Power Electronics, IEEE Transactions on - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 3, MAY 2002

379

Nonlinear Internal-Model Control for Switched Reluctance Drives Ge Baoming, Wang Xiangheng, Su Pengsheng, and Jiang Jingping, Senior Member, IEEE

Abstract—A nonlinear internal-model control (IMC) based on a suitable commutation strategy for switched reluctance motors (SRMs) is proposed. In the commutation strategy, a fixed critical rotor position is defined as the commutation point, which results in reduced computation. Combined the simplicity of the feedback linearization control and the robustness of IMC structure, the proposed drive has excellent dynamic and static performances for the torque and current control. The scheme is analyzed, and some important properties are obtained. Simulations and experiments were carried out on a 7.5 kW four-phase SRM, and the results show that the ripple of the output torque is very low in spite of model-plant mismatches.

robustness under the conditions of modeling uncertainties, inside and outside disturbances. It can effectively compensate for the nonlinearity of the plant. Moreover, by using the error between the plant and model outputs as a feedback signal, the nonlinear IMC has an inside integral action, which ensures the convergence of the plant output to the constant reference of the steady state. Based on the nonlinear IMC, this paper proposes a novel SRM control using a different commutation strategy.

Index Terms—High-performance drive, nonlinear internalmodel control, switched reluctance motors, torque control, torque-ripple minimization.

II. A COMMUTATION STRATEGY

I. INTRODUCTION

T

HE switched reluctance motor (SRM) has a simple structure (hence, low cost and high reliability) and a wide speed range; it can work at high speeds and in harsh environments [1]–[4]. However, its applications for variable-speed and servo-type drives were limited owing to the natural torque ripple. Early classical linear controllers could not deal properly with the nonlinearity of SRM electromagnetic characteristics, so the ripple in the torque profile was still pretty high [5], [6]. To improve the performance of the SRM drive, the feedback linearization control was investigated extensively [7]–[15]. This technique needs an accurate model of the motor though the control itself is simple. Using an approximate model does not produce a linear closed-loop system. An added integral term to ensure offset-free performance increases the relative degree of the system, which adversely affects the disturbance rejection properties of the nonlinear controller. Moreover, an approximate model will result in a degradation of the drive performances or even an unstable response. The stability analysis of the controlled system is indispensable, which in turn leads to the redesign of the control law to improve the stability robustness of the SRM drives, for example, see [10]. References [16] and [17] presented a nonlinear internal-model control (IMC) combining the feedback linearization control and IMC. The control system has strong Manuscript received September 27, 2001; revised January 26, 2002. This work was supported in part by the China Postdoctoral Science Foundation under Grant 200031. Recommended by Associate Editor J. Ojo. G. Baoming, W. Xiangheng, and S. Pengsheng are with the Electrical Machinery Group, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). J. Jingping is with the College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]). Publisher Item Identifier S 0885-8993(02)04639-2.

An appropriate commutation strategy is crucial in the control of SRM’s to reduce torque ripple. The single-phase conduction commutation in [8] does not essentially meet the expectation of torque ripple-free because it may cause voltage saturation. Commutation strategies of two phases that have a conduction overlap are therefore preferred. In [18] the critical rotor position is defined where two neighboring phases produce same torque at the same exciting current levels, which assumes low torque ripple and low copper losses. Reference [19] uses the same notion of , which meets the requirements of both low copper losses and low voltage. In [18] and [19], is varied with the required torque, which results in mass computation. fixed but the turn-on angle In fact, we can suppose varied when the objective is that the required phase voltage never exceeds the available inverter voltage, rather than minimum voltage or losses. As shown in Fig. 1, the corner is the position where the front edges of rotor poles meet the back edges of stator poles. The sharp “corners” of the inductance profile in idealized form result from the neglect of fringing [4]. For a typical -phase SRM, each inductance varies periodically with a , where is the period of the rotor pole-pitch number of rotor poles [20]. Each phase mainly produces torque . On-coming phase (phase in one stroke angle because the two in Fig. 1) is excited at an advancing angle commutation process cannot be instantaneous under the limit of to the current of the phase two the inverter voltage. From increases rapidly at the maximum rate under the full inverter voltage to minimize the magnetization period. Once the rotor and stator poles come to overlap (i.e., ), the phase two’s inductance starts to increase, which results in slow-down of the current rise. to , the As the current of phase two rises quickly from applied voltage for the out-going phase (phase one) would be adjusted down to maintain the total torque at the required constant value. As moves past , phase one is no longer the main torque contributor, which continues to decay until it drops to

0885-8993/02$17.00 © 2002 IEEE

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As phase is the main torque contributor and the total torque of the motor is given by (5) is the phase torque of phase . The dynamics of where can be obtained as (6) where Fig. 1. Commutation scheme with phase-inductance profile.

zero, simultaneously phase two takes over until . However, to reduce the control voltage burden on phase two, the rate of decay of remaining current should be limited (such as, the dragging out the process of control voltage on phase one is in favor of eliminating voltage saturation on phase two) [8]. Thus, the phase two’s voltage could be adjusted to compensate in real-time for the decrease of the phase one’s torque without difficulties. Using representation similar to [18], we have the expression for a 4-phase SRM as

(1)

and denote phases that produce the strong, where indices rising, and falling components of phase currents, respectively; .

(7) (8) The controller of phase does not need to be designed because its phase voltage is set to the maximum value of the in. verter voltage, that is, B. Nonlinear IMC for the Total Torque Due to the plant-model mismatch, unpredictable plant parameter variations, measurement noises and so on, the plant output be torque inevitably deviates from the model output. Let output of the plant, then the plant-model output error is given by (9)

III. NONLINEAR IMC OF SRM A. SRM Model Based on the proposed commutation strategy, the electrical dynamics of a four-phase SRM can be written as

To improve the system’s stability and robustness, and make the nonlinear IMC controller proper by eliminating the need for derivatives, a filter must be employed. For the first-order system, the filter can be selected as

(2)

(10)

denotes the model; and where the superscript are the voltage and current of phase and are is the current the angular position and velocity of the rotor; and are given by output of phase ; the functions

is a positive real. where The objective to control the torque is to keep the actual toque that converges to a constant track a desired trajectory even under disturbances or plant-model mismatches. According to the nonlinear IMC [17], an auxiliary variable is defined as (11)

(3) Then, the global system is obtained as (4)

(12) is the flux linkage of phase where winding resistance.

is the stator where

is the global output.

BAOMING et al.: NONLINEAR INTERNAL-MODEL CONTROL FOR SWITCHED RELUCTANCE DRIVES

381

If the feedback linearization control technique is applied to (12), the control law can be written as (13) is a positive real, and in is where is never available from the current controller of phase . singular on the condition of the proposed commutation. C. Nonlinear IMC for the Current of Phase Similar to that of the torque, the filter system of the current of phase is given by (14) Fig. 2.

is the plant output current of phase is a positive real. The objective to control the current of phase is to force the decay quickly and stably to zero following residual current the reference trajectory of the decaying current. Like that for the torque, an auxiliary variable is introduced as

Nonlinear IMC of SRM.

where

Set is solved from (19). Due to the to , the initial full utilization of the inverter voltage from of the turn-on angle can be derived as value (20)

(15) Then the control law of the system can be written as (16) is a positive real, and where into (2), we have

. Substituting (16) (17)

is the unaligned inductance. where , the computation procedure as follows can be Based on started: at the position using 1) calculate the phase voltage (13); , advance the turn-on angle by If and return to 1); , delay the turn-on If and return to 1); angle by is obtained. 2) the actual

which would be used in (13). E. Structure of the Control System

D. Computation of the Turn-On Angle To evaluate an initial value of , we assume is constant; 1) load torque 2) motor runs on steady state; 3) plant-model match; ; 4) ; 5) . 6) flux linkages satisfy , we require that the At the rotor position total toruqe produced by phases and be equal to the load torque, that is

The schematic diagram of the nonlinear IMC system for SRM is shown in Fig. 2. The system consists of four major parts, the actual SRM plant and its converter, the SRM model, the filters and the nonlinear IMC controllers. IV. PROPERTIES OF THE PROPOSED DRIVE A. Convergence As described in Section III, the systems of the auxiliary variand is linearized as ables (21)

(18)

and (22)

From (13), (17) and (18), we get

and and will both converge exponenIf tially to zero. To demonstrate the importance of the introducand , we define a torque tracking error as follows, tion of as an example taking (19)

(23)

382


Fig. 3. Dynamic responses of torque produced by the plant.

Fig. 5. Simulation results at 1200 r/min and 30 N1m.

If , then . Thus, the auxiliary variable can be used to estimate the tracking error of the plant output torque. The same interpretation can be applied to . The result is equals to the current tracking error of phase when . B. Effects of Control Parameters on Closed-Loop Performance For simplicity, we evaluate a special case where and is a step input signal. Using (10), (11) and (21), the closed-loop transfer function can be obtained as (25) where Fig. 4. Simulation results under different a . The solid line: a dash line: a = 900.

= 1300; the

(26)

(24)

It is clear to see that no control would be implemented as . On the contrary, the controller (13) performs a perfect . Hence, the control parameter plays an control as important role on both transient and static performances of the results in vigorous responses, while system. Large value of

Using (10), we obtain


Fig. 6. t

Responses of the proposed control system under varying model parameter R : t : R = 0:7 1:5 .

= 0 7 s–0 8 s, :

2

small value causes sluggish responses. Equations (25) and (26) also indicate that the closed-loop system is naturally stable, the actual torque will converge exponentially to the desired value as . long as Similarly, for the system of the current of phase , if and is a zero signal, we can get the conclusions as follows. ; the con1) No control would be implemented as . troller (16) performs a perfect control as 2) The residual current decays faster for a larger . 3) The steady-state output of the plant current is equal to zero. does not influence the transient and 4) The variation of static characteristics of the actual torque because the torque controller can compensate the effects. C. System’s Equilibrium Points , and It can be derived from the above analysis that , where and are the equilibrium values of the plant torque and current respectively. Using (10) and (14), we and . Therefore, on the steady state, get the equilibrium points of the control system can be expressed as

383

= 0 55 s–0 6 s, :

:

R

= 0 7 ; = 0 6 s–0 7 s, :

t

:

:

R

= 0 7 2 0 5 ; :

:

will not be zero given any plant-model mismatch. But, and ), owing to the SRM’s unipolar currents ( may equal to zero even if the plant and model is mismatched, because thus , according to 1) when the calculated the voltage input on the steady state (29) is calculated to be zero; , which may result from the 2) when the calculated , and since the plant-model mismatch, then back-EMF becomes negative (for the zone of decreasing inductance), its value is larger than the voltage drop on , so (actually owing to the fact the resistance that the current of phase is switched off on the steady is meaningless to the plant even though it is still state, the control input); , the actual will be set to 3) when the calculated , so and zero due to the limitation of will decay to zero faster than . V. SIMULATION RESULTS

(27) (28)

The simulations of the proposed nonlinear IMC drive -pole SRM which have been performed on a four-phase

384

Fig. 7.


Responses of the proposed control system under varying model parameter ; t = 0:7 s–0:8 s, b = 0:0454 1:2 A .

0:0454 2 0:8 A

2

parameters are listed in the Appendix. In the simulations V, and a well-established model [8] of magnetic saturation is used as (30) with

(31) , and are constants; is the where saturated flux linkage (1.1 Wb for the SRM used). According to the required accuracy in our investigation, only the first two . terms of the Fourier series in (31) are taken to calculate The parameters and can be determined by measuring the aligned and unaligned phase inductance.

b

:

t

= 0:55 s–0:6 s,

b

= 0:0454 A

;

t

= 0:6 s–0:7 s,

b

=

Fig. 4 shows the responses of the phase voltage, current and the output torque with different , where , 600 r/min and N m. Given a larger , the residual current decays more quickly to zero with the also gives a faster rise of voltage dropping deeper. A larger has little the on-coming current. However, the variation of influence on both transient and static performances of the torque response. With a double speed of the motor 1200 r/min, Fig. 5 shows the simulation results for the same load torque where and . The commutation process can be seen clearly with the voltages and currents of two neighboring phases shown together. At the start of commutation, the on-coming phase is excited with the entire inverter voltage, which makes the magnetization period minimal. As the motor speed rises, the is set advanced and the length (in the angle poturn-on angle sition) of the entire inverter voltage applied increases. It can be seen in Figs. 3–5 that the torque ripple keeps very small under various conditions of the simulations.

A. Transient and Static Performance In this part, we suppose that the model matches the plant perfectly. The dynamic responses of the torque with different values of are shown in Fig. 3 under a constant reference value of 30 N m. It can be seen that without any overshoot the electromag. netic torque rises to the set point faster with a larger

B. Robustness As it is difficult to get an accurate SRM model, it is very important for the control system to have a good performance of robustness under the condition of plant-model mismatches. To evaluate the robustness of the system, a simulation was carried and being constant; the out with the parameters


Fig. 8. Responses of the proposed control system under varying model parameter Wb; t = 0:7 s–0:8 s, = 1:1 1:2 Wb.

2

speed is 600 r/min, N m, and . Figs. 6–9 show the responses of the torque and the phase curby % by % by rent to the variations of %, and by % respectively from their original values that match those plant ones. causes the model current As shown in Fig. 6, a decreased higher than that of the plant. With the same decay rate, the plant current drops to zero firstly and after that the current controller sets zero to the decay rate of the model current so that it keeps at a small value above zero even in the afterward steady state. In makes the model current lower the opposition, an increased than the plant one, thus it decays to zero earlier. It can be seen in Fig. 6 that the model torque increases by is cut down by 50%. However the more than 2 N m when plant torque remains the reference value because the torque , controller eliminates the offset. Under a large enough can further reduce the ripple of the increasing the value of plant torque. Also shown in Fig. 6, the model torque decreases increases by 50%. Again the by more than 1 N m while torque controller suppresses the offset, and the plant torque keeps the balanced value with very little ripple. increases, It can be derived from (30) and (31) that and decrease as reduces. Fig. 7 shows a reduced rise rate of the model current at decreases. On the the beginning of commutation while

: t = 0:55 s–0:6 s,

385

= 1:1 Wb; t = 0:6 s–0:7 s,

= 1:1 2 0:8

contrary, with an increased , the current rises faster than that of the plant. After the commutation point, in the decreased situation the model current keeps rising quickly because is still much less than the supply the back-EMF . voltage at the low current level and the decreased as the back-EMF is greater than While with the increased the supply voltage due to the high level of the model current as the current begins to drop. well as the increased In Fig. 7 it can be seen that the model torque decreases by reduces by 20%, which is beseveral Newton meters while . With the cause of the decreased model current and , the model torque gets much higher than the balincreased anced value. Due to the peak current at the commutation point the torque ripple of the model gets significant. Anyway, in both , the plant torque keeps cases of the decreased and increased the reference value though it has some small ripple. Again, from (30) and (31) it can be derived that the decrease will result in the reduction of and . of causes a faster rise of the model current The decrease of makes the shown in Fig. 8. But the decrease of back-EMF of the model much less than the supply voltage, and the model current keeps rising quickly. As a result, the torque produced by the model is much greater than the plant one due to the larger rms current. results in a deIt is shown in Fig. 8 that an increased creased rise rate of the current. However with the increase of

386


Fig. 9. Responses of the proposed control system under varying model parameter a : t ; t = 0:7 s–0:8 s, a = 0:0545 1:16 A .

0:84 A

2

the back-EMF nearly balances the supply voltage after the commutation point, the current hardly increases. The result that the rms current of the model is less than the plant one makes the model torque decrease by several Newton meters. Anyway, the plant torque keeps the desired torque with very small ripple. Fig. 9 shows the reverse responses of the model torque and to the increased current in the situation of the decreased and the increased to the decreased . reduces resulting in the decrease of , the model As current rises faster at the beginning of commutation. The peak current at the commutation point makes the back-EMF greater does not change, than the supply voltage though unchanged the causing the current to decrease. With model torque gets higher given a larger model current. , the plant Both the same of the decreased and increased torque keeps the balanced value and its ripple is very small.

VI. EXPERIMENTAL RESULTS In our experiments, the applied motor was a 7.5 kW four-pole SRM fed by a classic asymmetrical half-bridge phase IGBT inverter; the torque and current controllers were implemented using a Motorola 56 001 digital signal processor (DSP); a controlled dc machine was used as the load.

= 0 55 s–0 6 s, :

:

a

= 0:0545 A

; t = 0:6 s–0:7 s, a

= 0:0545 2

As the output electromagnetic torque of the motor can be expressed as (32) where and are the viscous friction coefficient and the inertia constant of the motor-load sysem, respectively. The output torque can be obtained by (32) where the acceleration is calculated by the numerical differentiation of speed. A highly precise encoder with a frequency-multipling circuit was used to ensure measurement precision of speed. The load throughout experiments was kept at 30 N m; the maximum voltage of the dc-link supply was 460 V; and the conand trol parameters were set as . Figs. 10–12 show the experimental results at the steady-state speeds of 600 r/min, 1000 r/min and 1200 r/min, respectively. It can be seen from the results that, the width (in the angle position) of the full inverter voltage applied at the beginning of commutation increases with the speed. It means an earlier turn-on of the on-coming phase at a higher speed, which agrees with the simulation results. After the commutation point the phase voltage is adjusted by the PWM chopping. In Figs. 10–12 it is shown that the torque profiles are all very smooth but a significant increase of torque ripples compared to


Fig. 10. Experimental results at 600 r/min (measured total torque, phase voltage and current).

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Fig. 12. Experimental results at 1200 r/min (measured total torque, phase voltage and current).

VII. CONCLUSION Based on a suitable commutation strategy, a nonlinear IMC was developed for the SRM drive. The important properties of the control, such as the convergence, the equilibrium points and the effects of the control parameters on the performances of the system, were analyzed. It is demonstrated by simulations that the dynamic response of the output torque gets faster as the conincreases. However, the current control patrol parameter rameter has little influence on the torque performance though it plays a role on the determination of the turn-on angle. The results of simulations also show that the control system has very good robustness against the model-plant mismatches. To further evaluate the properties of the proposed control system, experiments were carried out on a 7.5 kW four-phase SRM. The results show that the controller can compensate for the nonlinearity very well and the outputs of the torque have very small ripples. Fig. 11. Experimental results at 1000 r/min (measured total torque, phase voltage and current).

the simulation results can be noticed though they all are limited % around the mean level. Theoretically, the to a range of frequency of the fundamental torque ripple is 24 times the mechanical frequency of the motor, since each of the four phases -pole is excited six times per revolution in the four-phase SRM. It can be seen in Figs. 10–12 that the frequencies of the fundamental torque ripple are right 240 Hz, 400 Hz, and 480 Hz for the speeds of 600 r/min, 1000 r/min, and 1200 r/min, respectively. For a SRM, the incremental inductance and the back-EMF of each phase are nonlinear. Moreover, the back-EMF will increase with the motor speed, which nonlinearities will become much more noticeable at the high speed. Therefore, a significant torque ripple will be produced at a high speed if the controller can not deal with the nonlinearities properly. In our results of experiments there is no obvious difference of the torque ripple among those three cases with various motor speeds. It means that the nonlinearities have been compensated effectively with the proposed controller.

APPENDIX Parameters of the SRM: Rated power Stator/rotor poles Phases Rated voltage Stator winding resistance Unaligned inductance Aligned inductance

kW

V mH mH

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Ge Baoming was born in Shanxi Province, China, in 1971. He received the B.Sc. and M.Sc. degrees from Liaoning University of Technology, Fuxin, China, in 1994 and 1997, respectively, and the Ph.D. degree from Zhejiang University, Hangzhou, China, in 2000, all in electrical engineering. From 2000 to 2002, he was a Postdoctoral Fellow in the Department of Electrical Engineering, Tsinghua University, Beijing, China. His research interests include the permanent magnet synchronous and switched reluctance motor drives, real-time control of electrical machines, power electronics systems, and nonlinear control theory and its applications to electric drives.

Wang Xiangheng received the M.S. and Ph.D. degrees from the Electrical Engineering Department, Tsinghua University, Beijing, China, in 1964 and 1986, respectively. He was with Dongfang Electric Machine Works, Sichuan Province, China, from 1968 to 1978. He is now a Professor at Tsinghua University. His field of interest is the analysis and control for electric machines and their system, electric drives and their automation, and fault analysis for electric machines and their protection.

Su Pengsheng was born in Sichuang Province, China, in 1946. He received the B.E. and M.E. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1970 and 1982, respectively. Since December 1992, he has been an Associate Professor in the Department of Electric Engineering, Tsinghua University. His research interests are in the areas of diagnosis of electrical machines, control of electrical machines, and power electronics.

Jiang Jingping (SM’88) was born in Zhejiang Province, China, in 1935. He has been on the Teaching Staff of Zhejiang University, China, since 1958. He held the rank of Associate Professor in 1978 and was appointed Full Professor in 1985. From 1979 to 1981, he was a Visiting Scholar at the University of Wisconsin, Madison. From 1988 to 1989, he was a Visiting Professor at the Department of Engineering, University of Reading, Reading, U.K. His special interests center on the application of minicomputers and microprocessors in real-time control applications. He serves as Vice Chief Editor of the Journal of Zhejiang University.