Nonlinear Internal-Model Control for Switched Reluctance Drive with Torque Ripple-Free

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1 ISSN ATKAAF 43(1 2), (2002) Nonlinear Internal-odel Control or Switched Reluctance Drive with Torque Ripple-Free UDK IFAC IA Original scientiic paper Based on the nonlinear internal-model control (IC), associated with the suitable commutation strategy, a novel control solution or switched reluctance motor (SR) is ormulated and designed. The commutation strategy uses a deinite critical rotor position as commutation point, which reduces the computational burden. The nonlinear IC-based voltage control scheme or SR extracts the simplicity o the eedback linearization control and the robustness o IC structure, which ensures the torque ripple-ree and the drive's robustness in spite o the plant- -model mismatch disturbances. Some important properties are presented. Simulation results show that the high-perormance control or SR has been achieved. Key words: nonlinear internal-model control, SR, torque-ripple minimization 1 INTRODUCTION Switched reluctance motor (SR) drives are suitable or many variable speed and servo type application, and gaining increasing attention in the motor drive industry. However, the SR presents a coupled nonlinear multivariable control structure, early classical linear controllers are not ast enough to take into account the nonlinearity o the SR electromagnetic characteristics, which results in ripples in the torque proile [1, 2]. The drawback has led to a search or complex nonlinear control methods to compensate or the machine nonlinearity and torque ripples. One approach that has been investigated most extensively is the eedback linearization control (FLC) [3 6]. The application o this technique is simple and straightorward, but it needs an accurate model. Using an approximate model does not produce a linear closed-loop system, and the perormance o the SR drives degraded, in some cases even led to an unstable response. Nonlinear internal-model control (IC) combines the simplicity o FLC technique and the robustness o IC structure [7, 8]. The control system is not only highly robust with respect to modeling uncertainties and to other kinds o disturbances, but also can eectively compensate or the nonlinearity o the plant. The nonlinear IC implicitly includes integral action, and can guarantee the convergence o the plant output to a steady state constant reerence. Although the nonlinear IC displays so many interesting properties, as ar as the authors are aware, it has not yet applied to the SR drives. In this paper, based on the nonlinear IC, associated with a suitable commutation strategy, a novel control solution or SR is ormulated and designed, which ensures the torque ripple-ree and robustness in spite o the plant-model mismatch or disturbances. 2 COUTATION STRATEGY In general, the single-phase conduction commutation strategy is widely used due to its simpleness, and but it does not essentially meet the expectation o the torque ripple-ree owing to its resulting in the phase voltage saturation. Thereore, it is important to propose a suitable commutation strategy to compensate or the eects o the saturation. As shown in Figure 1, θ c is deined as a commutation point, which is periodic with period θ p = 2π/mN r where m and N r are the numbers o stator phases and rotor poles, respectively. Due to the inite inverter voltage, the commutation process cannot be instantaneous. So the on-coming phase-b be excited at an advancing angle θ on. Thus, the phase conduction region is divided into two sections, namely θ on θ θ c and θ c θ θ c + θ P, respectively. From θ on to θ c, phase-a is the stronger phase, which is the main torque contributor, and the lux linkage in phase-b must be driven rom zero as rapidly as possible. During this period, the controller orces the entire inverter voltage to be applied to phase-b in AUTOATIKA 43(2002) 1 2,

2 Nonlinear Internal-odel Control... di dt y k2 k2= ik2 k2 k2 k2 = + g u (2) Fig. 1 Novel commutation scheme order to minimize the magnetization periods. eanwhile, the applied voltage or phase-a is calculated in real time taking into consideration the actual torque developed by the phase-b so that the desired total torque can be produced. As θ moves past θ c, phase-a is no longer the main torque producer, and it should be demagnetized quickly as possible, while the current in phase-b, which is now stronger, becomes the main torque contributor up to θ c + θ P. Remarks: Compared with the commutation strategies in [9] and [10], although there exist certain similitude in the one proposed by the present paper, but the ollowing distictions are obvious: a) In [9] and [10], the critical rotor positions θ c will change when the required torque changes. In our commutation strategy, it is invariable in any case. b) During the demagnetization period, unlike the approach reported in [9] where di/dθ was ixed, and also unlike in [10] where dψ/dθ = V max /ω. In our commutation strategy, di/dθ is controlled by the nonlinear IC. c) Turn-o angle θ o is not required in our commutation strategy. 3 NONLINEAR IC OF SR 3.1 Electrical dynamic model When the plant output is the total torque that is to be controlled to the desired reerence, the electrical dynamic model based on the novel commutation strategy can be deduced as: d y T e = Fk1+ Gk1uk1 dt Te= Te (1) with v G di = dt T k1 ψk1 k1 i k1 ik ψ l ψ l l = R i, 2, l + ω l i θ l g l di = + g u dt y = i 1 T k1 ψk1 ψk1 k1 k1 ω ik1 i θ k1 1 T k2 ψk2 ψk2 R i k2 ω uk2 ik2 i θ k2 F = R i j= 1 j = ψ l =, l 2, i l { k } m T T + ω + v θ i (3) (4) (5) (6) (7) (8) where the indices k1, k2 and denote the phases that produce the strong, rising, and alling components o phase current, respectively; u k1, u k2 and u are the respective controls o phase-k1, k2 and ; u k2 = V max ; the superscript denotes the model; in, ψ n, and T n (n {k1, k2, }) are the current, lux linkage and phase torque o phase-n; T e is the total torque; R is the stator winding resistance in each phase; θ and ω are the angular position and velocity o the rotor. 3.2 Nonlinear IC or total torque { k } Due to the plant-model mismatch and various uncertainties, the plant output inevitably deviates rom the model output. Let y p Te be the torque output o the plant, i.e., y p Te = T p e, and e P Te be the plant-model output error. Then, e P Te = y p Te y Te. (9) 14 AUTOATIKA 43(2002) 1 2, 13 20

3 Nonlinear Internal-odel Control... To improve the system's perormance, the ilter must be employed. For the irst-order system as the subsystem (1), it can be selected as the ollowing orm x = a x + a e F F yte = xte Te F Te F Te E Te F Te P (10) where the superscript F denotes the ilter; a F Te is a positive real. An auxiliary variable is deined as m F Te = Te + Te Te y y y y (11) where y * Te is a desired trajectory that converges to a constant yte *. Our main control objective is to make the actual toque T p e tracks y * Te in spite o disturbances or plant-model mismatch. According to the nonlinear IC [8], the obtained control law can be written as F F F P k1 Te Te Te Te Te Te Te F a x + a e y + a y uk1 = Gk1 where a Te is a positive real. (12) 3.3 Nonlinear IC or stator current o phase- Similar to that o the total torque, let y p be the stator current output o phase- in the plant, i.e., y p = i p, and e P be the plant-model current output error. Then, P p e = y (13) y. Corresponding with subsystem (3), the ilter is selected as x F = a F x F + a F e P (14) F F y = x where a F is a positive real. The objective o the current controller is to orce residual current i p decay quickly to zero, besides achieve the stabilization o the stator current dynamics. When the desired decaying current trajectory is y * that conveys to zero, an auxiliary variable is introduced as F y = y + y y (15). On the basis o the nonlinear IC, the current control law o phase- can be calculated as F F F P a x + a e y + a y u = g where a is a positive real. (16) 4 PROPERTIES 4.1 Convergence For any positive real a F Te and a F, which are large enough, given a Te and a such that a Te > 0 and a > 0. Then, the tracking errors o both torque and current will converge exponentially to zero in the nonlinear IC-controlled SR drive, in spite o the disturbances or plant-model mismatch. The tracking error o torque is taken as an example to interpret them. As described in Section 3, the systems o the auxiliary variable y Te is linearized as y Te = a Te y Te. (17) oreover, the tracking error o torque is written as P P F 1 F e = y y = y + ( a ) x. (18) Te Te Te Te Te Te I, 1/a F Te 0 then e P Te y Te. Thus, the auxiliary variable y Te can be used to estimate the tracking error o the plant output torque. From (17) it can be obtained that y Te will converge exponentially to zero i a Te > 0. The same interpretation can be applied to the tracking error o current. 4.2 Eects o control parameters 1) The control parameter a Te has a direct eect on the dynamic, static perormances and robustness o the closed-loop system, and it can be tuned to provide a compromise between the perormances and robustness. No control action is taken in the limit as a Te 0. On the contrary, the IC controller (12) provides perect control in the limit as a Te. 2) I the model is perect, then the eect o the control parameter a Te on the closed-loop response is particularly simple, large value results in vigorous responses, while small value causes sluggish responses. I a Te > 0 and y * Te is bounded, the torque produced by the plant is bounded. 3) For the control parameter, the similar results can be obtained in current control o phase-. But it does not aect the dynamic and static perormance o the torque response because the torque controller can compensate the eects. 4.3 Equilibrium points o the system At the steady state, the stable equilibrium points o the closed-loop system are, respectively P F P P e e Te = Te Te Te Te ( T, T, x ) ( y, y e, e ) (19) AUTOATIKA 43(2002) 1 2,

4 Nonlinear Internal-odel Control... P F P P ( i, i, x ) = (0, e, e ). (20) It is clear that, e Te P will not be equal to zero when there exists the plant-model mismatch. But, owing to the SR's unipolar currents e P may equal to zero even i the plant and model is mismatched. The details are given as the ollows: 1) I the theoretical equilibrium point o current i is greater than or equal to zero, Then its actual equilibrium point is identical to the theoretical results. 2) I the theoretical equilibrium point o current i is less than zero, Then its actual equilibrium point will be equal to zero due to being limited by i 0. As a result, e P = 0, and only i decay to zero more quickly than i P. 4.4 Nonlinear IC system The schematic diagram o the nonlinear IC structure or SR is shown in Figure 2. The drive mainly consists o our sections, i.e., the actual SR and its power converter, SR model, the ilters, and the nonlinear IC controllers. The actual plant torque T P e is calculated by using the measured phase currents I P = [i P 1, i P 2,, i P m ] and rotor position θ. The measured position and calculated speed as the time-varying parameters are directly ed back to the SR model and the controllers. The SR model runs in parallel with the plant, its currents I = [i 1, i 2,, i m ] are ed back to the controllers. oreover, the SR model output torque T e and residual current i are compared with the corresponding outputs o the plant, and the dierences are ed back through the ilters, where x F = [x F Te, x F ] are the outputs o the ilters. 5 SIULATION RESULTS A SR drive controlled by the nonlinear IC controller as shown in Figure 2 has been simulated on an IB PC using the sotware package AT- LAB. The test motor is a our-phase 8/6-pole SR and the details o its parameters are provided in the Appendix. The model suggested by Ilic'-Spong [3], which takes magnetic saturation into account, is used in this work. When using the phase inductance L(θ) at the phase current i 0, unknown parameters a and b can be obtained, i.e., a = A 1, and b = A Dynamic and static perormances Figure 3 shows the dynamic responses o the torque developed by the plant when the nonlinear IC-controlled SR drive runs at a constant torque reerence o 40 N m. It can be seen that, the electromagnetic torque can quickly track the set point without any overshoot, and the compensation or the torque nonlinearity and torque ripple-ree have been achieved by using the nonlinear IC. The parameter a Te directly aects the dynamics o the torque response, and the torque response is aster with a larger a Te. Fig. 3 Dynamic responses o torque produced by the plant Fig. 2 Nonlinear IC scheme o SR drive Fig. 4 Current responses under the dierent a 16 AUTOATIKA 43(2002) 1 2, 13 20

5 Nonlinear Internal-odel Control... Figure 4 shows the current responses o the SR drive when the drive runs at the speed o 60 rad/s and the load torque o 30 N m, where θ on = = 0.9º and θ c = 2º. It is clear that, the residual current decay more quickly to zero or the larger a, however, the rate o rise o current is correspondingly increased at the beginning o the conduction. But, the choice o a does not aect the dynamic and static perormances o the torque response. It only aects the choice o turn-on angle θ on. 5.2 Robustness Robustness tests are carried out by varying the model parameters such as b (that is used to model the phase inductance), R, and ψ s rom their original values. Fig. 6 Torque responses o the proposed drive under varying model parameter ψ s Fig. 5 Torque responses o the proposed drive under varying model parameter b Figure 5 shows the torque responses o the nonlinear IC-controlled SR drive when the parameter b is varied by ±5 % rom its original value, while the other parameters are held constant. Figures 6 7 show the corresponding torque responses when the model's saturated lux linkage ψ s and phase resistance R are varied by ±20 % rom the original value, respectively. Figures 8 10 show their respective current responses corresponding to Figures 5 7. The SR drive runs at the speed o 60 rad/s and the load torque o 30 N m. Fig 7 Torque responses o the proposed drive under varying model parameter R AUTOATIKA 43(2002) 1 2,

6 Nonlinear Internal-odel Control... Fig. 8 Current responses under varying the model parameter b. (a) b = A 1, (b) b = A 1. Fig. 10 Current responses under varying the model parameter R. (a) R = Ω, (b) R = Ω Fig. 9 Current responses under varying the model parameter ψ s. (a) ψ s = Wb, (b) ψ s = Wb It can be seen in Figures 5 7 that the parameter variations have little eect on the torque produced by the plant although the torque ripple o the model gets signiicant, and the plant torque keeps the desired torque with very little ripple. It shows that the nonlinear IC controller is highly robust against the parameter variations. oreover, as shown in Figures 8 10 the residual current o phase-o in the SR model cannot always decay to zero when the model parameters are varied, and its equilibrium point is greater than or equal to zero. In the latter case, the residual current in the model decays more quickly than one o the plant. But, in any cases, the residual current o phase-o in the plant always decays quickly to zero. For example, in Figure 10 (a) a decreased R causes the model current higher than that o the plant. With the same decay rate, the plant current drops to zero irstly and ater that the current controller sets zero to the decay rate o the model current so that it keeps at a small value above zero even in the aterward steady state. In the opposition, in Figure 10 (b) an increased R makes the model current lower than the plant one, thus it decays to zero earlier. 18 AUTOATIKA 43(2002) 1 2, 13 20

7 Nonlinear Internal-odel Control... Anyway, the torque controller can eectively suppress the torque ripple. oreover, increasing the value o a Te can urther reduce the ripple o the plant torque. 6 CONCLUSIONS Based on the suitable commutation strategy proposed to compensate or the eects o the phase voltage saturation, the nonlinear IC was applied to the SR drive, and an eective SR control method has been proposed to improve the overall perormance o the drive. The important properties o the nonlinear IC-controlled SR drive such as its convergence, equilibrium points and eects o the control parameters on the system's perormance were analyzed in detail. The results show that a plant-model mismatch can be tolerated in the drive, and the plant output will converge exponentially to the reerence in spite o the disturbances or uncertainties. In addition, the torque control parameter has a direct eect on closed-loop perormance, and the large values result in vigorous responses, while the small values cause sluggish responses. However, the current control parameter does not aect the dynamics and static state o torque response, and it only achieves stability task and aects the choice o turn-on angle in the commutation strategy. The simulation tests carried out on a 7.5 kw our-phase SR controlled by the nonlinear IC have veriied our novel control solution and its properties. The results proved that the nonlinear IC has completely compensated or the nonlinear torque characteristics even i there is a plant-model mismatch. A dramatic reduction in torque ripple and the insensitivity o the responses o the nonlinear IC-controlled drive to parameter variations and disturbances have been achieved. Fig. 11 Torque and current responses o the proposed drive operating at 157 rad/s and N m Figure 11 shows the responses o the torque and the phase current to the increases o R by 20 %, b by 5 %, and ψ s by 10 % rom their original values that match those plant ones, where the control system operates at the nominal speed 157 rad/s and nominal torque N m. From 0.09 s to 0.1 s the model matches the plant perectly, aterward the model parameters increase. As a result, the model torque ripples are signiicant in Figure 11 (a). However, the plant torque keeps the balanced value with very small ripple limited to a band o ± 0.2 % about mean, as shown in Figure 11 (b). APPENDIX The parameters o SR: Rated power 7.5 kw Rated voltage 460 V Stator winding resistance 0.7 Ω Stator/rotor poles 8/6 Phases 4 Rated speed 157 rad/s Rated torque N m aximum inductance 110 mh inimum inductance 10 mh ACKNOWLEDGEENT The authors would like to thank the China Post-doctoral Science Foundation committee or its supports. AUTOATIKA 43(2002) 1 2,

8 Nonlinear Internal-odel Control... REFERENCES [1] J. Ish-Shalom, D. G. anzer, Commutation and Control o Step otors. Proc. 14 th Symp. Incremental otion, Contr. Syst. and Devices, pp , June [2] B. K. Bose, T. J. E. iller, P.. Szczesny, W. H. Bicknell, icrocomputer Control o Switched Reluctance otor. IEEE Trans. Ind. Applicat., vol. IA-22, no. 4, pp , Aug [3]. Ilic'-Spong, R. arino, S.. Peresada, D. G. Taylor, Feedback Linearizing Control o Switched Reluctance otors. IEEE Trans. Automat. Contr., vol. AC-32, no. 5, pp , ay [4] S. K. Panda, P. K. Dash, Application o Nonlinear Control to Switched Reluctance otors: a Feedback Linearisation Approach. IEE Proc.-Electr. Power Appl., vol. 143, no. 5, pp , Sept [5] H. Cailleux, B. Le Pioule, B. ulton, Comparison o Control Strategies to inimize the Torque Ripple o a Switched Reluctance achine. Electric achines and Power Systems, vol. 25, no. 10, pp , Dec [6] Y. Haiqing, S. K. Panda, L. Y. Chii, Perormance Comparison o Feedback Linearization Control with PI Control or Four-Quadrant Operation o Switched Reluctance otors. Proc. IEEE Appl. Power Electron. Con. and Expo. (APEC), pp , [7]. A. Henson, D. E. Seborg, An Internal odel Control Strategy or Nonlinear Systems. AIChE Journal, vol. 37, no. 7, pp , Jul [8] J. Alvarez, S. Zazueta, An Internal-odel Controller or a Class o Single-Input Single-Output Nonlinear Systems: Stability and Robustness. Dynamics and Control, vol. 8, no. 2, pp , Apr [9] R. S. Wallace, D. G. Taylor, A Balanced Commutator or Switched Reluctance otors to Reduce Torque Ripple. IEEE Trans. Power Electron., vol. 7, no. 4, pp , Oct [10] P. C. Kjaer, J. J. Gribble, T. J. E. iller, High-Grade Control o Switched Reluctance achines. IEEE Trans. Ind. Applicat., vol. 33, no. 6, pp , Nov./Dec Nelinearno upravljanje s unutarnjim modelom za pogon s prekida~kim reluktantnim motorom bez oscilacija momenta. Predlo`eno je i razra eno novo rje{enje za upravljanje sklopnim reluktantnim motorom (SR) zasnovano na nelinearnom upravljanju s unutarnjim modelom (IC) i prikladnoj strategiji komutacije. Strategija komutacije koristi deiniranu kriti~nu poziciju rotora kao to~ku komutacije {to doprinosi smanjenju ra~unskih zahtjevnosti. Shema za upravljanje naponom SR-a zasnovana na nelinearnom IC-u osigurava linearizaciju zatvorenog sustava i robusnost IC strukture {to rezultira ukupnom robusno{}u pogona bez oscilacija momenta unato~ nepodudaranju modela smetnji sa stvarnim smetnjama. Opisana su neka va`na svojstva ovoga na~ina upravljanja. Simulacijskim se rezultatima pokazuje visoka kvaliteta upravljanja SR-a. Klju~ne rije~i: nelinearno upravljanje s unutarnjim modelom, SR, minimizacija oscilacija momenta AUTHORS ADDRESSES: Dr. GE Baoming Electrical achinery Group, Department o Electrical Engineering, Tsinghua University, Beijing , P.R.China Pro. WANG Xiangheng Department o Electrical Engineering, Tsinghua University, Beijing , P.R.China Pro. JIANG Jingping College o Electrical Engineering, Zhejiang University, Huangzhou , P.R. China Received: AUTOATIKA 43(2002) 1 2, 13 20