Automatic flux-weakening control of permanent magnet synchronous motors using a reduced-order controller

Automatic Flux-Weakening Control of Permanent

Magnet Synchronous Motors Using a Reduced-Order

Controller

Jiunn-Jiang Chen and Kan-Ping Chin, Member, IEEE

Abstract—This study presents a novel means of designing a simple and effective position and velocity controller for perma-nent magnet synchronous motors (PMSM). In contrast to the conventional two-loop control methods with full-state feedback, the proposed controller does not need current information of the motor for feedback purposes. However, under normal operation the steady-state -axis current can still be controlled to zero to minimize power dissipation. In addition, implementing a simple overmodulation strategy allows the controller to automatically generate a flux-weakening control to expand the range of op-erating speed when voltage saturation occurs. In addition to not depending on system parameters used by the controller, the automatically generated demagnetizing current is also optimal in the sense of minimum power dissipation that differs from the maximum output torque design or the constant power design of the general flux-weakening control methods. Simulation and experimental results show that the controller can achieve an effective speed and position control with near-minimum power dissipation, even when voltage saturation occurs.

Index Terms—Feedback, flux-weakening control methods, PMSM, torque.

I. INTRODUCTION

R

ECENT advances in power electronics and high-speed mi-croprocessors have led to considerable attention in ap-plying nonlinear control theory to electronically-commutated alternate current (ac) electrical machines. In general, the math-ematical model of an ac motor consists of coupled high-order nonlinear ordinary differential equations representing the dy-namics of electrical and mechanical subsystems. Hence, a fully digitally controlled ac motor is a multiinput nonlinear system where the inputs are the phase voltages and the outputs are the position, the velocity or the torque at the rotor shaft. Recent developments in nonlinear system analyzes and control tech-nologies suggest that the controllers for electrical motors should be designed directly from nonlinear models. To simplify con-troller design, the two-loop control method [1], as shown in Fig. 1, is the most frequently applied in small electrical ma-chine control. Sliding mode control has also been applied to the control of electrical machines, such as induction motors [2] and synchronous motors [3]. More recently, feedback lineariza-tion techniques have been extensively applied to the control of

Manuscript received October 25, 1999; revised May 30, 2000. Recommended by Associate Editor J. Ojo.

The authors are with the Department of Mechanical Engineering, Na-tional Chiao-Tung University, Hsinchu 30010, Taiwan R.O.C. (e-mail: [email protected]).

Publisher Item Identifier S 0885-8993(00)07313-0.

Fig. 1. Block diagram of a two-loop control system for the PMSM. variable reluctance motors [4], permanent magnet synchronous motors (PMSM) [5], induction motors [6], and hybrid motors [7], [8]. Other investigations have applied adaptive control with feedback linearization terms in various motors [9]–[11]. Al-though these nonlinear control technologies can be used for vari-able speed control design, intensive computations and full-state feedback, including position, velocity and phase currents, are required for such implementation.

Another relevant topic of motor controls is how to extend the speed ranges. As it rises with an increasing speed, the back emf cancels a portion of the input voltages and limits the injected currents, thus restricting the maximum speed of the motor to a certain level. To extend the range of operational speed of the motor, many authors have proposed different algorithms to accomplish flux-weakening control [12]–[17]. Flux-weakening control of PMSM’s employs a negative -axis current (demag-netizing current) to suppress the back emf. References [12]–[15] proposed several flux-weakening controls of PMSM’s. In these methods, a -axis current command is determined to achieve maximum torque output or constant power output. However, these methods are extremely sensitive to the accuracy of the motor parameters used in controllers. References [16], [17] presented flux-weakening control methods that do not require knowledge of the load torque or motor parameters. A method is that in which the -axis current command is proportional to the filtered -axis current error [16]. Another method is that in which the -axis current command is adjusted by taking a PI control of the voltage error between the saturated voltage and the output voltage command of current controller [17]. However, for both of these flux-weakening control methods, the control stability in flux-weakening regions always depends on the choices of the control gains which are used to tune the -axis current command. Consequently, how to select control gains is of primary concern when implementing these two flux-weakening control methods.

In this study, we present a nonlinear reduced-order model for a PMSM by approximating the original full-order model using 0885–8993/00$10.00 © 2000 IEEE

Fig. 2. Block diagram of a reduced-order control system for the PMSM.

a singular perturbation method [18], [19]. The motivation be-hind doing so is to exploit the two-time-scale properties of small electrical machines where the mechanical subsystem is signifi-cantly slower than the electrical subsystem. Under such circum-stances, the mechanical variables in the electrical subsystem are regarded as frozen and the electrical variables in the mechanical subsystem are regarded as capable of instantaneously arriving at their quasisteady-state. Consequently, the electrical variables can be expressed as functions of the mechanical variables; a re-duced-order dynamic model that only consists of the mechan-ical variables is derived as well. The resulting reduced model of the PMSM is accurate enough to capture the dominant dy-namic behavior of the PMSM, and can be used to develop effec-tive nonlinear PMSM reduced-order controllers. Fig. 2 schemat-ically depicts a reduced-order control system.

Because this reduced-order model is a second-order non-linear system, many standard nonnon-linear and adaptive control techniques can be applied to the design of the controllers. Moreover, with this control methodology, the current measure-ment, which is sensitive to noise corruption and required in any full-state feedback controller, can be eliminated. Although phase currents are not measured for control purposes, the steady-state -axis current can still be indirectly controlled to zero under normal operation to achieve near-minimum power dissipation. Furthermore, implementing an overmodulation strategy allows the controller to automatically generate a demagnetizing current to expand the speed range when input voltage saturation occurs. This flux weakening control is also optimal in the sense of minimum power dissipation, and differs from the maximum output torque design or the constant power design in the general flux-weakening control methods [12]–[15]. Stability analyzes based on the Lyapunov’s linearization method indicate that, to stabilize the closed-loop system, the control gains in the reduced controller must be both upper and lower bounded. In addition, when voltage saturation occurs, the closed-loop system is still stable, as long as the degree of saturation is not too much. Simulation and experiments are performed to verify the performance of the proposed controller. According to those results, the proposed controller can achieve an effective position and velocity control with near-minimum power dissipation, even when voltage saturation occurs.

The rest of this paper is organized as follows. Section II presents the mathematical model of a PMSM. In Section III, we derive a reduced model using the singular perturbation method. The reduced controller based on this reduced model, as well as the overmodulation strategy that we have adopted is also de-scribed. Section IV describes the stability analyzes. Sections V and VI summarize the simulation and experimental results, respectively. Conclusions are finally made in Section VII.

II. MOTORMODEL

The mathematical model of a surface permanent magnet synchronous motor (SPMSM) in a synchronous frame, or the so-called frame, can be described as follows:

sgn (1)

(2) where

mechanical state vector whose components are the rotor position and rotor velocity, re-spectively;

electrical state vector in the frame, and the components of are the direct and the quadrature-axis stator currents respectively; input vector whose components are input voltages in the frame;

phase inductance; phase resistance; number of pole pair;

torque (permanent magnet) constant; rotor inertia;

viscous damping coefficient; Coulomb friction coefficient.

Owing to that this study focuses on achieving position or ve-locity controls without using the current information, the input voltage is constrained to a function of time and the mechanical

states, .

III. CONTROLSTRATEGIES A. Reduced Order Modeling

The two-time-scale phenomenon of small electrical machines is manifested by the fact that the time constant of the elec-trical subsystem, , is significantly smaller than that of the mechanical subsystem, . Hence, assigning a small positive constant allows us to rewrite the motor model (1), (2) into a standard singular perturbation model [18], [19] by multi-plying to both sides of the electrical subsystem (2)

sgn (3)

(4)

Notably, in (4), although is a small number, the term may not be a small one when is large. Therefore, cannot be considered as a negligible parasitic term. Suppose the fast subsystem (4) reaches its quasisteady-state instantaneously when the electrical time constant is significantly smaller than the mechanical time constant. The

quasisteady-state of the electrical variables can be determined by substituting in (4)

(5)

where , and and are the

quasisteady-state values of and , respectively. By replacing in (3) with its quasisteady-state , a reduced model of the full system (3), (4) is obtained

sgn (6)

The reduced model (6) is described by a set of nonlinear second-order differential equations in which the states are the same as the states of the mechanical subsystem in the full mode (3), (4). For further development, we let , , and , where and are the desired position and the desired velocity respectively.

B. Control Law Design

Let be the desired steady-state -axis current. The relation-ship between and that produces this steady-state -axis current can be solved from (5) as follows:

(7) By substituting (7) into (5), becomes

(8) Furthermore, the reduced model (6) can be rewritten as the fol-lowing form by substituting (8) into (6):

sgn (9) Based on (9), a feedback linearization control law can be chosen as follows:

sgn (10)

where , , and , and

are control gains. By substituting the control laws (10) into the reduced model (9), the closed-loop error dynamics becomes

(11)

By selecting the control gains as , , and , we can assign the eigen-values of the closed-loop error dynamics (11) to , and . Consequently, the equilibrium point of the reduced model (6) is globally exponentially stable under the control law (7) and (10), as long as the control gains, , or equivalently, the real part of the values, , are all positive.

Fig. 3. Adopted overmodulation strategy. C. Overmodulation Strategy

Overmodulation occurs when the magnitude of the command voltage is greater than the saturated voltage, which is the max-imum magnitude of the output voltage of an inverter, and is lim-ited by the inverter dc link voltage. The inverter dc link voltage is denoted as . An overmodulation strategy determines the actual output voltage vector when overmodulation occurs [20]. In this study, the overmodulation strategy is chosen so that when the magnitude of the command voltage vector exceeds the sat-urated voltage, the magnitude of the command voltage vector is shrunk to be equal to the saturated voltage; meanwhile, the angle of the command voltage vector remains unchanged. The overmodulation strategy can be described as follows:

(12) (13) where

(14) is a scaling factor for the output voltages when the saturation oc-curs, denotes the magnitude of the saturated voltage, and and represent the actual output voltages after scaling. Fig. 3 illustrates the adopted overmodulation strategy. Theoretically, the boundary of all saturated voltage vectors of a PWM inverter will form a hexagon. In this study, the hexagon is replaced by a circle as the boundary of all saturated voltage vectors for conve-nience. Under such a condition, the relation between the actual output voltage and the saturated voltage can be described by the following equation:

(15) where the magnitude of the saturated voltage is a constant value and can be expressed as .

IV. STABILITYANALYSES

In the previous section, control laws (7) and (10) are designed to stabilize the reduced model, which assumes that the fast (elec-trical) subsystem reaches its quasisteady-state instantaneously. Nevertheless, (7) and (10) will also stabilize the original full system as demonstrated hereinafter. In this section, we analyze the stability of the closed-loop full system under both normal operating condition and voltage saturation condition using the Lyapunov’s linearization method.

TABLE I

THE RATINGS AND IDENTIFIED

PARAMETERS OF THESINANO#7CB30-2SE6F MOTOR

A. Normal Operation

Because the -axis current, , does not generate torque in a SPMSM system, it is commanded to become zero under normal operating conditions to reduce the power dissipation from the winding resistance. However, because is not a state in the reduced model, to achieve near minimum power dissipation in this study, we will let instead. By substituting (7) and (10) with and be a constant into the original model (1) and (2), the closed-loop error dynamics of the full system becomes

(16)

where and . The system described

by (16) is an autonomous system, and its origin is a unique equilibrium point. Consequently, the asymptotic stability of (16) also indicates the control goals, , , and

, are achieved. The local stability of the system (16) can be analyzed by linearizing (16) around its equilibrium point. The parameters of the motor listed in Table I are employed in the following eigenvalue analyzes. Fig. 4(a)–(c) illustrate the loci of eigenvalues of (16) by varying from 10 to 4000 rpm. On the other hand, the equivalent control gains are chosen to

have equal values, , at , , and

for Fig. 4(a)–(c), respectively. One set of two loci on the left-hand side of Fig. 4(a) consists of eigenvalues of the elec-trical subsystem, and the other set of three loci on the right-hand side of Fig. 4(a) consists of eigenvalues of the mechanical sub-system. In Fig. 4(b) and (c), where the equivalent control gains are increased to and , respectively, the set of

Fig. 4. Loci of eigenvalues when! = 10  4000 rpm and (a)  =  =

 = 5 2 2, (b)  =  =  = 35 2 2, and (c)  =  =  = 60 2 2.

loci representing eigenvalues of the electrical subsystem moves toward the imaginary axis and to the right of the set of loci repre-senting eigenvalues of the mechanical subsystem. Furthermore, when the equivalent control gains are further increased to above a certain value, the set of loci representing eigenvalues of the electrical system moves into the right-half plane and the system becomes unstable. From this, we can infer that the control gains in (10), which was developed from the reduced model, not only needs to be lower-bounded, but also upper-bounded to stabilize the full model.

B. Voltage Saturation Operation

During the control of an electrical motor, the back emf, which is proportional to the speed of the motor, cancels the input voltage and restricts the motor speed to a certain value corresponding to a given input voltage level. When the input voltage reaches the saturation voltage of the inverter, the motor reaches its maximum speed unless a demagnetizing current is generated to achieve the so-called flux-weakening control. In terms of speed control, the motor may even become unstable if the flux-weakening control is not implemented and the desired motor speed requires an input voltage level that exceeds the saturation voltage. In this study, because the current is only indirectly controlled in the reduced controller, what happens to the overall closed-loop system when the voltage saturation occurs is of interest. Let be zero and be kept at a constant speed as before. When the voltage saturation occurs, the actual output voltages are expressed as in (12) and (13). By substituting (12), (13) and the control law (7) and (10) into the original full model (1) and (2), the closed-loop error dynamics becomes

sgn (17)

where , is a constant, and

is the scaling factor at the equilibrium point of (17). By translating the coordinate axis of the system (16) to in (17), the equilibrium point is moved to the origin of (17). Con-sequently, the asymptotic stability of (17) also indicates the

con-trol goals, , , and , are achieved.

As in the previous section, the system described by (17) is lin-earized around its equilibrium point to analyze the stability of (17). Fig. 5 illustrates the loci of eigenvalues of (17) by setting the three equivalent control gains at , and by varying from 10–4000 rpm. Moreover, Fig. 5(a) and (b) illustrate the loci of eigenvalues corresponding to the condi-tions when the scaling factor at equilibrium point is

and , respectively. Notably, the loci of eigenvalues plotted in Fig. 4(b) are equivalent to the loci of eigenvalues when . These figures reveal that, when the value of de-creases, the set of loci representing eigenvalues of the mechan-ical subsystem moves to the right of the pole plane. These re-sults indicate that the closed-loop system operating under the voltage saturation region is stable as long as the degree of sat-uration is not too much. Moreover, following the loci of eigen-values move toward the imaginary axis, the settling time of the

Fig. 5. Loci of eigenvalues when! = 10  4000 rpm,  =  =  =

35 2 2: (a)  = 0:5 and (b)  = 0:3.

responses under voltage saturation operation is longer than that under normal operations. The longer settling time implies that the controller under voltage saturation operation has weaker abilities of trajectory tracking and disturbance rejection. Based on these analysis results, an additional control strategy is pro-posed in the Experimental Section to reduce these drawbacks.

Also of relevant interest is how voltage saturation affects the values of the currents. By substituting the overmodulation rules (12) and (13) into (5), the quasisteady-state -axis current under voltage saturation, , becomes

(18) Because is always less than zero when , it is a de-magnetizing current that realizes flux-weakening control auto-matically when voltage saturation occurs. Moreover, the scaling factor at equilibrium can be derived from (15) as

(19)

where

Consequently, by substituting (19) into (18), at equilib-rium can be expressed as

(20)

where .

The quasisteady-state -axis current in (20), , as gen-erated by our reduced controller when voltage saturation occurs, is an optimal that satisfies both the loading condition in the mechanical subsystem and the saturated voltage constraint in the inverter. To prove this result, we consider the full model (1) and (2) operating at constant speed and constant currents

sgn (21)

(22) and solve the following problem [21]:

maximize

subject to

sgn

According to our results, the two extremums are on the boundary of saturated voltage constraint, and the optimal value of is derived as follows:

(23) Because in (23) is equal to in (20), we can con-clude that the reduced controller proposed herein automatically becomes a flux-weakening controller when voltage saturation occurs. In addition, the generated steady-state demagnetizing current, , is also optimal in the sense of minimum power dissipation.

The effect of , as generated by the reduced controller when saturation occurs, can be further explained through a current figure. Consider the two sets of equations (21) and (22). For a given set of electrical parameters, ( ), and input conditions, ( , ), only a unique set of currents, ( , ), satisfies (22) when the motor speed is fixed at a constant. However, when the motor speed is sufficiently high such that voltage saturation occurs and , the solutions of (22) can be plotted as a parabola, as shown in Fig. 6 which employs the motor parameters listed in Table I. The shaded area under the parabola in Fig. 6 denotes all solutions of (22) that satisfy the voltage constraint . Notably,

Fig. 6. Voltage-limit curve for the PMSM.

the shaded area in Fig. 6 is generated from the model of the electrical subsystem (22) under the voltage constraint alone. On the other hand, the model of the mechanical subsystem (21) dictates that is a constant. Hence, the solutions of (21) and (22) can be plotted as a segment of a straight line, AB, in Fig. 6. The intersections of this line segment and the parabola at point A and point B depict the solutions of (21) and (22) when the voltage saturation occurs. From the results of (20) and (23) we can infer that when voltage saturation occurs continuously, control law (7), (10) combined with the adopted overmodulation strategy (12), (13) forces the operation point to stop at point A in Fig. 6 although the is set at zero. Furthermore, the value of point A in Fig. 6 depends only on the motor parameters and the saturated voltage. As a result, we can infer that under the prerequisite of stable control neither the parameters nor the gains in the controllers can affect the value of point A in Fig. 6.

V. SIMULATIONRESULTS

The performance of the proposed controller is first verified through velocity tracking simulations. The parameters of the model of the simulated motor are identified from the motor used in the experimentation in the next section. Table I lists the pa-rameters. Simulations are performed under four different con-ditions.

i) An ideal normal operating condition assumes that the motor parameters are identified correctly, and the input voltage is not saturated.

ii) Although an ideal operating condition assumes that the motor parameters are identified correctly, the input voltage is saturated in a high-speed region.

iii) Inexact motor parameters are used in the controller to examine the robustness of parameter uncertainty of the controller. In this case, the inertia, , the winding resis-tance, , and the motor constant, , in the controller are chosen to be 50%, 50% and 105%, respectively, of the values used in the simulated motor. In addition, this sim-ulation assumes that the motor is under normal operation and the input voltage is not saturated.

iv) The parameters in the controller are the same as in iii). However, the input voltage is saturated in a high-speed region.

The dc link voltage is set at 180 V when the simulation con-ditions dictate that voltage saturation should not occur, and at

Fig. 7. Trajectory of the velocity command.

Fig. 8. Results of the tracking simulation with the reduced-order controller which contains exact motor parameters. dc link voltage isV = 180 V. (a) position error, (b) velocity error, (c)d-axis current, and (d) q-axis current.

140 V for the simulations that voltage saturation is desired in a high-speed region. When the dc link voltage is set at 140 V, overmodulation continuously occurs at a high speed (exceeding 3311 rpm) for this simulated PMSM. The velocity command, as illustrated in Fig. 7, is a composite trapezoidal trajectory with two constant speed regions: one at 4000 rpm and the other at 3000 rpm. The position command is the integration of the velocity command. In all the simulations, the sampling rate is 5 kHz, and the three equivalent control gains are chosen to have

equal values at .

Figs. 8–11 summarize the simulation results under the above four conditions. According to Figs. 8(a) and (b) and 10(a) and (b), the position error and the velocity error have converged to zero under normal operation. However, the response of in these simulations must be more closely examined. The value of when its dynamics disappears is equivalent to the quasis-teady-state -axis current . According to (5) and (7),

as long as the parameters in the controller are equivalent to the real electrical parameters of the motor. This finding correlates with the simulation result in Fig. 8(c), where the controller parameters are assumed to be correct. Also,

Fig. 9. Results of the tracking simulation with the reduced-order controller which contains exact motor parameters. dc link voltage isV = 140 V. (a) position error, (b) velocity error, (c)d-axis current and estimated d-axis current, and (d)q-axis current and estimated q-axis current.

Fig. 10. Results of the tracking simulation with the reduced-order controller which contains inexact motor parameters. dc link voltage isV = 180 V. (a) position error, (b) velocity error, (c)d-axis current, and (d) q-axis current.

strays from zero as illustrated in Fig. 10(c), where the elec-trical parameters used in the controller are inexact. Additional sensitivity analysis, as shown in Fig. 12, indicates that among the three electrical parameters, is the most sensitive to the magnet constant . Consequently, if high precision control of the steady-state -axis current is demanded, accurate identifi-cation of the electrical parameters, especially the magnet con-stant , is required before implementing the proposed control. Nevertheless, according to [22], the magnet constant can be

Fig. 11. Results of the tracking simulation with the reduced-order controller which contains inexact motor parameters. dc link voltage isV = 140 V. (a) position error, (b) velocity error, (c)d-axis current, and (d) q-axis current.

Fig. 12. Sensitivity ofi with respect to electrical parameters.

identified the most accurately among the three electrical param-eters by using the batch least-square method. Additionally, if precise electrical parameters are given, the -axis and the -axis currents can be estimated by constructing a simple current esti-mator from (5):

(24)

where represents the index of the sampling sequence, and and are the estimated values of and , respectively. Fig. 9(c) and (d) illustrate the simulation results of this current estimator. It is possible to use these estimated currents as a soft-ware indicator for over current in the motor and the driver.

Fig. 13. Results of the tracking experiment with the reduced-order controller, whose steady-stated-axis current command is i = 0. dc link voltage is V =

180 V. (a) position error, (b) velocity error, (c) d-axis current, and (d) q-axis

current.

When voltage saturation occurs, the position error and the ve-locity error also converge to zero, as shown in Figs. 9(a) and (b) and 11(a) and (b). However, the transient errors are larger than that at normal operation. Fig. 9(c) reveal that, although the -axis current command is zero, decreases to a nega-tive value automatically at high speed and returns to zero at low speed. Moreover, by comparing at the voltage saturation re-gion in Figs. 9(c) and 11(c), the two negative steady-state , or the so called demagnetizing currents, have the same value. This finding implies that the steady-state demagnetizing current under our control law is independent of the system parameters used in the controller.

Although this study is focused on SPMSM’s, additional sim-ulations indicate that, by using the proposed control method, we are able to achieve automatic flux-weakening control for inte-rior permanent magnet synchronous motors (IPMSM’s) as well.

VI. EXPERIMENTALRESULTS

The experimental setup includes a Sinano #7CB30-2SE6F permanent magnet synchronous motor, the power stage of a Micro Trend UT90 driver, a proprietary control card made in-house, and a PC. The control card converts the analog phase current measurements into digital signals, decodes the encoder signals, and generates space vector pulse width modulation (SVPWM) switching signals to control the power stage. The PC is then used to compute the control algorithms and the coordinate transformations among the vector space, the stator reference frame, and the rotor reference frame. Notably, the current measurements herein are only used to monitor the current responses. They are not used for control purposes. The experimental conditions resemble those in the tracking simulation, except that the identified motor parameters, as shown in Table I, are used in the controllers. Figs. 13 and 14

Fig. 14. Results of the tracking experiment with the reduced-order controller, whose steady-stated-axis current command is i = 0. dc link voltage is V =

140 V. (a) position error, (b) velocity error, (c) d-axis current, and (d) q-axis

current.

summarize the experimental results under dc link voltage that is set at 180 V and 140 V, respectively. Under normal operation, both the position error and the velocity error converge to zero, and is also approximately zero, as exhibited in Fig. 13. When voltage saturation occurs, Fig. 14(c) indicates that the phenomenon of auto-flux-weakening indeed appears in the experiment. Consequently, both the position error in Fig. 14(a) and the velocity error in Fig. 14(b) can still converge to zero at voltage saturation operation. However, the transient errors are larger than that at normal operation. This phenomenon has been predicted from the eigenvalue analysis for voltage saturation operation in Fig. 5. According to that analysis, reducing the degree of voltage saturation can reduce the transient errors and increase the control robustness in voltage saturation region. The fact that approaches its optimal value automatically whenever voltage saturation occurs allows us to modify the command current in (7) to reduce the degree of voltage saturation. As a result, we set according to the following rules:

if (25)

where denotes a positive overmodulation tuning gain, and the initial value is set to zero. In (25), is modified as the integration of the degree of overmodulation when voltage satu-ration occurs. Consequently, varies until voltage saturation stops, or equivalently, controller returns to normal operation. Fig. 15 presents the new experimental results with the modified control law. Apparently, the control performance is improved.

Since phase currents are not measured in this study, the pro-tection of the motor and the driver from overloading must be

Fig. 15. Results of the tracking experiment with the reduced-order controller, whose steady-stated-axis current command is the auto-adjusting command. dc link voltage isV = 140 V. (a) position error, (b) velocity error, (c) d-axis current, and (d)q-axis current.

implemented using other methods. For example, the current es-timator (24) can be used as a software indicator for over cur-rent if the motor parameters are accurately known. Other hard-ware-oriented methods include the installation of a simple pro-tection device or circuit, e.g., fuse, or a current-sensing resis-tance, which is much cheaper than a Hall current sensor, in the dc link bus. This idea comes form the fact that the cur-rent flowing in the dc link bus at any time is always the max-imum phase current among the three phase windings when they are connected in a wye configuration. Consequently, there is no need to measure the current in the individual winding.

VII. CONCLUSION

This study applies the singular perturbation method to design a position and velocity controller for a SPMSM control system. The controller proposed herein is computationally simple and does not require the measurement of current signals for the feed-back purposes. Consequently, the cost of the motor driver can be reduced. On the other hand, although this controller is designed without current-loop control, the steady-state current can still be demanded indirectly under normal operation to reduce the copper loss. Moreover, this controller automatically gener-ates a flux-weakening control to follow the velocity command when the voltage saturation occurs, and the demagnetizing cur-rent is optimal in the sense of minimum power dissipation. As a result, this controller can always achieve near-minimum power dissipation both during normal operation and voltage saturation operation. Simulation and experimental results show that the controller can achieve effective variable-speed and position con-trol with near-minimum power dissipation, even when voltage saturation occurs.

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Jiunn-Jiang Chen was born in Kaohsiung, Taiwan,

R.O.C., on May 14, 1972. He received the B.S. de-gree from National Chiao-Tung University, Hsinchu, Taiwan, in 1995, where he is currently pursuing the Ph.D. degree.

His research interests include power electronics and high-performance electrical machine control.

Kan-Ping Chin (S’89–M’92) was born in Taipei,

Taiwan, R.O.C. He received the B.S. degree in mechanical engineering from National Taiwan University, Taipei, in 1982 and the S.M. and the Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1988 and 1991, respectively.

He is now an Associate Professor at the National Chiao-Tung University, Hsinchu. His research inter-ests include control of servo motors and the micro-electromechanical systems (micromotors).