Speed Sensorless Control of Induction Motor based on Indirect Field-Orientation

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1 Speed Sensorless Control of Induction Motor based on Indirect Field-Orientation Marcello Montanari a, Sergei Peresada b, Andrea Tilli a, Alberto Tonielli a a Dept. of Electronics, Computer and System Sciences (DEIS) University of Bologna Viale Risorgimento 6 Bologna, ITALY b Dept. of Electrical Engineering Kiev Polytechnic Institute Prospect Pobedy 7 Kiev 6, UKRAYNE Abstract - A novel speed sensorless vector control is presented for induction motor. Speed and rotor resistance estimator has been designed using Lyapunov-like technique. Speed and flux controller has been realized using indirectfield orientation concept. With this choice, robustness with respect to stator resistance uncertainties can be improved by means of rotor flux reference feedback in the speedrotor resistance estimator. The performance of the solution proposed has been tested by simulations and by experiments. I INTRODUCTION In recent years, a large number of speed sensorless vector control systems for induction motor (IM) have been proposed. Speed information is generally provided by a speed transducer on the motor shaft; recently, low cost and high performance digital signal processors (DSP) become available allowing to obtain speed by means of digital estimators integrated with motor control. This solution represents an advantage in terms of costs, simplicity and mechanical reliability of the drive. Several schemes of speed estimators have been proposed in the literature; among them, the model reference adaptive system (MRAS) approach is very attractive and gives good performance [] []. The classical MRAS method is based on the adaptation of the rotor flux [] []; with this scheme, some difficulties in terms of precise and robust speed estimation arise, especially at low speed. The need of a pure integration in the speed estimator represents a drawback in the low speed region, due to drift and low frequency disturbances; moreover, parameter sensitivity (in particular to stator resistance) represents a usual disadvantage for all model-based estimators []. To overcome these problems, alternative MRAS schemes based on back-emf or reactive power [6] have been presented, but it seems that they don t solve troubles at low speed. The common approach to increase dynamic performance and stability of speed sensorless field oriented control systems is the on-line parameter adaptation [7] [8] [9]. The main contribution of this paper is a novel speed sensorless vector control based on: a) a speed and rotor resistance estimator, designed using Lyapunov-like technique, b) indirectfield oriented control (IFOC), c) suitable adjustments to improve robustness with respect to parameter variations, measurement errors and plant non-ideality. Rotor resistance estimation and other compensations adopted allow enhanced dynamic performance. Simulation results and experimental tests illustrate the good performance of this solution, also in the low speed region. The paper is organized as follows: in section II induction motor model and estimator strategy are given. Speed and rotor resistance estimation analysis is presented by means of simulation and investigated by means of a physical approach. In section III sensorless IFO control performance is analyzed in presence of stator resistance uncertainties and some solutions for increased robustness are presented and discussed. In section IV, digital implementation of the estimator algorithm is considered and experimental results are given. II SPEED AND ROTOR RESISTANCE ESTIMATION Under the assumptions of linearity of the magnetic circuits and equal mutual inductances, neglecting iron losses, the equivalent two phase model of the induction motor, represented in a fixed stator frame (a; b) and expressed in state-space form, is the fifth order model: dψ a = ffψ a!ψ b + ffl m i a dψ b = ffψ b +!ψ a + ffl m i b di a = fli a + fffiψ a + fi!ψ b + ff u a di b = fli b + fffiψ b fi!ψ a + ff u b d! = μ(ψ a i b ψ b i a ) T L : () J In (), (i a ;i b ) are stator currents, (ψ a ;ψ b ) are rotor fluxes, (u a ;u b ) are stator voltages,! is the rotor speed, T L is the load torque; the other parameters depend on rotor and stator resistances and inductances R s ;R r ;L s ;L r, mutual inductance L m

2 and moment of inertia J, as follows: ff = L s L m L s L r fi = L m ffl r μ = L m L r ff = R r L r fl = R s ff + fffil m Introducing two auxiliary variables (z a ;z b ) defined as z a = i a + fiψ a, z b = i b + fiψ b, IM model expressed in the statespace base (z a ;z b ;i a ;i b ;!) is: dz a ff i a + ff u a dz b ff i b + ff u b di a = (fl + ff)i a!i b + ffz a +!z b + ff u a di b = (fl + ff)i b +!i a + ffz b!z a + ff u b d! = μ J fi (z ai b z b i a ) T L : () Note that (z a ;z b ) variables are proportional to stator fluxes, with coefficient =ff. For the design of the rotor resistance estimation, let us define the rotor resistance error a = R r R rn,wherer rn is the nominal value for R r. From now on, the subscript N on other variables stands for the nominal value calculated on the basis of R rn. The design of the speed and rotor resistance estimator is based on Lyapunov-like method. The following tenthorder dynamic system is designed, based on measured signal u a ;u b ;i a ;i b and the assumption of exactly known stator resistance and inductances: d^z a d^z b d^z a d^z b = R s ^i a + ff ff u a + K ~ i a ^i b + ff ff u b + K ~ i b ff i a + ff u a ff i b + ff u b d^i a = (fl N + ff N )^i a ^!^i b + ff N ^z a +^!^z b + ^a L r ^za ( + L m fi)^i a (a) (b) (c) (d) + ff u a + K ~ i a ^ b (e) d^i b = (fl N + ff N )^i b +^!^i a + ff N ^z b ^!^z a + + ^a L r ^zb ( + L m fi)^i b + ff u b + K ~ i b +^ a (f) d^a = fl L r ~ i a ^za ( + L m fi)^i a + ~ i b ^zb ( + L m fi)^i b d^! = ~ i a ^z b ^i b ~ i b ^z a ^i a fl (g) (h) d^ a = fl ~ i b (i) d^ b = fl ~ i a (j) where ^a and ^! are the estimated rotor resistance error and the estimated speed respectively, ( ~ i a ; ~ i b ) are current estimation errors defined as ~ i a = i a ^i a, ~ i b = i b ^i b. The stability proof based on Lyapunov-like technique is reported in the appendix. In the hypothesis of constant speed and rotor resistance, the stator current estimation errors tend asymptotically to zero; if the persistency of excitation (PE) condition is satisfied, also the exponentially convergent estimate of! and R r is provided. The adaptation laws (g),(h) are very similar to the ones proposed in [8], but they are derived using a different approach. As already stated in [8], with constant rotor flux amplitude it is not possible to obtain correct simultaneous speed and rotor resistance estimation, owing to lack of persistent excitation. The estimator has been analyzed by means of a physical approach, to understand this phenomenon. It can be noted that: ffl if the flux modulus is constant and the motor torque is zero, IM model doesn t depends on R r. Hence, rotor resistance error doesn t affect speed estimate and rotor resistance estimate is not updated, because rotor currents are zero, ffl if flux modulus is constant, stator current errors tend to zero, but! and R r could be wrongly estimated. The relation between the estimation errors in steady-state is the following:! ^! = R r L r L m i q ψ + ^R r L r L m i q ψ ; where i q is the torque current and ψ is the flux amplitude. In order to overcome these problems and to obtain simultaneous! and R r estimation, a small AC frequency is added to the magnetizing current command, with the purpose of obtaining a variable flux and the PE condition satisfaction. Note that problems due to the lack of persistency of excitation don t exist if the speed estimation only is implemented. Therefore, disabling the rotor resistance adaptation, speed estimation can be achieved imposing a constant reference flux amplitude.

3 8 6 Speed reference (Nm) 8 6 Load torque * Speed iq* Control * IFOC Ud* Uq* o^ d,q->a,b / id a,b->d,q iq / Ua* Ub* Uc* ia ib PWM inverter IM Figure : Speed reference. Load torque. Slip s^ Rr^ ^ Speed & Rr Estimator ia ib va vb Speed and estimated speed 6 Rotor resist. est. Figure : Block diagram of sensorless IFO control. 8 6 Speed estimation error (A) (Ω) Current a error. Figure : Speed and rotor resistance estimation with % initial error in R r (R rn = R r =:). Variable flux reference. motor, to test performance at high ( rad/s), low ( rad/s) and zero speed ; at t = :9 s: a load torque equal to 6Nm (% rated torque) is applied. An initial % error on R r is assumed, R r estimation is enabled at t =:9s. Adding to the flux a Hz sinusoidal signal whose amplitude is % of the rated flux value,! and R r estimation is correctly achieved, as shown in fig. :! and R r errors tend to, thanks to flux modulation that allows PE satisfaction. Note that in the ideal situation of correct IM model, good estimate is obtained also at zero speed with an applied load torque. From the speed estimation error in fig. it can be noted that during the speed transient starting at t =:s, the rotor resistance error introduce an error in the estimated speed; from t =:9s, thanks to correct R r estimation, also the speed error is negligible. The estimator has been designed under the hypothesis of constant speed and rotor resistance. Nevertheless, the speed varies with a fast dynamic, while the rotor resistance is a slowvarying variable dependent on thermal variations. Thus, rotor resistance estimation dynamic is imposed slower than speed estimation dynamic; in this way, considering also the concept of dynamic separation, the presented estimator achieves better performance in terms of estimation errors. Moreover, for the estimator stability proof, speed is supposed constant, hence rotor resistance estimation should be disabled during speed transients, to avoid drift errors in the estimate. Then, since rotor resistance adaptation law (g) depends on estimated rotor currents (i.e. motor torque), rotor resistance estimation is enabled only if rotor current amplitude is sufficiently high, in order to avoid incorrect estimation due to measurement noise and model errors. The proposed estimator has been tested by means of simulation under the operating conditions shown in fig., with an IM whose parameters are reported in Table I. After an initial time interval [; :s:] needed to drive the motor flux to the rated value, a speed reference profile is applied to the unloaded III SENSORLESS IFO CONTROL A sensorless IFO control, based on the feedback of estimated speed and rotor resistance, has been realized. The IFO control scheme proposed in [] is used. The structure of the IFO controller and of the estimator is reported in fig.. The use of ^! and ^R r in the vector controller allows correct synchronous speed estimation. Performance of the sensorless vector control in the hypothesis of correct R s value in the same operating conditions of fig. are reported in fig.. Good speed and torque control is achieved also in the low speed region. Sensorless control with rotor resistance adaptation ensures good performance in presence of load torque and during transients. In presence of stator resistance uncertainties, errors in (z a ;z b ) estimation cause incorrect speed estimation, mostly in the low speed region. Moreover, since pure integration is needed in equations (a)-(d), problems due to drift arise at low speed. In order to solve these problems and to increase robustness to stator resistance uncertainties, a feedback based on the error of flux estimation with respect to flux reference has been

4 introduced in equations (a)-(d), that is: Speed and estimated speed 6 Rotor resist. est. d^z a a + ff ff u a + K ~ i a + K (i a + fiψa Λ ^z a) (a) d^z b b + ff ff u b + K ~ i b + K (i b + fiψb Λ ^z b) (b) d^z a ff i a + ff u a + K (i a + fiψa Λ ^z a) (c) d^z b ff i b + ff u b + K (i b + fiψb Λ ^z b) (d) where (ψa Λ ;ψλ b ) are the reference flux signals, expressed in a fixed reference frame. (ψa Λ ;ψλ b ) are calculated by the reference flux amplitude ψ Λ and by the orientation angle " computed in the IFO control, that is: 8 6 Speed error (Ω).. Speed estimation error ψ Λ a = ψλ cos " ψ Λ b = ψλ sin " : Flux-reference-based error feedback can be applied in the hypothesis that the real flux (not measurable) is equal to the reference flux, thanks to IFO control. This solution seems to be more effective than other adjustments proposed in the literature to solve drift problems, such as the substitution of the pure integrator with a low pass filter []. In addition, this approach increases the robustness of the estimator with respect to motor model inaccuracies and parameter variations, measurement errors, actuator distortion. In fig. simulation results are reported. Even with a R s error, satisfactory tracking performance is achieved at low speed, but a zero speed condition cannot be guaranteed with load torque, owing to error in speed estimation (speed error is present also in the hypothesis of correct rotor resistance value in the estimator). IV EXPERIMENTAL RESULTS Digital implementation of the estimator requires discretization of equations (), that may introduce estimation errors and instability. Hence, particular attention must be paid to the choice of the sampling time and the discretization method []. The estimator () can be simplified removing the (i),(j) equations without significant performance degradation. Since a simple Euler discretization of the whole estimator leads to performance degradation, a different approach is followed. During a sampling interval, the rotor speed, the rotor resistance and the stator currents can be assumed constant if the sampling time is small enough with respect to the system dynamics. The stator voltages are structurally constant during a sampling interval since they are the outputs of a digital controller. Hence, the discrete-time model of equations (a)- (f) describing the electrical part of the observer can be computed by means of the exponential matrix function. Since the exponential matrix function is dependent on estimated speed Figure : Sensorless IFO control with % initial error in R r (R rn = R r =:). 8 6 Speed and estimated speed Speed error (Ω) 6.. Rotor resist. est. Speed estimation error Figure : Sensorless IFO control with % initial error in R r (R rn = R r =:) and% error in stator resistance (R sn = R s =:). and rotor resistance, the discrete-time model should be computed on-line. A second-order Taylor approximation of the exponential matrix function is adopted in the digital implementation proposed. In this way, the discrete-time estimator shows good performance and can be implemented on available DSP with sample time equal to T s = μs. The adaptation laws (g),(h) for speed and rotor resistance estimation are discretized with classical Euler method. The experimental setup is based on a power inverter with V DC-link voltage. A khz symmetrical PWM technique

5 Speed. ω * ω.. Current a error (A) ω ω^ ω * ω^ Figure 6: Four quadrant operation at high speed Speed ω * ω... Current a error (A) ω ω^ ω * ω^ Figure 7: Four quadrant operation at low speed is adopted to drive the inverter switches. Stator currents are measured by Hall-type sensors. Since voltage sensors are not available in actual equipment, applied voltages in the estimator equations are considered equal to reference voltages imposed by the controller to drive inverter. Motor speed is measured by an encoder with ppr. The control board adopted is based on a DSP TMSC. The sampling time is set to μs. The DSP performs data acquisition, generates speed and flux profiles, implements estimator and IFO control algorithms and generates PWM inverter commands: the control board is connected to a standard PC to program the DSP and to display acquisition data. A.kW induction motor (whose data are reported in Table I) is used connected to a DC motor that generate load torque. The estimator parameters are set at K = ;K = ;fl ==;K =. The experiments are performed to test the speed sensorless vector control proposed in different operating conditions: high and low constant speed, speed transient, resistive and regenerative load torque. In the first five tests, only the speed estimation is enabled, while the rotor resistance is kept equal to its nominal value. In the figures, the symbols! Λ and ^! represents the speed reference and the estimated speed respectively. Experimental results reported in fig. 6 and 7 are obtained without applied torque, but the DC motor is connected to the IM giving a large inertial load (J ' :Kg=m ). In both tests, a smooth speed profile is applied to the motor. Fig. 6 shows motor behavior during steady-state condition at high speed ( rad/s) and during transient with speed inversion. Fig. 7 shows motor performance in the low speed region (maximum reference speed is rad/s). From these tests, it can be noted that during transient and with inertial load, sensorless vector control shows good performance. In fact, a negligible steady-state error is obtained either at high or low speed and good tracking is achieved during transient. Current error do not tend to null value owing to model errors. In fig. 8 and 9 braking and regenerative load torque equal to ± 6Nm (rated torque) is applied when a constant speed reference of rad/s is imposed. Good torque disturbance rejection is achieved with an almost null steady-state error and a speed transient of about ms. Note that, in presence of active load, performance are worse; in fact, a longer and less damped arises, but stability is still preserved. In fig. resistive load torque equal to Nm (% rated torque) is applied at t =:s, with a constant speed reference of rad/s. The load torque rejection at low speed is inferior with respect to high speed, owing to model errors and system non-ideality. Anyway, the adjustments adopted to increase the estimator robustness substantially improve the estimator performance. Experimental results showing simultaneous speed and rotor resistance estimation are reported in fig. ; they are obtained with a constant speed reference of 8 rad/s and an applied Nm load torque. A :6Hz sinusoidal component is added to the nominal flux to obtain the persistency of excitation. A non-

6 6 Speed 9 ω ω^ Speed 9 ω ω^ ω * ω ω * ω^ ω * ω ω * ω^ Torque current (A) Torque current (A)... Figure 8: Dynamic response with resistive load Figure 9: Dynamic response with regenerative load sensorless vector control is used to drive the motor. An initial rotor resistance error equal to Ω (% of its nominal value) is present; estimator ensures that speed and rotor resistance estimation error goes to zero. Although the presented results show good performance, some problems still exist. As claimed in previous section, sensorless control with rotor resistance adaptation should give better performance with respect to the one without this kind of adaptation. Anyway, with the actual experimental equipment, sensorless vector control based on simultaneous speed and rotor resistance estimation has implementation problems and does not produce significant improvements, due to measurement errors; moreover, at very low speed some degradation have been noted. It seems that this mismatch between simulation and experimental results is due to measurements errors and model errors. In particular, the stator voltage used in the estimator is obtained by the imposed voltage, that is different from the actual voltage applied to the motor, due to actuator distortion, PWM dead-times and other non-ideality. Moreover, stator resistance uncertainties and drift introduce errors in the estimator, especially in the low speed region; although some remarkable improvements have been obtained with the feedback action based on the error of flux estimation with respect to flux reference, some problems still exist. Equipment improvement and solution for an increased robustness will be the topic of future work. V CONCLUSIONS A sensorless IFO control has been realized. A new speed and rotor resistance estimation MRAS scheme based on Lyapunovlike method has been used. Solutions for robustness improvement have been adopted. Simulation results show good performance of the sensorless vector control at high and low speed and good robustness to parameter variations. Some problems exist at low speed. Sensorless control with rotor resistance adaptation gives better performance with respect to the one without this adptation. Performance of the sensorless IFO control has been verified by means of experiments, which show results comparable with the simulated ones. The realization of the IFO control based on simultaneous speed and rotor resistance estimation has implementation problems, owing to model errors and lack of precise information on the actual voltages applied to the motor. The vector control performs well at high speed also with applied torque; problems exists at very low speed with high applied load torque, owing to model errors and non-ideality; nevertheless, the adjustments adopted to increase the estimator robustness allows to obtain performance improvement. APPENDIX Speed and rotor resistance estimator stability can be proved by Lyapunov-like method. Let define errors ~! =! ^!, ~a = a ^a, ~ i a = i a ^i a, ~ i b = i b ^i b, ~z a = z a ^z a, ~z b = z b ^z b, ~z a = z a ^z a, ~z b = z b ^z b, ~ a = a ^ a, ~ b = b ^ b,

7 7 Table I: Induction motor data Speed ω * ω...6 ω ω^ ω * ω^...6 p number of pole pairs! speed rpm rated T m motor torque 6:Nm rated P power :Kw rated R s stator resistance 6:7Ω R r rotor resistance :Ω L s stator inductance :7H L r rotor inductance :7H L m mutual inductance :H J rotor inertia :Kgm Torque current (A) d~z a = d~z b = d ~ i a = (fl N + ff N + K ) ~ i a! ~ i b + ff N ~z a ~ b + ~a L r ^za ( + L m fi)^i a +~!(^zb ^i b ) Figure : Dynamic response with resistive load at low speed Speed Speed est. error Ref. flux current (A) R r error estimate (Ω) 6 8 d ~ i b = (fl N + ff N + K ) ~ i b +! ~ i a + ff N ~z b +~ a + ~a L r ^zb ( + L m fi)^i b ~!(^z a ^i a ) d~a = fl L r ~ i a ^za ( + L m fi)^i a + ~ i b ^zb ( + L m fi)^i b d~! = ~ i a ^z b ^i b ~ i b ^z a ^i a fl d~ a = fl ~ i b d~ b = fl ~ i a : Let consider the function: V = (~ i a + ~ i b )+ (~z a +~z b )+ fl (~z a +~z b ) Figure : Simultaneous! and R r estimation where ( a ; b ) are defined as a =! ~z a ; b =! ~z b and can be proved to be constant. Assuming that! and R r are constant, from the IM model () and the estimator equations (), the following error dynamics results: d~z a Rs = ff + K ~i a d~z b Rs = ff + K ~i b + fl ~a + fl ~! + fl (~ a +~ b ) where fl > ;fl > ;fl > ;fl > are tuning parameters. Choosing fl = and ont he basis of the model error ff R s ff +K equations, it follows that the time derivative of function V is _V = (fl + ff + K )( ~ i a + ~ i b ) < : It can be proved that stator current errors tend asimptotically to zero. Moreover, if the persistency of excitation (PE) condition is satisfied, also the exponentially convergent estimate of

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