Achieving Beyond Flexible Dynamics Control of a Prototype Lightweight Positioning System: A Theory-Driven Experimental Approach F.B.J.W.M.

Transcription

1 Achieving Beyond Flexible Dynamics Control of a Prototype Lightweight Positioning System: A Theory-Driven Experimental Approach F.B.J.W.M. Hendriks DCT Master of Science Thesis Committee: Prof. ir. O.H. Bosgra (Graduate Professor) Dr. ir. A.A.H. Damen Ir. T.A.E. Oomen (Supervisor) Prof. dr. ir. M. Steinbuch Dr. ir. M. M. J. van de Wal Eindhoven University of Technology Department of Mechanical Engineering Control Systems Technology Group Eindhoven, April 2009

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3 Abstract In high tech industry, research is being done on accurate control of electromechanical systems. An example of such a system is a wafer stage, used for the production of integrated circuits. To meet the increasing demands of higher accelerations and accuracy, the stages have become more light-weighted, and as a result, more flexible. These flexible modes are inherently multivariable. Introducing the necessity of model-based multivariable control. The goal of this project is to acquire a proof of principle of the implementation of a novel robust beyond rigid body controller. To achieve this goal, the entire controller design procedure, from identification to implementation, has to be very accurate. Attention has been paid to acquire measurement data with a high quality. To estimate very accurate models, improved modeling techniques have been used. In a validation step, the model quality has been quantified. Finally, optimal robust, performance optimized, µ s controllers have been calculated using a modified controller synthesis algorithm. The theoretical controller design approach has been validated with experiments on a prototype of a flexible light-weight system. The improved model quality has been confirmed. Implementation of the controllers shows that the controller design approach results in superior control performance. All synthesized controllers stabilized the system when implemented. The controllers achieved a bandwidth, beyond the first flexible mode of the test setup. The strong theoretical foundation of the controller design approach used, together with the successful implementation on a prototype flexible motion system, is promising for next-generation motion control. I

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5 Contents Contents III 1 Introduction Background Motivation Problem Formulation Outline of the Report Experimental Setup System Description System Properties Delay Disturbances Linearity Aliasing Motivation for Periodic Input Design Using the DFT Discretization in Time Domain Windowing Discretization in Frequency Domain Multisine Crest Factor Internal Stability and Motivation for Dual-Youla Parametrization Internal Stability Coprime Factors Properties Advantages Dual-Youla Parametrization Control Interpretation Bezout Identity P Stabilized by C All Stabilized P Parameterized by P III

6 5 Discrete Time Controller Design Discrete Time Control Loop w-domain Experimental example Dynamic Weighting Filters Signal Based Weighting Filters Loop-Shaping Based Weighting Filters Closed Loop Loop-Shaping Open Loop Loop-Shaping Simulation Example Identification for Control Control Goal Control-Relevant Identification Control-Relevant Coprime Factors Experimental Example Model Validation Frequency Domain Approach Deterministic Disturbance Model Reducing Optimism Validation Test Experimental Example H Robust Control µ Synthesis Controller Design Setup Skewed-µ Synthesis Algorithm Aspects Experimental Results Conclusions and Recommendations Conclusions Recommendations A Estimated Plants and Coprime Factors 101 B Achieved Closed-Loop Transfer Functions and Maximum Singular Values 111 C Step Responses 121 Bibliography 127 IV

7 Chapter 1 Introduction 1.1 Background For quite some years, research is being done on accurate control of electromechanical systems. Electromechanical systems are widely used in industry. These systems range from single input singly output (SISO) systems, to very demanding, high performance, multi input multi output (MIMO) systems. Although the high performance systems are far more demanding, and use more sophisticated control approaches (feed-forward, ILC, complex trajectory design), the bandwidth of the feedback controller is limited by the flexibilities of the system. A typical example of a high performance electromechanical system is a wafer stage, which is a part of a wafer scanner. A wafer scanner is used to produce Integrated Circuits (ICs) by means of a photolithographic process on a silicon disc, called a wafer. There are several ICs on a wafer, and only one IC can be processed at a time. Because the imaging optics are fixed to the environment, the wafer stage has to be moved continuously. The wafer stage is a six degrees-of-freedom electromechanical system, which achieves high accelerations and accuracy. 1.2 Motivation Demands on high performance industrial systems are ever increasing. Systems become faster (higher throughput) and/or more accurate (more sophisticated products). In order to meet demands, companies in the IC industry have designed systems that achieve very high accelerations. To achieve these accelerations, the wafer stages have become more light-weighted, and as a result, more flexible. With classical controller design procedures, it is hard to cope with the coupled dynamics in MIMO systems. Due to the MIMO nature of the stage and the increased flexibility, classical controller design guidelines can not be applied straightforwardly. To overcome this problem, more sophisticated, model based, controller design approaches are needed. Several controller design approaches exist, that can cope with MIMO systems naturally. The coupled dynamics do not cause a fundamental problem. Readily available algorithms exist to calculate an optimal controller for a given control problem. However, in reality, the model of the system, used in model based control, always differs from the system. Any realistic system is too complex to be described accurately by a model. This model mismatch may result in performance degradation when the controller is implemented 1

8 on the true system. It is not even guaranteed to be stabilizing the true system. To introduce some robustness against model errors, weights are used. Some rough, ad hoc, guidelines to design weights for robustness exist. However, these guidelines do not guarantee stability, whereas they can limit performance when chosen too conservative. Controller performance could improve if the pragmatic approach could be removed or replaced. Instead of using ad hoc assumptions, it would be beneficial to estimate the uncertainty of the model. Then stability can be guaranteed and too much conservatism can be avoided. Theory on how to handle uncertainty is readily available. However, practical examples on how to handle uncertainty and at the same time optimize for performance are not found in literature. In this research project it will be tested whether flexible modes can be controlled, such that Beyond Rigid Body control is possible. A useful and accurate model is acquired using the identification procedure described in [14] and [13]. With sophisticated controller design algorithms, an optimal robust performance controller is calculated and implemented on a test setup. The setup consists of a flexible beam, which acts as a prototype of a flexible light-weight system. 1.3 Problem Formulation Considering the above, the goal of this project is formulated as: Design a high performance, Beyond Rigid Body, robust controller and implement and test it on the test setup. With respect to the project goal there are two main questions that should be answered: Q1) How can an accurate model set be obtained for subsequent robust controller design? Q2) How can a robust controller, optimized for performance, be designed and implemented? The procedure to acquire a model consists of several steps. Not every step is straightforward. This leads to the following sub-questions in order to answer question 1: Q1a) What is an appropriate input signal for an identification measurement? Q1b) What is a suitable model structure to model the uncertain plant? Q1c) What is the quality of the estimated model? Q1d) How can the uncertainty be estimated? Question 2 addresses the design and implementation of the controller. Although most controllers used nowadays are discrete time controllers, controller design methods often use a continuous time approach. Also, optimization for performance in a robust control framework is not straightforward. This leads to the following sub-questions, regarding question 2: Q2a) How to deal with discrete time controller implementation for continuous time plants? Q2b) How to quantify performance for next generation flexible systems? Q2c) How can performance optimization explicitly be addressed in a robust control design approach? 2

9 1.4 Outline of the Report The report is organized as follows. In Chapter 2, the experimental test setup is discussed. First, the working principle of the setup will be explained. Then the hardware and software will be presented. Finally, some properties of the setup will be discussed on the basis of experimental data. This setup will be used for all the measurements throughout this report. In Chapter 3, the effects of the Discrete Fourier Transform, resulting from discretization and windowing, will be discussed. Here, a distinction will be made mainly between periodic and non-periodic signals. Finally, the multisine signal will be discussed in more detail and an optimization of this signal will be considered. A useful model structure will be introduced in Chapter 4. First, the notion of internal stability will be discussed briefly. The notion of internal stability is important for the discussion of model structures. Then general coprime factorization and its properties will be presented. This is followed by the presentation of the Dual-Youla parametrization and its advantages. Discrete time controller design will be discussed in Chapter 5. Here, it will be shown that a discrete time controller can directly be designed in discrete time domain using continuous time design tools. This is possible by means of a bilinear transformation. In an experimental example, the practical applicability of the discrete time controller design approach will be shown. Chapter 6 will discuss several dynamic weighting filter design approaches. Signal based, as well as open-loop and closed-loop loop-shaping based weighting filters will be discussed. The chapter will be concluded with a simulation example of how weighting filters can be used in a controller design procedure. The link between identification and control will be presented in Chapter 7. Here it will be shown how the identification can be linked to the control goal, such that control relevant identification is possible. An experimental example will show the result of a control relevant identification procedure. Chapter 8 will discuss the validation of the uncertain model. It will also present a method to separate disturbances and uncertainty, such that a realistic estimation of the uncertainty is possible. Also this chapter will be concluded with an experimental example, to show the applicability of the proposed method. The calculation of the robust controller will be discussed in Chapter 9. It will be shown that a performance goal can be included in the problem as an uncertainty. Also, a novel designed robust performance optimization will be presented, together with some implementation issues of the algorithm. Finally, conclusions will be drawn and recommendations will be given in Chapter 10. 3

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11 Chapter 2 Experimental Setup During this research project, a robust controller design procedure is investigated. To test the practical applicability of this approach, it is tested on an experimental setup. A description of the system and the hardware is given in Section 2.1. In Section 2.2, several properties of the system are discussed. 2.1 System Description The main element of the system is a flexible beam. The beam is a prototype for flexible positioning systems. The beam is made of steel and measures mm. It is fixed to the environment by 5 leaf springs. The springs fix 4 DOFs of the beam, the x and ϕ DOFs are free. The maximum displacement is x-direction is approximately ±1.5 mm. Due to the leaf springs, the system is open-loop stable. As depicted in Figure 2.1, the system is equipped with 3 actuators and 3 sensors. As only the x and ϕ DOFs are free, the system is over-actuated. Over-actuation will not be used during this project. Only the SISO situation, where actuator 1 and 3 apply the same input and the output consists of the summed signals of sensors 1 and 3, is considered during this research project. The actuators are current-driven voice-coil actuators from Bei Technologies. The sensors are contactless fiberoptic displacement sensors from Philtech with an approximate accuracy of 1 µm. 2.2 System Properties In this section some system properties will be discussed. These system properties are very important, as they could explain some unexpected behavior of the system during measurements and give an indication of the quality of the measurements. Properties that will be discussed are: delay, noise level, linearity and aliasing Delay Delay is very important for control. In general, the more delay the system to control exhibits, the lower the bandwidth of the controlled system will be. The way a controller is implemented 5

12 Sensor 1 Sensor 2 Sensor 3 x ϕ Actuator 1 Actuator 2 Beam Actuator 3 Figure 2.1: Overview of the experimental test setup Angle [ o ] 0 Angle [ o ] Figure 2.2: Approximate fit of the delay ( ) and the angle of the measured plant ( ) with a sampling frequency of 1 khz (left) and 1.5 khz (right). in the control loop has influence on the delay, irrespective of the controller implemented. All plants considered during this project are continuous time plants. All implemented controllers are digital controllers. To enable the transition from discrete time signals to continuous time signals and back, a hold circuit and a sampler are needed (see Section 5.1). From a control point of view, the sampler, the continuous time plant and the hold circuit can be seen as a discrete time plant, SP H = P d. This is depicted in Figure 5.2. Without a plant, the discrete time signal can exactly be reconstructed: SH = I. With a dynamical system, the sampler and hold circuit introduce 0.5 samples delay [12]. After some measurements, the identified system turned out to have more than 0.5 samples delay. To estimate the amount of delay, a fit of the slope of the angle of the identified system is made. The delay is estimated with e jωt. This is shown in Figure 2.2 for a sampling frequency of 1 and 1.5 khz. The delay is approximately 1.5 samples, 1 sample more than expected. This raised the question whether the data acquisition system performes well compared to other readily available systems. To check this, a comparison is made with the TUeDACS system. To measure the delay of the data acquisition systems, the output is directly connected to the input. The sampling frequency is 1 khz and a multisine with frequencies from 1 to 500 Hz is used as input signal. The measured delay is shown in Figure 2.3. It can clearly be seen that 6

13 Angle [ o ] Figure 2.3: Comparison between the delay of the TUeDACS ( ) and the real time data acquisition system of the experimental setup ( ). the TUeDACS system has much more delay, 3 samples. Our initial data acquisition system only has 1 sample delay. The exact source of the delay is unknown, but compared to the TUeDACS, the real time implemented dedicated data acquisition system performs well Disturbances Estimating the size of disturbances and noise is important when performing measurements. First of all, if measurements contain a lot of noise, and the measured signal is relatively small, the signal can not be distinguished from the noise. As a result, no (useful) input output relation can be determined. A measure for the noisiness of a signal is the Signal to Noise Ration (SNR). The SNR is defined as: SNR = P s P n = ( As A n ) 2, (2.1) where P s and P n are the power of the signal and noise, respectively, and A s and A n are the amplitude of the signal and noise, respectively. In decibels the SNR is defined as: ( ) ( ) Ps As SNR db = 10 log 10 = 20 log P 10. (2.2) n A n Also, by analyzing the noise, the frequencies can be determined for which the system is sensitive to disturbances and/or a lot of disturbances are present. In Figure 2.4, the measured output of the system is depicted when no input is applied. Between 15 and 20 seconds, disturbances are added to the system by shaking the table on which the system stands. A close up of the noise is shown to the right. The maximum amplitude of the noise is approximately 3 µm. This is very close to the resolution of the sensors, as quantization steps can be distinguished in the graph. Comparing the noise level to a measured output, used for identification, results in a SNR of (SNR db =50dB). 7

14 Displacement [mm] Time [sec] Displacement [mm] x Time [sec] Figure 2.4: Response of the system when no input is applied. Between 15 and 20 seconds disturbances are added (left) and a close up of the response between 5 and 10 seconds is shown (right). A frequency domain representation of the noise is depicted in Figure 2.5. The part where disturbances are added, is clearly larger at lower frequencies. The system is thus sensitive to low frequent disturbances coming from the table. At 100 Hz a very sharp peak can be detected in the noise signal. According to [20], the second flexible mode of the system occurs at 100 Hz, leading to an increased sensitivity to disturbances. The same can be seen for the third flexible mode at 185 Hz, although this peak is not as high, but much wider Linearity Controller design considered in this report is based on linear control theory. Investigating whether the system is linear, and if not, where non-linearities occur, can help to explain phenomena later on. Consider Figure 2.4, where it can be seen that the system does not go back to 0, after the disturbances are applied. As the system is open-loop stable, due to the leaf springs, this could indicate hysteresis. To look into this a bit further, a very slow input signal is used, which is depicted in Figure 2.6. The output is also shown in Figure 2.6 as a function of the input. The response in the positive and the negative direction are not overlapping, thus hysteresis is clearly present. Also, small steps can be distinguished in the response. Apparently, the system suffers from stick-slip, which could be the cause of the hysteresis. Another linearity test is performed by means of several sinesweeps. In Figure 2.7, a time domain representation of an upsweep is shown. The amplitude of the sweep increases with time, such that the sweep exhibits a relatively flat spectrum. This is depicted in Figure 2.8. The downsweeps used are exactly the same, except that the signal is applied in opposite direction in time. The systems identified with the sinesweeps are depicted in Figure 2.8. Overall the systems are very similar. The peaks of the system tend to be a little larger, when a higher amplitude sinesweep is used. This is best noticeable at low frequencies, see Figure 2.9. Around 125 Hz, a small amplitude related non-linearity seems to occur. This is depicted in Figure

15 Figure 2.5: Frequency domain representation of the response of the system when no input is applied. The 3 lines represent the first part ( black), the disturbance part ( red) and the end ( green) of the response. Input [V] Time [sec] Displacement [mm] Input [V] Figure 2.6: Input used to detect hysteresis (left) and the response of the system (right). 9

16 Input [V] Time [sec] Figure 2.7: Time domain representation of the sinesweep P 10 4 Figure 2.8: Frequency domain representation of the sinesweep (left) and the plants identified with the upsweeps ( ) and downsweeps ( ) with amplitudes of 0.4 Volts (black), 0.2 Volts (green) and 0.1 Volts (red) P 10 1 P Figure 2.9: Close up of the plant FRF for the frequency ranges 1-15 Hz (left) and Hz (right). 10

17 2.2.4 Aliasing Aliasing can occur when continuous time signals are discretized, as is being done when performing measurements on a digital computer. Although aliasing will not be a problem during this project as only discrete time performance and behavior will be analyzed. However, knowing of the existence of aliasing can help understanding the system. As mentioned before, all measurements during this project will be performed with a sampling frequency of 1 khz. To check for aliasing an identification experiment has been performed with a sampling frequency of 1024 Hz, see Figure If an aliased resonance would be present, it would be shown in the figure 24 Hz shifted compared to the 1 khz measurement. No shift of a dynamic phenomena can be detected. As a double check, another identification measurement is performed, but now with a sampling frequency of 10 khz. This measurement is also shown in Figure If an aliased resonance would be present in the 1 khz measurement, it would most probably be gone in the 10 khz measurement. This cannot be seen in the figure. All 3 identified plants are very similar. It is concluded that no significant aliasing occurs. 11

18 P P Figure 2.10: Comparison of the FRF of the plant acquired with a sampling frequency of 1 khz ( black), 1024 Hz ( red, top) and 10 khz ( red, bottom). 12

19 Chapter 3 Motivation for Periodic Input Design Input design is very important for identification experiments. Periodic inputs have several advantages over stochastic inputs, and therefor, periodic input design will be briefly discussed in this chapter. The Fourier transform of a discrete time signal will result in a more accurate discrete frequency domain signal, when periodic inputs are used, as compared to stochastic inputs. This is discussed first. With periodic inputs, it is also possible to distinct disturbances from uncertainty. This is part of the identification and validation procedure used during this project, and will therefor be treated in Chapter 8. In Section 3.2, a specific periodic input, multisine, will be discussed. 3.1 Using the DFT The Fourier Transform is a most important tool in signal analysis. It transforms time domain signals into frequency domain signals. Nowadays most measurements are performed on digital computers. Computers can only handle finite data sets with a finite resolution. Therefor, the signals have to be discretized and quantized. In practice the Discrete Fourier Transform (DFT) is used. Next, the most important steps are discussed to give an insight in the consequences of each step Discretization in Time Domain Most real-life systems are continuous in time, but computers can only handle discrete time signals. For a computer to be able to handle the signals, continuous signals have to be discretized, i.e., sampled. The signal is sampled at an equidistant time grid. The discrete time signal is then given by: u d (n) = u(nt s ), with T s the sampling time. The sampling process can be interpreted as a multiplication with a Dirac train: ũ d (t) = u(t)δ Ts, with δ Ts = δ(t nt s ). (3.1) n= 13

20 spectrum continuous time signal spectrum discrete time signal f s /2 0 f s /2 f s /2 0 f s /2 Figure 3.1: Continuous spectrum of continuous and discrete signal. Let the spectrum of the discrete time signal be defined as: U d (e jωt s ) = n= u d (n)e jωnt s. (3.2) Then the following holds using a Fourier Transform: U d (e jωt s ) = Ũd(jω) = F(ũ d (t)) = ũ d (t)e jωt dt. (3.3) The spectra of the discrete and continuous time signals are related. A multiplication in time domain corresponds with a convolution in frequency domain [11]: u(t)δ Ts corresponds with U(jω) f s δ fs (f). δ fs is, similar to δ Ts, a periodically repeated Dirac impulse with period f s = 1/T s. Considering all the above, the relation between the continuous spectra of the discrete and continuous time signals is given by: U d (e jωt s ) = U(jω) f s δ fs (f) = 1 T s k= U(j(ω kω s )). (3.4) As can be seen in Figure 3.1, this results in a repeated spectrum with period f s. If the original continuous signal contains frequencies f > f s /2, the repeated spectrum will overlap. This is called aliasing. If aliasing occurs, information at the overlapping frequencies is lost. If no aliasing occurs, the spectrum of the continuous time signal can uniquely be recovered from the discrete time signal for f s /s < f < f s / Windowing All signals considered in the previous section are of infinite length. It is practically impossible to acquire an infinite length signal and computers can only cope with finite signals. Mathematically, this can be represented by multiplying the time domain signal with a measurement window w(t): { = 1 0 t < T w(t) = 0 elsewhere (3.5) The spectrum of this window is given by: W (jω) = T e jωt/2 sinc(ωt/2), with sinc(x) = sin(x). (3.6) x 14

21 T = 1s Figure 3.2: Spectrum of a rectangular window ( ) and a Hanning window ( ). The spectrum of the signal is shown in Figure 3.2. The spectrum has zero-crossings a multiples of 1/T, except for 0. Looking at Figure 3.2 it is quite obvious that the power of a frequency is smeared over neighboring frequencies due to this window. This is called leakage. There are other windows, like the Hanning window: w(t) { = 1 cos(2 πt/t ) 0 t < T = 0 elsewhere (3.7) This window is also shown in Figure 3.2. It has a leakage effect that is more localized than the rectangular window. But this and other windows have other drawbacks, like less zerocrossings, which will be discussed in the next section Discretization in Frequency Domain All spectra considered are continuous spectra. A computer can only calculate a finite number of frequencies. An equidistant grid with spacing 1/T is selected to discretize the spectrum. Sampling in the time domain led to a repeated spectrum. Sampling in the frequency domain has the same effect, it leads to a periodically repeated time signal. In general, sampling in the frequency domain will lead to unwanted results. This is shown in Figure 3.3. This is because the measured frequencies are in general not on the DFT grid. The unwanted results can already be seen from the time domain signal. If the periodically repeated time domain signal has a discontinuity at the border of the window, it contains frequencies that are not on the DFT grid. If the measurement time T is a multiple of the periods of the frequencies in the signal, no discontinuity appears at the border of the window, and the spectrum can be reconstructed exactly with a rectangular window. The reason for this is that the DFT grid exactly coincides with the zero-crossings of the spectrum of the window, except for the true frequency. This is shown in Figure 3.3. Other windows, like the Hanning window, can not reconstruct the signal exactly, even if T is a multiple of the periods of the frequencies in the signal, because the main lobe is wider, see Figure 3.2. This window always has a leakage effect. It may seem unrealistic to measure a signal which has no discontinuities at the border of the measurement window, but it is not. During most measurements, the operator is free in his choice of input. When periodic inputs are chosen, the output will also be periodic, 15

22 T = 1s; u = sin(2π*5.5) T = 1s; u = sin(2π*5) Figure 3.3: Continuous ( ) and discrete ( ) spectrum of rectangular windowed discrete signals. and no leakage will occur, resulting in a perfect reconstruction of the spectrum of the signal. Windows, other than the rectangular, are only needed with low SNR or if the input cannot be chosen periodic. For more information on the DFT, it is referred to [18], [9] and [11]. 3.2 Multisine A specific example of a very useful periodic signal is a multisine. As the name implies, a multisine is the sum of multiple sinewaves: u(t) = F A k cos(2πkf o t + φ k ). (3.8) k=1 The frequency resolution of this signal is f o and has a signal period of T o = 1/f o. This is a periodic signal, thus no leakage occurs when the DFT is performed with a rectangular window. A major advantage of this signal over other signals is that the amplitude spectrum can be freely chosen with the parameter A k. No dynamic filter is needed to get the desired spectrum, as is the case with stochastic signals. With the parameter A k it is even possible to exclude frequencies by setting the amplitude to 0. This can be very useful when the energy of a signal should be concentrated around several specific important frequencies. When sinewaves are summed, without any thought, the resulting signal will most probably consist of a major part with low amplitude and a minor part with high amplitude. This is unwanted. Parameter φ k can be used to distribute the amplitude evenly over the period, while maintaining the desired spectrum. This will be discussed next Crest Factor Input design does not only consist of choosing a suitable signal and frequency content. The amount of energy in a signal is also an important property of the signal. To achieve a good SNR, an input signal, used for identification, should contain as much energy as possible. In practice, the amplitude of the input signal is often upper bounded. To determine how 16

23 much energy the signal contains compared to its amplitude, a quality measure is needed. An important measure is the crest factor [18]. The crest factor is defined as: Cr(u) = u peak u RMSe = max u(t) t [0,T ] u RMS Pi /P T, with u 2 RMS = 1 T T 0 u 2 (t)dt, (3.9) u(t) the input signal and P i and P T the useful (in the desired spectrum) and total power of the signal u(t), respectively. The interpretation of the crest factor is straightforward. The amplitude of the input signal is bounded (u peak ), and the useful energy content (u RMSe ) should be as large as possible. So, the smaller the crest factor is, the better the signal. There exist several algorithms to construct multisine signals with a crest factor that is as small as possible. Also, in [18] snowing techniques are discussed. These techniques increase the amount of useful energy by adding some extra frequencies to the signal. During this project, no snowing will be used. The crest factor can then be calculated easily by: Cr(u) = u peak u RMS = u pow(u), (3.10) because the useful power is now equal to the total power in the signal. 17

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25 Chapter 4 Internal Stability and Motivation for Dual-Youla Parametrization The Dual-Youla parametrization will be used to construct a control relevant uncertainty structure. It guarantees that the real system P o will be in the model set P, and that the entire model set is stabilized by the controller C exp used during the measurements. The Dual-Youla parametrization uses coprime factors. These will be discussed in Section 4.2. Then, the Dual-Youla parametrization is discussed in more detail in Section 4.3. But first, in Section 4.1, internal stability will be discussed. A notion of internal stability is needed to proof properties of the Dual-Youla parametrization, which will be used in Chapter 7 to acquire a control relevant identification criterion. 4.1 Internal Stability Consider the negative feedback control loop depicted in Figure 4.1, with controller C and plant P. For P to be internally stabilized by C, every signal in the control-loop should be bounded [26]. This is equivalent to the requirement that all closed-loop transfer-functions from the inputs to the outputs are stable. The closed-loop transfer-function matrix T (P, C) is given by: [ y u ] [ ] [ ] P SC P S r2 = SC S r 1 [ P (I + CP ) = 1 C P (I + CP ) 1 [ P = I =T (P, C) (I + CP ) 1 C (I + CP ) 1 ] (I + CP ) 1 [ C I ] [ ] r 2 [ r2 r 1 r 1 ] [ r2 r 1 ] (4.1) (4.2) (4.3) ]. (4.4) Thus, if T (P, C) RH, the control loop is internally stable. 19

26 r 2 C r 1 u P y Figure 4.1: Control Loop. 4.2 Coprime Factors A system can be represented in different ways. A coprime factorization is one of them, and has some advantages over other representations, such as a more flexible uncertainty modeling for closed-loop systems. But before the advantages of a coprime factorization are discussed, a short overview will be given of what coprime factors are and what their properties are Properties A right coprime factorization (rcf) of P is given by: P = ND 1, (4.5) where N and D are stable coprime transfer functions, i.e., N, D RH. Coprimeness implies that N and D do not have common RHP-zeros, which would result in pole-zero cancelation in ND 1. Mathematically, coprimeness means that there exist stable X r and Y r, i.e., X r, Y r RH, such that the following Bezout identity is satisfied: X r D + Y r N = I. (4.6) Similarly, a left coprime factorization (lcf) of P is given by: P = D 1 Ñ, (4.7) where Ñ and D are stable coprime transfer functions, i.e., Ñ, D RH. Thus there exist stable X l and Y l, i.e., X l, Y l RH, such that the following Bezout identity is satisfied: DX l + ÑY l = I. (4.8) A normalized right coprime factorization (nrcf) P = N D 1 is a special case of a rcf. A rcf is called normalized if the following Bezout identity is satisfied: D D + N N = I. [ ] Defining X r =, then X D N r satisfies Xr X r = I and is called inner. (4.9) There also exists a normalized left coprime factorization (nlcf), P = D 1 Ñ, which satisfies the following Bezout identity: D D + Ñ Ñ = I. [ Defining X l = D Ñ ], then X l satisfies X l X l = I and is called co-inner. (4.10) 20

27 4.2.2 Advantages A major advantage of coprime factors is the flexibility of uncertainty modeling for closed loop systems. For coprime uncertainty, it can be guaranteed that the plant is in the model set, P o P, and that the entire model set is stabilized by a controller used during measurements, F l (P, C exp ) RH. General coprime uncertainty is given by P = ( ˆN + N )( ˆD + D ) 1 = ( ˆ D + D ) 1 ( ˆÑ + N ) for a rcf and a lcf of ˆP, respectively. Uncertainty structures, as described in [6], are not as flexible. During this project, identification from closed-loop measurements for robust control is considered. In closed-loop, a plant can behave very different from open-loop operation. This can result in a plant model that behaves the same as the plant during closed-loop operation, but different during open-loop operation. To account for the discrepancy between the plant, P o, and the model, ˆP, a stable uncertainty model, RH, is used in robust control. Then, it is assumed that the plant is in the model set, P o P, where the model set P is described by ˆP and. It turns out that sometimes P o P. To illustrate this, consider the following example. Example 4.1. Assume the plant, P o, is given by: P o = 1 s 0.8 (4.11) It is clear that the plant exhibits a RHP pole, and is unstable. To perform measurements in closed-loop, a controller, C = 150, is used. From these measurements, the model, ˆP, is estimated, which is given by: ˆP = 1 s (4.12) This model is stable. The open-loop and closed-loop step responses of the plant and the model are given in Figure 4.2. Although the model and the plant behave different in open-loop, the closed-loop responses are almost the same, and one would expect to have a fairly good model. However, if the uncertainty is represented by an additive ( ˆP + ) or multiplicative ( ˆP (I + ) or (I + ) ˆP ) uncertainty structure, the plant will never be in the model set. When coprime uncertainty is used, N = 0 and D = 1.6 will include the plant in the model set. The other uncertainty structures cannot include the plant in the model set due to the fact that the uncertainty in these structures cannot introduce RHP poles, which is needed to include the plant in the model set. Thus the number of RHP poles of the model set will always be the same as the number of RHP poles of the nominal model for additive and multiplicative uncertainty. In a similar way, it can be shown that inverse additive ( ˆP (I ˆP ) 1 ) and inverse multiplicative ( ˆP (I ) 1 or (I ) 1 ˆP ) uncertainty structures cannot introduce RHP zeros. The number of RHP zeros of the model set will always be the same as the number of RHP zeros of the nominal model for these uncertainties. With coprime factor uncertainty, it is possible to vary the number of RHP poles and zeros. As can be seen from the example, not all model errors can be overcome by increasing. Another drawback of the mentioned uncertainty structures is that the model set can contain candidate models that are not stabilized by the controller used during the measurements. Because the controller stabilized the plant during the measurements, these candidate models 21

28 Response Time [s] Response Time [s] Figure 4.2: Open loop (left) and closed loop (right) step responses of the stable model ˆP ( ) and the unstable plant P o ( ). are known not to be the plant. In a later stage, the robust controller has to stabilize all models in the model set, also the models that are known to be not to be the plant. This is clearly unwanted. To overcome this problem, a Dual-Youla parametrization can be used. This will be discussed next. 4.3 Dual-Youla Parametrization The Youla-Kucera parametrization is used to find all stabilizing controllers for a given plant P, which is internally stabilized by a controller C, see [5]. The dual form parameterizes all stabilized plants for a given controller C, which internally stabilizes a plant P. During this project the right coprime Dual-Youla parametrization is used. Given a plant and its rcf, P = ND 1, and a stabilizing controller and its rcf, C = N c Dc 1, then all plants stabilized by C are parameterized by: P = N D 1 = (N + D c )(D N c ) 1, RH. (4.13) Stability of is required to guarantee the stability of the coprime factors N and D. The Dual-Youla parametrization only uses one uncertainty block ( ), as opposed to the general coprime factor parametrization ( N, D ). Still, the Dual-Youla parametrization parameterizes all stabilized plants. To proof this, the proof will be split up into several parts. In subsection 4.3.1, it will be shown that every stable pair {P, C} satisfies the Bezout identity. This result will be used in subsection and subsection Then, in subsection 4.3.2, it will be shown that the entire set, defined by P, is stabilized by a controller C. Finally, in subsection 4.3.3, it will be shown that every plant, stabilized by C, can be described by P Control Interpretation Bezout Identity In this section it will be shown that the solution of the Bezout identity can be given a control interpretation. In section 4.1 it is explained that internal stability is achieved if T (P, C) RH. Consider a rcf of a plant, P = ND 1, and a lcf of a stabilizing controller 22

29 C = D 1 c Ñ c. Then, the following holds [23]: [ ] P T (P, C) = (I + CP ) 1 [ C I ] (4.14) I [ ND 1 = I [ ND 1 = I [ N = D ] (I + ] 1 D c Ñ c ND 1 ) [ 1 D 1 c Ñ c I ] (4.15) D( D c D + ÑcN) 1 [ Dc D 1 c Ñ c I ] (4.16) ] ( D c D + ÑcN) 1 [ Ñ c Dc ] Coprime factors are stable by definition, thus: [ N D (4.17) ] ( D c D + ÑcN) 1 [ Ñ c Dc ] RH (4.18) ( D c D + ÑcN) 1 = Z RH (4.19) Consider a new rcf of the plant, P = N z D 1 z = (NZ)(DZ) 1. Then: ( D c D z + ÑcN z ) 1 =( D c DZ + ÑcNZ) 1 (4.20) =Z 1 ( D c D + ÑcN) 1 (4.21) =I (4.22) and also D c D z + ÑcN z =I (4.23) This is a Bezout identity. The solution of a Bezout identity of a rcf of a plant is the lcf of a stabilizing controller. Also, the solution of a Bezout identity of a lcf of a controller is the rcf of a stabilized plant. The same can be shown for a lcf of a plant and a rcf of a controller. Note that the only assumption that is made, is that P is stabilized by C. Thus every stable pair {P, C} satisfies a Bezout identity, and vice versa P Stabilized by C In this section it will be shown that the entire set, parameterized by P, as given in Equation (4.13), is stabilized by a single controller C. Let a rcf and a lcf of a controller be given by C = N c Dc 1 = D 1 c Ñ c, respectively. And let a rcf of a plant, stabilized by C, be given by P = ND 1. Then, all plants, parameterized by P are stabilized by controller C, if the following Bezout identity is satisfied: I = D c D + ÑcN (4.24) = D c (D N c ) + Ñc(N + D c ) (4.25) = D c D + ÑcN D c N c + ÑcD c (4.26) = D c D + ÑcN + (ÑcD c D c N c ) (4.27) 23

30 Recall that C = N c Dc 1 = D 1 c Ñ c, thus N c = D 1 c Ñ c D c. Substituting N c gives: D c D + ÑcN + (ÑcD c D c N c ) = D c D + ÑcN + (ÑcD c D D 1 c c Ñ c D c ) (4.28) = D c D + ÑcN (4.29) =I (4.30) The pair {P, C} satisfies the Bezout identity for every. Thus the entire set is stabilized by controller C All Stabilized P Parameterized by P It is quite cumbersome to proof that P parameterizes all plants, stabilized by C, using T (P, C). By using S + T = I, the proof simplifies a little. An equivalent expression for internal stability can be calculated: [ ] [ ] I 0 So P S T (P, C) RH T (P, C) = i RH 0 0 S i C, (4.31) with S i and S o the input and output sensitivity, respectively. Now consider the following: [ ] [ ] 1 So P S i I P = (4.32) S i C S i C I [ I N = D 1 ] 1 N c Dc 1 (4.33) I ([ ] [ ]) Dc N = D 1 1 c 0 N c D 0 D 1 (4.34) [ ] [ ] Dc 0 D Ñ = (4.35) 0 D Ñ c Dc ([ ] [ ]) ([ ] [ Dc D Ñ Ñ = + + c D ]) c 0 D 0 N c Ñ c Dc 0 0 (4.36) [ ] [ ] [ Dc 0 D Ñ Dc Ñc D = + c D ] c 0 D Ñ c Dc N c Ñc N c D (4.37) c [ ] 1 [ I P Dc Ñc D = + c D ] c C I N c Ñc N c D (4.38) c [ ] 1 [ ] I P Dc = + [ ] Ñ C I N c Dc (4.39) c Consider any stabilized plant P 1, for which should be found. Then the following holds: [ ] 1 [ ] 1 [ ] I P1 I P Dc = + [ ] Ñ C I C I N c Dc (4.40) c [ ] 1 [ ] 1 [ ] I P1 I P Dc = [ ] Ñ C I C I N c Dc (4.41) c [ ] ( [ ] 1 [ ] ) 1 [ ] I P1 I P N Ñ D = (4.42) C I C I D S i 24

31 For every stabilized plant, a RH can be calculated. Every plant that is not stabilized by C, i.e., T (P, C) RH, requires RH. This can easily be seen from Equations (4.39) and (4.42). Thus, the Dual-Youla parametrization parameterizes every plant, stabilized by C. 25

32 26

33 Chapter 5 Discrete Time Controller Design Several decades ago, controlled systems were implemented as analog systems consisting of resistors, capacitors, etc. With the introduction and the huge development of the digital computer, most controlled systems are nowadays implemented as digital systems. Digital systems have lower costs, increased flexibility and superior accuracy, compared to analog systems. Although most analog controllers are replaced by digital controllers, controller design is still mainly performed for analog controllers. This is because for continuous time systems, the Bode phase-gain relationship exists, and it does not in discrete time. This enables the use of easy-to-use design tools. In this chapter, a discrete time controller design is introduced, as all controllers used in this project will be discrete time controllers. First, in Section 5.1, the control loop is stated in discrete time. Then commonly used controller design is briefly discussed. In Section 5.2, the w-domain is introduced. This domain enables the use of continuous time controller design techniques for discrete time controllers. 5.1 Discrete Time Control Loop Classical analog control design could easily be stated in continuous time. A one-degree-offreedom continuous time feedback interconnection setup is depicted in Figure 5.1. All signals in this setup evolve in continuous time. Also, the controller model C and the plant model P are continuous time models. Nowadays, digital computers are mainly used to control systems. Thus, the controllers implemented are discrete time controllers. And signals are sampled and evolve in discrete time. This is depicted in Figure 5.2. A discrete time controller C d still controls a continuous time plant P, of course. To apply continuous inputs to the plant, a hold circuit H is needed. The conversion from continuous time signals to discrete time signals is done by a sampler S. From r e C u P y Figure 5.1: Continuous time control loop. 27

34 r d e d u d u y y d P C d d H P S Figure 5.2: Discrete time control loop. a control point of view, SP H can be seen as a discrete time plant P d. If the continuous time plant P is LTI, the discrete time plant P d is also LTI when considered in discrete time. P d is not LTI when considered in continuous time. Thus, the implementation of S and H presents no problem for the implementation of a discrete time controller. It is common practice to use continuous time plant models to design a continuous time controller. Then the controller is discretized for implementation. This approach has two shortcomings [16]. Firstly, the plant model is identified from discrete time signals. It is not possible to reconstruct an accurate continuous time model form discrete time signals [3]. Hence, a discrete time plant model should be used. Secondly, discretization of a continuous time controller involves an approximation. The approximation error typically decreases with increasing sampling frequency. In practice, however, the sampling frequency is upper bounded. This approximation introduces errors and limits performance. The first shortcoming can easily be avoided. Discrete time modeling algorithms are readily available in commercially available programs like MATLAB. The second shortcoming can be avoided by directly designing a discrete time controller. However, controller design in discrete time is more difficult than in continuous time. Techniques used in classical loop-shaping and filter design do not apply. They are based on asymptotic behavior and easy calculation of the cut-off frequency. This is more difficult in discrete time than in continuous time, as the following example shows. Example 5.1. [12] Consider the Laplace transform of a continuous time first order transfer function: K(s) = P + Ds, (5.1) where P, D R. The frequency response is then given by K(jω) = P + jdω. (5.2) The asymptotes of this transfer function in the Bode magnitude plot are given by lim K(jω) = P ω 0 (5.3) lim K(jω) = Dω. (5.4) ω And the cut-off frequency is given by ω cut off = P D. (5.5) 28

35 A discrete time transfer function of the same order is given by K d (z) = a + bz, (5.6) with frequency response K d (e jωh ) = a + be jωh. (5.7) The asymptotic behavior or this transfer function is given by lim ω 0 K(ejωh ) = a + b (5.8) lim K(e jωh ) = a b. (5.9) ω ωs 2 A cut-off frequency is more difficult to determine in the discrete time case. To be able to use continuous time loop-shaping and filter design techniques, the w-domain is introduced and will be discussed next. 5.2 w-domain A LTI continuous time system can be described exactly in the s-domain. A characteristic element of the s-plane is the imaginary axis. To get the frequency response of a continuous time transfer function it is evaluated on the imaginary axis by substituting s = jω. Stability of a system is determined by the position of its poles. Poles in the left half plane are stable, whereas poles on the other side of the imaginary axis, the right half plane, are unstable. If the position of a pole is given by s = σ + jω, (5.10) then the pole is stable if σ < 0. Poles of the continuous time system are mapped into the discrete time ones according to z = e sh [12]. Thus: z = e sh = e σh+jωh = e σh e jωh = e σh e jωh+j2πl, l Z. (5.11) It can easily be seen that discrete time poles are stable if z < 1. Note that the imaginary axis (σ = 0) is mapped onto the unit circle ( z = 1). Poles that are separated by an integer multiple of the sampling frequency ω s are mapped onto the same point in the z-plane. This is the same aliasing effect that occurs with a discrete Fourier transform. The mapping is said to be not injective or not one-to-one. No unique inverse transformation exists. A graphical overview of the mapping is shown in Figure 5.3. The goal of the mapping from the z-domain to the w-domain is that the w-domain has the same properties as the s-domain, such that continuous time loop-shaping techniques and filter design can be applied. Furthermore, the mapping should be onto and one-to-one, i.e., for every point on the plane a unique (inverse) transformation should exist. Such a mapping is available, and is known as the bilinear, Möbius, or linear fraction transformation. The mapping is given by [12],[17] w = W (z) = az + b cz + d ; z = W 1 (w) = dw + b cw a 29 (5.12)