 94 Indirect Self Control Remark: Note that the choice of controller parameters in (6.37) means that the feedback controller F is chosen as /(st ISC )P - at zero torque. If we neglect that P in general is nonzero (although rather small), we may consider this choice of controller parameters as a special case of Internal Model Control (IMC), where the process dynamics are inverted (compare with the tuning of FOC in Section 5.). Remark: With FOC, the interaction between i sd and i sq due to the back EMF was suppressed by introducing so called active damping. Similarly, we could introduce an additional degree of freedom in the torque control loop as suggested in Figure 6- to reduce the influence of the cross coupling. Still, the same effect may be achieved by modifying the controller parameters and the prefilter. Figure 6-: Additional degree of freedom to suppress interaction with ISC similar to active damping with FOC. Here d represents the disturbance from the flux loop, cf. Figure Loop Shaping An alternative approach to setting the controller parameters is to apply loop shaping ideas. This will be done in this subsection for the torque control loop, where the loop gain is given by Kp + Ki s K sk p / Ki + da i PDF T TA= e = e st + s st + σ T s Tdas σ. (6.4) To obtain large stability margins, we want a slope of the loop gain of - around the crossover frequency ω c. As we normally have that ω c >>/T σ, this implies that the following condition should be satisfied Ki < ωc. (6.43) K p

107 Indirect Self Control 95 It then follows that the crossover frequency ω c and the proportional gain K p are (approximately) related through K p = PDF T TA. (6.44) ωct σ That is, to reach a desired crossover frequency ω c, the proportional gain should be selected as K p = ω c T σ. By making the quotient in (6.43) a factor eight less than the crossover frequency, the proposal for controller parameters then becomes Tσ ωc T Kp = ωctσ =, Ki = Tσ =, (6.45) T 8 8T ISC where ω c = /T ISC is the desired crossover frequency. Remark: In [56] also the effect of the feed-forward controller was considered when deriving the controller gains. Here the following suggestion was given Tσ Tσ Kpi =, Ki =. (6.46) TISC 4TISC With these parameter settings, the actual crossover frequency becomes and the phase margin can be derived as σ ISC + 5 ωc = ωcd ωcd =, (6.47) ( ) T ISC + 5 Tda φm = arctan + 5. (6.48) T ISC Gain Scheduling Often the switching frequency of the inverter is varied as a function of the stator frequency to avoid low frequency torque oscillations and to generate predictable harmonics. This in turn means that also the time delays vary with the stator frequency, see further Chapter. Varying time delays mean that the controller bandwidth and/or the stability margins change with the operating points. To keep a certain phase margin φ m with the zero-pole

108 96 Indirect Self Control cancellation method, it follows from (6.4) that the crossover frequency in the torque loop should satisfy π ωc = φm T da. (6.49) From the expression (6.37) we see that the controller parameters then should be updated according to π π Kp = φ m Tσ, Ki = φ m. (6.5) Tda Tda / T ISC For flux control it follows from (6.37) and (6.38) that ψ π K p = φ m Tdψ s. (6.5) Remark: If the sampling time is much less than the pulse period, the time delays in the two control loops are proportional to the pulse period. In the flux loop we have T d = T s +½T p ½T p and in the torque loop T da = T s +T p T p, see (6.36). From expression (6.3) it then follows that the corresponding discretetime controller parameters to (6.5) are constant. That is, Tp π Tp π Tp π Pψ = φm, Knr = φm Tσ, Tnr = φm. (6.5) T d ψ s T da T da Simulations In this section a few simulation results are shown to evaluate the performance of the ISC controller. Here the original non-linear equations of the induction motor and the controller are simulated and the ISC is tuned according to (6.46). Actually, the discrete-time controller is used and the controller parameters are recalculated using expression (6.3). We first examine step responses in torque and flux and then give an example of disturbance rejection. Step responses with the ISC are shown in Figure 6-3 for the three operating points defined in Chapter 4, i.e., at speeds of %, 5% and 9% of base speed. The responses are scaled such that in the plot corresponds to 5% of nominal torque and flux. The simulation consists of:

109 Indirect Self Control 97 Torque steps between and 5 Nm (83% of nominal torque) A flux step from % to 9% of nominal flux at time.5 s From Figure 6-3 we see that the responses are more or less independent of the operating point and that the cross coupling is very small, which fits with the derived expressions for the closed-loop system. Step Responses Scaled Torques and Fluxes Time [s] Figure 6-3: Step responses with ISC at three different operating points. The upper curves represent the torque responses are the lower curves represent the flux responses where the signals are scaled such that in the plot corresponds to 5% of nominal torque and flux. The solid lines show the responses at % of base speed, the dashed curves show the responses at 5% of base speed and the dotted curves show the responses at 9% of base speed. We see that the responses look similar at the different operating points and that the cross coupling is very small. The investigation on disturbance rejection performed in Section 4.9 showed that suppression of DC-link voltages is limited by input constraints at higher speeds. The reasons for this is that, at higher speeds, the stator voltage magnitude approaches its upper limit and disturbances (adding to the stator voltage magnitude at the input of the plant) can not be completely compensated for. Rotor speed disturbances on the other hand should not be a problem. In Figure 6-4, simulation results are shown when introducing disturbances in the DC-link voltage and the rotor speed. The sampling time is set to.5 ms and the controllers are tuned to give maximum bandwidth with 45 phase margin, i.e., 8 Hz for torque control and 6 Hz for flux control. Figure 6-4.A shows the responses in torque and flux at a step in the DC-link voltage from 75V to 6V. The responses at % of base speed are shown as the solid lines, the responses at 5% of base speed are shown as the dashed lines

110 98 Indirect Self Control and finally the responses at 9% of base speed are shown as the dotted lines. As seen, the torque control error is large at 9% of base speed. Remember that the performance requirement is to keep the scaled control error less than one. It also follows that the stator voltage magnitude saturates, as the controller is not able to keep the flux at its reference value. The DC-link voltage acts in a high gain direction and give large errors in flux and torque. High frequency components of the stator voltage magnitude, caused by the DC-link voltage mainly affect the torque, whereas low frequency components affect the flux, cf. the gains in the two input directions in Figure 4-5. The responses in torque and flux at a step in rotor speed of half the pull-out slip frequency are shown in Figure 6-4.B for the three operating points. As seen, this disturbance can be efficiently suppressed. 5 Torque and flux response at DC-link voltage step Torque. Torque and flux response at rotor speed step Torque Flux.6.4. Flux Time [s] Time [s] Figure 6-4: (A) Control errors due to DC-link voltage step and (B) control errors due to rotor speed step. Here the signals are scaled such that in the plot corresponds to 5% of nominal torque and flux. At higher stator frequencies, DC-link voltage disturbances give large errors in flux and torque. 6.6 Summary In this chapter we have derived a continuous-time linear model of the closedloop system when ISC is applied to an induction motor (excluding the input filter dynamics). The effects of time delays and disturbance rejection have here been carefully treated. It has been shown that the orientation to the stator flux used with ISC effectuates a partial decoupling of torque and flux control. At zero torque, the decoupling is perfect. In general, however, the flux loop affects the torque loop and the coupling increases with the torque level. Based on the linear models, expressions for the ISC controller parameters have been derived to achieve a given crossover frequency with reasonable phase margins.

111 Chapter 7 ISC with Observer In practice, torque and the stator and rotor fluxes are usually not directly measured. In order to implement the ISC control law in Figure 6-4, these signals therefore have to be estimated by an observer, i.e., a model-based estimator using measurements. In this chapter we will discuss how to design a so called full-order observer. We will also derive models of the closed-loop system including observer and show that the steady-state errors of torque and stator flux magnitude with reasonable model errors are small at higher speeds. Estimation of torque and stator flux is discussed in Section 7. and a full-order observer for the induction motor is presented in Section 7.. Here also steady state errors due to incorrect motor parameters are investigated. Finally, Section 7.3 deals with dynamic properties of the closed-loop system including observer. Note that there is no intention to give a complete description of flux and torque estimators for induction motors in this chapter. We only give some insight into the estimation problem. 7. Torque and Stator Flux Estimation From the closed-loop system in Figure 6-4, it follows that an implementation of the ISC control law requires the motor torque, the stator flux, and the rotor flux magnitude. The rotor flux magnitude is used for conversion from torques to slip frequencies and an accurate estimate is not very critical for good closed-loop performance. The stator flux is needed for decoupling, i.e., for orientation of the outputs of the feedback and feed-forward controllers, see (6.). The stator flux is also used to estimate the torque through (3.8), which also can be written as ˆ 3 * T t = n Im i t ψ ˆ t. (7.) { } () () () p s s

112 ISC with Observer Here the asterisk means complex conjugate and we note that torque estimation according to (7.) does not directly depend on any motor parameters, only via the estimated stator flux. Hence, we may conclude that good performance of ISC requires an accurately estimated stator flux, which via (3.5) can be obtained as t ( ˆ ) ψˆ () t = u ( τ) R i ( τ) dτ. (7.) s s s s The only involved motor parameter in (7.) is the stator resistance. The stator resistance is relatively easy to estimate and the stator winding temperature may be measured to adapt the resistance to temperature variations. Also, the influence of R s is only significant at low speeds as u s >> R s i s at higher speeds. Stator flux estimation using (7.) should therefore be fairly independent of parameters at higher speeds. As discussed in Section 5., pure integration as in (7.) causes problems in practice. The integration may then be replaced by a low-pass filter as done with the Statically Compensated Voltage Model (SCVM), see [7]. An alternative to avoid the pure integration is to estimate the stator flux using a full-order observer of the induction motor (which however requires the motor speed). This approach is treated in the following section. Remark: Proper operation with ISC relies on a good estimate of the stator flux. This is opposed to the need for a good estimate of the rotor flux when using the control scheme FOC, presented in Chapter 5. Estimation of the rotor flux was briefly discussed in Section 5., where the so called voltage model (VM) was given by Equation (5.6). This equation corresponds to the stator flux estimator (7.). The only involved motor parameter in (7.) is the stator resistance, compared to the stator resistance and the leakage inductance in (5.6). Therefore it is often argued that stator flux orientation is less sensitive to parameter variations compared to rotor flux orientation, see e.g. [38] and [9]. At lower speeds, the so called current model (CM) is preferred for rotor flux estimation. Similar models exist also for stator flux estimation but will not be considered here, where focus is on sensitivity to parameter errors at higher speeds. 7. Full-Order Observer In this section a full-order observer for the induction motor is presented. The observer is based on a mathematical model of the motor and consequently

113 ISC with Observer depends on the motor parameters. These parameters therefore need to be estimated, but also correctly adapted to the operating points. We will briefly describe normal parameter variations, but also examine the effects of using incorrect parameter values for the steady-state estimates of the control variables stator flux magnitude and torque. 7.. Observer Equations One way of avoiding pure integration of the stator flux estimator (7.) is to replace the measured stator current by an estimated stator current. According to (3.7), the stator current can be estimated through a linear combination of the fluxes as ˆ i () ˆ () ˆ s t = + ψs t ψ r () t. (7.3) Lm Lσ Lσ Hence, the stator flux estimate now also appears on the left side of (7.), which removes the pure integration. As the stator current equation (7.3) also involves the rotor flux, this method requires both motor equations (3.5) and (3.6) to be simulated, i.e., the full order induction motor model (note that we consider the speed to be measured and that the rotor flux magnitude is also needed by the ISC control law to compute u T in (6.)). The observer can then be represented by the following state space model ˆ x = A ω x ˆ + B u, (7.4) where the state vector is given by xˆ () t ( ) m () t () t s ψˆ s =. (7.5) ψˆ r From the motor equations (3.5) and (3.6) it follows that the system matrices A and B in the state equation (7.4) equal A ( ω ) m R s Rs + L m L σ L σ =, B =. (7.6) R r R r jnpωm() t Lσ L σ

114 ISC with Observer With perfect parameter estimates, the equation for the estimation error is given by x t = A ω x t, (7.7) where the estimation error is defined by () ( ) () m () t = () t ˆ () t x x x. (7.8) Convergence of the estimation error for the general case with a varying speed is considered e.g. in [88] using Lyaponov techniques. Here we only examine properties of the observer at fixed speeds. The convergence properties of the estimation error can then be studied through the eigenvalues of the matrix A, which are shown in Figure 7- at zero torque for a number of constant rotor speeds. The locations of the poles at % and 5% of base speed are marked with * s and o s, respectively. The appearance of the eigenvalue plot of the matrix A depends on the relation between the stator and rotor resistances, see for example [3]. In our example R s >R r. Also compare with the pole-zero map for the polar representation in Figure 4-. The plot in Figure 7- is for a model in stator coordinates, whereas the plot in Figure 4- is for a model in polar coordinates. 6 Poles as a function of the speed 4 Im Re Figure 7-: Poles of the observer as a function of speed at zero torque (a non-zero torque only slightly affects the locations of the poles). The locations of the poles at % and 5% of base speed are marked with * s and o s, respectively. In Figure 7- we see that the eigenvalues are in the left half plane for all speeds, which means that the estimation error goes to zero for all operating points. At low speeds, the convergence may be rather slow, however, as the

115 ISC with Observer 3 real parts of the poles are close to the origin. Further, the poles of the observer are poorly damped at higher speeds. These properties can be improved by adding a correction term to the state equation as ˆ x = A( ω) xˆ + Bu ( ˆ s + kobs is is ) (7.9) iˆ = Cxˆ, where (cf. (7.3)) s C = +. (7.) Lm Lσ Lσ The observer gain k obs in the modified state equation (7.9) is a x vector (with in general complex-valued elements). Remark: Sometimes an observer gain is required in order to use the term observer. The term estimator is then used with zero observer gain. Here we use the term observer in both cases though. We will also refer to an observer with zero observer gain as an open-loop observer and hence to an observer with non-zero gain as a closed-loop observer. With the observer according to (7.9) (and perfect parameters), the system matrix for the estimation error is now given by A(ω)-k obs C. Here k obs can be selected to affect the properties of the observer. In general, the observer gain should be set to obtain: Well damped poles of the observer Fast convergence of the estimation error Low sensitivity to parameter errors Selection of the observer gain k obs is for example discussed in [88] (with rotor flux and stator current as state variables). In [88] an observer gain is proposed for the full-order observer to move the poles of the observer to ( / ω) and ( / ω) T ± j p T ± j p, (7.) r r where p and p can be chosen arbitrarily and T r =L m /R r is the rotor time constant. The focus is then on convergence of the initial estimation error under the assumption of perfect motor parameters. In [37], the influence of the observer gain on parameter sensitivity is discussed for rotor flux estimation. For low sensitivity of rotor flux estimation to parameter errors, it is well known that the so called current model (CM) is preferred at low speeds and

116 4 ISC with Observer the voltage model (VM) at high speeds. The CM and VM are reduced order observers and the observer gain of a full-order observer can be used to shift between the two models to minimize parameter sensitivity over the entire speed range, see e.g. [5] and [37]. It should also be noted that the design goals of the observer stated above may be conflicting. For example, in [89] an observer is designed for low sensitivity to rotor and stator resistance variations, where the convergence is worse than with zero observer gain. The pole placement of another strategy, which will be used during simulations in this thesis is shown in Figure 7-. Except at low speeds, the convergence rate (real part of poles) as well as damping has now been improved compared to using zero observer gain. 6 Eigenvalue as a function of the speed 4 Im Re Figure 7-: Poles of error dynamics with an observer that uses a full observer gain. The dashed curves show the pole locations with zero observer gain. The locations of the poles at % and 5% of base speed are marked with * s and o s, respectively. Remark: An alternative to stabilizing the stator flux estimator (7.) is to directly add a correction term in the following way ψ = Rˆ i + B u + k i i ˆ. (7.) ( ) ˆ s s s s obs s s Note, however, that by writing k = k + Rˆ, (7.3) obs obs s where k obs is a general gain, it follows that (7.) can be written ψ = Rˆ i ˆ + B u + k i i ˆ, (7.4) ( ) ˆ s s s s obs s s

117 ISC with Observer 5 which equals (7.9) (the rotor equation is not affected). Hence, the two approaches (7.) and (7.9) are equivalent. Practical experience shows that an observer gain k obs in the range of a least R s is needed for proper behavior and we are hence back to the full-order observer. Remark: A further detail that is left out here is implementation issues when realizing the observer equations on a microprocessor. It is of course essential to have a numerically stable algorithm. For example, direct discretization of the equations by using Euler forward may give an unstable system with long sampling times, see [8], [9]. Here it is proposed to integrate the rotor flux in rotor flux coordinates before the transformation to stator coordinates. 7.. Parameter Variations This subsection discusses parameter variations to be considered during performance analysis with model errors. The motor resistances R s and R r strongly vary with temperature. With a temperature coefficient of.39 (copper) and a nominal temperature of C, a temperature interval of for example -4 C - 5 C (which is realistic for a traction motor) gives a resistance variation between 77% and 9% of the nominal values. In practice, the stator temperature may be measured or temperature models may be used to estimate the motor temperature, which decreases the temperature deviation between the model and the motor. However, model errors for temperature estimation and dynamic variations still give temperature errors, particularly in the rotor. The main inductance L m is subject to saturation, which is illustrated in Figure 7-3.A. The variation of the parameter value depends on which nominal value is chosen. With the nominal inductance chosen at nominal flux, it follows from Figure 7-3.A that the main inductance in this example may change between 66% and 4% of the nominal value for flux levels up to 3% of nominal flux. The leakage inductance L σ may vary with the load (for example with cast rotors) as shown in Figure 7-3.B. (Due to the inverter switchings, the lower limit of the total RMS current is larger than zero. Consequently, the large values of the leakage inductance corresponding to very small rotor currents may not appear in practice.) With the nominal value of L σ chosen as.79 mh (see Appendix A), the inductance may at least vary between 7% and % of the nominal value.

118 6 ISC with Observer Main Inductance [mh] Main Inductance as a function of stator flux Flux/Rated Flux Leakage inductance Lσ [mh] Leakage inductance as a function of rotor current Rotor current [ARMS] Figure 7-3: (A) Inductance L m as a function of the stator flux. (B) Total leakage inductance L σ as a function of the rotor current. The marked points represent actual measurements. For the analysis of sensitivity to parameter errors, we will consider (independent) variations in the motor parameters between 5% and % of the nominal values (unless anything else is specified). From the figures for the inductance variations above we see that this range is quite large. Also for the resistance values with temperature measurement, these variations are larger than what would be expected in practice. We hence test the closed-loop performance under fairly tough conditions Steady State Stator Flux Estimation By using the full-order observer we made the observer stable compared to directly estimating the stator flux through (7.). On the other hand, the fullorder observer depends on all motor parameters, not only the stator resistance. In this subsection, the steady-state estimation errors of the stator flux magnitude are examined when using the full-order observer (7.9). We restrict the analysis to the case with zero observer gain. In Appendix C it is shown that with zero observer gain, the real flux magnitude is related to the estimated stator flux magnitude as ψ ψˆ s s = ( ω ˆ Tσ ) ( ω Tˆ σ ) ˆ ˆ ˆ ˆ Rs ωtσ Rs Rs + ω ˆ + + ˆ ˆ L ( ˆ ) Lm L σ + ω T σ σ +. (7.5) Rs ωt Rs Rs ( ωt σ σ ) + ω + + Lσ + ( ωt ) Lm L σ σ + ( ωtσ )

119 ISC with Observer 7 Ideally, integrations in the control loops with ISC make the estimated flux match the reference flux and the estimated torque match the reference torque at steady state (see Appendix B). This means that the stator flux estimate equals the flux reference and the torque estimate equals the torque reference. Hence, for example the flux quotient (7.5) then corresponds to the ratio Ψ /Ψ ref. We start analyzing the flux quotient (7.5) at zero slip frequency, where it reduces to ψ ψˆ s s = ( npωm) ( npωm) Rˆ s + Lˆ m R s + Lm. (7.6) Hence, at zero slip frequency only the stator resistance and the main inductance have influence on the flux estimation error. The rotor parameters are not important as no current is generated in the rotor circuit. As the speed increases, the stator flux estimation error decreases (the quotient (7.6) goes to one). This seems reasonable as the parameters enter the flux equation through the term R s i s, which has small influence at higher speeds. In Figure 7-4.A the factor (7.6) is shown for parameter variations (of the real motor) in R s and L m. For example, using a too small value of the stator resistance in the observer gives a too large stator flux estimate (which with flux control leads to a too small real flux). With non-zero slip frequencies, also the rotor parameters R r and L σ influence the stator flux estimate. At increasing slip frequencies, the influence of the main inductance decreases. The influence of the leakage inductance is also small, but incorrect values of the two resistances have large effect on the flux estimate at low speeds. This is shown in Figure 7-4.B at 5% of nominal pullout slip frequency (which corresponds to % of nominal torque with nominal parameters), where the resistances of the motor are varied. The worst combinations are when the resistances are estimated in opposite directions. An underestimation of the stator resistance gives a too large flux estimate (too small real flux), whereas the opposite holds for the rotor resistance. Remark: A similar analysis is performed in [37] for rotor flux estimation using so called Frequency Response Functions (FRF).

120 8 ISC with Observer 4 Flux estimation error factor 4 Flux estimation error factor Speed [pu] Speed [pu] Figure 7-4: (A) Flux estimation factor at zero slip frequency with variations in R s and L m. (B) Flux estimation factor at 5% of the nominal pull-out slip with variations in R s and R r. The speed is here normalized by the base speed Steady-State Torque Estimation With torque estimation according to (7.), and stator flux estimation using a full-order observer with zero gain, the estimated torque at zero slip frequency can be evaluated as (see Appendix C) Tˆ R Rˆ n ω s s p m 3 ˆ np Lm L m = ψˆ s L m R s ( npωm). (7.7) + L m Expression (7.7) shows that the torque estimate at zero slip frequency may in fact be nonzero. The estimated torque (normalized to nominal torque) at zero slip frequency is shown in Figure 7-5 for variations in the motor parameters R s and L m. We see that rather large errors may appear at low speeds but that errors decrease with increasing speed. Again, the reason for this is that the parameter dependencies enter via the estimated stator current, whose influence only is significant at low speeds. At non-zero slip frequencies, the quotient between the estimated torque and the real torque is given by (see Appendix C) tan ( χis χs ) ( ( ) ) T ψ s = Tˆ ψˆ cos χ tan χ χ + tan χ s s is s s, (7.8)

121 ISC with Observer 9 where χ = χ ˆ χ and s s s tan tan ( χ ) ( ˆ u χs tan χu χs) ( χ χ ) ( χ ˆ χ ) tan + tan χs =, (7.9) + tan tan ( χ ˆ χ ) u u s u s Rˆ ˆ s ωtσ + ω + npωm Lˆ σ + s =, (7.) Rˆ s ˆ ˆ Lm L Tˆ ( χ χ ) is ( ω ˆ Tσ ) ˆ ( ˆ R ωt ) s σ + σ + ( ω σ ) Tσ ω Lσ s = + + Lm Lm Lσ ( T ω ) σ. (7.) Torque/nominal torque Torque estimation error Speed [pu] Figure 7-5: Torque estimation error at zero slip frequency with varying stator resistance and leakage inductance. The speed is here normalized by the base speed. The expression for tan(χ u -χ s ) in (7.9) is given by (7.) but without hats. We see that the amplitude errors as well as angle errors of the estimated flux contribute to the torque estimation error. Remark: For (7.8) to be well defined, we need to assume that the estimated torque is non-zero. It is then not sufficient that the slip frequency is non-zero. This inverse of (7.8) is, however, well defined for non-zero slip frequencies.

122 ISC with Observer The reason why we still choose to use the ratio (7.8) and not the inverse of it, is that with an integrating controller, the estimated torque equals the torque reference at steady state. The quotient (7.8) then shows the steady-state torque as a factor of the reference torque. Figure 7-6.A shows the torque quotient (7.8) for variations in the rotor resistance R r and the leakage inductance L σ at 5% of nominal pull-out slip frequency. The torque estimation errors are mainly due to errors in the rotor resistance. Figure 7-6.B shows the corresponding torque quotient for variations in the stator and rotor resistances. Again we see that the errors decrease with speed. We have large errors at low speeds, however..5 Torque estimation error factor.5 Torque estimation error factor Speed [pu] Speed [pu] Figure 7-6: (A) Torque estimation quotient with variations in the rotor resistance and the leakage inductance. (B) Torque estimation quotient with errors in the stator and rotor resistances. The results are shown at 5% of nominal pull-out slip frequency. The speed is here normalized by the base speed. Just for comparison, we also check torque estimation with an estimated stator current in (7.). At zero slip frequency the estimated torque now is zero, cf. (7.7). At nonzero slip frequencies, the corresponding quotient to (7.8) becomes T ˆ R r ψ + s = Tˆ R ˆ r ψ s + ( Tˆ σω ) ( T ω ) σ. (7.) At non-zero slip frequencies, we see from (7.) that torque estimation errors due to errors in the rotor resistance (and to some extent the leakage inductance) do not decrease with speed. In Figure 7-7 the torque quotient (7.) is shown with variations in the rotor resistance and the leakage inductance, cf. Figure 7-6.A. The largest effect is due to errors in the rotor resistance. Overestimating the rotor resistance gives a too large torque, cf. (7.).

123 ISC with Observer.5 Torque estimation error factor Speed [pu] Figure 7-7: Torque estimation quotient with pure simulation. The results are shown at 5% of nominal pull-out slip frequency for variations in R r and L σ. The speed is here normalized by the base speed. 7.3 Model of Closed-Loop System with Observer In this section we derive linear models of the closed-loop system obtained when ISC including an observer is applied to an induction motor. These linear models will be used to analyze robustness of ISC with respect to model errors in Chapter 8. With the observer given by (7.9) and torque estimation according to (7.), the closed-loop system can be drawn as in Figure 7-8 if time delays are neglected. Figure 7-8: Block diagram of the closed-loop system with ISC control including observer.

124 ISC with Observer The block G T in Figure 7-8 indicates that the control variables y are generated as a mixture of measured and estimated signals; the torque is calculated using measured current and estimated stator flux, see (7.). From the figure we also note that the measured current is used for cancellation of the voltage drop across the stator resistance, cf. the control law (6.). Finally, the observer uses the measured current for correction of the estimated quantities, i.e., the observer gain k obs in general. In order to simplify the analysis of the closed-loop system in Figure 7-8, we introduce the following approximations: The control variables torque and stator flux magnitude are measurable. According to the steady state analysis in Section 7. (with k obs = ) we know that this is not quite true, although the stationary estimation errors at higher motor speeds are rather small. The estimated stator current is used to cancel the voltage drop across the stator resistance in the ISC control law (6.). Simulations show that this has no major effect on performance, at least not at higher speeds. Both approximations have largest effect at low speeds, where the voltage drop across the stator resistance does not give a negligible contribution to the stator voltage. The results of the analysis may therefore not be accurate at very low speeds. However, from the analysis of the RGA elements in Section 4.4, we know that the system is sensitive to uncertainty around the stator frequency and that this sensitivity increases with the motor speed. With the assumptions above, we may still adequately analyze these expected problems at higher speeds. In Subsection 7.3. a model of the closed-loop system with zero observer gain is derived and in Subsection 7.3. models with non-zero observer gains are discussed Model with Zero Observer Gain With zero observer gain (and the approximations above), the closed-loop system in Figure 7-8 can be represented as in Figure 7-9. In Figure 7-9 all the motor quantities needed for control, except the torque and stator flux magnitude for the feedback controllers, are estimated by the (open-loop) observer. This means that we can represent the closed-loop system as in Figure 7-, which follows from Figure 6-4 and Figure 6-7.

125 ISC with Observer 3 Figure 7-9: Closed-loop system including observer with zero observer gain. Figure 7-: Linear model of closed loop system with (open-loop) observer. Remark: In Figure 7- we use hats also above the controller blocks as these involve estimated values of the rotor resistance and the steady-state rotor flux magnitude, see (6.3). By introducing the following notation ˆ ˆˆ ˆ K = G PF K = G PF ˆˆ, (7.3), fw fw and adding the time delay, we may also represent the system as in Figure 7- (again the time delay is assumed to not affect the decoupling). K fw r K + D G y Figure 7-: Feedback representation of the closed loop system including time delays.

126 4 ISC with Observer With the linear feedback model in Figure 7-, we may now examine robustness (robust stability) to model errors. For that purpose we introduce a perturbed model G p of the motor with relative input uncertainty as p ( ) G = G I + w, (7.4) where is an uncertainty block (representing time delays as well as model errors) and w is a positive real scalar weight function. The system in Figure 7- can then be drawn as in Figure 7-. Figure 7-: Closed-loop system with relative input uncertainty. The feedback loop in Figure 7- can also be represented as in Figure 7-3, where the input complementary sensitivity function Q I is given by I ( ) ( ) Q = I + KG KG = G I + PF PFG. (7.5) Figure 7-3: Equivalent representation of the feedback loop including the uncertainty block. Here we assume that the nominal system (without uncertainty) is stable. If we consider a full uncertainty block, i.e., we only require ( ( j )), σ ω ω, (7.6) then the closed-loop system in Figure 7-3 is stable as long as ( ) ( ) ( ) ( ), w jω Q jω = w jω Q jω ω. (7.7) I For stability, the weight function w hence has to satisfy I

127 ISC with Observer 5 ( ω) w j, ω. (7.8) Q I ( jω ) Note that the selection of the feedback controller F affects the robust stability properties. If for example F is chosen as /(st ISC )P -, then PF = /(st ISC )I, where I is the identity matrix. This matrix then commutes with G. This means that the input complementary sensitivity function in (7.5) equals the output complementary sensitivity function, which is given by Q = (I+PF) - PF. For other choices of F, the factors G - and G in (7.5) may not cancel. Since G has large RGA elements, see Section 4.4, this may give Q I large values, which means that smaller uncertainty blocks are tolerated in order to keep the closedloop system stable. /σ max (Q I ) Frequency [rad/s] Figure 7-4: Size of full uncertainty allowed for stability. The solid limit corresponds to tuning of the torque controller as (6.37), whereas the dashed and dotted limits correspond to tuning according to (6.45) and (6.46), respectively. We now check the robust stability condition (7.8) with the three different proposals of controller tuning given in Section 6.4. For these choices, the limit (7.8) is shown in Figure 7-4 at 5% of base speed (64 rad/s) and zero torque. The solid line corresponds to the limit with the zero-pole cancellation approach (6.37). At zero torque, this actually means that F = /(st ISC )P -. Consequently, the limit is the inverse of the maximum singular value of the output complementary sensitivity function, which is well damped. The dashed curve corresponds to the controller parameters (6.45) and the dotted curve to tuning according to (6.46). Here we see that the maximum singular value of the input complementary sensitivity function increases as the factors G - and

128 6 ISC with Observer G in (7.5) do not cancel. This means that less uncertainty is tolerated for stability. At zero torque, we hence conclude that the parameter selection (6.37) is best concerning robust stability. It should be noted that the full uncertainty considered here is conservative and the difference with a more realistic uncertainty may be smaller. Above we examined robust stability, i.e., which model errors could be tolerated without destabilizing the closed-loop system. However, we also want the performance to stay unaffected with model errors. Here we measure performance in terms of the output sensitivity function. By choosing F = /(st ISC )P -, the actual sensitivity function with relative input uncertainty can be written as where ( ) ( ) Ŝ = I + GK = SG I + Q G, (7.9) stisc S = I, Q = I. (7.3) st + st + ISC As shown in [7], large RGA elements of G may make the actual sensitivity function large. We hence conclude that F = /(st ISC )P - seems like a good choice from the robust stability point of view, but may be less fortunate with respect to robust performance. To consider robust stability as well as robust performance, we will apply µ-analysis to evaluate the controller proposals (6.37) and (6.46) in Section 8.5. An option to design the PI controller F could also be to use µ-synthesis to optimize robust performance. Using µ-synthesis to tune a PI controller has been considered in [47]. ISC 7.3. Model with Non-Zero Observer Gain The linear model in Figure 7- was derived under the assumption of zero observer gain. As was discussed in Section 7., a non-zero observer gain may be desired to improve the convergence properties of the estimation error or to decrease sensitivity to parameter errors. We will now derive a linear model of the closed-loop system with a non-zero observer gain. We will, however, not do a complete investigation but restrict the analysis to observer gains k obs chosen as k obs = (k obs ) T. In this case, it is shown in Appendix D that the loop gain can be modeled as

129 ISC with Observer 7 where ( ) ˆ L = G I + G PF ˆ, (7.3) G ( ) ( ˆ ˆ ) ( ˆ ) ˆ ( ) = k I G PΨ G I + k G G I + k G. (7.3) obs s i obs i i obs i Here G i is the transfer function from stator voltage to stator current and Ψ s is a steady-state matrix that links the outputs of the feedback controllers to the stator voltage, see (B.9) in Appendix B. The actual expression of Ψ s then depends on the representation of the stator voltage. The closed-loop system (without feed-forward controller) may hence be represented by the simple control structure in Figure 7-5, where K is still given by Expression (7.3). From the expression for the loop gain (7.3) we may interpret the effect of using a non-zero observer gain with incorrect parameters as adding additional model errors to the system G. We can always represent the system G as G = G I + = G I +, (7.33) ( ) ( ) for some. From (7.3) we see that the additional model error increases with the observer gain k obs. A large gain hence amplifies the additional model errors. From the RGA analysis we know that (even diagonal) errors at the operating point stator frequency may severely degrade performance and may even cause instability. We hence conclude that a large non-zero observer gain may be bad for stability. r + + K G ~ - y Figure 7-5: Closed-loop system with ISC and observer with non-zero observer gain (feedforward controller neglected). The conclusion of this subsection is that, additionally to the requirements on the observer gain given in Subsection 7.., we also have to consider closedloop performance when choosing the observer gain.

130 8 ISC with Observer 7.4 Summary In this chapter we have considered stator flux and torque estimation, which is needed for practical implementation of ISC. By using a full order model of the induction motor, a stable observer is obtained. For the special case with zero observer gain, expressions for the steady-state estimation errors in stator flux magnitude and torque were derived. These show low sensitivity to model errors as higher speeds, whereas fairly large error may be obtained at low speeds. At low speeds, a non-zero observer gain may therefore be needed. Under the assumption of measured control variables, i.e., torque and stator flux magnitude, linear models of the closed-loop system including observer were derived. With zero observer gain, selection of the PI controller parameters was discussed with respect to robust stability. It was also motivated that non-zero observer gains may lead to stability and performance problems in case of model errors at higher speeds.

131 Chapter 8 Sensitivity and Robustness Analysis This chapter investigates sensitivity and robustness of ISC to parametric and non-parametric (unstructured) model errors. By parametric model errors we mean errors in the motor parameters. Focus in this chapter is on dynamic properties, whereas steady-state effects due to model errors were considered in Section 7.. Here we will show that fairly large model errors must be introduced to significantly degrade (dynamic) performance. With parametric errors, the leakage inductance is the single most sensitive parameter. Overestimation of the leakage inductance leads to decreased stability margins. Performance requirements to be fulfilled by the closed-loop system are stated in Section 8.. Basically, we require the maximum singular value of the sensitivity function to stay below a given upper bound, for all considered model errors. From the previous chapters we know that an implementation of ISC needs knowledge of the motor parameters (for the observer as well as the control law). The motor parameters may be incorrectly estimated but errors may also be introduced by incorrect or neglected adaptation to the operating conditions. Here we consider motor parameter variations between 5% and % of the nominal values (which are used by the controller), see the discussion in Subsection 7... In order to gain insight into the effects of parameter mismatches, the investigation with parametric uncertainty is done in steps with increasing complexity. We will consider: Perfect field orientation in Section 8.. Flux estimation with open-loop observer (k obs = ) in Section 8.3. Flux estimation with closed-loop observer (k obs ) in Section 8.4. Results with parametric uncertainty will be presented at zero torque, but look similar also at non-zero load. Further, non-parametric uncertainties are examined in Section 8.5. For the analysis of both parametric and nonparametric uncertainty we rely on the linear feedback models derived in

132 Sensitivity and Robustness Analysis Section 7.3. This means that the model in Figure 7- is used in Section 8. and Section 8.3 and that the model in Figure 7-5 is used in Section 8.4. For convenience, the system in Figure 7- is repeated in Figure 8-. The induction motor is here represented with the stator voltage magnitude and stator voltage frequency as inputs. This is advantageous when imposing constraints on the inputs. Note that we use the scaled representation of the system that was introduced in Section 4.5. K fw r K + D G y Figure 8-: Feedback representation of the closed loop system. 8. Performance Requirements The requirements on the closed loop system will be set in terms of the transfer functions S and KS. This is the so called mixed-sensitivity approach [7]. Requirements on the output sensitivity function S = (I+GK) - guarantee control performance, whereas the requirements on the transfer function KS secure that the generated control inputs are not too large. 8.. Sensitivity Requirements Performance requirements on the closed loop system will be given in terms of the output sensitivity function S, which for example is the transfer function from references to control errors in Figure 8-. Thus, we demand that σ ( S( jω) ) <, ω, (8.) w jω PS ( ) for some weight function w PS (jω). By defining W PS = diag(w PS, w PS ), we can equivalently state the requirement (8.) as ( WPS ( j ) S( j )),, or WPS S σ ω ω < ω <. (8.)

133 Sensitivity and Robustness Analysis A common choice of w PS, which also is used here, is w PS ( jω ) = jω + ω B M jω+ ω Q B, (8.3) where ω B represents the desired bandwidth, Q defines tolerable values for the sensitivity function at steady state, and M sets the upper limit at high frequencies. In this thesis the parameters are set to M =, Q = e-6 and ω B = 5 rad/s, which result in the limit of the singular values of S shown as the bold dotted curve in Figure 8-. This figure also shows the maximum singular value of the nominal sensitivity function S with a time delay of.5 ms, where the controller parameters are set according to the zero-pole cancellation method in (6.37) with T ISC =.4. According to Subsection 7.3., tuning according to (6.37) is well suited for robust stability at zero torque. We will not explicitly examine stability margins in connection with parametric models errors at zero torque. All parametric model errors examined in this chapter lead to stable closed-loop systems. Upper limit of maximum singular value of S M = ω B = Frequency [rad/s] Figure 8-: Performance requirement for the maximum singular value of the sensitivity function (bold dotted line) and the maximum singular value of the nominal sensitivity function. 8.. Input Requirements The transfer function from references to control inputs in Figure 8- is given by KS. The input to the induction motor is the stator voltage, which here is represented through its magnitude and frequency. From Section 4. we know

134 Sensitivity and Robustness Analysis that the magnitude of the stator voltage is limited by the DC-link voltage through expression (4.), whereas the stator frequency is only restricted in steady state, see (4.). We will neglect the steady state constraint on the frequency and therefore impose the following requirement in the scaled representation ( WPKS ( j ) K( j ) S( j )), σ ω ω ω < ω, (8.4) where the weight matrix W PKS is given by WPKS ( jω ) =.. (8.5) Note that we neglect the voltage contribution from the feed-forward controller K fw in Figure 8- (or F fw in Figure 7-). Clearly, the output of this controller also adds to the control input and is hence also subject to the input limitations. However, at torque steps the contribution from the feedback controller dominates the control input. Remark: Analysis with parametric uncertainty is mostly performed at 5% of base speed, where input constraints do not cause any problems. We therefore do not explicitly verify this requirement during treatment of parametric uncertainty. One can show that the input constraints are not violated. 8. Perfect Field Orientation We start the analysis of parameter sensitivity by assuming that the signals needed to implement the ISC actually are measurable. If we further assume that the stator resistance is known, the controller outputs will be correctly oriented to the stator flux. The linearized closed-loop system may then be represented as in Figure 6-8, which is repeated in Figure 8-3 but without disturbance rejection. Figure 8-3: Linear model of the closed-loop system with ISC under the assumption of perfect field-orientation.

135 Sensitivity and Robustness Analysis 3 In Figure 8-3 we use hats above the controller blocks to illustrate that the design of the PI controllers rely on a model of P. Robust stability of this system was addressed in Section 6.4 through phase margin requirements. The loop gain in Figure 8-3 can be evaluated as ψ ψ sk p s std L = PDF = e stσ + TT, (8.6) σ ˆ K p + K st R i σ + ψ r K s p stσ + Rr stσ + which follows from the expressions of the decoupled system P and the feedback controller in (6.9) and (6.3), respectively. Note that we do not explicitly show the additional time delay due to the average in the torque loop. From the loop gain expression (8.6), we see that perfect field orientation gives perfect decoupling only at zero torque. In general, one of the off-diagonal elements is non-zero (although rather small), which gives cross coupling from flux to torque. We also see that the flux control is independent of all motor parameters. The gain in the torque loop, however, is affected by the rotor resistance R r and the leakage inductance L σ (through T σ ). If we assume that the torque controller parameters are chosen to cancel the pole at T σ, i.e., according to the expression (6.37), then the gain in the torque control loop is given by stda ˆ ˆ stda e R ˆ ˆ r stσ + e slσ + Rr L = =. (8.7) st R st + st sl + R ISC r σ ISC σ r From (8.7) we see that errors in the rotor resistance affect the loop gain at low frequencies, whereas errors in the leakage inductance affect the loop gain at high frequencies (above /T σ ). If we assume that the nominal crossover frequency ω c is chosen such that ω c = /T ISC >>/T σ, then only errors in the leakage inductance affect the stability margins. With this assumption of the crossover frequency, we may approximate the loop gain (around ω c ) as st e da Lˆ σ L. (8.8) st L Hence, the real crossover frequency (provided ω c >> /T σ ) is given by ISC σ

136 4 Sensitivity and Robustness Analysis Lˆ σ ωc =. (8.9) T ISC Lσ ω This means that the phase margin φ m is given by φ = arg L π ( jω ) + π = ω T Lˆ L ω T φ σ σ m c c da c da Lσ c m, (8.) where φ m is the nominal phase margin. From (8.) it follows that the phase margin is reduced if the leakage inductance is overestimated (provided ω c >> /T σ ). This observation is also visible in Figure 8-4.A, which shows the Bode plot of the torque loop gain (8.7). The dotted curve represents an overestimation and the dashed curve an underestimation of the leakage inductance L σ. Figure 8-4.B shows the corresponding loop gains for errors in the rotor resistance R r. (In this example ω c T σ =.4 with nominal T σ.) 6 Bode Diagram 6 Bode Diagram Magnitude (db) 4 Magnitude (db) Phase (deg) -9 - Frequency [rad/s] 3 Phase (deg) -9 - Frequency [rad/s] 3 Figure 8-4: Bode plots of the torque loop gain with (A) model errors in the leakage inductance L σ and (B) the rotor resistance R r. Underestimation corresponds to the dashed lines, whereas overestimation corresponds to the dotted lines. Hence, we conclude that overestimation of the leakage inductance gives reduced phase margin if ω c >> /T σ. Underestimation of the leakage inductance gives increased phase margin but lower bandwidth. Errors in the rotor resistance mainly affect the low frequency properties and hence the shape of step responses. They are not critical for stability, though. We may also evaluate robustness through the amplitude margin A m, which can be evaluated as (provided ω c >> /T σ )

137 Sensitivity and Robustness Analysis 5 A m π L = = L ( jω ) T ˆ pc dωc L A m σ σ, (8.) where the phase-crossover frequency ω pc is given by ω pc = π / /T d. We see from (8.) that an overestimation of the leakage inductance also gives a reduced amplitude margin (cf. discussion of phase margin). We now check the performance requirements stated in Section 8.. With perfect field-orientation, the only parameters that affect the loop gain in (8.6) are the rotor parameters L σ and R r. From (8.6) it also follows that the cross functions S and S are zero (at zero torque) and that S equals the nominal transfer function, which is plotted in Figure 8-. Hence, S is the only part of the sensitivity function that depends on parameter errors and is shown in Figure 8-5.A. The bold dotted line represents the performance requirement stated in Section 8... Small peaks are seen in S, which all correspond to overestimation of the leakage inductance. We also have errors at low frequencies, which are due to errors in the estimate of the rotor resistance. The maximum singular values of the sensitivity functions are shown in Figure 8-5.B. S Maximum singular value of S Frequency [rad/s] -3-4 Frequency [rad/s] Figure 8-5: (A) Magnitude plot of the sensitivity function S with parameter errors in the leakage inductance and the rotor resistance. (B) The corresponding maximum singular value of the sensitivity function S. The bold dotted lines show the performance requirement. Figure 8-6.A shows results when simulating the system in Figure 8-3 with a linear controller as well as a linear model of the induction motor (the upper curves show the torque). The simulations show no cross coupling between torque and flux. The torque, but not the flux, is affected by parameter errors, as expected from the expression for the loop gain in (8.6).

138 6 Sensitivity and Robustness Analysis In Figure 8-6.B the corresponding results are shown from simulations with actual non-linear implementations of the motor and the controller for the same parameter errors. We see that the linear analysis predicts the results very well Step Responses Time [s] Step Responses Time [s] Figure 8-6: (A) Simulated step responses with linear feedback controller and linear motor with incorrect values of the rotor resistance and the leakage inductance. The upper curves show the torque responses and the lower curves show the flux responses. (B) Simulated step responses with incorrect values of the rotor resistance and the leakage inductance. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. At non-zero torque, the loop gain has a small cross-coupling term, see (8.6). This is also visible in the simulation in Figure 8-6.B. The linear motor model used in the simulation in Figure 8-6.A is evaluated at zero torque and hence shows no cross coupling. Remark: The analysis of the closed-loop performance was done for the polezero cancellation approach (6.37). The main conclusions, however, also hold with other choices of controller parameters. With a general PI controller, we still need K p = T σ ω c to reach a nominal crossover frequency ω c. It then follows from (8.6) that the torque loop gain may be approximated by L ( s) ˆ Ki Rr, small s ˆ Kp + K R i r s s Rr = (8.) R ˆ r stσ + ωc Lσ, large s, s L which shows that errors in the rotor resistance affects the loop gain at small frequencies and the leakage inductance affects the loop gain at high frequencies. σ

139 Sensitivity and Robustness Analysis Flux Estimation with Open-Loop Observer In the previous section, sensitivity and robustness to incorrect motor parameters were investigated under the assumption of measured signals. Here we relax the conditions by using an observer to estimate the signals needed for field-orientation. Note that the stator flux magnitude and the torque in the outer feedback loop in Figure 7-9 still are assumed to be measured. With zero observer gain, the closed-loop system can then be represented as in Figure 7-, which is repeated in Figure 8-7 where also the time delay has been added. Figure 8-7: Linear model of closed loop system with (open-loop) observer. From Section 4.4 we know that the induction motor has large RGA elements at higher speeds, which may cause problems together with a decoupling controller in presence of uncertainty. This issue may also be understood from Figure 8-7, where the ISC is modeled as a feedback controller in series with a decoupling part G - P. With an open-loop observer (k obs = ), the decoupling, which tries to cancel the dynamics of the system G, is purely based on a model of G. From the pole plot in Figure 4-, we know that the induction motor has a poorly damped pole pair with imaginary part approximately equal to the operating point stator frequency (at higher speeds). With incorrect motor parameters, the cancellation of these poorly damped poles will not be perfect and we expect error components with frequency around the operating point stator frequency. These are hence consequences of imperfect fieldorientation. In this section it will be shown that field-orientation with an estimated stator flux indeed affects the closed-loop performance in presence of model errors. It will be shown that L σ is the most sensitive parameter (it affects the model at higher speeds) but also R s is important (it was pointed out in Section 4.3 that the stator resistance affects the location of the poorly damped pole pair). However, unrealistic errors have to be introduced in order to substantially degrade closed-loop performance. The errors mainly increase coupling from the flux reference to torque and to suppress this effect we will use a prefilter on the flux reference to avoid exciting the critical frequencies.

140 8 Sensitivity and Robustness Analysis Note that the realistic simulations actually use an observer with non-zero observer gain (giving the pole-plot in Figure 7-). The gain is still relatively small and it will be seen that it does not significantly affect the result. The feedforward term F fw of the ISC controller is not simulated in this section Rotor Parameter Errors With the same parameter variations as in the previous section (R r and L σ ), the non-perfect estimation of the process signals now introduces cross coupling in the loop gain matrix L, also at zero torque. Figure 8-8.A shows the sensitivity function S (at zero torque), where the largest peaks correspond to overestimation of the leakage inductance (note that S affects cross coupling from flux to torque). The peaks are also seen to violate the performance criterion in Figure 8-8.B. By comparing Figure 8-8.B with Figure 8-5.B, we see additional peaks at the operating point stator frequency, which hence are due to the deteriorated decoupling. The peaks in Figure 8-8.B well correspond to the peaks of S. It turns out that there are no significant peaks in S. S Maximum singular value of S Frequency [rad/s] -3-4 Frequency [rad/s] Figure 8-8: (A) Magnitude plot of S at variations in the rotor resistance and the leakage inductance. (B) Maximum singular value of S with model errors. Remark: Compare the results of flux control robustness performed in Section 4.7, where it was indicated that flux control is sensitive. Here we see that with model errors, a step in the flux reference also affects the torque. In Figure 8-9.A, simulation results with the linear controller in Figure 8-7 and a linear motor model are shown. Compared to Figure 8-6.A, we now additionally have cross coupling from flux reference to torque. Figure 8-9.B shows the results with a non-linear discrete-time ISC and non-linear motor. Time delays are included in both simulations. We see that the linear model

141 Sensitivity and Robustness Analysis 9 well predicts the results of the non-linear simulation, apart from steady-state errors which are not considered by the linear model. Note that we only have steady-state errors at non-zero torque, which was predicted in Section Step Responses Time [s] Step Responses Time [s] Figure 8-9: Simulations with errors in the rotor resistance and the leakage inductance with (A) linear models and (B) non-linear models. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. In Figure 8- the corresponding plots to Figure 8-9.B are shown at % and 9% of base speed. Compared to the simulation at 5% of base speed, we see that cross coupling is less at % and worse at 9% of base speed, which fits with the RGA plot in Figure 4-3. On the other hand, the steady-state errors are larger at % and smaller at 9% of base speed, which fits with the steadystate analysis in Subsections 7..3 and Step Responses Time [s] Step Responses Time [s] Figure 8-: Simulations with non-linear models with errors in the rotor resistance and the leakage inductance at (A) % and (B) 9% of base speed. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux.

142 3 Sensitivity and Robustness Analysis Remark: In all simulations we use a first order prefilter with cut-off frequency 5 rad/s on the reference signals. This filter reduces the effect of the cross coupling at higher speeds. Without this filter, changes in the flux reference cause larger disturbances in the torque. Usually the flux reference is kept fixed, which means that the cross coupling from the flux reference to torque is not a problem. From Figure 6-9 we also see that disturbances approximately only affect the torque control loop. This means that the large peaks in S are not excited by disturbances Additional Parameter Variations By investigating one motor parameter at a time, one finds that the single most sensitive parameter for decoupling is the leakage inductance. Especially in combination with errors in the stator resistance, large sensitivity peaks arise. We discussed the importance of a correct estimate of the leakage inductance at higher frequencies with perfect field-orientation. We may also motivate the sensitivity of the stator resistance as this parameter was shown to influence the real part of the poles of the linear motor motor at higher speeds in Section 4.3. In Figure 8-.A, the sensitivity function S is shown where the real motor parameters L σ and R s are varied. The worst case occurs for minimum real leakage inductance (overestimation) and maximum stator resistance (underestimation). The maximum singular values of the sensitivity functions are shown together with the performance requirement in Figure 8-.B. S Maximum singular value of S Frequency [rad/s] -3-4 Frequency [rad/s] Figure 8-: (A) Magnitude plot of S at variations in the stator resistance and the leakage inductance. (B) Corresponding maximum singular values of S. The result of a simulation for this case is shown in Figure 8-.A. The corresponding simulation results with the non-linear controller are shown in

143 Sensitivity and Robustness Analysis 3 Figure 8-.B. Even in this worst case, we still see relatively small effects (note that cross coupling increases without the prefilter on the flux reference) Step Responses Time [s] Step Responses Time [s] Figure 8-: Simulations with errors in the stator resistance and the leakage inductance with (A) linear models and (B) non-linear models. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. To really push the system, we increase the parameter variations in L σ and R s to 5% and 4% of the nominal values. It should be stressed that these are unrealistically large errors. The resulting simulations are shown in Figure 8-3, from where it follows that the non-linear system is better damped compared to the linear approximation. Again, the worst cases occur when the leakage inductance is underestimated Step Responses Time [s] Step Responses Time [s] Figure 8-3: Simulations with 5% and 4% errors in the stator resistance and the leakage inductance with (A) linear models and (B) non-linear models. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux.

144 3 Sensitivity and Robustness Analysis Remark: We may also point out that a non-linear simulation with measured torque gives identical responses as in Figure 8-3.B, except that there are no steady state torque errors. Hence, the simplification of assuming measured control variables seems valid when analyzing dynamic effects. In Figure 8-3 we now also see larger steady-state torque and flux errors. For comparison with the simulations, the flux and torque estimation factors (7.5) and (7.8) are plotted in Figure 8-4 with the slip frequency set to 5% of pullout slip, which corresponds to 7 Nm compared to 5 Nm used in the simulations. Flux estimation error factor Speed [pu] Torque estimation error factor Speed [pu] Figure 8-4: (A) Flux estimation quotient at 5% of pull-out slip frequency and (B) torque estimation quotient at 5% of pull-out slip frequency. As the torque controller contains a pure integration, we may assume that the estimated torque at steady state equals the torque reference. The flux controller is only a P controller, but the loop gain in the ideal case of no parameter errors still contains an integration. We therefore consider also the flux estimate to match the reference. The quotients plotted in Figure 8-4 may then be interpreted as Ψ /Ψ ref and T /T ref. At 5% of base speed, we see from Figure 8-4 that we may expect a worst case stator flux magnitude of 98.5% of the reference and a worst case torque equal to 9% of the torque reference. As in the figure corresponds to 5% of nominal flux, the largest flux error in the simulation with torque equal to 5 Nm in Figure 8-3.B gives ψs ψs ψ.3*.5ψ = = =.985. (8.3) ψˆ ψ ψ s ref Similarly, as corresponds to 5% of nominal torque, the largest torque error in Figure 8-3.B gives

145 Sensitivity and Robustness Analysis 33 T T 4.5*.5T.9 T ˆ = T = ref 5*.5T =. (8.4) The theoretical results hence match the results from the simulation. The conclusions from the analysis of using an open-loop observer are: R r : The rotor resistance affects the low frequency properties of the torque loop gain and therefore also for example the shape of the step responses. This parameter is not very critical for stability though. L σ : The leakage inductance has large effect on decoupling. The leakage inductance is the most sensitive individual parameter. On the other hand, it does not affect the steady-state accuracy of stator flux magnitude or torque. R s : The stator resistance effects the steady-state errors but also the decoupling. L m : The main inductance has no significant effect neither on steadystate estimates nor decoupling Estimated Currents for Torque Estimation By using the estimated stator current in the equation for the torque (7.), the results in Figure 8-5 are achieved in presence of rotor resistance and leakage inductance variations. Step Responses Time [s] Figure 8-5: Torque and flux step responses where the torque is estimated using the estimated stator current. The upper and lower curves show the torque and flux responses, respectively. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux.

146 34 Sensitivity and Robustness Analysis The steady state problems are due to errors in the rotor resistance, whereas errors in the leakage inductance cause transients errors. Clearly it is critical to use the measured current for torque calculation (at higher speeds). In Section 7..4 it was concluded that torque estimation with estimated stator currents would be very sensitive to errors in the rotor resistance also at higher speeds. With zero observer gain this is pure feed-forward control, which is not robust to plants with large RGA elements, see [7]. 8.4 Flux Estimation with Closed-Loop Observer Using a non-zero observer gain makes the analysis of the closed-loop system more difficult. For the special case with observer gains on the form k obs = (k obs ) T it was shown in Section 7.3. that the system can be modeled by Figure 7-5, which is repeated in Figure 8-6. r + + K G ~ - y Figure 8-6: Closed-loop system with ISC and observer with non-zero observer gain (feedforward controller neglected). The expressions for the system G in (7.3) and (7.3) show that the observer gain may amplify model errors with non-perfect motor parameters. Especially around the operating point stator frequency this may be devastating for performance. In Figure 8-7.A results are shown from a simulation with an observer gain k obs = (7.5R s ) T at 5% of base speed. The solid curve corresponds to perfect parameters, whereas the dashed curve corresponds to an overestimation of the leakage inductance by a factor. From the simulation we see that the increased non-zero observer gain has a bad influence on performance with model errors. To show that the problems are not restricted to observer gains with the special structure assumed above, a simulation with k obs = (7.5R s -5R s ) T is shown in Figure 8-7.B. This observer gain is obtained by multiplying the gain used to generate the pole plot in Figure 7- by a factor 5. We conclude that a zero observer gain is positive for good dynamic performance at higher speeds. At higher speeds an open-loop observer also gives accurate steady-state torque and flux estimates. To improve the convergence rate or to decrease sensitivity to model errors, a non-zero observer gain may be needed at low speeds.

147 Sensitivity and Robustness Analysis Step Responses Time [s] Step Responses Time [s] Figure 8-7: Two simulations with increased observer gain. The solid curves correspond to perfect parameters, whereas the dashed curves correspond to an overestimation of the leakage inductance by a factor. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. 8.5 Non-Parametric Uncertainty In this section we analyze sensitivity and robustness of ISC with respect to non-parametric model errors. Here µ-analysis is applied to the linear feedback model of the system shown in Figure 7-, which is repeated in Figure 8-8. We hence restrict the analysis to cases where the observer gain is set to zero, which was seen in the previous section to be a realistic choice in practice (apart from low speeds). Figure 8-8: Closed-loop system with relative input uncertainty. Already in Subsection 7.3. we analyzed robust stability of this feedback system with unstructured relative input uncertainty. We saw that the choice of the feedback controller F influences robust stability. It was shown that F = /(st ISC )P - would give a large stability margin, but this choice might also give increased sensitivity, i.e., worse robust performance. In this section we will simultaneously evaluate robust stability and robust performance for two

148 36 Sensitivity and Robustness Analysis choices of feedback controllers F. This is done be examining robust stability of an augmented feedback loop, where the uncertainty block contains structure. For this purpose the so-called structured singular value µ is used instead of the maximum singular value to calculate robustness margins. In this section we also use diagonal relative input uncertainty (often common in practice), which is less conservative compared to the full uncertainty block considered in Subsection Structured Singular Value In this subsection we introduce the structured singular value, which we use to evaluate robust performance and robust stability with structured uncertainty (for details, see a textbook on robust control, such as [7]). Consider a feedback loop as in Figure 8-9, where is a set of norm-bounded blockdiagonal perturbations. Figure 8-9: M -structure for robust stability analysis. If we assume that the nominal system M and the perturbations are stable, then the M -system in Figure 8-9 is stable for all allowed perturbations with σ ( ), for all ω, if and only if the structured singular value µ of M is less than one for all frequencies, i.e., (see e.g. [7]) ( M ( j )), µ ω < ω. (8.5) The structured singular value depends on the structure of, which is explicitly shown in the notation in (8.5). To compute the structured singular value of M we find the smallest structure (measured in terms of σ ( ) ), which makes det(i-m ) =. Then µ(m) = /σ ( ), i.e., { = } ( M) ( ) ( I M ) µ min σ det for structured. (8.6) If no such structured exists, then µ(m) =. Note also that for a full unstructured uncertainty block, we have that µ(m) = σ (M).

149 Sensitivity and Robustness Analysis Uncertainty Description Here we introduce non-parametric model uncertainty to be considered during the analysis. We could have used knowledge from the parametric model errors examined in the previous sections to try to shape also a non-parametric uncertainty. This is, however, non-trivial for MIMO systems. We will therefore follow the proposal in [85] and introduce uncertainty due to an uncertain time delay. The time delay will be modeled as uncertainty at the plant input, see Figure 8-8. That is, the induction motor is modeled as where W I = diag(w I, w I ) and w ( ) jωt G e d G I W I I +, (8.7) jωt,. (8.8) + d max I = I < ωt d max j The weight w I is proposed in [7] to approximate a time delay and the maximum time delay, T dmax, is set to ms, which is reasonable for a traction application. A Bode plot of w I in (8.8) is shown in Figure 8-. For unstructured uncertainty, we will use a desired bandwidth of rad/s. Note that the uncertainty at this frequency is fairly small. Magnitude [abs] Phase [deg] Bode diagram Frequency [rad/s] 3 4 Figure 8-: Bode plot of uncertainty weight w I.

150 38 Sensitivity and Robustness Analysis Model for Controller Evaluation In Section 8. we stated the performance requirements (8.) and (8.4). Here we combine these to the following stacked performance requirement p ( ω) ( ω) WPKS KS j WPKS KSp maxσ ω W W PS Sp j <, (8.9) PS Sp where S p is the perturbed sensitivity function, W PKS is set by (8.5) and W PS = diag(w PS, w PS ). Here we use w PS defined by (8.3) but with a bandwidth requirement of rad/s. Note, however, that the stacked requirement is not exactly equivalent to the two original requirements as ( jω ) ( jω ) Φ = Φ + Φ ( j ) ( j ) σ ω ω Φ. (8.) If for example the magnitudes of the two components are both equal to one, i.e., the two individual requirements are fulfilled, then the H norm of the stacked requirement equals >. The stacked requirement is still used for mathematical convenience. In order to evaluate the performance criterion, we rewrite the closed-loop system in Figure 8-8 as in Figure 8-. Here a fictitious input d and two outputs z and z have been added, such that the transfer function from the input to the outputs correspond to the vector in the performance criterion (8.9). From Figure 8- it follows that () t () t z W KS d t PKS p = z W PS Sp z F p (). (8.) Remark: Note that the feedforward term F fw in Figure 8-8 is neglected here. It does not affect stability or the sensitivity function but should be considered when treating the input constraints. However, compared to the corresponding feedback controller F, it gives a rather small contribution and is neglected for simplicity. By closing the loop with the controller K in Figure 8-, we obtain the model in Figure 8-, where the system N can be expressed in terms of a lower fractional transformation (LFT) as (, ) + ( ) N F P K P P K I P K P. (8.) l µ µ µ µ µ

151 Sensitivity and Robustness Analysis 39 I P µ u I W I y I d W PKS z z W PS u G v K Figure 8-: Plant model for µ-analysis. The system P µ, which is defined in Figure 8-, can be represented as WI WPKS Pµ = WPSG W PS WPSG, (8.3) -G -I -G and the system N in Figure 8- can then be evaluated to wq I I wks I N = WPKSQI WPKS KS, (8.4) WPSGS I WPS S where Q I = KG(I+KG) - is the input complementary sensitivity function and S I = (I+KG) - is the input sensitivity function. We note that the nominal performance requirement is fulfilled if (cf. (8.9)) ( N ( j )) σ ω <, ω. (8.5) Further, the element N corresponds to the system used for robust stability analysis in Subsection If the system is nominally stable, then the perturbed system is stable if the feedback loop N I is stable. Since we here use a diagonal uncertainty block I, the stability condition on N is given in terms of the structured singular value µ instead of the maximum singular value. That is, the system is robustly stable if

152 4 Sensitivity and Robustness Analysis ( N ( j )) µ ω <, ω. (8.6) I Figure 8-: Model of the closed-loop system on N -structure for robust performance analysis. We may now express the transfer function F p defined in (8.) as an upper LFT, i.e., (, ) ( ) F = F N N + N I N N. (8.7) p u I I Our robust performance requirement can then be stated as ( ) F = p Fu N, < I. (8.8) We now introduce a fictitious full complex performance perturbation P and define the new block-diagonal uncertainty = diag( I, P ). It then follows that the robust performance condition (8.8) is equivalent to (see [7]) ( N( j )), µ ω < ω. (8.9) The augmented uncertainty block is shown in Figure 8-3. I P u y N Figure 8-3: Plant model for controller evaluation including fictituous performance uncertainty. We may now summarize the following four criteria in terms of the structured singular value µ, [7]: Nominal Stability (NS): N internally stable Nominal Performance (NP): ( N P ( j )) Robust Stability (RS): ( N ( j )) µ ω <, ω (and NS) µ ω <, ω (and NS) I

153 Sensitivity and Robustness Analysis 4 Robust Performance (RP): ( N( j )), µ ω < ω (and NS), where = diag( I, P ). Note that the NP requirement is equivalent to (8.5) as P is a full uncertainty block Controller Evaluation This subsection shows the evaluation criteria defined in the previous subsection evaluated for the system in Figure 8-8 with controller according to (7.3). Two sets of controller parameters are considered, namely (6.37) and (6.46) to give a bandwidth of rad/s, i.e., with T ISC =. (lower than in Section 8.3 to fit with the µ-synthesis design in Section 9.). The analysis is done for three operating points with nominal flux and zero torque, where the stator frequency is set to %, 5% and 9% of base speed ω base The sampling time is.5 ms for all operating points and the results are presented in Figure 8-4 to Figure 8-6 with controller parameters according to (6.46) and in Figure 8-7 to Figure 8-9 with controller parameters set as (6.37). Under the assumption of perfect parameters, the analysis in Section 6.3 indicated no difference in behavior between the different operating points. This also follows from Figure 8-4.A. The reason that the nominal performance at operating point three deviates from the first two operating points is that the maximum singular value of KS gets close to the limit. It however stays less than the limit. Note that this is possible as ( ) T =, which is larger than one..6 Structured Singular Value - NP.6 Structured Singular Value - NP Angular Frequency [rad/s].4 - Angular Frequency [rad/s] 4 Figure 8-4: Nominal performance criterion with controller parameters according to (6.46) at (A) Nm and (B) 5 Nm. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed.

154 4 Sensitivity and Robustness Analysis Structured Singular Value - RS Structured Singular Value - RS Angular Frequency [rad/s] - Angular Frequency [rad/s] 4 Figure 8-5: Robust stability criterion with controller parameters according to (6.46) at (A) Nm and (B) 5 Nm. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed. 3 Structured Singular Value - RP 3 Structured Singular Value - RP Angular Frequency [rad/s].5-4 Angular Frequency [rad/s] Figure 8-6: Robust performance criterion with controller parameters according to (6.46) at (A) Nm and (B) 5 Nm. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed. With controller parameters according to (6.46), the robust stability and robust performance criteria for the ISC all show peaks at frequencies corresponding to the operating point stator frequencies, see Figure 8-4 and Figure 8-6. This was to be expected as a linearized model of the induction motor has large relative gain array (RGA) elements at higher stator frequencies, see Section 4.4. Inverse-based controllers, such as the ISC (at zero torque), are usually not recommended for plants with large RGA elements, [7]. If we use controller parameters according to (6.37), the product PF is given by a scalar function times an identity matrix at zero torque. From (7.5) we see that the system G is cancelled in the expression for the input complementary sensitivity function. This gives larger robustness compared to e.g. using (6.46),

155 Sensitivity and Robustness Analysis 43 compare Figure 8-5.A and Figure 8-8.A. At non-zero torque, the controller F does not invert the system P. This gives peaks in the robust stability criterion, but the robustness is still better compared to the other tuning, cf. Figure 8-5.B and Figure 8-8.B. Also robust performance looks slightly better with the tuning (6.37) compared to (6.46), see Figure 8-6 and Figure Structured Singular Value - NP.6 Structured Singular Value - NP Angular Frequency [rad/s] Angular Frequency [rad/s] 4 Figure 8-7: Nominal performance criterion with controller parameters according to (6.37) at (A) Nm and (B) 5 Nm. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed. Structured Singular Value - RS Structured Singular Value - RS Angular Frequency [rad/s] 4 - Angular Frequency [rad/s] 4 Figure 8-8: Robust stability criterion with controller parameters according to (6.37) at (A) Nm and (B) 5 Nm. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed.

156 44 Sensitivity and Robustness Analysis 3 Structured Singular Value - RP 3 Structured Singular Value - RP Angular Frequency [rad/s].5-4 Angular Frequency [rad/s] Figure 8-9: Robust performance criterion with controller parameters according to (6.37) at (A) Nm and (B) 5 Nm. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed Simulations The RP criterion in Figure 8-6 indicates problems with robust performance at the operating point stator frequency. This result was verified by simulations with the ISC with incorrect motor parameters in Section 8.3. In this subsection, we will not investigate the consequences of a varying time delay, which was used to define the uncertainty. Instead we will interpret the difference in the two versions of the ISC control laws (6.8) and (6.9) as an unstructured model error (not necessarily according to (8.7) and (8.8)). A discrete-time ISC is used with tuning according to (6.46) and the simulation consists of: Torque steps between and 5 Nm A flux step from % to 9% of nominal flux at time.5 s Step responses with the ISC implementations (6.8) and (6.9) are shown in Figure 8-3.A and Figure 8-3.B, respectively. For the latter implementation, we see large disturbances in the torque at a step in the flux reference. Although the deviation in flux is small (only magnitude deviation), the error in torque is large and the error increases with the rotor speed. This fits with the results of Figure 8-6. In this case, however, the error is not caused by erroneous parameters or time delays but by an approximation in the control law, which gives an error in the magnitude of the stator flux.

157 Sensitivity and Robustness Analysis 45 Step Responses Step Responses Scaled Torques and Fluxes 5 5 Scaled Torques and Fluxes Time [s] Time [s] Figure 8-3: Step responses with ISC (A) according to (6.8) and (B) according to the alternative proposal (6.9). The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. 8.6 Summary In this chapter we have examined the effects of model errors on the closedloop performance achieved with ISC. The analysis was done with both parametric and non-parametric uncertainty. With parametric model errors, the analysis was performed in steps. It was shown that with perfect field orientation, the closed-loop torque control is only affected by errors in the estimated rotor resistance and leakage inductance. Errors in the estimate of the rotor resistance affect the torque loop gain at low frequencies and errors in the estimate of the leakage inductance affect the torque loop gain at higher frequencies. As long as the desired crossover frequency is much larger than the rotor leakage time constant, stability is therefore only influenced by errors in the leakage inductance. In this case, the stability margin of the torque loop is reduced with an overestimation of the leakage inductance. Perfect field orientation requires a perfect estimate of the stator flux (and a perfect estimate of the stator resistance). Usually, the stator flux is not measured but is estimated by an observer. The effect of using an observer with zero observer gain was first examined. In this case, the field orientation can be modeled as an inverse-based controller. With incorrect estimates of the motor parameters, cancellation of the motor dynamics is not perfect. Due to the poorly damped pole pair of the induction motor at higher speeds, cross coupling in the closed-loop system is introduced around the operating point stator frequency. In particular, coupling from flux reference to torque increases. Still, even with large parameter errors, the deterioration of

158 46 Sensitivity and Robustness Analysis performance compared to perfect field-orientation is not very large as estimation of the stator flux is fairly independent of parameters at higher speeds. The cross coupling from flux reference to torque can be reduced by using a prefilter on the flux reference. Through simulations it was shown that a non-zero (relatively large) observer gain may cause poor closed-loop performance. This fact was motivated in Subsection 7.3., where it was shown that a non-zero observer gain may amplify the model uncertainty. The conclusion is to use a small observer gain at higher speeds for stability. At higher speeds, pure simulation of the motor equations also gives good steady-state estimates of the control variables torque and stator flux magnitude. Finally, the influence of non-parametric uncertainty was examined using µ- analysis (with zero observer gain). Again, reduced robust performance was indicated around the operating point stator frequency at higher motor speeds. This fact was here also used to motivate the large cross coupling from flux reference to torque obtained with the modified ISC control law proposed in [3], where the reference flux magnitude is used for field-orientation instead of the actual flux magnitude.

159 Chapter 9 Linear Controller Design This chapter applies linear design techniques to the problem of controlling the stator flux magnitude and torque of an induction motor. Just as during the analysis of the ISC in the previous chapter, we will consider the control variables to be measurable. In Section 4.4 we showed that a linearized model of an induction motor has large Relative Gain Array (RGA) elements at higher stator frequencies. This means that the ISC may not be the optimal choice of controller as inverse based controllers are usually not recommended for such plants [7]. Better robustness may be achieved with a controller not inverting the process dynamics, which could be designed using for example H techniques. Another motivation for applying robust linear control to an inverter fed induction motor is to properly handle saturation of the stator voltage magnitude at base speed. If the motor flux is kept constant, the stator voltage magnitude increases with the stator frequency and the possibility to manipulate the stator voltage amplitudes is gradually lost, as the stator voltage magnitude approaches the limit set by (4.). This effect is normally not considered but can be incorporated in robust linear design by increasing the cost on large control signals. In Section 9. linear controllers are designed using µ-synthesis based on the representation of the system in Figure 8-. That is, one-degree-of-freedom controllers are designed that generate the stator voltage directly from control errors in torque and stator flux. This design is done with respect to robust performance for one operating point at a time. The possibility to extend the synthesis by making the actual stator frequency available to the controller is considered in Section 9.. Here so called Linear Parameter Varying (LPV) controllers are designed and evaluated. Such controller designs are described in for example [], [9] and [65]. In [6] and [67] LPV controllers are designed for an induction motor to cope with a varying rotor speed. The designs are showed to work over the entire operating range but saturation of the stator voltages is not considered. The robust design also only covers parts of the total control law. Here we investigate the possibilities to consider the entire

160 48 Linear Controller Design control law in the robust design, given only estimates of torque and stator flux magnitude. The design uses the method described in [9], where the LPV controller is given as a Linear Fractional Transformation (LFT) of a fixed linear controller and the varying parameter, which here is the stator frequency. The LFT design has for example been applied to aircraft applications, see [79]. 9. µ-synthesis In this section linear µ-synthesis controllers are designed for the three operating points treated in Section 8.5. The design is based on the representation of the system in Figure 8-, with performance requirements as given in Section 8. and the uncertainty description in Subsection The controller K is hence chosen to minimize the robust performance criterion. The resulting structured singular values with these controllers are shown in Figure 9- and Figure 9-. The µ-synthesis design shows that it is possible to reduce the peaks of the robust performance (RP) structured singular values at dispense of nominal performance..6 Structured Singular Value - NP Structured Singular Value - RS Angular Frequency [rad/s] - 4 Angular Frequency [rad/s] Figure 9-: (A) Nominal performance criterion and (B) robust stability criterion achieved with the µ-synthesis controller. The solid lines correspond to % of base speed, the dashed lines correspond to 5% of base speed and the dotted lines correspond to 9% of base speed. In Figure 9-3 simulation results with the µ-synthesis controller are shown under the same conditions as with the simulations with the ISC shown in Figure 6-3. By comparing the two figures, we see a worse nominal performance with the µ-synthesis controller as is indicated by the nominal performance criterion in Figure 9-.A. Especially the cross coupling from torque reference to flux is large.

161 Linear Controller Design 49 3 Structured Singular Value - RP Angular Frequency [rad/s] Figure 9-: Robust performance criterion: µ-synthesis. The solid line corresponds to % of base speed, the dashed line corresponds to 5% of base speed and the dotted line corresponds to 9% of base speed. Step Responses:µ-Synthesis Controller Scaled Torques and Fluxes Time [s] Figure 9-3: Step responses with the µ-synthesis controller. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. To check robust performance, model errors are introduced by changing the parameters of the motor while keeping the parameters used by the controller fixed. In Figure 9-4 the corresponding results to Figure 8-9.B are shown, i.e., with errors in the rotor resistance and the leakage inductance. When also varying the leakage inductance and the stator resistance, as done when generating the simulation results in Figure 8-.B, some parameter combinations lead to an unstable system with the µ-synthesis controller. From this section we hence conclude that the optimal linear controller designed with respect to the robust performance criterion perform worse than the ISC (for

162 5 Linear Controller Design the tested cases). The robust performance criterion seems very conservative for the cases simulated Step Responses Time [s] Figure 9-4: Simulations with errors in the rotor resistance and the leakage inductance with a linear (discrete-time) µ-synthesis controller. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. 9. LFT Controller In this section we try to design a linear controller that handles a wide range of operating points. The idea is to use a so called linear parameter varying (LPV) controller, which is modified according to the operating conditions. The problem of analytically guaranteeing stability for systems with such controllers is rather complicated and the results depend on the assumptions made on the varying parameters. One may either assume that: The parameters are constant but unknown The parameters change with time but the rate of change is limited The parameters may change arbitrarily fast The first alternative gives the least H norms, whereas alternative three gives the most conservative results. For the LFT controller design, the third assumption is made and the synthesis starts by representing the process as an LFT system, where the varying parameters are pulled out in a feedback loop as shown in Figure 9-5. These parameters are then made available to the controller.

163 Linear Controller Design 5 δ w u G z y Figure 9-5: Plant with varying parameters on LFT form. If the system matrices of G can be written A B A B A B = + δ, (9.) C D C D C D where δ is the varying parameter and A B U = ( V V ), (9.) C D U it follows that the system in Figure 9-5 is described by [9] () () () () () () x t A U B x t. (9.3) z t = V V w t y t C U D u t After including additional uncertainties and weighting functions for performance requirements, the model in Figure 9-6 is obtained. diag(δ, ) u d u P µ y z v K I R (δ) Figure 9-6: Process model for LFT controller design. The augmented system with weighting functions for uncertainty and performance is denoted by P µ. The controller K is also designed as an LFT

164 5 Linear Controller Design and the notation I R (δ) indicates that there may of course be parameters that are not known to the controller. By including the δ-block of the controller in the upper uncertainty block, the optimum controller, minimizing the H norm of the closed-loop system, can be solved for using Linear Matrix Inequalities (LMIs), [9]. With a state space description, the scaling described in Section 4.5 affects the system matrices in the following way ˆ A = A, B = BD ˆ, C = D Cˆ, (9.4) where the hats indicate the original unscaled system. Note that the scaling matrix D u is a non-linear function of the stator frequency. We then use (here we neglect the lower limitation of the stator voltage magnitude in (4.)) u = max Ud us Ud ψ sω π π, (9.5) where the approximation u s Ψ s ω in (9.5) is valid at higher speeds. By adding the additional restriction of only considering stator frequencies less than base speed ω base, the expression for u max becomes linear in the stator frequency. We hence exclude the so called fieldweakening range from the design. u e 9.. Linear Process Model To simplify the analysis, operating points are restricted to zero torque. In this case the system matrices become Rs Rs Rs ψsω Lm L L σ σ Tσ Tσ A = ω Rs Rs ψ s Lm Lσ ω Rs R s ψ s Lm Tσ L σ ( ω ) g( ω ) g f ( ω ) B = ψs ψs T (9.6) (9.7)

165 Linear Controller Design 53 where f 3 p ψ s C = L σ, (9.8) = = ( ω ), g( ω ) ω L R m s ωl m ω L m + + Rs Rs. (9.9) The two functions in (9.9) are plotted against normalized stator frequency in Figure 9-7. The normalization quantity, ω base, is the stator frequency where the stator voltage amplitudes saturate and is called the base speed. g(ω ) and f(ω ) g(ω ) f(ω ) ω /ω base Figure 9-7: Functions f(ω ) and g(ω ). The chosen design method requires the functions of the varying parameters to be approximated by rational functions. The complexity of the model increases with the order of the rational functions. To cover all stator frequencies, the two functions in (9.9) would have to be approximated by relatively high order rational functions. To simplify the design problem, we therefore restrict attention to operating points with stator frequencies larger than 5% of base speed. In this case we may approximate the functions f(ω ) and g(ω ) by zero and one respectively, i.e.,

166 54 Linear Controller Design T B ψs ψ s. (9.) To cover all stator frequencies, either more complex approximations may be used or one could try combining controllers designed for different operating ranges. This is, however, not done in this thesis. 9.. Extracting the Stator Frequency The operating point parameter δ is chosen to measure stator frequency deviations from half the base speed ω base, i.e., ω ω δ = base. (9.) ω For stator frequencies between 5% and % of ω base, it now follows that the system matrices A, B, C and D (approximately) satisfy (9.). One can further show that relation (9.) is satisfied for T U =. (9.) U ωbase ωbase ψ s ( V V) = (9.3) ψsωbase base 9..3 Model for Controller Synthesis The performance requirements given in Section 8. lead to the system shown in Figure 9-8. This model is used for controller design.

167 Linear Controller Design 55 δι Ι u δg y δg P µ u Ι W I y Ι d u + + G W PKS z W PS - v z K δι Figure 9-8: Process model used for LFT controller design Results The controller minimizing the H norm of the system in Figure 9-8 was solved for using SeDuMi, [78], together with the Matlab interface YALMIP, [48]. To consider the additional structure of the input and performance uncertainties, first order scaling matrices were used during controller synthesis, similar to what is done during DK-iterations in ordinary µ-synthesis (see [7] for details on DK-iterations). Without further model reduction, the produced linear controller is of th order. In Figure 9-9 the structured singular value achieved with the LFT controller is shown as a function of the stator frequency. Shown in Figure 9-9 are also the resulting structured singular values with pointwise µ-synthesis (three points only, see Section 9.) and µ-synthesis with a constant controller, i.e., a fixed controller trying to deal with the varying parameter δ. As could be expected, the LFT controller is beat by pointwise µ- synthesis, but the information about the varying parameter makes the LFT controller better than a fixed µ-synthesis controller.

168 56 Linear Controller Design 3.5 Structured singular values 3.5 µ-synthesis controller.5 pointwise µ-synthesis controller LFT controller ω /ω base Figure 9-9: Structured singular values for robust performance as functions of the stator frequency. To check robust performance, simulations with rotor resistance and leakage inductance errors are shown in Figure 9-. The simulation is performed at 5% of base speed and hence corresponds to Figure 8-9.B with ISC Step Responses Time [s] Figure 9-: Simulations with LTV controller with errors in R r and L σ. The upper curves show the torque responses and the lower curves show the flux responses. The signals are scaled such that in the plots corresponds to 5% of nominal torque and flux. From the figures we conclude that torque steps give larger disturbances in flux with the LFT controller compared to the ISC. On the other hand, cross coupling in the other direction is less with the LFT controller.

169 Linear Controller Design Summary In this chapter the possibilities to design linear feedback controllers for the induction motor were examined. The design was done by assuming that the control variables stator flux magnitude and torque are measurable. This assumption was motivated by the accurate steady-state estimates of these quatities at higher speeds, which were established in Section 7.. First controllers were designed for robust performance at a fixed operating point, using so called µ-synthesis. It was shown that the high peaks in the maximum structured singular value obtained with ISC in the previous chapter could be reduced. This, however, led to worse nominal performance. It was also shown through µ-analysis and simulations that the µ-synthesis controllers gave worse robust stability compared to the ISC. A practical controller for the induction motor has to handle a large variety of operating points. From Chapter 4 we know that especially the motor speed strongly affects the motor dynamics. To handle such variations, a linear parameter varying (LPV) controller was designed assuming the operating stator frequency to be available. It was shown that this controller would work over a larger speed range compared to a µ-synthesis controller designed for one single operating point. Still, not the entire speed range was covered. Furthermore, not only the speed varies in practice, but also for example the load, the temperature and the flux level. By including also these variations, the controller synthesis would become very complicated.

170

171 Chapter DC-Link Stability In Section.4 it was shown that an induction motor drive with poorly damped input filter may become unstable due to power oscillations between the inverter and the input filter. It was shown that the problems increase with the operating point power and that the instability could be attributed to efficient suppression of DC-link voltage disturbances. An intuitive approach to solve such stability problems is to modify disturbance rejection to give the inverter a resistive behavior as seen from the DC link. One such scheme is suggested in [35] and [8]. Under the assumption of perfect torque control it was shown that this method would stabilize the drive at all operating points. However, perfect control is never achieved in practice and it was illustrated through two examples that applying stabilization as proposed in [35] and [8] (as well as [86]) does not guarantee a stable drive. A more flexible stabilization algorithm, containing a general stabilization controller, was therefore proposed in Subsection.4.3. Further, a model of the drive to be used for accurate synthesis of stabilization controllers was presented in Figure -9, where torque control was modeled by the expression for the transfer function in Equation (.3). By using the linear models derived in the previous chapters for FOC as well as ISC, we are now ready to return to the treatment of DClink stability. After reviewing the expressions for the torque transfer functions G c and G d in (.3) with ISC and FOC, a linear model of the drive with simplified feedback structure is presented in Section.. Stability of the drive is then treated in Section. using the Nyquist stability criterion, where it is shown that the improved model predicts instability of the examples simulated in Section.4. Through the stability analysis, also general conditions for stability of the drive are derived in terms of the inverter input admittance. In Section.3, it is shown that the ideas for stabilization presented in [3] and [6], together with some other possibilities, all fit into the model in Figure -9 with different expressions for the stabilization controller. In Section.4, finally, stabilization controllers are designed to stabilize the drive. Here we also consider the trade-off between stability and efficient torque control.

172 6 DC-Link Stability. Linear Model of the Drive In this section we summarize the linear models G c and G d in the torque equation (.3), which were derived for FOC in Chapter 5 and for ISC in Chapter 6. Thereafter a model of the drive with simplified feedback structure using the inverter input admittance is presented. As discussed in Section 3., different models of the induction motor are used with ISC and FOC. With ISC, motor quantities and parameters refer to the so called Γ-model of the induction motor, whereas the inverse Γ-model is used with FOC. To keep the presentation simple, this difference is not reflected in the notation. For example this means that L σ is used to denote the leakage inductance in both models although the actual values of L σ in the two models differ by a scaling factor. The same principle is adopted for the block diagrams of the closed-loop systems. Here also the same notation for the two cases is used although the actual systems differ... Indirect Self Control (ISC) In Section 6.3 it was shown that, apart from low speeds, the internal closedloop torque and flux control with ISC may be modeled as in Figure 6-9. If we neglect variations in the motor speed, the closed-loop torque control of the induction motor may hence be represented as in Figure -, where subindices have been dropped for simplicity. Figure -: Block diagram of the induction motor controlled with ISC. The blocks in Figure - are given by (6.9) and (6.3), i.e., n ψ F F s K K P s n n σ Rr Rr 3 p r fw =, ( ) =, p + i ( ) = 3 pψr 3 pψr s Rr st ( + ).(.)

173 DC-Link Stability 6 From Figure - it then follows that the transfer functions G c and G d in the torque equation (.3) can be written as ω G ( ), c = SPD Ffw + F Gd = SP( DA), (.) U where the sensitivity function S is defined by S = + PDFA. (.3) d.. Field-Oriented Control (FOC) In Section 5. it was shown that, except at very low speeds, the internal torque feedback loop with FOC can be modeled by Figure 5-7, which is repeated in Figure - where sub indices have been dropped for simplicity. Figure -: Block diagram of the induction motor controlled with FOC. The controller F and the system P in Figure - are given by F( s) = kp + ki, P =. (.4) s L s+ α σ ( ) It now follows from Figure - that the transfer functions in the torque equation (.3) with FOC satisfy 3np ψrω Gc = SPDF, Gd = SP( DA), (.5) U where the sensitivity function S is given by d c

174 6 DC-Link Stability S = I + PDF. (.6)..3 Feedback Representation of the Drive To simplify the feedback structure due to the power supply in Figure -9, we introduce the inverter input admittance Y as the transfer function from DClink voltage to DC-link current. The system in Figure -9 may then be redrawn as in Figure -3. E G E U d m T ref G c m U d i d Z DC U d T U d + + T + Y m U d ^ G d T m U d G d G c K Figure -3: Linear feedback model of the drive, where the feedback structure has been simplified by introducing the inverter input admittance Y. The feedback loop due to the power supply now consists of the input filter block Z DC and the inverter input admittance Y. From Figure -3 it follows that Y is given by ωm Tωm Y = ( Gd + GcK), (.7) U d Ud Gˆ d where Ĝ d is the transfer function from DC-link voltage to torque after stabilization has been applied. From Figure -3 it also follows that by including the power supply, the torque equation (.3) is modified to (with the supply voltage E as disturbance instead of the DC-link voltage U d )

175 DC-Link Stability 63 Tωm Tωm Y + Y U + d Ud Ud T () t = Z DC GcTref () t + GE E() t. (.8) + YZ DC ωm + YZ DC Note that the term Y+T ω m /U d appearing in (.8) is proportional to the modified disturbance transfer function Ĝ d. With perfect disturbance rejection, i.e., Ĝ d =, it then follows that the torque equation (.8) reduces to T(t) = G c T ref (t). The drive would, however, be internally unstable (although not directly visible in the torque). With a non-zero stabilization controller, the drive may be internally stabilized, but a non-zero stabilization controller also modifies the transfer function from torque reference to torque. The stabilization controller should hence stabilize the drive while keeping this influence small. If we assume that the transfer functions G d and G c and the stabilization controller K are all stable, it then follows from (.7) that also the inverter input admittance Y is stable. Further, since the input filter Z DC is stable, stability of the transfer functions in (.8) consequently only depends on whether (+YZ DC ) - is stable or not. Stability of the drive may therefore be analyzed by solely plotting the Nyquist curves of the loop gain YZ DC. This approach is taken also with analysis of stability of a DC-DC converter in e.g. [4] and for stability analysis of a traction power supply grid in [46]. Remark: By explicitly using the inverter input admittance, the block diagram in Figure -3 and Equation (.8) seem not well defined at zero motor speed. This is however not a practical problem, as the inverter input admittance is proportional to ω m, see (.7). The notational dilemma can be solved by introducing a scaled admittance as Ud ˆ T Y = Y = Gd, (.9) ω U m which is well defined for all operating points. Still, we will use the physical quantity Y, which means that we sometimes have to allow expressions with the operating point speed in the denominator. Remark: Compared to stability analysis using eigenvalues as in [6], robustness issues are more easily handled using the Nyquist criterion. For example also the effects of time delays may be analyzed. The feedback representation also facilitates stability analysis using measurements. This introduces the possibility to accurately evaluate the stability margins of a real d

176 64 DC-Link Stability drive. This fact will be used in Chapter to verify the performance of a proposed stabilization controller.. Stability Analysis In this section we will show that the instabilities presented in Section.4 are predicted by the model in Figure -3. From this analysis we will also be able to derive stability and performance requirements of the drive in terms of properties of the inverter input admittance Y... Unstable Examples Revisited We now return to the examples shown in Figure -8, where stabilization according to (.7) with ρ = was applied without stabilizing the drive. In Section.4 we analyzed the drive under the assumption of perfect torque control of the induction motor. The inverter could then be modeled by the equivalent resistance R inv given by expression (.9). The inverse of this expression, which corresponds to the equivalent inverter input admittance Y, can then be written as Tω m Y = ( ρ sgn ( Tωm) ). (.) U d With ρ =, the inverter input admittance in (.) is a positive constant at each operating point. Hence, it then follows from the expression for the damping factor ζ drive in (.4) that the drive is stable. We can also verify this by plotting the Nyquist curves of the loop gain YZ DC with Y according to (.). We now relax the unrealistic assumption of perfect control and instead use the general expression for the admittance given by (.7). In Section.4 it was shown that the linearized effect of stabilization according to (.7) can be modeled by the stabilization controller (.). By inserting this expression into (.7), it follows that the inverter input admittance becomes ω Tω m m Y = Gd + ( ρgc sgn ( Tωm) ). (.) U U d d Note that the expression for the admittance in (.) reduces to (.) with perfect torque control, i.e., with G c = and G d =. The Nyquist plots of the loop gain YZ DC using (.) (with ρ = ) are shown at zero torque and torque

177 DC-Link Stability 65 close to pull-out torque in Figure -4 (here magnitudes are limited to 5 for better visibility). With zero torque the input filter with 8.8 Hz resonance frequency in Appendix A is used, whereas the input filter with 53 Hz resonance frequency is used for the other case. In Figure -4 we see that the Nyquist curves encircle the point - at several operating points. In particular this holds for the bold dashed curves corresponding to the cases simulated in Figure -8. The more detailed model of the drive hence predicts the instabilities presented in Section.4. Imag(YZ DC ) 5 Nyquist plot of loop gain YZ DC Real(YZ DC ) Imag(YZ DC ) 5 Nyquist plot of loop gain YZ DC Real(YZ DC ) Figure -4: (A) Nyquist curves of the loop gain YZ DC at zero torque with the input filter with 8.8 Hz resonance frequency. The bold dashed curve corresponds to 6% of nominal speed. As it encircles the point - it indicates an unstable drive. (B) Corresponding Nyquist curves at pull-out torque with the input filter with 53 Hz resonance frequency. The bold dashed curve corresponds to 5% of nominal speed and encircles the point -. Note that the magnitudes are limited to 5 for visibility reasons. To understand the reason for the instabilities, we now examine the loop gains more in detail. To form the loop gains YZ DC plotted in Figure -4, the inverter input admittances are multiplied by the input filter transfer function Z DC. The Bode diagram of Z DC with resonance frequency 8.8 Hz is shown in Figure -5. We see that the filter has a very sharp magnitude peak at the resonance frequency ω. We also see that the phase of Z DC changes from 9 to -9 within a narrow frequency region around ω. These properties of Z DC imply that the loop gain YZ DC sweeps 8 clockwise, centered around arg(y(ω )), with a large magnitude. If the real part of the input admittance is negative at ω, it then follows that the Nyquist curves of the loop gain may encircle the point - in the complex plane. It is only required that the gain of YZ DC is large enough.

178 66 DC-Link Stability Magnitude [abs] Bode Diagram of Z DC Phase [deg] Frequency [Hz] 3 Figure -5: Bode plot of the input filter Z DC with 8.8 Hz resonance frequency. Nyquist curves of the inverter input admittances Y are plotted in Figure -6 for the same operating points as used in Figure -4. At zero torque it follows from (.) that the admittance with perfect torque control would be zero, whereas the admittance with realistic torque control is shown in Figure -6.A. Here we see that, although the admittance is not zero, the magnitude of Y is indeed very small at low frequencies with a phase shift close to 8. This is due to the feedback control and the feed-forward compensation of DC-link voltage disturbances, see the expression for G d in (.). With increasing frequency of the DC-link voltage oscillations, disturbance rejection deteriorates and the magnitude of Y increases. This also means that the phase decreases and eventually drops below 9. Below the frequency where this happens, the admittance hence has negative real part. For example this holds at 8.8 Hz, which is the resonance frequency of the input filter used to generate Figure -4.A. With large operating point torque, the inverter input admittance is shown in Figure -6.B. The behavior is now to a large extent determined by the second term of Y in (.). (Due to the band-pass filter discussed in Subsection.4.3, the stabilization is not active at very low frequencies and the real part of the admittance is therefore negative at these frequencies.) Without stabilization, the admittance is pushed to the left in the complex plane at increasing power by the second term in (.7). With stabilization, this effect is balanced within the torque control bandwidth, where G c, see (.). Above the torque control bandwidth, however, the real part of G c decreases and the admittance is shifted to the left in the complex plane (the effect increases with increasing

179 DC-Link Stability 67 frequency). We for example see that the real part of the admittance at 53 Hz is negative at some operating points. Imag(Y) Nyquist plot of admittance Y ω m Real(Y) Imag(Y) Nyquist plot of admittance Y ω m Real(Y) Figure -6: (A) Nyquist curves of the inverter input admittance Y at zero torque for several rotor speeds. The values of the admittance at 8.8 and 53 Hz are marked with o s and * s, respectively. The bold dashed curve corresponds to 6% of nominal speed. (B) Nyquist curves of inverter input admittance Y at torque close to the pull-out torque for several rotor speeds. The values of the admittance at 8.8 and 53 Hz are marked with o s and * s, respectively. The bold dashed curve corresponds to 5% of nominal speed. From the analysis above we conclude that the special properties of the transfer function Z DC implies that Re{Y(jω )} < may cause instability (it is only required that the gain of Z DC (jω )Y(jω ) is large enough). From the plots of the admittances we see that this condition is most likely to be fulfilled with small values of ω at small torques and with large values of ω at large operating point torques. From the studied examples, we see in Figure -6 that the real parts actually are negative at the resonance frequencies at some operating points. As the input filters used in these examples have high gains (small resistances), the gain of the specific input filters are large enough to make the Nyquist curves of the loop gain encircle the point -, see Figure -4. (The other combinations of torque levels and input filters give stable drives). We may now interpret the presented problems with stabilization according to (.7) (derived under the assumption of perfect torque control) through the corresponding inverter input admittance (.). Neglecting that G d may give problems at zero power with small resonance frequencies, whereas the approximation G c = may be a problem at high powers and high resonance frequencies. Note that the first term of (.) has positive real part at 53 Hz, which is seen in Figure -6.A. This term does therefore not contribute in a negative way here, although it is not considered by the stabilization.

180 68 DC-Link Stability.. Normalization of Coupling Vector The normalization of the coupling vector to compensate for variations in the DC-link voltage to a large extent contributes to the large phase shift of the inverter input admittance at zero torque. The normalization corresponds to the feed-forward compensation in Figure - and Figure -, and hence to the factor -DA in the expressions for G d in (.) and (.5). By approximating the effect of the combined time delay and average operator, i.e., DA, by an equivalent time delay of T da seconds, we have I-DA st da in the Laplace domain (for small time delays). The normalization hence gives a magnitude reduction but also a phase shift of 9. These effects are evident by comparing the two figures Figure -6.A and Figure -7, which show the inverter input admittance with and without the feed-forward compensation. We see that the normalization considerably reduces the magnitude, i.e., improves disturbance rejection (G d is proportional to Y at zero torque), but makes the phase of the admittance exceed 9 at the lower resonance frequency 8.8 Hz. The increased phase shift leads to an unstable drive, which was shown in Figure -4. Imag(Y) Nyquist plot of Y without feed-forward compensation Real(Y) Figure -7: Inverter input admittance at zero torque without feed-forward compensation of DClink voltage disturbances. The values of the admittance at 8.8 and 53 Hz are marked with o s and * s, respectively. The feed-forward compensation reduces the gain of the admittance but also introduces large phase shifts which may be bad for stability.

181 DC-Link Stability Stability and Performance Requirements From the examples examined above we conclude that, for stability, it is desirable that the inverter input admittance at the resonance frequency of the input filter is in the right half plane, i.e., arg(y(ω )) should be between -9 and 9. In order to obtain large stability margins, we strengthen the condition to require that arg(y(ω )) is small. Ideally, the phase should be zero. With positive power, we see from the expression for Y in (.7) that the Nyquist curves of the admittance are shifted to the left in the complex plane. Consequently, the importance of proper stabilization increases with the power. Braking on the other hand increases the real part of the admittance and hence improves the situation from a stability point of view. This also fits with the expression for the damping factor (.4) with ideal control. From the expression for Y in (.7), we see that modifying the inverter input admittance really means modifying the disturbance transfer function G d (into Ĝ d ). For maximum stability margins, we should hence ideally modify G d to make the inverter input admittance positive real. From a disturbance rejection point of view, we further want the gain of Ĝ d still to be small. Combining these two requirements, we find that the ideal inverter input admittance is real with a small positive value. Remark: In this section it was argued that stability of the drive requires the real part of the admittance to be positive at the resonance frequency of the input filter. The validity of this simple stability measure was demonstrated by examples. It should, however, be noted that the input filters in these examples have very small resistances. With larger resistance, the drive may be stable also with a negative real part of Y. The stability measure may thus be conservative. As the trend in traction applications is to reduce the filter capacitance to save space and weight, and hence to increase the resonance peak of the input filter, this condition is still used as a guideline for stabilization treated below. Indeed, with a negative real part of the admittance at ω, input filters exist that destabilize the drive, even if they may not be realistic from a practical point of view. We also note that Re{Y(jω )} > is used as stability criterion at all resonance frequencies of a traction power supply grid in [46]..3 Alternative Stabilization Methods So far we have assumed that stabilization is done by modifying the torque reference. In the literature other strategies for stabilization have been proposed,

182 7 DC-Link Stability see [3], [6] and [7]. In this section we will show that these and a few other alternatives actually are equivalent to torque reference modification. We may then, without loss of generality, restrict synthesis of stabilization controllers to the scheme (.). If we assume that the input filter is fixed, the feedback loop can only be stabilized by modifying the inverter input admittance through the disturbance transfer function Ĝ d. With the torque reference modification (.), the disturbance transfer function G d is modified as Gˆd = G d + G c K, (.) which follows by inserting the torque reference modification (.) into the torque equation (.3). We will use the expression for the modified disturbance transfer function Ĝ d when comparing the different stabilization schemes. Remark: From the expressions for G d in (.) and (.5), it follows that G d also can be affected through the torque/current controller via the sensitivity function S, or via a modification of the average filter A. As discussed above, stability problems at zero torque are caused by the large phase shift of the inverter input admittance due to efficient disturbance rejection (the inverter behaves like a negative resistance). Part of this large phase shift is caused by the sensitivity function, which gives a phase shift of 9 below the bandwidth of the torque/current control. To improve stability, the bandwidth of the sensitivity function should hence be reduced. One can, however, show that even in the extreme case with no feedback control at all, the drive is only stabilized for small torques. This approach will therefore not be pursued further. The possibility to modify the average filter will be considered in Subsection.3.4 though..3. Current Reference Modification If current control is applied, such as the classical FOC, then stabilization may be performed as () () () i t = i t + K U t, (.3) sqref sqref is d where i sq (t) is the torque producing quadrature component of the stator current and K us is a stabilization controller. Stabilization according to (.3) is for example proposed in [3] and [7], and modifies the disturbance transfer function G d as (cf. (.5))

183 DC-Link Stability 7 ˆ 3npψ r Gd = Gd + GcKis. (.4) By comparing the two expressions for the modified disturbance transfer function (.) and (.4), it follows that current reference modification (.3) is equivalent to torque reference modification (.) with 3npψ r K = Kis. (.5) This expression naturally corresponds to the relation between the torque and the quadrature component of the stator current, see (5.)..3. Stator Voltage Modification In [6], stabilization with a field-oriented controller is proposed by adding a correction term directly to the stator voltage space vector in rotor flux coordinates, i.e., () t = () t + K U () t u u, (.6) sref sref us d where K us has equal d and q-components. This means that the equation for the stator voltage (.8) is changed to usref us () t = Du sref () t + DKus + ( DA) Ud ( t). (.7) U d From the analysis in Chapter 5, we realize that the component of the stator voltage adding in the d-direction will have little influence on the torque and hence also on the inverter input admittance (at higher frequencies). We therefore only consider the q-component of the stabilization controller. From Figure - it then follows that stabilization with (.6) modifies the disturbance transfer function as ˆ 3npψ r Gd = Gd + SPDKusq, (.8) where S is the sensitivity function (.6). By using the expressions for G c and G d in (.5), the second term in (.8) can be written as 3npψr 3npψr SPDKusq = Gc F Kusq. (.9)

184 7 DC-Link Stability Stabilization with (.6) is hence equivalent to torque reference modification (.) with 3npψ r K = F Kusq. (.) As the inverse of the PI controller F is proper, the modification (.) poses no practical limitations. There are no large positive phase shifts needed at high frequencies, which may be difficult to realize in practice. Hence, the effect obtained with stabilization according to (.6) with a certain stabilization controller K us, can also be achived with torque reference modification. The Bode plot of the inverse of the PI controller F=K p +K i /s is shown in Figure -8. Bode plot of F - - /K p - K i /K p -3 - Frequency [Hz] Figure -8: Bode plot of the inverse of the PI controller F. Remark: In so called six step operation, see Subsection.., the magnitude of the coupling vector is fixed. Exact modification of the reference stator voltage as in (.6) is then no longer possible..3.3 Stator Frequency Modification Also the frequency of the stator voltage may be modified for stabilization purposes. From the analysis above it follows that, with ISC, a disturbance added to the reference stator frequency affects the torque through the transfer function SPD, which can be written G c (F+F fw ) -, see (.). Hence, the modification

185 DC-Link Stability 73 ˆ ω () t ω () t K U () t = + (.) gives the following modified disturbance transfer function ω d ( ) Gˆd G d G c F F fw K ω = + +. (.) With ISC, stabilization according to (.) is hence equivalent to torque reference modification (.) with ( ) fw K = F + F K ω. (.3) As noted in the previous subsection, realization of (F+F fw ) - imposes no practical problems..3.4 Modification of Average Filter As mentioned in the beginning of the section, the disturbance transfer function G d can also be affected by replacing the average filter in (.8) by a general filter Â. As discussed in Section.., the normalization of the coupling vector (where A is used) reduces the gain of the disturbance transfer function G d, but introduces a phase shift, which may destabilize the drive. In order to simultaneously reach the two requirements of a small magnitude of Y (for disturbance rejection) but also a small phase shift of Y at the resonance frequency of the input filter (for stability), we may look for a different filter Â. With the ISC method, the modified disturbance transfer function with the filter  can be written as ˆ ˆ ω ( ˆ ω A A Gd = SP DA) = Gd + Gc. (.4) U U F + F d d fw Hence, (.4) can be interpreted as torque reference modification (.) with ω ˆ A A K =. (.5) U F + F d With FOC we may similarly write 3n ψ ω 3n ψ ω ˆ ˆ p r ( ˆ) p r A A Gd = SP DA = Gd + Gc. (.6) U U F d fw d

186 74 DC-Link Stability That is, the effect of a modified average filter may also be achieved with (.) and 3n ˆ pψrω A A K =. (.7) U F d Remark: It should be noted that the normalization of the stator voltage is not possible with a fixed modulation index (magnitude of the coupling vector). This for example holds in six-step operation. However, as time delays with six step operation usually are rather long (low switching frequency), the gain of the factor -DA may be larger than one. Hence, even if normalization would have been possible in six step operation, it would not be advantageous from a disturbance rejection point of view..4 Stabilization Controller Design In the previous section it was shown that the different stabilization strategies proposed in the literature are equivalent to stabilization with torque reference modification (.), with different expressions for the stabilization controller. It is therefore sufficient to only consider this type of stabilization. In Section. we stated requirements on the closed-loop system in terms of the inverter input admittance. In order to achieve a certain desired inverter input admittance Y d, it follows from the expressions for the admittance in (.7) that the stabilization controller K should be chosen as U d Tω m K = Gc Yd + G c Gd. (.8) ωm Ud In Subsection..3 we discussed selection of Y d based on requirements on stability and disturbance rejection. A third requirement we impose on Y d is that the resulting stabilization controller (.8) should be simple to implement. These three design goals are partly conflicting and a good compromise has to be found. In order to do this, the design process is performed in three steps. In a first step, a simple controller making the admittance real and positive, i.e., obtaining optimum stability margin, is derived. While keeping the simple expression, the stability margin is then slightly reduced in a second step to improve the disturbance rejection properties. This controller, however, still requires the inversion of the torque transfer function G c, which is not possible in practice. In a final step, appropriate approximations of this inversion are therefore discussed.

187 DC-Link Stability 75 The explicit expression of the stabilization controller K depends on the specific torque control method used. As we consider two schemes, ISC as well as FOC, we will derive two expressions for K, which we denote K DTC and K FOC, respectively..4. Design for Maximum Stability Margin With ISC we use the expressions for G c and G d given by Equation (.) to rewrite the second term of the ideal stabilization controller in (.8) as ( DA) ( ) ω G c d ( Gc ) d G G = U GDF A GDF Similarly, with FOC we use the expressions in (.5) to obtain p r c d c U d ( ) ( ) fw. (.9) 3n ψ ω G G = G G I DA. (.3) For small pulse periods (small time delays) we can approximate D and -A in the denominator of (.9). If we further approximate the product DA by an equivalent time delay of T da seconds, we can also use -DA st da in (.9) and (.3). This means that we after some rearrangement can write U 3 d ψ npωmω r Tda Tωm 3 npψr DTC = c d + + ω da ω m Ud Lσ U d UdLσ K G Y T and (.3) U 3 npω d r mωt ψ ad Tω m KFOC = G c Yd + + ω m Ud Lσ U d 3 n ψ s G + U L s p r + c αc ωtda d σ + αc. (.3) To reach a positive real inverter input admittance (for maximum stability margins) with simple stabilization controllers, we propose to choose

188 76 DC-Link Stability Y, Tω m > 3 ψ n ω ω T (.33), Tω m. r p m da d = + Tω m Ud Lσ U d The admittance (.33) is positive real apart from cases where the stator frequency and rotor speed have different signs, which may happen around zero speed. This is not a problem as the magnitude of the admittance is small enough not to cause any problems at those speeds ( YZ DC <). In order to obtain the admittance (.33), it follows from the expressions of the stabilization controllers (.3) and (.3) that and K T n ψ Gc, Tω m > = + U d, Tω m. 3 p r DTC ω Tda UdLσ (.34) K G 3 n c, m pψ r s+ Gc αc ω > U FOC = ωtda d UdLσ s + + αc Tω m T T,. (.35) By choosing the inverter input admittance as (.33), we have minimized the influence of G c - in (.3) and (.3), which is positive as the inversion is not possible to exactly compute in practice. The remaining appearances of G c - in (.34) and (.35) are required to make the resulting admittance positive real, which is our goal in order to reach maximum stability margins..4. Improved Disturbance Rejection In this subsection we reduce the gain of the stabilization controllers K DTC and K FOC to improve disturbance rejection. To keep the expressions for the controllers simple, the gain reduction is done by multiplying the first terms of the expressions for the controllers in (.34) and (.35) by a constant factor K. The resulting inverter input admittance then becomes, Tω m > 3 ψ npω r mωtda Y = ( K ) Gc FOC T m U ω T d L + σ ωm U (.36) d

189 DC-Link Stability 77 where, with ISC ( FOC = s+ Gc α s + α c c) αc, s + αc ( s + αc ) with FOC. (.37) In (.37) the inverse of G c was approximated by neglecting time delays and by using controller parameters according to suggestions in [6], i.e., (5.). As the admittance (.36) is no longer real with K <, the stability margins are reduced. This is the price we pay for reducing the gain of the stabilization controllers and hence improving disturbance rejection. To derive an appropriate value for K, we for simplicity assume that in (.36) has constant magnitude independently of the phase (of course this is not true for analytic functions). If we demand that the phase of the admittance Y stays between ± 45, some geometry gives us that the magnitude of has to be less than /. This is illustrated in Figure -9, where the argument of the vector -, which equals the argument of Y at zero torque, should be within ±45. From the figure it follows that this requires that the radius of the circle is less than or equal to the length of the dotted vector, which is /. Figure -9: Phase requirement of -. Note that arg(- ) = argy. The maximum gain of FOC in (.37) over all frequencies is given by / 3 and we will use / 3 (-K ) as an upper limit for in (.36) also with ISC (to allow a certain resonance peak of G c ). Hence, the requirement on the magnitude of can be summarized as / 3(-K ) < /, which approximately means that K >.4 and we therefore propose to set K =.5.

190 78 DC-Link Stability Imag(Y) Nyquist plot of Y with K = Real(Y) Imag(Y) Nyquist plot of Y with K = Real(Y) Figure -: (A) Nyquist curves of the inverter input admittance with stabilization (.34) (K =) at zero torque for several rotor speeds. Small amplitudes correspond to low speeds. The values of the admittance at 8.8 and 53 Hz are marked with o s and * s, respectively. (B) The corresponding Nyquist curves with K =.5. Imag(YZ DC ) Nyquist plot of loop gain YZ DC Real(YZ DC ) Voltage [V] DC-link voltage Time [s] Figure -: (A) Nyquist curves of the loop gain with stabilization with K =.5 and the input filter with 8.8 Hz resonance frequency. The bold dashed curve represents the loop gain at 6% of base speed. As it does not encircle the point -, the linear model of the drive now is stable, cf. Figure -4.A. (B) Corresponding simulation at 6% of base speed that verifies that the drive has been stabilized, cf. Figure -8.A. In Figure -.A, the inverter input admittance (.36) is shown at zero torque with K =. (The non-zero phase of Y at 8.8 Hz is due to the bandpass filter, see Section.4). Compared to the result without stabilization, which is shown in Figure -6.A, we see that stabilization has made the inverter input admittance passive also at the lower resonance frequency. This has been achieved without increasing the magnitude of Y, which for zero torque is directly proportional to the disturbance transfer function Ĝ d, see (.7). The corresponding admittance with K =.5 is shown in Figure -.B, where it is seen that the disturbance rejection is further improved

191 DC-Link Stability 79 (the magnitude of Ĝ d is decreased). However, the phase of the admittance at the resonance frequencies is increased. It still stays less than 45, which was the design goal. In Figure -.A, the Nyquist curves of the loop gain of the drive with input filter with 8.8 Hz resonance frequency and Y according to Figure -.B are shown (K =.5). Compared to Figure -4.A we see that the curves now do not encircle the point -. This indicates that the drive has been stabilized. Stability is also verified by the simulation shown in Figure -.B at 6% of base speed. The corresponding simulation without stabilization was shown in Figure -8.A. (The drive with the input filter with 53 Hz resonance frequency is stable as well with K =.5)..4.3 Approximation of G c - The selection of the stabilization controllers (.34) and (.35) was made to give the drive maximum stability margins, i.e., to make the inverter input admittance positive real. For positive power, however, this requires the inversion of G c, which is not possible to exactly compute in practice due to for example time delays. Therefore an approximation of the inverse has to be used. If we denote the approximation F lead, the inverter input admittance becomes (cf. (.33)) 3 ψ npωmωt r da Y = ( K ) G c FOC + Ud L σ Tω m ( GF c lead, ) Tω m > + U d Tω m T. ω m U d (.38) In [56] and [58], the inverse of G c is simply neglected, i.e., F lead =. From (.38) we see that this means that the second term of the admittance is no longer perfectly real above the bandwidth of the torque control. With a real part of G c less than one, the real part of the inverter input admittance decreases with increasing positive power. However, if the phase shift of the transfer function G c at ω is moderate, the phase of the inverter input admittance is normally small enough to keep the drive stable. Neglecting G c - also decreases the gain of the stabilization controller above the bandwidth of G c. As long as stability is preserved, the reduction of stability margins due to

192 8 DC-Link Stability the non-real admittance may hence be desirable from a disturbance rejection point of view. Although F lead = works well in many situations, there are also cases where a better approximation of the inverse of G c is needed. For the example with large torque given in Section.4, the admittance (.38) with F lead = shows the same problem as the admittance (.) shown in Figure -6.B (compare the last terms of the admittances (.38) and (.)). Hence, the inverse of G c in the expression for the stabilization controller (.34) cannot simply be neglected in this example with high power and high resonance frequency of the input filter. A simple solution is then to replace G - c by a lead filter F lead as proposed in [6]. The idea is then to compensate for the phase shift of G c where it is really needed, i.e., at the input filter resonance frequency ω. By using the stabilization controller in (.34) with a lead filter giving a phase advance of 5 degrees at ω, the Nyquist curves of the resulting inverter input admittance Y are shown in Figure -.A. The corresponding Nyquist plot of the loop gain YZ DC is shown in Figure -.B (input filter with 53 Hz resonance frequency), which indicates that the system has been stabilized. This is also confirmed by the simulation shown in Figure -3. Imag(Y) Nyquist plot of admittance Y ω m Real(Y) Imag(YZ DC ) Nyquist plot of loop gain YZ DC Real(YZ DC ) Figure -: (A) Nyquist curves of inverter input admittance with torque close to pull-out torque with stabilization (.34) where G c - is approximated by a lead filter giving 5 phase increase at 53 Hz. The values of the admittance at 53 Hz are marked with * s. The bold dashed curve corresponds to 5% of nominal speed, which now is in the right half plane at 53 Hz. (B) Nyquist curves of the corresponding loop gain, where the point - is not encircled. In the expression for the stabilization controller used with FOC (.35), the inverse of G c also appears in the term independent of the operating point torque. Here we suggest to approximate G c - by simply neglecting the time delay, see the expression for G c in (.5). For example, with current controller parameters according to (5.) it follows that

193 DC-Link Stability 8 s+ Gc αc s+ αc. (.39) s+ α s+ α Note that the resulting stabilization controller K FOC is still proper. c c DC-link voltage 78 Voltage [V] Time [s] Figure -3: Simulation at pull-out torque at 5% of base speed with input filter with 53 Hz resonance frequency. Here stabilization with a 5 lead filter is used, which stabilizes the system..4.4 Practical Stabilization Controllers Summarizing the previous subsections, i.e., the expressions for the stabilization controllers giving maximum stability margins, the gain reduction K =.5, and the approximation of G - c by a lead filter, we obtain the following final expressions for the stabilization controllers and K T n ψ F lead, T ωm > = + Ud, Tω m 3 p r DTC ω Tda 4 UdLσ (.4) K T n ψ + F lead, T ωm > = + U d +, Tω m. 3 p r s αc FOC ωtda 4 UdLσ s αc (.4) The stabilization controllers (.4) and (.4) meet a good compromise between damping and disturbance rejection, but are also simple to implement. Note for example that, without the lead filter, the stabilization controller K DTC

194 8 DC-Link Stability is simply a constant (at a given operating point), independent of the torque controller parameters. This expression equals the static gain derived in [56]. 8 DC-link voltage Torque 7 5 Voltage [V] Torque [Nm] Time [s] Time [s] Figure -4: (A) Magnitude plot of transfer function from T ref to T in (.8) at zero torque. (B) Magnitude plot of transfer function from E to T in (.8) at zero torque. (C) Response in DC-link voltage to a step in the line voltage at % (solid line) and 8% (dashed line) of nominal speed at zero torque. (D) Response in torque to a step in the line voltage at % (solid line) and 8% (dashed line) of nominal speed at zero torque. To illustrate the trade-offs involved in choosing the inverter admittance, Bode plots of the transfer functions from torque reference and line voltage to torque in (.8) with the stabilization controller (.4) are shown in Figure -4. Here the input filter with 35.5 Hz resonance frequency (realistic resistance) in Appendix A is used and the transfer functions are evaluated at zero torque (very low speeds have been excluded). With the chosen stabilization controller, the damping of the drive increases with speed (cannot avoid poor damping at low speeds as Y is proportional to ω m ). This is seen in Figure -4.B, where the resonance peak decreases with speed. The damping property is also seen in Figure -4.C, where the responses in DC-link

195 DC-Link Stability 83 voltage and torque are shown at steps in the line voltage at % (solid) and 8% (dashed) of base speed. We see that damping is improved at the higher speed. The improved damping is, however, paid with increased torque transients as seen in Figure -4.D (but also by an increased overall level in the transfer function shown in Figure -4.B compared to without stabilization). The admittance Y should hence be chosen large enough to give good damping but small enough to give acceptable peaks in transients. The trade-off was here also affected by striving for simple expressions for the stabilization controllers. Remark: Disturbances added directly to the DC-link voltage have much smaller effect on the DC-link voltage as well on the torque compared to distuturbances in the line voltage..5 Summary In this chapter we have analyzed stability problems of an induction motor drive due to the interaction between the inverter and the input filter. Analysis of these problems is usually done under the assumption of perfect control. In Section.4 it was shown that design under such assumptions may fail. To improve synthesis of the stabilization controller, more accurate models of the torque control have been introduced, including effects of time delays and limited control bandwidth. With these models, the instabilities of the examples given in Subsection.4.3 could be analytically predicted. By using a feedback representation of the drive, performance requirements of the closed-loop drive could be stated in terms of the inverter input admittance. It was concluded that for large stability margins and efficient torque control, the phase of the inverter should be close to zero and the magnitude should be small at the resonance frequency of the input filter. This way the drive is stabilized with small influence on the torque control performance. Shaping of the inverter input admittance was obtained by modifying the torque reference. In this chapter it was also shown that this approach is equivalent to other proposed stabilization schemes. That is, a certain inverter input admittance achieved with any of the other methods can also be obtained with torque reference modification. Explicit expressions for the stabilization controller were derived to give a good trade-off between damping of the drive and efficient torque control. The design was done for FOC as well as ISC.

196

197 Chapter Verification of Stability Margins from Measurements In Chapter we derived stabilization controllers to stabilize an induction motor drive with a resonant input filter. In this chapter the performance of the proposed stabilization controller K DTC in (.4) is verified on a real system. The stabilization controller, as well as modulation and ISC, are therefore implemented on real control hardware (digital signal processor). Instead of letting the controller operate a real process, the inverter and induction motors are simulated with a HW-in-the-loop simulator, which saves time and money and also makes the identification process easier. The simulated process is run on with sampling time of 4µs, which should be fast enough to capture all relevant process dynamics from the stability point of view. The hardware-inthe-loop simulator is shown in Figure -. This kind of simulation for a traction application has prior been presented in for example [84]. Figure -: HW-in-the-loop simulator used to validate the proposed stabilization controller.

198 86 Verification of Stability Margins from Measurements Stability and robustness analysis will be performed at three torque levels; maximum positive torque, zero torque and maximum negative torque at rotor speeds between % and 65% of nominal speed. Section. describes how to evaluate stability and robustness of the drive and Section. gives details on the SW implementation. Section.3 and Section.4 then present stability results of the linear dynamics and the nonlinear feedback loop due to linearization effects, respectively.. Stability and Robustness Analysis In Chapter, the linear feedback model of the induction motor drive in Figure -3 was derived. However, if we start from the model of the drive in Figure -3, we may more generally introduce the following non-linear relation between the DC-link voltage and the DC-link current d () = ( )() i t f U t d. (.) Together with the linear input filter, the feedback loop of the drive may then be represented as in Figure - (cf. Figure -3 where we used a linear model of f, which we denoted Y). E G E U d f(.) i d Z DC Figure -: Non-linear feedback model of an induction motor drive. To examine stability of the non-linear feedback structure in Figure -, we will use the small-gain theorem (input-output stability). Since no phase information is used, this test may be quite conservative. However, by combining the small gain theorem with stability tests for linear systems, less conservative conditions can be obtained. In the following subsections we will show how this can be done for the system in Figure -, inspired by the work of Schoukens et al [69]. We start by reviewing the small-gain theorem in Subsection... Subsection.. then describes how to examine stability of the linear dynamics of the feedback loop in Figure -, and robustness issues are finally treated in Subsection..3.

199 Verification of Stability Margins from Measurements 87.. Input-Output Stability Generally, input-output L stability of two interconnected systems as in Figure -3 can be studied using the small-gain theorem. u G y y G u Figure -3: Two interconnected systems. Suppose both systems are L stable, that is where y y β u + α (.) β u + α, (.3) ( τ) y = y dτ. (.4) Then the feedback connection u (t)=y (t) and u (t)=y (t) is L stable if β β, see [4]... Stability Analysis Let us represent the non-linear relation between DC-link voltage and DC-link current in (.) as d () ( ) () () i t = Y p U t + v t. (.5) Here Y is a linear model and the additive error term v(t) accounts for model errors as well as noise. We will consider the following model error model () NL () () () = ( )() d v t = v t + e t (.6) vnl t g Ud t (.7) v β U + α, α, β. (.8) NL v d v v v

200 88 Verification of Stability Margins from Measurements The error term has been decomposed into one non-linear input signal contribution v NL (t) and additive noise e(t). The factor β v is the gain of the nonlinear system g and the constant α v can be used to model offsets and external L signals, see [45]. The L gain of the nonlinear system g is formally defined as vnl g = max. (.9) Ud U With the representation of the non-linear system f given by (.5) and (.6), it follows that Figure - is equivalent to Figure -4. d g(.) ~ E + U d Z + E Y v NL i d Z DC Figure -4: Feedback model of an induction motor drive, where the inverter is modeled by a linear part Y and an additional non-linear contribution g. If the linear part of Figure -4 is stable, i.e., the system obtained with v NL (t) =, we may close the linear feedback loop and stability of Figure -4 is equivalent to stability of the system in Figure -5. ~ g(.) U d v NL -Z DC +YZ DC Figure -5: Equivalent representation of the non-linear feedback loop of the induction motor drive. If we assume that ZDC ( jω ) ( ) ( ) βz, ω + Y jω Z jω < DC (.) g = β v, (.)

201 Verification of Stability Margins from Measurements 89 the small gain criterion then implies that the feedback system in Figure -5 is input-output stable if β Z β v (provided Z DC /(+YZ DC ) is stable). An alternative approach to prove input-output stability is to directly estimate the total loop gain in Figure -5. We therefore introduce the signal w(t) as where () ZDC ( p) ( ) ( ) w t = vnl t, (.) + Y p Z p NL DC () ( ) d () v t = g U. (.3) We may now estimate the gain from the input U d (t) directly to the signal w(t). If w β U + α, (.4) w d w where β w <, we know from the small-gain theorem that the closed loop system is stable. To summarize, the proposed procedure to experimentally verify stability of the induction motor drive in Figure - is hence given by the following two steps: Identify the linear model Y in (.5) and check stability of the linear dynamics in Figure -4. Use the linear model Y to estimate the non-linear gain β w in (.4). If the linear dynamics are stable and the gain β w is less than one, then the system is stable...3 Robustness The previous subsection described a procedure to validate stability of an induction motor drive from measurements. In practice we not only want stability, but also a certain amount of robustness, i.e., we want the system to remain stable also if the actual system deviates from the one we use for stability analysis. For analysis of robust stability we will add linear perturbations to the linear model Y. We will represent the perturbed system Y p with relative uncertainty as

202 9 Verification of Stability Margins from Measurements ( ω) ( ω) ( ( ω) ( ω) ) Y j = Y j + w j j, (.5) p I I where w I is a weight function and I is a stable linear system such that I ( jω), ω. (.6) We hence measure robustness as the size of the weight w I, which makes the drive unstable (frequency by frequency). With the representation (.5) of the perturbed inverter input admittance, the feedback loop in Figure -5 is modified to Figure -6 (here we assume that Z DC /(+YZ DC ) is stable). Figure -6: Equivalent representation of the induction motor drive, with added relative linear model uncertainty I. The signal w(t) defined by (.) is now given by DC ( ) ( ) ( ) ( ) DC ( ) ( ) ( ) Z p Y p Z p w() t = vnl () t + wi I ( p) Ud ( t).(.7) + Y p ZDC p + Y p ZDC p w t () w() t By using the triangle inequality, the input-output stability requirement of the system in Figure -6 can be written as w w w YZ max max + max = + <,(.8) DC βw wi Ud U Ud Ud d U d U d + YZ DC where β w is defined as the gain in the loop including the nonlinearity, cf. (.4). For stability we hence need w I YZ DC < ( βw), ω. (.9) + YZ DC

203 Verification of Stability Margins from Measurements 9 As a single measure of robustness we introduce w I as the smallest value of w I over all frequencies, i.e., w I YZ DC ( βw). (.) + YZ DC The linearization errors hence reduce the stability margin by the factor -β w, which is illustrated in Figure -7. Figure -7: Illustration of robustness of the closed-loop system. At each frequency, the uncertainty that can be added to the nominal value of the admittance without destabilizing the drive is represented by the inner circle. The effects of the non-linear model errors reduce the circles by a factor -β w at all frequencies.. Implementation This section describes some relevant details of the software implementation of the torque and stabilization controllers. Also details of the used pulse patterns are given... Pulse Patterns In the example used in this chapter, realistic pulse patterns are applied. The inverter switching frequency is then varied with the stator frequency according to Table - and Figure -8.A, where f s denotes the stator

204 9 Verification of Stability Margins from Measurements frequency (measured in Hz). As described in Section., the motor voltages are generated by switching the potentials of the phases of the voltage source inverter. The quality of the voltage is for example determined by the inverter switching frequency. Generally, high switching frequencies generate high frequency harmonics, which have little influence on the electromechanical system. High switching frequencies also give short time delays. From these points of view, it is advantageous to use as high switching frequency as possible. On the other hand, large switching frequencies generate more losses. Further, to generate the large fundamental stator voltages, which are needed at higher speeds to keep the flux constant, the switching frequency must be reduced. This for example follows by comparing Figure -6 and Figure -7 in Section.. To efficiently use the inverter, the switching frequency is therefore varied with the stator frequency. At low stator frequencies a large switching frequency is used and as the stator frequency increases, the switching frequency is decreased in order to increase the stator voltage magnitude. For the practical example in this chapter, seven different pulse patterns are applied. where so-called six step operation is used above base speed (pulse pattern number 7 in Table -). With pulse patterns nr - nr 7, the switching frequency is proportional to the stator frequency (synchronized pulse patterns). The reason for synchronizing the switching frequency to the stator frequency is to avoid low frequency oscillations of the torque. Synchronized pulse patterns are recommended when the quotient between switching frequency and stator frequency is lower than, see [73]. As only the low frequency components of the stator voltages were modeled in Section., the only property of the actual pulse pattern that is needed is the pulse period, which influences the time delay. The varying switching frequency of the different pulse patterns leads to a varying pulse period as shown in Figure -8.B. The total time delay, estimated by expression (6.36), contains the sample time T s and the pulse period T p. In the real implementation T s varies with the operating points, which, however, is not considered by the linear models. Remark: With six-step operation (see Subsection..) the stator voltage magnitude saturates, which has two effects on the control. First, the stator flux decreases as the motor speed increases, and second, the stator voltage magnitude is fixed and may not be modified by the control. Six-step operation is here modeled only by adjusting the operating point flux and during the evaluation it turned out that the theoretical models deviated from the models obtained through identification. The identified loop gains indicated a lower bandwidth than what was modeled. This was actually the case also for the pulse pattern next to six-step operation (pulse pattern number 6 in Table -).

205 Verification of Stability Margins from Measurements 93 The theoretical models were therefore adjusted ad hoc by multiplying the torque controller parameters by the factors.5 and.4 in pulse patterns number 6 and 7, respectively. Pulse Pattern Switching Frequency Pulse Period No f sw = 55 Hz T p = /(f sw ) No f sw = f s T p = /(4f s ) No 3 f sw = 5f s T p = /(3f s ) No 4 f sw = 7f s T p = /(f s ) No 5 f sw = 5f s T p = /(f s ) No 6 f sw = 3f s T p = /(6f s ) No 7 f sw = f s T p = /(6f s ) Table -: Switching frequencies and pulse periods in the different pulse periods applied in this example. Switching frequency [Hz] Switching frequency as a function of stator frequency *f s 4 3 *f 7*f s s 5*f s 3*f s f s Stator frequency [Hz] Pulse period [ms] Pulse period as a function of stator frequency Stator frequency [Hz] Figure -8: (A) Switching frequency as a function of stator frequency and (B) pulse period as a function of stator frequency (which here is denoted by f s).

206 94 Verification of Stability Margins from Measurements.. Stabilization and Torque Controllers The lead filter F lead of the stabilization controller K DTC in (.4) is set to one and the first term of the expression in (.4), which is shown as the solid line in Figure -9, is approximated by the dashed line in Figure DC-link stabilization gain Stator frequency [Hz] Figure -9: Approximation (dashed curve) of the first term of the DC-link stabilization gain (.4) (solid curve) as a function of stator frequency. The band-pass filter described in Subsection.4.3 is implemented with cut-off frequencies 3 and 3 Hz and the parameters of the discrete-time torque controller are set to K nr =.9 and T nr =.3..3 Linear Stability Analysis In this section the linear models Y in (.5) are identified for a number of operating points. These linear models are used to evaluate stability and robustness of the specific drive, but also to verify the linear models of the inverter input admittance derived in Chapter. For the latter purpose, linear models are identified both with and without stabilization. Subsection.3. discusses how to identify the inverter input admittance from measurements. As we use a hardware-in-the-loop simulator to generate the measurements, the loop in Figure -4 can be opened such that the DC-link current does not affect the DC-link voltage through Z DC. This way the models can be identified in open loop. The results of the identification are presented in Subsections

207 Verification of Stability Margins from Measurements Identification of Inverter Input Admittance The theory and practice of system identification of linear time invariant (LTI) models are well developed and advanced methods and tools are available. The problem gets more difficult when considering non-linear systems. When applying standard identification methods for estimating linear models to data from a nonlinear system, the best linear approximation in a certain sense is obtained, see e.g. [9]. Consider a nonlinear dynamical system with input sequence {u(t n )} and corresponding output sequence {y(t n )}, t n =nt s, n=,,,n, where T s is the sampling time. The best linear model of this inputoutput relation, in the sense that it produces the same second order statistics, is given by the frequency response G jωts ( e ) jωts ( e ) jωts ( e ) Φ yu = Φ uu causal where subscript causal denotes the causal part, Φ ( j T s uu e ω ) spectral density of the input, and ( j T s yu e ω ), (.) is the power Φ is the cross power spectral density of the input and output signals. This linear approximation depends of course on the specific input signal. This is closely related to linearization using Taylor expansion of the nonlinear system, which often is done around a given operating point. We will consider variations around a specific operating point u(t n ) = u and y(t n ) = y. It is often easier to remove the DC components before forming the model, i.e., to work with the signals u(t n )=u(t n )-u and y(t n )=y(t)-y. We will assume that the DC components have been removed, but still use the notations u(t n ) and y(t n ). A common choice of input signal is u(t n ) = C sin(ω t n ), i.e., a sinusoidal signal with frequency ω and amplitude C. Let y(t n ) be the corresponding output. From observations {y(t n ),u(t n )}, the (finite) Fourier transforms of the signals can be calculated as N jωt s jωts n UN ( e ) = u( tn) e, (.) N n= N jωt s jωts n YN ( e ) = y( tn) e. (.3) N n=

208 96 Verification of Stability Margins from Measurements The Fourier transform of u(t n ), is then approximately equal to NC /4 for ω = ω and almost zero otherwise. The empirical transfer function estimate equals Gˆ jωts ( e ) jωts ( e ) ω s ( e ) Y =. (.4) N N UN j T It is possible to show, see [44], that ˆ j Ts GN ( e ω j T ) converges to ( s ) G e ω, under certain assumptions on the system and the disturbances, as N tends to j T G e ω s is the best linear frequency response infinity. In this case ( ) approximation of the nonlinear system for a sinusoidal input signal with amplitude C and frequency ω (around the operating point u ). The continuous-time Fourier transform can be approximated by ˆ ( ) ˆ jωt ( s π GN jω GN e ), ω <<. (.5) T s.3. Coasting Nyquist curves of the identified admittances without stabilization for the operating points with zero torque are shown in Figure -. imag(y) Nyquist plot of Y real(y) Figure -: Inverter input admittance obtained through simulations at zero torque and several motor speeds (without stabilization).

209 Verification of Stability Margins from Measurements 97 Due to the different time delays in the different pulse patterns (see Figure -8.B), the trajectories of the inverter input admittance now vary more between the different operating points compared to e.g. Figure -.A. The corresponding Nyquist curves obtained with the linear model from Chapter are shown in Figure -.A. Compared to Figure - we see large deviations of the Nyquist curves with largest magnitudes, which correspond to six-step operation. To model the limited control authority in six-step mode, where the modulation index of the inverter is fixed, the controller parameters are simply reduced, see Subsection... The resulting inverter input admittance is shown in Figure -.B. By comparing Figure - and Figure -.B, we see that we now have similar shapes of the identified and theoretical models, although there are deviations in phase and amplitude at several operating points (which motivates stabilization controller design for large stability margins). During the rest of the evaluation, only the results with reduced controller gains are presented. imag(y) Nyquist plot of Y real(y) imag(y) Nyquist plot of Y real(y) Figure -: Nyquist plots of the (A) original and (B) modified theoretic inverter input admittances Y at zero torque (without stabilization). To evaluate stability of the linear dynamics of the drive, the loop gains YZ DC are plotted in Figure -. Figure -.A shows the Nyquist curves of the loop gain with identified inverter input admittance and Figure -.B shows the corresponding Nyquist curves using the theoretical admittance. The theoretical and simulated results match rather well. We also see that for this particular system, the drive is stable at coasting also without stabilization. The stability margins are however small and result in a poorly damped system. By adding the stabilization controller, the Nyquist curves of the loop gain changes from Figure - to Figure -3. We see that the stabilization increases the distances from the Nyquist curves to the point -. This is an indication of improved robustness.

210 98 Verification of Stability Margins from Measurements Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Figure -: Nyquist plots of the (A) simulated and (B) theoretic loop gains YZ DC at zero torque without stabilization. Magnitudes are limited to 5 for better visibility. Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Figure -3: Nyquist plots of the (A) simulated and (B) theoretic loop gains YZ DC at zero torque with stabilization. Magnitudes are limited to 5. In Figure -4 the robustness measure w I defined by (.) (with β w = ) is shown as a function of speed. The figure to the left shows w I obtained with the identified inverter input admittance, whereas the figure to the right shows the corresponding result with the theoretical model. The dashed curves show the robustness margin without stabilization and the solid curves show the robustness measure when stabilization has been applied. We see that the stabilization improves the stability margin and we also see that the predicted results with the theoretical models match the measured results well. (The strange behavior of the robustness measure around 35 Hz stator frequency obtained from measurements is due to identification problems in this frequency range poor signal to noise ratio).

211 Verification of Stability Margins from Measurements 99 4 Stability Margin 4 Stability Margin Stator frequency [Hz] Stator frequency [Hz] Figure -4: Robustness measure w I in coasting as a function of speed using (A) the identified inverter input admittance and (B) the theoretical inverter input admittance. The dashed curves are obtained without stabilization and the solid curves are obtained with stabilization. To see that the stabilization has been achieved without making disturbance rejection worse, we have plotted Nyquist curves of the inverter input admittance with stabilization in Figure -5. Compared to the corresponding figures without stabilization, i.e., Figure - and Figure -.B, we see that the magnitude of Y (which directly corresponds to disturbance rejection at zero torque) has not been increased. Only the phase has been affected to improve the stability margins. imag(y) Nyquist plot of Y real(y) imag(y) Nyquist plot of Y real(y) Figure -5: Nyquist plots of the (A) simulated and (B) theoretic inverter input admittances at zero torque with stabilization..3.3 Driving As noted in Section., driving is the most critical case when it comes to DC-link stability as stability margins are reduced with increasing power. This

212 Verification of Stability Margins from Measurements for example follows from the simplified stability measure (.5) and the expression for the damping factor (.4). In Figure -6, Nyquist plots of the loop gain at driving are shown without stabilization. As the Nyquist curves encircle the point -, we conclude that the drive is unstable. We also see that the theoretical and simulated results match fairly well. Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Figure -6: Nyquist plots of the (A) simulated and (B) theoretic loop gains YZ DC at driving without stabilization. Magnitudes are limited to 5. The corresponding Nyquist curves with stabilization are shown in Figure -7. Here it is seen that the Nyquist curves of the loop gain no longer encircle the point. Hence, the drive has been stabilized. Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Figure -7: Nyquist plots of the (A) simulated and (B) modified theoretic loop gains YZ DC at driving with stabilization. Magnitudes are limited to 5. With stabilization, we may again compute the stability measure w I with β w = (note that the drive is unstable without stabilization). The result is shown in Figure -8 as a function of speed. Around base speed, the stability margin

213 Verification of Stability Margins from Measurements obtained from measurements looks better compared to w I reached with the theoretical models. Again, the noisy behavior of the stability margin obtained from measurements around 35 Hz stator frequency is due to bad signal to noise ratio during the identification. 4 Stability Margin 4 Stability Margin Stator frequency [Hz] Stator frequency [Hz] Figure -8: Robustness measure w I in driving as a function of speed using (A) the identified inverter input admittance and (B) the theoretical inverter input admittance. The result is obtained with stabilization. Without stabilization the drive is unstable..3.4 Braking The Nyquist plots of the loop gains in braking are shown in Figure -9. As expected, the drive is stable and we see that the stability margins are larger in braking compared to coasting and driving. We also see that the theoretical and simulated results match well. Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Figure -9: Nyquist plots of the (A) simulated and (B) theoretic loop gains YZ DC at braking without stabilization. Magnitudes are limited to 5.

214 Verification of Stability Margins from Measurements The corresponding Nyquist curves with stabilization are shown in Figure -. Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Imag(YZ DC ) Nyquist plot of YZ DC Real(YZ DC ) Figure -: Nyquist plots of the (A) simulated and (B) theoretic loop gains YZ DC at braking with stabilization. Magnitudes are limited to 5. Figure - shows the robustness measure w I obtained with the identified as well as with the theoretical inverter input admittance. The dashed curves show the robustness margin without stabilization and the solid curves show the robustness measure when stabilization has been applied. We see that the predicted result with the theoretical models match the measured results well. In this example the stabilization has small effect on the stability margin in braking. 4 Stability Margin 4 Stability Margin Stator frequency [Hz] Stator frequency [Hz] Figure -: Robustness measure w I in braking as a function of speed using (A) the identified inverter input admittance and (B) the theoretical inverter input admittance. The dashed are obtained without stabilization and the solid curves are obtained with stabilization.

215 Verification of Stability Margins from Measurements 3.4 Linearization Errors In the previous section, linear models of the inverter input admittance Y were identified from measurements. It was shown that, with stabilization, the linear dynamics of the drive were stable at all examined operating points. In this section, these models will be used to also estimate the non-linear gain β w in (.4). For the drive to be stable also when considering the linearization effects, this gain should be less than one. From the expression for the robustness measure w I in (.), we see that a non-zero β w (between and ) reduces the stability margin established in the previous section. The problem of estimating β w using the definition in (.4) is of course to find the input, which gives the maximal gain. For linear systems, this corresponds to a sinusoidal input signal and the problem simplifies to finding the correct frequency. For our non-linear system we will first try to estimate the gain via an ad-hoc method in Subsection.4., but we also apply a more systematic approach called power iterations in Subsection.4.3. We start in Subsection.4. with a motivation to why we chose to directly estimate the total loop gain β w in Figure -5, instead of only estimating the gain of the non-linear part..4. Linearization Errors From Section. we know that, due to the inverter, the stator voltages contain high frequency components related to the switching frequency. This also holds for the DC-link current. Hence, feeding the inverter with smooth DClink voltages still generates high frequency components in the DC-link current. These signals are not generated by the linear model and consequently appear in the model error v NL (t). This makes the gain β v of the non-linear model error large and it is not possible to prove stability using β Z β v <. In Figure - an example of the estimation error in open loop is shown, where the input to the block Y in Figure -4 consists of sinusoids. Figure -.A shows the measured and estimated DC-link currents, whereas Figure -.B shows the difference between them, i.e., the estimation error. Here we see how the unmodeled switching frequency harmonics lead to a large model error, and hence also to a large gain β w. In Subsection.. we also proposed to directly estimate the total loop gain β w from U d (t) to w(t). The signal w(t) is obtained by letting the error signal v NL (t) through the filter Z DC /(+YZ DC ) in Figure -5, whose magnitude response is shown in Figure -3 (at zero torque). For low speeds, the

216 4 Verification of Stability Margins from Measurements magnitude of Y is small and is approximately equal to Z DC. This explains the large resonance peak in Figure -3. However, at higher frequencies the gain of the filter is small, which means that the influence of the switching frequency harmonics shown in Figure -.B is suppressed. 4 Measured and estimated DC-link current 4 DC-link current estimation error, v NL (t) 3 3 Current [A] - Current [A] Time [s] Time [s] Figure -: (A) Measured and estimated DC-link currents and (B) estimation error. The measured current contains high frequency components due to the inverter switchings. These components are not considered by the linear model and hence appear also in the estimation error, which makes the estimated non-linear gain very large. Figure -3: Magnitude of Z DC /(+YZ DC). By applying the filter Z DC /(+YZ DC ) to the error signal v NL (t), the high frequency components dominating in Figure - are suppressed. The

217 Verification of Stability Margins from Measurements 5 resulting filtered currents and current errors are shown in Figure -4. The estimation error in Figure -4.B has considerably been reduced, which motivates to directly estimate the total loop gain. 4 Measured and estimated filtered DC-link current 4 Filtered DC-link current estimation error, w(t) 3 3 Current [A] - Current [A] Time [s] Time [s] Figure -4: (A) Filtered DC-link currents and (B) filtered estimation error..4. Experimental Result If we neglect the additive noise, the error signal v NL (t) in (.6) can be calculated as () () ( ) () v t = i t Y p U t, (.6) NL d d using measurements of U d (t) and i d (t). By passing v NL, obtained with (.6), through the filter Z DC /(+YZ DC ), an estimate of β w defined in (.4) can be achieved. The gain from U d (t) to w(t) has been calculated for a large range of input signals. For example test signals with random frequencies and phases of sinusoids were used to excite the system. The conclusion is that a sinusoidal signal with frequency equal to the resonant frequency of Z DC /(+YZ DC ) gives a reasonable estimate of the maximum gain of the nonlinear system. In particular, this is true for low motor speeds, while a more wide band signal gives slightly better results for higher motor speeds. This makes sense from the magnitude plot of Z DC /(+YZ DC ) in Figure -3. Due to the large resonance peak, it seems reasonable that the worst case input signal will have most of its energy around the resonance. (The choice of amplitude is also important to obtain a good signal to noise ratio, but also to take the non-linear effects into account.)

218 6 Verification of Stability Margins from Measurements The calculation of v NL (t) involves the linear model of the inverter input admittance. As only a frequency domain model was identified in Section.3, all calculations are done in the frequency domain. This is possible due to Parsevals relation, which in discrete time says j T where ( s N ) N N j n/ N wt ( ) ( ) n = WN e π, (.7) n= n= W e ω is the Fourier transform of w(t), as defined by (.). Figure -5 shows the experimental results of the estimated gain β w at zero torque as a function of motor speed (here a single sinusoid with a frequency corresponding to the maximum gain of the filter in Figure -3 is used for every speed). This result shows that the gains are well below one, and that we thus have quite good stability margin for most motor speeds. The main problem is for the case when the resonance frequency of Z DC and the stator frequency are close. Here the signal to noise ratio is low, which implies larger estimation errors. Figure -6 gives the experimental results for driving and braking, showing gains smaller than one for all operating points. Note, however, that the obtained results only give a lower bound of the gain. β w Motor speed [Hz] Figure -5: Nonlinear gain β w in coasting as a function of motor speed.

219 Verification of Stability Margins from Measurements 7 β w β w Motor speed [Hz] Motor speed [Hz] Figure -6: Nonlinear gain β w in (A) driving and (B) braking as functions of motor speed..4.3 Power Iterations In the previous subsection an estimate of the non-linear loop gain β w in Figure -5 was calculated using the properties of the filter Z DC /(+YZ DC ), whose Bode plot is shown in Figure -3. In this subsection we try a more systematic approach to estimating β w, which is called power iterations. With power iterations, input sequences are generated iteratively through experiments on the system. Power iterations monotonically increase gain estimates for linear time invariant (LTI) systems and the convergence point can be made arbitrarily close to the system gain by using long enough experiments. In [3] it is illustrated through an example that power iterations also may be useful for certain non-linear systems. The method is called power iterations due to its close connection with the power method, used in linear algebra to iteratively compute an approximation of the largest eigenvalues of a symmetric matrix. The power method appears in many standard books on matrix analysis, e.g. []. Note that we only obtain a lower limit of the nonlinear gain by using power iterations. In the experiments performed to estimate β w, a white initial DC-link voltage disturbance sequence is applied with zero mean and a standard deviation of 5% of nominal DC-link voltage. In Figure -7 the gain with the initial white input sequence and the gain after 9 iterations are shown as the dashed and solid lines, respectively. Here we see that the gain with the initial white input sequence is quite similar to the gain obtained in the previous subsection. By using power iterations, we see that the gain is increased at higher speeds. At lower speeds, however, the iterations do not give an increased gain. An explanation to this is that the error signal here is dominated by the effect of a disturbance (poor signal to noise ratio). During the simulations, a quite large

220 8 Verification of Stability Margins from Measurements stator frequent oscillation in the DC-link current was observed that might be caused by offsets in the phase current measurements. At lower speeds, the gain is strongly influenced by the resonance peak of the filter shown in Figure -3. What we see in Figure -7 (and in the other figures showing the estimated gain) is the effect of the stator frequent disturbance as it passes through the resonance peak of the filter. At higher speeds, the effect of the disturbance is less significant and the error signal is to a larger extent determined by the input. Here the power iterations result in an increased gain. For further details, see [4]. We also note that the distinct peaks in the estimated gain appear where the added disturbance has caused the modulator to change pulse patterns. At these points the model used to predict the current response is evaluated at an incorrect operating point. β w Motor speed (Hz) Figure -7: Nonlinear gain β w in coasting as a function of motor speed..5 Summary In this chapter we have verified the performance of the stabilization controller proposed in the previous chapter using measurements from a HW-in-the-loop simulator. Here real control HW and SW are used, whereas the inverter and induction motors are simulated in real-time. HW-in-the loop simulations save time and cost compared to power lab or train experiments and experience shows that they give very reliable results. Further, with simulations, the inverter input admittance used for robustness analysis can be estimated in open-loop, whereas closed-loop identification would be necessary in a power lab (cannot prevent the DC-link current from influencing the DC-link voltage).

221 Verification of Stability Margins from Measurements 9 During stability analysis, not only the linear dynamics but also non-linear errors due to linearization were considered. The linear dynamics show good robustness and good agreement with the results obtained from the linear models. To also consider linearization effects in the stability analysis, the gain of the non-linear model error was estimated. Here only lower bounds on the gain can be estimated, which was done by using a physically motivated approach but also by through so-called power iterations. The estimated lower bounds all indicate a stable system.

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223 Chapter Summary and Suggestions for Future Work This thesis has considered robust control of an induction motor drive. Control design was performed in two steps. First a torque controller was constructed for a stiff DC-link voltage, and thereafter an outer stabilization controller was designed to stabilize oscillations in the DC link with minimum influence on the torque control. This chapter briefly summarizes the main contributions and presents suggestions for future work.. Summary This section summarizes the thesis. Controllability Analysis of an Inverter Fed Induction Motor A controllability analysis of the inverter fed induction motor was first performed to better understand fundamental properties and limitations of the plant from a control point of view. The controllability analysis showed that: The motor has large RGA elements for large operating point stator frequencies, which lead to high sensitivity to model errors with a decoupling controller. Variations in the DC-link voltage affect the magnitude of the stator voltage, which acts in a high gain direction around the resonance frequency of the motor. Due to input constraints and limited control bandwidth, disturbances in the DC-link voltage therefore have a large effect of the motor outputs at high stator frequencies. Robust Control of an Induction Motor with Ideal Power Supply This thesis has focused on the control method Indirect Self Control (ISC), where stator flux magnitude and torque are controlled through PI controllers

224 Summary and Suggestions for Future Work in a coordinate system oriented to the stator flux space vector. It was shown that: The stator flux orientation results in partial decoupling of torque and flux control. The decoupling is partial in that changes in the reference flux affect the torque at non-zero operating point torque. The influence is rather small, however, but increases with the load. At higher speeds, estimation of the stator flux with a full-order observer with zero observer gain shows little dependence on the motor parameters. Here large parameter errors have to be introduced to significantly affect performance due to incorrect orientation, although the system according to the RGA analysis is sensitive to uncertainty. The errors increase the coupling from flux reference to torque and it may be advantageous to use a pre-filter on the flux reference to avoid exciting the critical frequencies around the operating point stator frequency. An example was also given where a different type of model uncertainty caused large coupling at higher motor speeds. These errors were generated by an approximation in the ISC control law. At lower speeds, stator flux estimation with the open-loop observer (zero observer gain) is sensitive to parameter errors, but errors in the fieldorientation also have less effect on performance at those speeds. With correct field-orientation, it was shown that flux control is independent of parameters but that: The rotor resistance affects the torque control loop gain at low frequencies and the leakage inductance affects the loop gain at higher frequencies. With a large bandwidth, stability is hence only affected by errors in the leakage inductance. An overestimation of the leakage inductance leads to a reduced stability margin. An accurate estimation of the stator flux is not only required for fieldorientation (decoupling), but also for torque estimation. It was shown that steady state torque estimation with the open-loop observer is very accurate at higher speeds, whereas large errors may appear at lower speeds. Here a nonzero observer gain is required for satisfying performance. At higher speeds, experience and simulations show that large observer gains may lead to stability problems. It was shown that large observer gains can amplify models errors. The stability problems were therefore interpreted as a consequence of the large RGA elements, which make the closed-loop performance sensitive to model errors.

225 Summary and Suggestions for Future Work 3 DC-Link Stability With efficient suppression of variations in the DC-link voltage, the inverter behaves like a negative resistance as seen from the DC-link of the drive. The inverter then decreases the already small damping factor of the input filter and eventually destabilizes the drive. With a fixed input filter, this stability problem must be solved by modifying the inverter input admittance. In this thesis, this was accomplished by modifying the torque reference as a function of the DC-link voltage. Other approaches exist but it was shown that an inverter input admittance achieved with any of these schemes also can be achieved with torque reference modification. Stability was analyzed using a model of the drive containing the impedance of the input filter and the inverter input admittance in a feedback loop. With this representation, stability of the drive can be evaluated using the Nyquist stability criterion. This way: Stability margins may be established also in presence of time delays (as opposed to eigenvalue analysis of the drive). Stability margins of the drive may be checked through measurements (or hardware-in-the-loop simulations as in this thesis). This way stability margins of a real system may be computed.. It was concluded that, with a large resonance peak of the input filter: A stable drive (approximately) requires the real part of the inverter input admittance to have positive real part at the resonance frequency of the input filter. For maximum stability margins, the inverter input admittance should be real and positive at the resonance frequency. With perfect rejection of DC-link voltage disturbances, which usually is assumed during treatment of DC-link stability, this condition is only violated at positive power. However, due to inevitable time delays, disturbance rejection is never perfect in practice and it was shown that: The drive may become unstable also at zero torque. Stabilization is used to modify the inverter input admittance, which means that disturbance rejection is worsened. With non-perfect disturbance rejection, also the dynamics from torque reference to torque are affected by the input filter dynamics and the stabilization. Hence, it was concluded that: Stabilization is a trade-off between damping of the drive and tight control.

226 4 Summary and Suggestions for Future Work This is usually not recognized in connection with stabilization controller design, where the goal often is just to give the drive as good damping as possible. It is not noted that a very good damping actually may lead to nonsatisfactory performance of the drive. To minimize the influence of stabilization on torque control, while still reaching acceptable stability margins, we require that, in addition to the requirement for stability: The magnitude of the inverter input admittance at the resonance frequency should be small. With linear models of the drive including the effects of time delays and limited control bandwidth, explicit expressions for stabilization controllers to use with ISC as well as FOC were derived to meet a good compromise between damping and efficient torque control. The stability margins reached with the stabilization controller were verified through hardware-in-the-loop simulations using real control HW and SW. The general ideas used during design of the stabilization controllers also apply to other types of motors such as permanent magnet motors.. Future Work This section gives proposals for future research within the topics treated in this thesis. Robust Control of an Induction Motor with Ideal Power Supply It has been shown that the steady-state estimates of torque and stator flux are very sensitive to parameter errors with a full-order observer with zero gain at low speeds. For this reason, a non-zero observer gain is required. Non-zero observer gains could also be used to speed up error convergence. However, it was also shown that large observer gains may give an unstable closed-loop system at higher speeds. It is desirable to find gains that allow fast error convergence with well-damped poles or decreased parameter sensitivity but still guarantee closed-loop stability at all speeds. Here further work is needed. In this thesis we have considered the speed of the motors to be measured. In an application without speed sensors, the full-order observer applied in this thesis cannot be used. The consequences on closed-loop performance with ISC for this case are still to be examined.

227 Summary and Suggestions for Future Work 5 DC-Link Stability Design of stabilization controllers was based on simple requirements on the inverter input admittance at the resonance frequency of the input filter. From the equation for the torque given by (.8), design of the stabilization controller could be expressed in terms of an optimization problem, with requirements on robust stability and robust performance, similar to the problem statement used for torque and flux controller design in Chapter 9. Here also the effects of the bandpass-filter could be explicitly considered. Throughout the treatment of DC-link stability, we have considered the mechanical speed to be constant. This assumption may be justified as the inertia of the mechanical system often is very large and fast changes in the speed are not realistic in traction applications. However, at slippery tracks fast changes of the speed may actually happen and in such cases also the mechanical system should be considered during stability analysis. The added dynamics due to the mechanics are shown as the shaded block in Figure - (cf. Figure -9). Here the generated torque affects the mechanical speed, which in turn influences the torque. m E G G M G E T ref + G c T m U d + + i d - Z DC U d G d T m U d Figure -: Induction motor drive including the mechanical dynamics. By adding the dynamics of the mechanical system, the transfer functions from torque reference and DC-link voltage to torque change to (cf. Equation (.3)) () = ( ) () + ( ) () T t GωGM GcTref t GωGM GdUd t. (.) G G c The design of the stabilization controller could now be repeated with the new transfer functions G c and G d. d

228 6 Summary and Suggestions for Future Work To reach acceptable results with the mechanics included, one could also think of using a new feedback path from the mechanical speed to the torque reference. This could also be an approach to damp oscillations due to mechanical resonances. Mechanical oscillations affect the power and hence also result in oscillations in the DC-link voltage. In this case, the joint problem of damping the electrical and mechanical dynamics has to be considered. The stabilization controllers were designed to optimize the inverter input admittance with respect to the input filter dynamics. With different dynamics on the DC-side of the inverter, other strategies for shaping the admittance may be preferred. Consider for example a system with an auxiliary converter in parallel with the induction motor drive. The inverter then sees the dynamics of the auxiliary converter in parallel with the input filter. By using a model of the auxiliary converter, the stabilization controller could then be optimized for this configuration.

229 Appendix A Drive Data This appendix presents relevant details of the induction motor drives used in the examples in this thesis. As different drives are used to examine effects of torque and flux control with a stiff DC-link voltage and DC-link stability, two data sets are given below. The first data set is used with torque control with ideal power supply, whereas the second data set is used when examining DClink stability. Drive Data Set During analysis of torque and flux control of an induction motor with constant DC-link voltage in Chapter 3 - Chapter 9, the following motor parameters are used (Γ-ECD): Stator resistance R s = 8.5 mω Rotor resistance R r = 7.3 mω Stator inductance L m = 6. mh Leakage inductance L σ =.79 mh Number of pole pairs n p = Rated DC-link voltage U d = 75 V Base speed ω base = 58 rad/s Rated flux Ψ =.9 Vs Rated torque T = 6 Nm Maximum torque T max = 4 Nm Drive Data Set This section gives details of the drives used in connection with the treatment of DC-link stability. This includes motor and filter data but also torque levels.

230 8 Drive Data Motor Parameters The induction motors used during the treatment of DC-link stability are defined through the following parameters (Γ-ECD): Stator resistance R s = 89.4 mω Rotor resistance R r = 65.8 mω Stator inductance L m = 43.8 mh Leakage inductance L σ =. mh Number of pole pairs n p = Rated DC-link voltage U d = 7 V Base speed ω base = 369 rad/s Rated flux Ψ =.59 Vs Maximum torque: T max = 78 Nm Note that the resistances are given for C but the analysis is done at a temperature of C. The temperature coefficient is.39. In the examples used in connection with DC-link stability we also use 4 motors in parallel. Input Filter Data During the treatment of DC-link stability, three different input filters are used: Input filter with 35.6 Hz resonance frequency: Filter inductance: L = 5 mh Filter capacitance: C = 4 mf Filter resistance: R = 4 mω Input filter with 8.8 (53) Hz resonance frequency: Filter inductance: L = 6 () mh Filter capacitance: C = (8) mf Filter resistance: R = mω Note that the last two input filters have a very small resistance. This small resistance is used to clearly motivate the derived requirements on the inverter input admittance in Subsection..3. Tractive Effort Three different driving scenarios are considered during the treatment of DClink stability: Coasting (zero torque)

231 Drive Data 9 Driving (maximum torque) Braking (minimum torque) During driving and braking, the torque reference and the power per motor vary with speed as shown in Figure A-. Torque [Nm] Driving and braking torque as functions of speed Motor speed [Hz] Power [kw] Driving and braking power as functions of speed Motor speed [Hz] Figure A-: (A) Torque and (B) power as functions of speed in driving and braking.

232

233 Appendix B Closed-Loop Equations In this appendix we derive a linear model for the closed-loop system when ISC is applied to an induction motor. We will consider transfer functions from the reference signals to the outputs but also from disturbances to outputs. The disturbances we consider are the DC-link voltage and the mechanical speed. During the derivation we will use a real-valued space vector representation and in a first step neglect time delays in the control loop. To handle time delays we need to also consider prediction of certain motor quantities. Real-Valued Space Vector Representation When discussing ISC in Chapter 6 we mainly used complex-valued space vectors. Also the equations for the induction motor in Chapter 3 were given on space vector form. The states of the motor could then be expressed with transfer functions with the stator voltage space vector as input. Hence, for example the stator current may be expressed as i s () = ( α ( ) + α ( )) G ( p) s () t G p jg p u t, (B.) cf. equations (4.8) and (4.9). With ISC, however, we need signals like the magnitude of the stator flux space vector. The equations are then no longer linear (not even at constant speed) and we need to linearize. Still it is not possible to use a transfer function representation between complex-valued space vectors as for example { ψ * } Re{ * s ψs ψsg us} Re ψ s = = Gψ u s, (B.) ψ ψ s s for some transfer function (with complex-valued coefficients) G Ψ. Note that we cannot simply differentiate the stator flux magnitude as a function of the stator flux as

234 Closed-Loop Equations df ψ ψ ψ, (B.3) f ( ) dψ s ( ψ ) s s s s since the derivative of the magnitude of complex-valued signal Ψ s in (B.3) only exists at Ψ s =, see [5]. In order to represent the needed signals we therefore will use real-valued state space models or transfer functions with real-valued coefficients in this appendix. G Closed-Loop System without Time Delays In this section the closed-loop equations of the controlled induction motor are derived when using ISC. We start by analyzing the system without time delays, which corresponds to the system in Figure 6-4. Non-Linear Equations of Decoupling The goal is to derive a linear model of the closed-loop system in Figure 6-4. In order to reach this goal we start by linearizing the effect of the ISC control law given by (6.). We hence derive a linear model from the signal u c to the outputs of the induction motor. Inserting the control law (6.) into the stator equation (3.5) gives the following non-linear equation for the stator flux space vector ( ψ () ()) u () t ψ () t = u t + ju t ψ t. (B.4) s T s c The control variables are the stator flux magnitude and the torque. To express these quantities we use a polar representation with magnitudes and angles of the space vectors as introduced by (3.) (note that the torque can be computed using magnitudes and angles as in (3.)). The derivative of the stator flux space vector in (B.4) may then be expressed as () d jχs() t ψ s t jχs() t ψ s() t = ( ψs() t e ) = e + j χs() t s() t dt ψ s () t ψ. (B.5) By also representing the rotor flux equation (3.6) with magnitudes and angles, the motor equations can be put on the following non-linear state space form s () t = ψ () t u () t s ψ () ψ (B.6)

235 Closed-Loop Equations 3 R R ψ ψ δ ψ (B.7) r r () t = () t cos () t () t r s r Lσ Lσ () t () t R ψ r s δ () t = χs() t χr () t = ut () t sin δ () t npωm( t), (B.8) L ψ where δ(t) = χ s (t) -χ r (t) is the load angle. Before linearizing these equations we determine the stationary operating points around which we describe the system. Steady State Relations At steady state, the derivatives of the flux magnitudes and the load angle in equations (B.6) - (B.8) are constant. It then follows from the stator flux equation (B.6) that σ r ( ) ψsu sk ψ ψ ψ p ψsref ψs = =, (B.9) which means that the stator flux equals the reference flux at steady state, i.e., ψ s = ψ. (B.) Further, from the rotor flux equation (B.7) we can link the stationary fluxes as sref ψ = ψ cosδ. (B.) r s This relation can be used together with the equation for the load angle in (B.8) to get Rr ut = tanδ + npωm. (B.) L σ A steady state condition on u T may also be derived from Expression (6.) in Section 6., which states that Rr ut () t = npωm() t + T ref () t + 3 npψ r () t (B.3) Rr + K pi ( Tref () t T() t ) + KiX pi 3 n ψ t where p r ()

236 4 Closed-Loop Equations R X pi () t = T t T t. (B.4) 3 n ψ p ( ref () ()) () t r r From (B.4) we conclude that the steady state torque matches the reference, i.e., T = T. (B.5) ref Inserting the torque relation (B.5) into the expression for u T in (B.3) gives Rr ut = npωm + T ref + X, (B.6) 3 n ψ where X is the steady state contribution from the PI controller. To determine X we note that the torque equation (4.4) and the expression for the steady state rotor flux magnitude (B.) give that 3 np T = ψ r tanδ. (B.7) L σ Together with the expression for u T in (B.) and the stator flux equation (B.5) this means that Rr Rr Rr X = tanδ T ref ( T Tref) L 3 n ψ = 3 n ψ =. (B.8) σ p r p r p r Now, from (B.4) and (B.5) it also follows that ω = u T. From (B.) we then conclude that Rr Rr ω = ω npωm = tanδ = T, (B.9) L 3 n ψ where the last equality follows from the steady state expression for the torque in (B.7). Linearization of Decoupling By linearizing equations (B.6) - (B.8) and using the steady state relations in the previous section we get the following linear state equation σ p r

237 Closed-Loop Equations 5 ψ s () t ψs () t Rr ψ r Rr R r ψ r () t = ψr tanδ ψr () t Lσ ψ s L L σ σ δ () t δ () t Rr tanδ Rr tanδ R r Lσ ψs Lσ ψr L σ A ψ s uψ () t + ut () t n p ωm() t B and the corresponding output equation (B.) ψs () t () ψs t 3 np ψ r 3 np 3 np T () t = tanδ ψr tanδ ψ r ψr () t Lσ ψ s Lσ L. (B.) σ ψr () t δ () t C We can now use the state space matrices defined in (B.) and (B.) to form the transfer functions from the inputs to the outputs, i.e., () () () t ( ) G ( p) () () () t ψ t u t y () t u () t. (B.) s ψ c T t = C pi A B ut t ψ r ω m Note that all linearized quantities represent deviations from an operating point, although this is not explicitly shown in the notation. From the expression for the matrix B in (B.) (and the fact that there is no D- matrix) we note that the rotor speed ω m enters the equations as u T but scaled with n p. We may therefore use a transfer function representation with the following notation

238 6 Closed-Loop Equations y G y = uc + ωm ψ n r G, (B.3) ψ r p D ω where D ω was defined as D ω = ( n p ) T and (cf. (B.)) G G G y =, G ψ r = ( G 3 G 3 ). (B.4) G G The transfer functions defined in (B.) can be evaluated as G y ( s) = ψ s s (B.5) T( s Tσ + Tσs+ + tan δ) 3 npψr( Tσs+ tan δ) s( s T T s tan ) σ + σ + + δ Rr ( s Tσ + Tσs+ + tan δ) and G ψ r ( s) ( stσ + + ) ( σ + Tσs+ + tan δ ) ψ tan δ T ψ tanδ = s s T r σ r stσ + Ts σ + + tan Transfer Function Representation. (B.6) δ We have now derived a linear model from the signals u c and ω m to the control variables stator flux magnitude and torque, see (B.3), (B.5) and (B.6). In order to properly treat time delays, but also to investigate the influence of disturbances, we here derive symbolic transfer function expressions for the transfer functions in (B.3). To also include the effects of DC-link voltage variations, we linearize the stator voltage equation (.4). If we neglect compensation of variations of the DC-link voltage as discussed in Subsection.., then the stator voltage is represented as

239 Closed-Loop Equations 7 () () () () s u () s t = Ud k t + kud t = usref t + u Ud t. (B.7) U () d usref t GUd U d Linearizing the control law (6.) and inserting it into the stator voltage equation (B.7) gives us us() t = Rsis() t + ucψs() t + ψ suc() t + Ud() t. (B.8) U d u () t sref We now introduce a real-valued description to represent the relation in (B.8) as where u () t = R i () t + U ψ () t +Ψ u () t + G U () t, (B.9) s s s c s s c Ud d and usd isd ψ sd uψ us =, is =, ψ s = u c = usq isq ψ, (B.3) sq ut U u u ψ ψ u. (B.3) cd cq sd sq sd c =, Ψ s =, GUd = ucq ucd ψsq ψsd U u d sq We have here chosen to use the space vector components in synchronous coordinates (relative to the stator flux space vector) to give the matrices constant elements. The notation in (B.3) is used to keep the similarity with the space vector equation (B.8). Hence, here Ψ s does not represent the magnitude of the stator flux space vector as in (3.). Equation (B.9) is illustrated in Figure B-. In Figure B- also disturbances in the motor speed have been added. Note, however, that the entrance of the rotor frequency now has been moved out of the block G, which used to denote the entire dark shaded block in Figure B-. From (B.3) we know that the motor speed disturbance adds to the signal u c through the transfer function D ω. It then follows that the block G ω in Figure B- must be given by Gω =Ψ sdω =Ψs. (B.3) np

240 8 Closed-Loop Equations Figure B-: Linear model of the decoupling part of ISC. From Figure B- it now follows that the stator voltage reference u sref can be expressed as ( ψ )( ) usref () t = RsGi + UcG usref () t + GUd Ud () t + Gωωm () t +Ψ suc () t.(b.33) By introducing the following notation for the different transfer functions of the model G in Figure B- (note that Ψ r here represents the rotor flux magnitude and not a vector as Ψ s defined in (B.3)) yt () Gy ( p) ψ s () t Gψ ( p) = us () t, (B.34) ψ r() t Gψ r( p) is() t Gi( p) the expression for u sref in (B.33) can now be written as u () t = sref I RsGi UcG Ψ s uc () t +Ψ sgud Ud () t +ΨsGωωm () t (B.35) DUd D ω () + (). ( ψ ) ( G U t G ω t ) Ud d ω m Note that we here moved the entrance of the disturbances to the other side of the block Ψ s, i.e., outside the shaded block, in Figure B-. We may now form transfer functions from the inputs u c, U d and ω m to the outputs y and Ψ r in Figure B-. By using the expression (B.35) for the reference stator voltage it follows that

241 Closed-Loop Equations 9 ( ψ ) ( () () ()) y = Gy I RsGi UcG Ψ s uc t + DUdUd t + Dωω m t G y (B.36) and ( ) ( () () ω ()) ψr = Gψr I RsGi UcGψ Ψ s uc t + DUdUd t + Dω m t G ψ r. (B.37) In these two equations we have identified expressions for the two transfer functions from the signal u c to y and Ψ r given by (B.5) and (B.6). Linearization of Feedback Controllers The signal u c in (B.4) is composed of the two signals u Ψ and u T given by (6.) and (6.). Linearizing the expression for u c and putting the result on the realvalued representation defined in (B.3) gives ( ) uc() t = rψr() t Dωωm() t + Ffwr() t + F r() t y() t, (B.38) ut () where the signal u(t) was defined and ψ K p r = ω, fw F = Rr, F R. r = K pi K i ψ 3n r pψ + r 3npψ r s (B.39) Here we used that T ref = T (see (B.5)) as well as the expressions for the steady state torque in (B.7) and the steady state slip frequency in (B.9). Closed-Loop Equations The linearized closed-loop system may now be represented as in Figure B-, where we introduced the notation P for the system in the lightly shaded block. By also moving the entrances of the disturbances outside the block P, the block diagram then only contains the feed-forward and feedback controllers and the open-loop system P. This representation can be used to for example tune the feedback controllers.

242 3 Closed-Loop Equations Figure B-: Linear model of closed-loop system with ISC. From Figure B- it follows that the reference stator voltage can be written usref () t = Rsis () t + Ucψs () t +Ψ s ( rψr () t + u() t ). (B.4) u () t By using the notation for the transfer functions in (B.34) it follows that usref () t =.(B.4) R G + U G +Ψ rg u () t + G U () t + G () t +Ψ u() t ( ψ ψ )( ωω ) s i c s r sref Ud d m s By collecting terms we may rewrite (B.4) as ( ψ ψ ) ( ωω ) usref () t = I RsGi UcG ΨsrG r Ψ s u() t + DUdUd () t + D m () t + (B.4) G U () t G ω (), t Ud d ω m where we also used the definitions of D Ud and D ω in (B.35). The output y of the induction motor in Figure B- can then be represented as yt () = Gy ( usref () t + GUdUd () t + Gωωm () t) = (B.43) P u() t + D U () t + D ω () t, where ( ) Ud d ω m ( ) ( ) ψ ψ ψ P = G I R G U G Ψ rg Ψ = G I rg. (B.44) y s i c s r s y r By using the last equality in (B.44), we may use the expressions (B.5) and (B.6) to evaluate the transfer matrix P as c

243 Closed-Loop Equations 3 where ψ s s P = stσ + TT, (B.45) σ st 3 np rm ( s) σ + ψ stσ + M ( s) Rr ( stσ + M ( s) ) stσ + tan δ M ( s) =. (B.46) st + The transfer function M depends on the torque through the load angle δ, defined by (4.9). In order to estimate the size of tan (δ ) we observe that from the torque expression (3.) and the steady state rotor flux (B.), together with the expression for the pull-out torque (3.3), the following equality holds T sin δ =. (B.47) T σ pull out The expression tan (δ ) can hence be evaluated as a function of the steadystate torque normalized by the pull-out torque. The result is shown in Figure B-3, where it is seen that the factor tan (δ ) is very small for most torque levels tan (δ ) as a function of T /T pullout T /T pullout Figure B-3: tan s (δ ) as a function of T /T pullout.

244 3 Closed-Loop Equations In Appendix A (data set ) the maximum used torque in driving is set to 78 Nm whereas the pull-out torque at nominal flux can be calculated to 46 Nm, see (3.3). Hence, at nominal flux the ratio between torque and pull-out torque does not exceed.4 and a reasonable approximation is therefore to set tan (δ )=. This in turn means that the transfer function M can be approximated by one, which leads to the following approximation of P in (B.45) ψ s s P( s). (B.48) TT ( st ) 3 n σ σ + pψ r ( st ) Rr ( stσ + ) σ + The transfer function D ω in (B.43) is defined in expression (B.3), whereas D Ud is given by D Ud u s Ri s scosϕ Re su d ψ sω ψ ω = =, (B.49) U d Ri s ssinϕ s u Im + su ψ d sω ψ which follows from the expression for the reference stator voltage in (6.) and the steady state values of the components u Ψ and u T in (6.) and (6.). The constant ϕ in (B.49) represents the steady state angle between the stator current and stator flux space vectors, see Figure 6-6. The closed-loop system (without time delays and stator voltage normalization) may hence be represented as in Figure B-4. Note that the disturbances now add at the input to the system P. Here we also see that rotor speed disturbances are perfectly compensated for. Figure B-4: Linear model of closed-loop system without time delays.

245 Closed-Loop Equations 33 Closed-Loop System with Time Delays In this section we extend the analysis of the previous section to also include time delays. Linear Model of Prediction In Subsection 6.. the continuous-time ISC control law was extended to also handle time delays due to modulation and real-time control. The extension relied on predictions of the quantities needed for decoupling. In this subsection we will derive linear models of these predictions and we start by rewriting the linear model of the stator flux in (B.34) as ψ ( ω ) () t G u ( t T ) G U () t G ω () t = + +. (B.5) s ψ sref d Ud d m The prediction of the stator flux at time t+t d, given information up to time t k, is assumed to be performed as where ĝ () t ψˆ ( t T t) ψ () t s d s t+ Td ( ) ˆ d sref ( d ) + gˆ t+ T τ u τ T t dτ + t t+ Td ( ) ˆ ( ) + gˆ t+ T τ U τ t dτ + t t+ Td ψud d d ( ) ˆ d m( ) + gˆ t+ T τ ω τ t dτ t + = + ψ, gˆ ψ Ud () t and ĝ () t systems in (B.5). Further Uˆ d ( τ t) and ˆm ( t) ψ ωψ (B.5) ωψ are estimated impulse responses of the ω τ are predictions of the DClink voltage and motor speed respectively at time τ, given information up to time t, where τ t. Note that for t τ t+t d ( τ ) ( τ ) u T t = u T, (B.5) ˆsref d sref d as the information is available. We will assume that no predictions of the DClink voltage and motor speed are available, which means that (B.5) and (B.5) lead to (only first integral is non-zero) ( t T t) () t ( t T t) ( t t T ) ψˆ + = ψ + ψˆ + ψˆ, (B.53) s d s sref d sref d

246 34 Closed-Loop Equations where ( + ) = ˆψ ( ) ( ) ψˆ t T t g t τ u τ dτ, (B.54) sref d sref t T d t ( ) = ˆψ ( ) ( ) ψˆ t t T g t τ u τ dτ. (B.55) sref d sref The prediction of the stator flux can hence be represented as in Figure B-5. If we assume that the estimated model in Figure B-5 equals the true model, then the prediction of the stator flux may be written ψ ( t T t) ψ () t Gψ ( I D) u () t ˆ s d s sref + = +. (B.56) Similarly, the prediction of the stator current is given by ( ) () ( ) () i t+ T t = i t + G I D u t. (B.57) s d s i sref Figure B-5: Linear model of stator flux prediction. Although we assumed that the quantities u Ψ (t) and u T (t) are not predicted, we will actually use a predicted value of the rotor flux to simplify future calculations. Hence, we model the predicted rotor flux as ψ ( t T t) ψ () t Gψ ( I D) u () t + = +. (B.58) r d r r sref As the rotor flux magnitude is a relatively slowly changing quantity, the predicted and true values should be more or less equal (short time intervals).

247 Closed-Loop Equations 35 Closed-Loop Equations With time delays included, the stator current and stator flux may be predicted as in (B.56) and (B.57) We here also assume that the rotor flux magnitude is predicted as in (B.58). The stator voltage reference is then generated as illustrated in Figure B-6. Figure B-6: Generation of reference stator voltage with time delay. From Figure B-6 it follows that the expression for the stator voltage reference is still given by (B.4), i.e., usref () t = Gy P( u() t + DUd Ud () t + Dωω m () t ) (B.59) GUdUd () t Gωωm (). t However, the output y is now given by ( ) y() t = Gy Dusref () t + GUdUd () t + Gωωm () t, (B.6) where the reference stator voltage is delayed. Expression (B.6) can be rewritten as

248 36 Closed-Loop Equations yt () = P( Dut () + DUdUd () t + Dωω m () t) + ( D I)( PΨ G )( G U t + G ω t ) () (). s y Ud d ω m (B.6) For small time delays we may neglect the second term and approximate the expression for the output in (B.6) by ( ) y( t) P Du( t) + DUdUd ( t) + Dωω m ( t). (B.6) If we also add rejection of disturbances in the DC-link voltage according to Figure -5, the corresponding picture to Figure B-4 with time delays is then given by Figure B-7. Figure B-7: Linear model of the controlled system when ISC is applied to an induction motor.

249 Appendix C Steady State Analysis In this appendix expressions for the steady state estimates of torque and stator flux magnitude are derived with a full-order observer with zero observer gain. Stator Flux Estimation Error In this section we will derive an expression, which relates the estimated and real stator flux magnitudes at steady state. We start the derivation by inserting the stator current equation (3.7) into the stator flux equation (3.5), which gives that s s ψ s() t = Rs + ψs() t + Rs ψ r () t + us() t. (C.) Lσ Lm Lσ By using the polar representation introduced in (3.), we can write (C.) as jχs() t jχs() t ( ψs() t e ) = ( ψ s() t + j χs() t ψs () t ) e d dt R = R + t + t e + u e e s jδ() t j( χu() t χs() t ) jχs () t s ψs() ψr () s Lσ Lm L σ The real part of (C.) then has to satisfy R ψ s s ψs ψr δ s χu χs Lσ Lm Lσ s () t = R + () t + () t cos () t + u cos( () t () t ). (C.).(C.3) At steady state, the derivative of the flux magnitude in (C.3) is zero. We may then also use the steady state relation for the rotor flux magnitude (3.5) to rewrite (C.3) as

250 38 Steady State Analysis Now we use the trigonometric identity ( χ χ ) us cos u s ψ s =. (C.4) sin Rs + δ L m Lσ tan δ sin δ = tan δ cos δ =, (C.5) + tan δ together with steady state relation for the slip frequency given by (3.6) to express the steady state stator flux magnitude in (C.4) as ( χ χ ) ( Tσ ω ) + ( ω ) u cos ψ s = Rs Rs + L L T s u s m σ σ. (C.6) To evaluate the cosine appearing in (C.4), we use also the steady-state expression for the imaginary part of the stator equation (C.). Together with the expression for the real part we can then express tan(χ u χ s ) as tan Rs ωtσ ω + Lσ s = Rs Rs + L L T ( χ χ ) u + ( ωtσ ) ( ωtσ ) + ( ω ) m σ σ, (C.7) where we used the identity (C.5) and the steady state relations for the rotor flux magnitude and slip frequency in (3.5) and (3.6). If we consider the square of the stator voltage magnitude, we may use that cos = s + tan, (C.8) ( χ χ ) u ( χ χ ) which means that the square of the stator flux magnitude in (C.6) may be written as us ψ s =. (C.9) Rs R ( ω s T ) σ Rs ωt σ + + ω + Lm Lσ + ( ωt ) L σ σ + ( ωtσ ) u s

251 Steady State Analysis 39 With a full-order observer with zero observer gain, the relation for the stator flux magnitude in (C.9) holds also for the estimated flux magnitude. As the same stator voltage is applied to the motor and the observer (and the rotor speed is measured), the steady state values of the stator frequency ω and the slip frequency ω in (C.9) are the same for the motor and the observer. The other quantities however differ and we use hats to represent quantities related to the observer. We may then express the quotient between the estimated flux magnitude and the real flux magnitude as ( ω ˆ Tσ ) ω σ ω + ( ω ˆ ) ( ˆ Tσ σ + ωtσ ) ˆ ˆ ˆ ˆ Rs Rs Rs T ˆ ˆ ˆ Lm L L σ ψ s = ψˆ s Rs Rs ( ωtσ ) Rs ωt σ + + ω + Lm Lσ + ( ωt ) L σ σ + ( ωtσ ). (C.) Torque Estimation Error We now examine the estimation error of the torque, where torque is estimated using Equation (7.). Note that this equation contains the estimated stator flux, obtained through the full-order observer, but also the measured stator current. By using a polar representation also for the measured stator current it then follows that the estimated steady-state torque may be written as 3 Tˆ ˆ ( ˆ = npisψs sin χis χs ) = 3 = ni ˆ p sψs ( sin χiss cos χs + sin χs cos χiss ), (C.) where i s and χ is are the magnitude and angle of the stationary stator current space vector. In (C.) we also defined the following two angles χ = χ χ, χ = χ ˆ χ. (C.) iss is s s s s From the stator current equation (3.7) it follows that the magnitude and angle of the stator current can be represented through is = ψ r sinδ + + ψs ψr cosδ, (C.3) L σ Lm Lσ L σ

252 4 Steady State Analysis and tan ψr sinδ Lσ s =. (C.4) + ψs ψr cosδ Lm Lσ Lσ ( χ χ ) is By using the steady-state expressions for the rotor flux magnitude and the slip frequency given by (3.5) and (3.6), the equations (C.3) and (C.4) can be written as ψ s s = + ( ωtσ ) + + ( ω ) Lm L Lm T σ σ i (C.5) and tan χ iss ωt Lσ = + + Lm Lm Lσ σ ( ω T ) σ, (C.6) at steady state. To evaluate the estimated torque in (C.), we need expressions for the sine and cosine of the angle χ iss. An example of the angle χ iss is shown in Figure 6-6. As long as π / χ iss π / we may evaluate cosχ iss as / (+tan χ iss ) and sinχiss as tanχ iss / (+tan χ iss ), where tanχ iss is given by (C.6). It then follows that and cos χ = iss = = + tan χiss + + Lm Lm Lσ ( ω T ) σ + + ( ωtσ ) + ωtσ m m σ σ L L L L (C.7)

253 Steady State Analysis 4 sin χ = iss = tan χiss = + tan χ iss Tσ ω L σ + + ( ωtσ ) + ωtσ m m σ σ L L L L. (C.8) Further, for the angle χ s in the expression for the steady-state torque (C.) we use that ( ˆ s tan u s ( u s )) ( χ ˆ u χs ) tan ( χu χs ), ( χ ˆ χ ) ( χ χ ) tan χ = χ χ χ χ = tan (C.9) = + tan tan u s u s where χ u is determined by the supply voltage. Note that the involved angles in (C.9) are specified by expression (C.7). For error angles χ s in between ±π /, we again use that tan χ cos χs =, sin χ = + tan + tan χ s s χs s. (C.) For example, the sine expression in (C.) may then be evaluated as tan ( χ ˆ u χs ) tan ( χu χs ) + tan( χ ˆ u χs ) tan( χu χs ) sin χs =. (C.) ( + tan ( χ ˆ u χs ))( + tan ( χu χs )) + tan χ ˆ χ tan χ χ ( ( u s ) ( u s )) At zero slip frequency, the angle χ iss = and the expression for the estimated torque in (C.) then becomes ˆ 3 T ˆ = npisψssin χs. (C.) From (C.7) it follows that (at zero slip frequency) Lm tan ( χu χs) = ω (C.3) R s

254 4 Steady State Analysis and (C.) can hence be evaluated as sin χ = s R ˆ s R s ω L ˆ m L m Rˆ s R s ω ω Lˆ + + L m m. (C.4) At zero slip the flux quotient in (C.) reduces to ψˆ ψ s s = R s + ω Lm Rˆ s + ω Lˆ m, (C.5) and the expression for the stator current magnitude in (C.5) becomes i ψ s s =. (C.6) Lm By inserting the expressions (C.4), (C.5) and (C.6) into the expression for the steady-state torque in (C.), it follows that Tˆ 3 n ˆ p = ψ s Lm R ˆ s R s n pωm L ˆ m L m. (C.7) ( npωm) R s + Lm With non-zero slip frequency, tanχ iss and we can rewrite (C.) as ˆ 3 tan χ s T ˆ = npisψs sin χiss cos χs +. (C.8) tanχiss From the torque expression (C.) it also follows that the actual torque is given by 3 T = npisψssin χiss. (C.9)

255 Steady State Analysis 43 We may hence form the quotient between the two torques as (note that T if ω ) as T ψs tan χiss =, (C.3) Tˆ ψˆ cos χ tan χ + tan χ ( ) s s iss s where hence the flux quotient is given by (C.), the cosine and tangent of the angle χ s by (C.) and (C.9), and finally tanχ iss by (C.6). If the estimated stator current is used instead of the measured to estimate the torque in (7.), then torque is estimated through pure simulation and we may just as well use the expression (3.) for torque estimation. The steady-state estimated torque then equals 3 n ˆ ˆ 3 n ω T ˆ ˆ ˆ L R + T p p = ψsψrsinδ = ψs ˆ ˆ σ r ˆ ( σω ). (C.3) The corresponding torque quotient at non-zero slip to (C.3) then becomes T ˆ R r ψ + s = Tˆ R ˆ r ψ s + ( Tˆ σω ) ( T ω ) σ. (C.3)

256

257 Appendix D Non-Zero Observer Gain In this appendix we derive a closed-loop model when using ISC and an observer with non-zero observer gain to control an induction motor. We restrict the analysis to observer gains on the form k obs = (k obs ) T. From the linearized control law (B.4) we have that the stator voltage with ISC is generated as (here we neglect disturbances) u t = Rˆ iˆ t + Uˆ ψˆ t +Ψ ˆ rˆ ψˆ t + u t, (D.) () () () ( () ()) s s s c s s r where the hats indicate that the voltage now is calculated from an estimated model and estimated signals. With perfect measurements, the signals in (D.) are given by iˆ t = Gu t, ψˆ t = G u t, ψˆ t = G u t. (D.) () () () () () () s i s s ψ s r ψr s If also the model of the induction motor is correct, then the stator voltage in (D.) can be evaluated as () = ( Ψ ) Ψ () = () us t I RsGi UcGψ srg ψr su t G Pu t G P, (D.3) where the last equality in (D.3) follows from the definition of the system P in (B.44). With an observer with model errors and a correction term, the estimated signals in the control law (D.) also depend on the measured stator current. We may then represent the estimated signals in the control law as iˆ = Gu ˆ + Gˆ i, ψˆ = Gˆ u + Gˆ i, ψˆ = Gˆ u + Gˆ i, (D.4) s i s ii s s ψ s ψi s r ψr s ψri s where Ĝ ii, Ĝ Ψi and Ĝ Ψri are some transfer functions. If we restrict the analysis to observer gains on the form k obs = (k obs ) T, then the estimation error is added just as the stator voltage to the state equation in (7.9). We may then consider the input to the observer to be

258 46 Non-Zero Observer Gain ( ) () () () ˆ () u t = u t + k i t i t, (D.5) ˆs s obs s s and the estimates in (D.4) can be written iˆ t = Gu ˆ ˆ t, ψˆ t = Gˆ uˆ t, ψˆ t = Gˆ uˆ t. (D.6) () () () () () () s i s s ψ s r ψr s By using these expressions, the stator voltage in (D.) can be expressed as u t = Rˆ Gˆ + Uˆ Gˆ +Ψ ˆ rg ˆ ˆ uˆ t +Ψ ˆ u t. (D.7) () ( ) () () s s i c ψ s ψr s s If we replace u s (t) in (D.7) by using the expression in (D.5) and collect the terms containing û s, we may through the definition of P in (D.3) rewrite (D.7) as ( ( ) ()) () () ˆ () u t = Gˆ Pˆˆ Ψ k i t i t +Ψ ˆ u t. (D.8) ˆs s obs s s s We continue by writing the estimated stator current in (D.8) as which means that iˆ t Gu t G u t k i t i t () = ˆ ˆ () = ˆ () + () ˆ () Gu () t s i s i s obs s s ˆ s i obs i obs i i s () ( ˆ ) ( ˆ ˆ ) () i s, (D.9) i t = + Gk G + k GG u t. (D.) By using the expression for the estimated stator current in (D.) we may now write ( ) () ( () ˆ ()) = ( + ˆ ) ( ˆ + ˆ ) kobs is t is t kobs Gi I Gk i obs Gi kobsgg i i us t,(d.) where the system was defined. By inserting the expression (D.) into (D.5) we get () () () ˆ () ( ) ( ) ( ) ˆ s obs s s s u t = u t + k i t i t = I + u t. (D.) We may hence interpret the system defined in (D.) as the relative change of the stator voltage used by the observer due to the correction term. By using equations (D.) and (D.), Equation (D.8) can be rewritten as

259 Non-Zero Observer Gain 47 () ( ) () ( ˆ ˆˆ ) () ˆ ˆ () u t = I + u t = G PΨ u t + G Pu t, (D.3) ˆs s s s which means that the stator voltage can be expressed as () ( ˆ ˆˆ ) ˆ = + Ψ ˆ () us t I I G P s G Pu t. (D.4) By comparing with the transfer function between the signals u(t) and u s (t) with zero observer gain shown in Figure 7-, we see that a new factor appears in (D.4) due to the non-zero observer gain. By using the expression for the stator voltage in (D.4) it follows that the output of the induction motor now is given by () () ( ˆ ˆˆ ) ˆ = = + Ψ ˆ () y t Gus t G I I G P s G Pu t, (D.5) G where we defined a new fictitious system G. By including the feedback controller F as in Figure 7-, we may then represent the feedback loop of the controlled induction motor with (the special) non-zero observer gain as in Figure D-, where K is still given by Expression (7.3). r + + K G ~ - y Figure D-: Closed-loop system with ISC and observer with non-zero observer gain (feedforward controller neglected).

260

261 Appendix E Space Vectors For electrical three-phase quantities Q(t)=(Q A (t) Q B (t) Q C (t)) T, the complexvalued space vector is defined as [39] π 4π j j s 3 3 Q () t = Qα + jqβ K QA() t + e QB() t + e QC () t, (E.) 3 where K q is a scaling factor. In this thesis we use K q =, which means that the magnitude of the space vector equals the amplitude of the three phase signals. To show this we consider a symmetrical three-phase quantity as Qˆ () t cos χ () t QA () t () ˆ π QB t = Q() t cos χ () t. (E.) 3 QC () t ˆ π Q() t cos χ () t + 3 By using the definition in (E.), the corresponding space vector then becomes () t = () s ˆ j () t Q KqQ t e χ, (E.3) which shows that the magnitudes of the space vectors and the three-phase signals are equal with K q =. A three-phase quantity where the components add to zero, i.e., () () () Q t + Q t + Q t, (E.4) A B C is said to have no zero sequence. This for example holds for the three-phase quantity given by (E.). For such signals the space vector transformation in (E.) is invertible, i.e., the space vector contains the same information as the original three-phase quantity. For general three-phase quantities, however, the space vector representation looses the information about the zero sequence.

262 5 Space Vectors The space vector transformation in (E.) can also be represented on vector form as QA () t Qα () t K q QB() t Q () t =. (E.5) β QC () t 3 3 Here we see that the rows of the transformation matrix in (E.5) are orthogonal and satisfy the constraint (E.4). In a three-dimensional space, the constraint (E.4) describes a plane and we may hence interpret the two row vectors as two orthogonal vectors in this plane. With K q set to 3/, the two vectors would also be of unit length and we could interpret the transformation in (E.5) as a change of coordinates, where the third component in the new coordinate system is dropped. This hence corresponds to a projection onto the plane defined by (E.4). With other values of K q than K q = 3/, we hence interpret (E.5) as a projection followed by a scaling. Note that the part of the threedimensional vector that is lost in the projection corresponds to the component along the normal of the plane. The normal is parallel to the vector ( ) T and the component of the three-dimensional vector in this direction is therefore given by the (scaled) sum of the elements, i.e., the zero sequence. For the induction motor, the electrical three-phase quantities satisfy the constraint (E.4) and may hence completely be described with space vectors. From the space-vector representation of the stator voltage in (.4), we see that the zero sequence of the coupling vector has no influence on the motor (as the stator voltage is completely determined by the space vector of the coupling vector). Note that the coupling vector is not restricted to satisfy the constraint (E.4). In general, each component of the average coupling vector can take all values between ±.5 independently of the other components. The set of average coupling vectors possible to generate by the inverter is then defined by a cube in three dimensions. This cube, as well as the plane defined by (E.4), is illustrated in Figure E-. By projecting the cube onto the plane we obtain the set limited by the outer hexagon in the plane in Figure E-. From the stator voltage representation (.4), it then follows that the set of stator voltages possible to generate with the inverter is given by this set followed by a scaling (by / 3 since we use the space vector definition with K q = ) and a multiplication by the DC-link voltage. This results in the area shown in Figure -3. We now interpret the voltage vectors u si in Figure -3 as the projections (plus scaling and multiplication by U d ) of the corners of the cube in

263 Space Vectors 5 Figure E- onto the plane. It then is clear that two of the corners are projected at the origin. These corners hence generate the zero voltages. By using the three-dimensional illustration in Figure E- we may also derive the voltage limit with so called sinusoidal modulation. With sinusoidal modulation, the three-dimensional coupling vector for the inverter is generated from a space vector. However, this means that the coupling vectors in the three-dimensional representation are confined to the plane in Figure E-, as the space vectors contain no zero sequences and hence satisfy (E.4). In this case, the set of admissible coupling vectors consist, not of the entire cube as before, but only of the intersection of the cube and the plane. This is a subset of the plane, which is limited by the inner hexagon in Figure E-. To reach the set of possible stator voltages we therefore only have to scale this set (depending on the choice of K q ) and multiply by the DC-link voltage. Figure E-: Geometrical interpretation of maximum stator voltage obtained with sinusoidal modulation and space vector modulation. Figure E- shows the maximum (average) stator voltage set, which can be reached with the inverter as well as the set obtained with sinusoidal modulation. Here we see that with sinusoidal modulation, the maximum stator voltage is limited to U d / compared to the maximum value of U d / 3. (We here use K q =, which means that the lengths of the space vectors correspond to the amplitudes of the three-phase signals). To overcome this limitation with sinusoidal modulation, a zero sequence component may explicitly be added to the individual coupling vector components before determining the switching times, see [3].