Gain-scheduling control of the Switched Reluctance Motor

Control Engineering Practice 6 (1998) 181—189

Gain-scheduling control of the Switched Reluctance Motor W.K. Ho*, S.K. Panda, K.W. Lim, F.S. Huang Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Received May 1997; in revised form October 1997

Abstract Switched Reluctance Motor (SRM) drive technology has developed significantly over the last few years. The simplicity in both motor design and power converter requirement, along with the availability of high-frequency, high-power semiconductor devices, has made SRMs competitive with conventional adjustable-speed drive technologies. Control of an SRM is completely different from that of conventional AC or DC drives. Some advanced control techniques, such as variable structure and feedback linearization, have been applied, but they are not easily implementable and hence not widely used in industries. This paper presents an easily implementable gain-scheduling PI controller for the switched reluctance motor. A first-order model is derived for the SRM through a series of approximations. The model is useful for simple gain-scheduling control. The simplicity of the design makes it highly practical and implementable. The technique described in this paper was successfully implemented on a 400 W, 4-phase switched reluctance motor for a speed-tracking application. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: PID control; gain-scheduling control; switched reluctance motors; nonlinear systems; self-tuning control

1. Introduction Variable-speed drives play an important role in industrial automation. Good control helps to improve the quality as well as the quantity of the end product, and aids in conserving electric energy (Sen, 1990). The motors commonly used in variable-speed drive applications are induction motors, DC motors and synchronous motors. While DC motors and induction motors have dominated the field of adjustable speed drives for the past few decades, the availability of high-speed power semiconductor devices and digital controllers has brought motor technologies that were thought to be obsolete into the forefront. Doubly salient SRMs belong to this class of machines which have been revived in recent years (Lawrenson et al., 1986). While the DC motor and the induction motor do not need a power converter for their basic operation, the SRM needs a power converter that is able to switch the supply voltage to different phases, depending on rotor position, as the shaft rotates. Though the principle of SRM operation was known some 150 years ago (Miller, 1989), it is only recently that it could be

*Corresponding author. Tel: (65) 8746286; fax: (65) 7791103; e-mail: [email protected] 0967-0661/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved PII S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 0 1 2 - 4

practically and economically implemented. This has been made possible by advancements in power electronics technology. The availability of high-frequency switching devices, and improvements in machine design, have made SRMs competitive with conventional DC and induction motors in industrial and consumer applications. There is now a growing interest in the design, performance, control and application of switched reluctance motors (Lawrenson, 1989). Because of their simplicity and controllability, the motors are being evaluated for applications ranging from low-power, high-performance servo drives (Ray et al., 1985) to high-power traction drives. The SRM is competitive with conventional machines in that it is rugged and reliable, and has a high power capacity per unit mass and volume. However, the SRM operates in a highly nonlinear mode in order to maximize its output torque and hence the efficiency of the drive system. To date SRM drives have mostly been controlled by using proportional-integral (PI) speed/ position controllers tuned manually through trial and error. Furthermore due to the nonlinear characteristic of the motor, a single PI controller setting is insufficient to give good control over the whole range of operation. Several nonlinear control methods, such as feedback linearization (Illic’-Spong et al., 1987; Panda and Dash, 1996) and sliding-mode control (Rossi and Tonielli, 1994;

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Buja, 1993), have been applied to SRMs for position/speed control applications. These methods are complicated, and require an accurate dynamic model of the motor. A relatively simple and easily implementable control method is needed by industrial design engineers for controlling the speed/position of an SRM. This paper describes a set of first-order models for the highly nonlinear SRM, and presents an implementation of a well-known nonlinear control technique—gainscheduling of the PI controller (As stro~ m and Wittenmark, 1989; Hang et al., 1993). The simplicity of the design makes it highly practical and implementable.

2. The Switched Reluctance Motor Fig. 1 illustrates the cross section of a 4-phase SRM. Concentrated windings on the stator poles, along the same magnetic path, are connected in series to form a phase. The excitation is switched sequentially from phase to phase as the rotor moves. The electromagnetic torque in an SRM is produced by exploiting the rotorposition-dependent reluctance of the magnetic path associated with each phase. When a phase is energized, a reluctance torque, which tends to align the stator and rotor poles, is produced. 2.1. The model The mathematical model of the SRM is a set of differential equations which are obtained by using standard electric machine theory. The differential equations which describe the SRM are (Filicori et al., 1993) dW j"u !Ri , j j dt

(1)

dh "u, dt

(2)

du 1 " (¹!¹ ) l dt J

(3)

where u is the average DC voltage applied to the stator j terminals of the jth phase, W is the flux linkage of the jth j phase, R is the stator winding resistance, u is the angular velocity of the rotor, ¹ is the load torque, and J is the l total rotor and load inertia. For the motor considered, the following assumptions can be made (Filicori et al., 1993) d Magnetic linearity d The phases are almost completely magnetically decoupled. Because of the symmetrical construction of the SRM, it is sufficient to consider only one phase in the following analysis. With the assumptions above, the torque, ¹, is given by Miller (1989) 1 d¸ ¹" i2 2 dh

(4)

where ¸, i and h are the self-inductance, the instantaneous phase current of the motor and the rotor position respectively. The phase inductance ¸(h) can be expressed as (Illic’Spong, 1987) (5) ¸(h)"( Ma#b sin(N h)N s r where N and W are the number of rotor poles and the r s saturated flux linkage respectively. Since torque is a nonlinear function of current and position, the system described by Eqs. (1)—(5) is clearly a nonlinear dynamic system. It can be linearized to derive a first-order transfer function model. The relationship between the change in the angular velocity Du and the change in control signal Du can be given as Du(s) K p " Du(s) s¹ #1 p

(6)

where K "u6/u6 and ¹ "[ WJ bN cos(N h)]uN 3/u6 2. p p s r r The detailed derivation is given in the appendix. 2.2. Controller design Consider the PI controller given as

A

Fig. 1. Cross-section of a 4-phase SRM.

B

1 G (s)"K 1# (7) c c s¹ i where K and ¹ are the proportional gain and integral c i time respectively. Note that the proportional gain K and c the integral time ¹ , for this controller are functions of i K and ¹ , which in turn are functions of the operating p p point, determined by the average velocity, u6, and average control signal, uN . In practice, K and ¹ can be chosen as c i K uN K " 1"K (8) c K 1 u6 p

W.K. Ho et al. / Control Engineering Practice 6 (1998) 181—189

and u6 3 ¹ "K@ ¹ "K 2 p 2 uN 2 i

(9)

where K and K can be tuned experimentally on-line for 1 2 a particular motor and application. The goal here is to adjust the PI controller coefficients automatically, to obtain an approximately uniform closed-loop dynamic response over the entire operational range. 3. Experimental results 3.1. Experimental setup Experimental work was undertaken using the setup shown in Fig. 2. The corresponding SRM drive system is shown in block diagram form in Fig. 3. In this evaluation, the controller is implemented on a digital signal processor (DSP TMS320C50) with a sampling period of 1 ms for the speed loop. In addition to computing the control signal, the DSP serves as an interface to the position sensor, computes the velocity from position signals, implements the function of an electronic commutator and serves as a data logger. Note that the computational requirements of this algorithm are not significantly more than those for conventional fixed-gain PI control, particularly if it is noted that much of the on-line computation can be replaced by table lookup functions with appropriate discretisation. Thus the proposed algorithm can be realised using a conventional microprocessor-based con-

183

troller. However for the purpose of this experiment, a digital signal processor provides a convenient evaluation tool. The SRM used has the following parameters: Number of phases 4 Voltage 160 V DC Peak output 400 Watts Rated speed 3000 rpm Max. speed 15,000 rpm Rated torque 0.66 Nm Max. RMS current 3.0 Amps Terminal resistance 1.1 Ohm/phase Efficiency 80% Weight 5.5 LBS Encoder type HP HEDS-5000 Resolution 360 Line/rev. Commutation 24 per revolution or 6 per phase. In the experimental SRM, the data measured and provided by the manufacturers are N "6, W " r s 169 mVs, ¸ "28.961 mH and ¸ "3.621 mH. At aliga u ned position, h"15°; at unaligned position, h"!15°. So from Eq. (5), ¸ "W (a#b), a s ¸ "W (a!b). u s From the above two equations, a"0.0964, b"0.075.

Fig. 2. Photograph of the experimental setting for the SRM.

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Fig. 3. Block diagram of the control system.

The switching components used in the converter are fastspeed insulated gate bipolar transisters, which have higher current densities than comparable bipolar transistors, while at the same time having the simpler gatedrive requirements of the familiar power MOSFET. They provide substantial benefits to a host of higher-voltage, high-current applications. The gate-drive signals are implemented by using a high-voltage MOS gate driver integrated circuit chip, IR2110. As shown in Fig. 3, the gain-scheduled PI speed controller has a single input, i.e. the speed error, and a single output; that, is the reference current I . This reference 3%& current is then amplified by the bridge power converter and applied to various phase windings, depending on the electronic commutator output. At every sampled instant of time the electronic commutator gets its input signal; that is; the position of the rotor from the position sensor as shown in Fig. 3. Then, based on the position information, the electronic commutator decides which phase winding has to be excited with the current I . The phase 3%& on and off angles can be kept constant, or varied. In the experimental results presented in this paper the on and off angles are kept constant at the optimum on and off angle respectively. The optimum angles were obtained by trial and error. The on and off angles are maintained at the optimum values throughout. The current sensor is used to limit the motor phase current within the window of I "I #0.1I and I "I #0.1I respec.!9 3%& 3%& .*/ 3%& 3%& tively. The choice of the window size of $10% of I is chosen from a practical point of view, to limit the 3%& switching frequency of the power semiconductor devices to be less than 10 kHz. The rotor position h is measured using an encoder. Rotor position information is required to generate the appropriate commutation signals to the bridge converter. In addition, the time interval between en-

coder pulses is measured, to provide velocity feedback information. The voltage source to the chopper » is provided by dc a bridge diode rectifier from an AC mains supply. The fixed DC voltage is switched on and off repeatedly, in conjunction with the rotor position information and measured phase current, to produce an average DC voltage u, which is effectively applied across the phase winding. 3.2.

Experimental results

The gain-scheduling PI controller described in the previous section has been implemented on-line. Note that the proportional gain and the integral time for this controller are functions of the operating point, determined by an average velocity and average control signal. In practice, a moving average can be used. Fig. 4 shows a typical current waveform. When the phase is switched on, the current settles rapidly into an approximately constant mean value for a period of the order of 5 ms. The ripple seen in the figure is due to the chopping action of the switches, and occurs at a frequency an order of magnitude higher than the mechanical bandwidth. When the two power semiconductor devices S and S (as shown in Fig. 1) of a given winding 1 2 are switched on, the supply voltage directly appears across the phase winding, and the phase current starts to increase. When the phase current I where j"1, 2, 3, 4 j is more than I the top switch S as shown in .!9 1 Fig. 1 is switched off, while the bottom switch S is still 2 allowed to remain on. This causes the current to freewheel between the switch S and freewheeling diode D . 2 2 This causes the phase current to drop, and when it drops below I the top switch S is again switched on to .*/ 1 increase the phase current. This cycle is repeated until the phase-on period is over, and when that happens both the

W.K. Ho et al. / Control Engineering Practice 6 (1998) 181—189

185

Fig. 4. Current waveform.

Fig. 5. Step response of SRM.

switches S and S , are switched off and the phase cur1 2 rent is allowed to flow back to the source, resulting in regeneration. Fig. 5 shows a closed-loop step response for a drive controlled by this gain-scheduling PI controller when the reference speed changes from 200 rpm to 700 rpm. It is important in industrial applications that dynamic performance is preserved despite different load conditions. The theory in Section 2 has demonstrated that with minor assumptions, the dynamics of the SRM drive can be represented as a first-order model with operatingpoint-dependent gain and time constant. If a conventional fixed-parameter PI controller is tuned for this drive, the controller must be tuned so that the perfor-

mance is acceptable over a wide range of operating conditions. The simple gain-scheduling proposed in this paper reduces the need for that compromise. Fig. 6 compares load regulation performance. A fixed-gain PI controller is tuned for operation at no load and a velocity of 1000 rpm. The PI parameters were chosen as K "0.2 and c ¹ "0.3. The effects of applying the rated load, and its i subsequent removal, are shown in the figure. Overlaid on the same figure is the performance using a gain scheduled PI controller. The simple gain-scheduling proposed in this paper is capable of superior load-disturbance rejection. Initially, the drive system is operating at a constant speed of 1000 rpm and delivering only frictional and

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Fig. 6. Load torque disturbance response under gain-scheduling control and conventional fixed PI control.

Fig. 7. Step input speed responses of SRM under gain-scheduling control and conventional fixed PI control with no load.

windage losses. Suddenly, a load of rated value is applied to the drive. This causes a drop in the motor speed, and subsequently the speed controller is able to respond to the speed error and bring the error down to zero. Similarly, when the load torque is withdrawn, the motor speed shoots up and is eventually brought back to the reference value by the speed controller. The fall and rise in the speed due to torque disturbance is superior with the gain-scheduled PI controller than with the fixed-gain PI controller. For the gain-scheduling PI controller, K and K were chosen to give K "9.5/K "9.5 u6 /u6 1 2 c p and ¹ "0.78¹ "0.14 * 10~3 u63/u6 2. i p

Using the same fixed-gain PI and gain-scheduling PI controller, Fig. 7 compares the response to setpoint changes with no load, and Fig. 8 compares the response to setpoint changes at full load. The fixed-gain PI controller was tuned at 1000 rpm; at that operating point, it can be seen that its performance is very similar to that of the gain-scheduled controller. However when the velocity is changed to other operating points, the fixed-gain PI controller is unable to preserve the same transient response, whilst the gain-scheduled controller maintains a similar step response, despite the changes in the operating point.

W.K. Ho et al. / Control Engineering Practice 6 (1998) 181—189

187

Fig. 8. Step input speed responses of SRM under gain-scheduling control and conventional fixed PI control with full load.

Fig. 9. Sinusoidal input speed responses of SRM under gain-scheduling control and conventional fixed PI control with no load.

These results are also reflected in Figs. 9 and 10, which show the response at no load and at full load to a sinusoidal velocity reference. The fixed-gain PI controller has been tuned for a nominal model obtained at 1000 rpm. Clearly, the gain-scheduled PI controller shows better tracking performance.

4. Conclusions The SRM is the latest entrant to the expanding and increasingly competitive field of variable-speed drives.

The control of the SRM is completely different from that of conventional AC or DC motors, and necessitates the introduction of new concepts with regard to control variables and strategies. An important characteristic of the SRM is its inherent nonlinearity, such that a conventional fixed PI controller with only one set of parameters is not suitable for the whole range of operation. A firstorder model was derived for the SRM. A gain-scheduling PI controller was also designed and tested experimentally. The results confirm the improved performance of the gain-scheduling controller over the conventional fixed-gain PI controller. The proposed gain-scheduling

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Fig. 10. Sinusoidal input speed responses of SRM under gain-scheduling control and conventional fixed PI control with full load.

controller is very easy to implement in practical, realtime applications, making it suitable for industrial use.

Appendix: Linearized model of SRM Assuming magnetic linearity W"¸i (Miller, 1989), and considering only one phase, Eq. (1) can be written as dW di d¸ dh "¸ #i "u!Ri. dt dt dh dt

(A.1)

di dW "W Ma#b sin(N h)N s r dt dt #iN W bu cos(N h) "u!Ri. (A.2) r s r Substituting Eq. (5) and Eq. (4) into Eq. (3), gives

C

D

u i" (A.5) W bN u cos(N h) s r r where cos(N h) is the average value of cos(N h) during r r the active period. Here, it is assumed that R is such that the voltage drop Ri is negligible compared to the average voltage. Substituting Eq. (A.5) into Eq. (A.3) gives

C

D

du 1 1 u2 " !¹ ¢g. (A.6) l dt J 2 W bN u2 cos(N h) s r r Linearizing around operating point u6 and uN gives

Substituting Eq. (5) into Eq. (A.1), gives

du 1 1 " i2N W b cos(N h)!¹ . r s r l dt J 2

state given by

(A.3)

du dg "gDu6, u6 # dt du

For a 4-phase SRM operating on a one-phase-on scheme, each phase is active for a quarter of a cycle. The current dynamics are much faster than the dynamics associated with the mechanical state variables, and the inner current control loop is at least 5 times faster than the outer speed loop. Consequently, shortly after each voltage pulse is applied, the current i reaches a steady

K

uN ,u6

) Du

(A.7)

where

K

dg du

Rearranging Eq. (A.2), gives di 1 " Mu!Ri!iW N bu cos(N h)N. s r r dt W Ma#b sin(N h)N s r (A.4)

K

dg ) Du# du u6,u6

u6,u6

K

!2u2u~3 " 2JW bN cos(N h) uN ,u6 s r r uN 2 "! JW bN u6 3 cos(N h) s r r

(A.8)

and

K

uN " . uN ,uN JW bN u6 2 cos(N h) s r r At steady state,

dg du

du gDuN ,uN " dt

K

C

(A.9)

D

uN 2 1 1 " !¹ "0. l 2 J uN ,uN W bN u6 2 cos(N h) s r r (A.10)

W.K. Ho et al. / Control Engineering Practice 6 (1998) 181—189

Therefore the steady state u6 is given by

S

u6 "

References

uN 2

. (A.11) 2W bN ¹ cos(N h) s r l r Substituting Eq. (A.11) into Eq. (A.8) and Eq. (A.9) gives

dg du dg du

K

K

S

2¹ "! l J uN , uN

2W bN ¹ cos(N h) s r l r uN 2

2¹ " l. JuN uN ,uN

189

(A.12) (A.13)

Substituting Eq. (A.12) and Eq. (A.13) into Eq. (A.7) gives du 2¹ 2¹ "! l J2W bN ¹ cos(N h) ) Du# l ) Du. s r l r JuN dt JuN (A.14) Taking the Laplace Transform gives 2¹ l JuN Du(s) " 2¹ Du(s) s# l J2W bN ¹ cos(N h) s r l r JuN u6 uN " u6 3 s) [JW bN cos(N h)]#1 s r r uN 2 K p " (A.15) s¹ #1 P where K "u6/u6 and ¹ "[JW bN cos(N h)]uN 3/u6 2. p p s r r

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