Experimental Nonlinear Torque Control of a Permanent ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 44, NO. 5, OCTOBER 1997

Experimental Nonlinear Torque Control of a Permanent-Magnet Synchronous Motor Using Saliency Damien Grenier, L.-A. Dessaint, Senior Member, IEEE, Ouassima Akhrif, Member, IEEE, Yvan Bonnassieux, and Bruno Le Pioufle, Member, IEEE

Abstract— In this paper, a new nonlinear control strategy is proposed for a permanent-magnet salient-pole synchronous motor. This control strategy simultaneously achieves accurate torque control and copper losses minimization without recurring to an internal current loop nor to any feedforward compensation. It takes advantage of the rotor saliency by allowing the current (id ) to have nonzero values. This, in turn, allows us to increase the power factor of the machine and to raise the maximum admissible torque. We apply input–output linearization techniques where the inputs are the stator voltages and the outputs are the torque and a judiciously chosen new output. This new output insures a well-defined relative degree and is linked to the copper losses in such a way that, when forced to zero, it leads to maximum machine efficiency. The performance of our nonlinear controller is demonstrated by a real-time implementation using a digital signal processor (DSP) chip on a permanent-magnet salient-pole synchronous motor with sinusoidal flux distribution. The results are compared to the ones obtained with a scheme which forces the id current to zero. Index Terms—Copper losses, input–output linearization, minimization, nonlinear control, salient-pole ac motors.

I. INTRODUCTION

P

ERMANENT-MAGNET (PM) synchronous motors are progressively replacing dc motors in applications that require variable-speed drives. While relatively easy to control, the separately excited dc motor does, indeed, present some inherent drawbacks, such as the mechanical wear of its brushes (particularly for high-speed motion), its relatively low torqueto-inertia ratio, and the fact that the heat produced by the copper losses in the rotor windings must be dissipated through its stator. The PM synchronous motor, on the other hand, offers several advantages for applications which require high

Manuscript received April 19, 1996; revised April 14, 1997. D. Grenier is with the Laboratoire d’Electrotechnique et d’Instrumentation (LEI), Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium. L.-A. Dessaint and O. Akhrif are with the Groupe de Recherche en ´ ´ ´ Electronique de Puissance et Commande Industrielle (GREPCI), Ecole de Technologie Supérieure, Montréal, P.Q., H2T 2C8 Canada. ´ Y. Bonnassieux is with the Laboratoire d’Electricit´ e, Signaux et Robotique ´ ´ (LESiR) U.R.A. C.N.R.S. D1375, Ecole Normale Supérieure de Cachan, 94235 Cachan, France, and also with the University Technological Institute (IUT) of Vitry, Vitry, France. B. Le Pioufle is with the Laboratoire d’électricité, Signaux et Robotique ´ ´ (LESiR) U.R.A. C.N.R.S. D1375, Ecole Normale Supérieure de Cachan, 94235 Cachan, France. Publisher Item Identifier S 0278-0046(97)06526-X.

acceleration and a good torque quality (i.e., no ripples), namely, a high torque-to-inertia ratio and an excellent power factor close to unity, since the copper losses are essentially located only in the stator. In addition, for the same delivered mechanical power, a PM synchronous motor needs a smaller line current value, which is favorable for the design of the electronic power converter feeding this drive. Nevertheless, the PM synchronous machines are strongly nonlinear systems. Salient-pole motors, in particular, present an additional complexity. Specifically, their torque can be decomposed into two components. The first component is the so-called reluctance torque, which is due to the saliency effect in the machine and is expressed as a nonlinear product of the “ ” and “ ” components of the currents. The second component, referred to as the hybrid torque, is produced through the interaction between the stator rotating magnetic field and the rotor magnets. It represents the largest component of the two and is a linear function of the current . Still, the progress accomplished recently in the area of nonlinear control [1] and microprocessor technology [2], [3] has allowed the implementation of sophisticated and performing nonlinear control laws to sinusoidal synchronous motors [4]–[7] and to various other types of electrical motors [8], [9]–[11], in order to improve their dynamic performances.These control strategies are either based on a direct scheme, in which the Park transformation or feedback linearization techniques are used in order to linearize directly the voltage–torque relationship [4], [7] (or voltage–speed or voltage–position in the case of speed or position control, respectively), or on an indirect and more classical currentcontrol scheme, in which the appropriate current repartition is imposed in stator phases by the use of internal current loops [11]–[14]. In order to simplify the control strategy of the machine, however, most of the direct and indirect scheme algorithms take only the hybrid part of the torque into account, even for salient-pole motors [4], [14]. The direct component of the stator currents is then forced to zero, which consequently orientates the stator magnetic field perpendicularly to the rotor field. Thus, the reluctance torque is canceled. As a result, the maximum torque obtainable for a given current is not maximized, and the copper losses are not minimized and, thus, nor is the power consumption of the motor.

0278–0046/97$10.00  1997 IEEE

GRENIER et al.: TORQUE CONTROL OF A PERMANENT-MAGNET SYNCHRONOUS MOTOR

On the other hand, some control strategies allowing to have nonzero values are proposed in the literature [11], [12]. They are all indirect current-control schemes and, as such, present several drawbacks. First, they assume a highperformance internal current loop, since it is always difficult to impose arbitrary currents in inductive stator windings [13]. Second, the design of the internal current loop is based on the assumption of a time-scale separation between the mechanical and electrical time constants. For PM ac motors, the mechanical and electrical time constants can be of the same order of magnitude, and the assumption of time-scale separation may, therefore, not be satisfied. Another drawback of this type of scheme is that torque regulation is achieved through an open-loop feedforward torque controller, which converts the desired torque into the required stator phase currents. Aside from the robustness issues that feedforward controllers raise, the choice of the phase currents references is not an easy task, since there is an infinite number of candidates for a given desired torque. In [12], the saliency effect is used, and the current references which insure a maximum torque/ampere are found graphically. This lookup table approach has to be reapplied, however, for every different motor. In [11], an internal current loop scheme is used for torque tracking. The motor used is a hybrid step motor, and the general model in which the inductance and back EMF functions are nonsinusoidal is considered. Exact feedback linearization is used to design the internal current controller, while feedforward compensation is used to decouple the electrical and mechanical subsystems. The arbitrariness in choosing the current references is formally expressed in terms of a free function, which can be selected in such a way as to minimize the power loss in the motor. The resulting scheme is interesting in terms of its generality. It is, however, computationally complex. The computation of the current references is performed in a feedforward manner and may involve time derivatives of the input torque command. This, as the authors point out, may be circumvented by augmenting the order of the controller. In this paper, a nonlinear controller based on a direct scheme which directly controls the torque instead of the currents is proposed. Our nonlinear controller achieves both objectives used in [11] and [12], namely, torque regulation and minimization of power losses without recurring to an internal current loop nor to any feedforward compensation. It is based on an input–output linearization scheme where the inputs are the stator voltages. The challenge at this level consists in finding two outputs for the nonlinear state model, which are physically linked to our two control objectives, while at the same time satisfying the input–output linearization conditions. Indeed, we will first show that choosing the motor torque and the copper losses as outputs does not allow us to apply input–output linearization techniques, since the vector of relative degrees is not defined at the desired operating point. We will thus define a new output linked to the copper losses which, when forced to zero, leads to maximum machine efficiency. Our scheme therefore combines the advantages of the direct scheme (not

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having to impose current profiles in inductive stator windings and no feedforward compensation) with the advantages of the indirect scheme, namely, to take into account the saliency effect in order to obtain a maximum torque/ampere. The experimental implementation of the controller was done for a machine with a moderate saliency effect (see the Appendix), but it will be more interesting with a motor with higher saliency effect. Among the different losses in the motor, we will particularly try to reduce the copper losses, since the other ones (essentially the iron losses) are neglected in our analytical model. Nevertheless, the proposed method could easily be extended to the case where the iron losses are also taken into account. The proposed nonlinear control has been tested experimentally. Through these experiments, we shall prove that such a strategy permits one to raise the power factor of the machine and to provide a higher maximum torque than the one obtained with the classical nonlinear controller, which maintains the current to zero. This paper is organized as follows. In Section II, the salientpole synchronous motor model expressed in the rotor frame is recalled. The choice of the outputs which permit one to define the nonlinear torque controller that minimizes the copper losses is discussed in Section III. Finally, Section IV is dedicated to the experimental implementation of this new controller using a floating-point digital signal processor (DSP96000). The obtained performances are compared to the ones obtained with a more classical nonlinear controller where .

II. ANALYTIC MODEL OF THE SALIENT-POLE SYNCHRONOUS MACHINE The model considered in this paper is obtained under the following hypotheses. 1) The variation of the flux induced by the rotor magnets in the stator phases must be sinusoidal, which implies that the induced electromotive forces are sinusoidal. 2) The variation of the inductance as a function of the rotor position must be sinusoidal. Moreover, in addition to this sinusoidal approximation, cogging torque, magnetic saturation, and iron losses are neglected in our modeling [14]–[16]. The “ ” and “ ” components of the current vector are calculated through a state transformation of the stator currents phases and , and of the position :

(1)

The three phase voltages and applied to the motor by the pulsewidth modulation (PWM) inverter, are obtained from the and voltages (computed by the controller) by

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A. Copper Losses as Outputs

means of the reverse Park transformation:

In order to reach the control objectives, we first propose to choose as outputs the motor torque and a variable proportional to the copper losses . The outputs are then (9) (2) Note: There are different variants for the Park transformation. Here, an orthonormal form has been used, in order to insure the equality of the power expressed in both “ ” and “ ” frame. With this orthonormal form, we have

(10) Because one of the objectives is to minimize the copper must be forced losses, the total differential of the function to zero for the operating point (11)

consumed power (3) for balanced systems (no homopolar currents). Under the assumptions mentioned above, the machine equations expressed in the rotor frame are the following:

and variations are not In this last expression, independent of each other. Indeed, it is not the absolute minimum value of the function which needs to be reached (this absolute minimum value corresponds evidently to , thus, to a null motor torque), but the minimum value for a given torque. The total differential must thus be calculated along a constant torque trajectory, so that

(4) (12)

(5) (6) The symbol significations and their values for the studied motor are given in the Appendix. In particular, we note that, for the studied motor which is a permanent-magnet motor with flux concentration, the inductance in the quadrature axis is higher than the one in the direct axis . Thus, the saliency ratio is higher than one. In the expression of the torque (6), we can recognize, in particular, the reluctance torque , caused by the saliency phenomenon: (7) as well as the hybrid torque rotor and stator fluxes:

From (11) and (12), we get that the reference operating point must verify (13) Let us now apply the input–output linearization techniques to the system [1]. The outputs and have to be successively derived with respect to time, until one of the components of the control vector appears. We get (14) where, notably,

due to the interaction between

(8) III. NONLINEAR TORQUE CONTROL USING THE SALIENCY EFFECT Since the system possesses two independent inputs and , two independent objectives can be achieved for the control of the machine. One objective is mechanical, i.e., position, speed, or torque control and the other one which, in our case, will be the reduction of the copper losses, while maximizing along the way the conversion of the electrical energy (limited by the maximal values of the voltages and currents delivered by the inverter feeding the motor) to mechanical energy.

The determinant of equals zero (i.e., is singular) for any operating point for which (13) is verified. Consequently, the relative degree vector is not defined for the reference operating point. The nonlinear system, the outputs of which are the torque and the copper losses, cannot be linearized and new outputs have to be defined. We note, nevertheless, that we did not use in the previous consideration the explicit form of , but only the fact that this output is minimum for the expected operating point. Any other energetic criterion optimized for a given constant torque (copper losses, iron losses, input power, efficiency, etc.) would verify a relationship like (13), which would lead to the same deadlock (a singular decoupling matrix).


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order to cancel any static error. Thus, the imposed linear control signals and are

B. Derivative of the Copper Losses Along a Constant Torque Trajectory as Outputs Next, we propose to use as output the total differential of the copper losses along a constant torque trajectory. We define (15) and The new outputs for our system are now . The objective of the nonlinear controller is to force this last output to zero, so that the reached operating point corresponds to an extremum of along a constant torque is a positive quadratic function of the stator trajectory. As currents, this extremum is unique in this operating area and actually corresponds to a copper-losses minimization. The new system outputs are, thus, (16)

(20) (21) Remark: Here, the same dynamic has been chosen for the two outputs. Choosing different dynamics could, nevertheless, be an alternative to avoid voltage saturation during transients (this point is currently being studied in our laboratory). In order to obtain the desired closed-loop behavior, the following voltages have to be provided to the motor: (22) which leads one to impose

(17) Let us remark that, in the case of a nonsalient-pole machine , this new output is directly proportional to the current , and we then get the classical constraint . Thus, the classical strategy forcing to zero corresponds to a strategy minimizing copper losses for a constant airgap motor. If we now apply to this new system the input–output linearization techniques, we get (18) and are defined at the bottom of the page. where is nonsingular for any operating point which can be reached by the motor. Indeed, the determinant

(19) could only reach zero, in the case of the studied machine (see the Appendix), if A, which is not an admissible current value for the motor and the PWM inverter. IV. EXPERIMENTAL RESULTS As the system is input–output linearizable, any desired dynamic can be imposed by means of a linear controller. Integrators are introduced in the chosen linear controller, in

(23)

(24) The corresponding control algorithm is summarized in Fig. 1. These control laws have been tested on our experimental setup. This benchmark is built around a synchronous motor (the characteristics of which are given in the Appendix), fed by a three-phase PWM MOS inverter (switching frequency 20 kHz, low dead time 200 ns), and controlled through a floating point DSP (Motorola DSP96000) development board (see Fig. 2) [17]. The sampling frequency has been set to the same value as the switching frequency (20 kHz). We can see in Fig. 3(a) the torque step response obtained when using our new control law (23) and (24). The 6.0 N m torque reference has been set so that the line current does not exceed the maximum value allowed by the inverter A). ( Fig. 4(a) represents the measured line currents in phases “ ” and “ ” and Fig. 5(a) their Park components. We realize in this last figure that becomes negative in order to maximize the torque must be negative to get positive because we have . These results must be compared to the ones obtained when using the classical nonlinear controller with forced to zero. In this last case, we can see in Fig. 4(b) the effect of the saturation on the line currents. When the line current value in a phase overreaches the maximal admissible value,

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Fig. 1. Block diagram of the nonlinear controller minimizing copper losses.

(a)

Fig. 2. Experimental setup.

the current limiter mechanism sets the corresponding phase voltage to zero. This introduces nonlinear phenomena which lead to oscillations on the line current. Oscillations on the Park components are observed in Fig. 5(b). In addition to these oscillations, Fig. 3(b) shows that the reference average value of the torque is not reached. Indeed, we observe in Fig. 5(b) that the current component becomes positive, which reduces the torque/current ratio. We can conclude from these results that working with minimized copper losses allows us, for a given maximal available electrical power, to obtain a higher maximal torque value. Indeed, it can be shown that the locus of the theoretically attainable currents for any rotor position in the “ ” frame is limited by the limit circle shown in Fig. 6. This circle

(b) Fig. 3. (a) Torque response with the new algorithm minimizing the copper losses compared to (b) the one obtained by a classical control strategy forcing id to zero.

corresponds to the equation (25) tracked by the proposed In Fig. 6, the trajectory algorithm, as well as the classical trajectory, are represented.


(a)

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(a)

(b) Fig. 4. (a) Line currents with the new algorithm minimizing the copper losses compared to (b) the one obtained by a classical control strategy forcing id to zero.

The torque reference value of 6.0 N m could, theoretically, be achieved for any position by either the nonlinear controller which minimizes the copper losses or the controller which forces to zero. Indeed, both corresponding operating points are inside the limit circle mentioned above. Nevertheless, the controller tracking to zero leads to higher line currents than the one minimizing the copper losses and, experimentally, we found that the controller tracking to zero leads to saturations on the line currents [see Fig. 4(b)]. The obtained operating point is then outside an experimental limit circle respecting a security margin due to saturation. As the current limiter mechanism brings instabilities on the currents, a security margin due to saturation has to be respected when the quantification effect of digital acquisitions is taken into account. This second limit circle is also shown in Fig. 6, and it appears that the operating point corresponding to a 6.0 N m torque and is outside this experimental limit circle. In addition, the equitorque curves being tangent to the limit circle at the intersection point with the trajectory , our method permits us to obtain the maximal torque for a given maximal line current value [12]. In the case of the studied motor, the gain is not very high (1.2%). Fig. 7 shows the maximal torque value increase, obtained by the use of our algorithm, versus the value of the salient ratio . The higher is (and it can easily reach four or five for a PM synchronous motor with flux concentration), the more interesting is the gain on the maximal torque value. We note, however, that this gain remains low for a machine with

(b) Fig. 5. (a) “dq ” components of the currents with the new algorithm minimizing the copper losses compared to (b) the one obtained by a classical control strategy forcing id to zero.

Fig. 6. Locus of the operating points in the “dq ” frame.

wounded rotor, where (the use of the is justified in this case). controller forcing The main drawback of our new controller is that it needs more on-line computation time, as shown in Table I. Nevertheless, the growth of the time needed for our DSP to compute the control algorithm in “ ” frame leads only to a 5% increase of computation time for the total control task (measure acquisitions Park transformation of the measured

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extended to the case of machines with a nonsinusoidal flux distribution. Finally, compared to the classical control schemes with internal current loop, the proposed method can easily be adapted to a speed or position control scheme. As mentioned before, the mechanical and electrical time constants of PM synchronous machines can be of the same order of magnitude. A scheme based on an internal current loop and singular perturbation techniques would not necessarily provide the desired electrical and mechanical performances. These can only be achieved if the outputs chosen reflect equally the electrical and mechanical control objectives, as is the case for our control strategy.

Fig. 7. Maximal torque increase versus salient ratio. TABLE I COMPARISON OF THE ON-LINE COMPUTATION TIMES Algorithm tracking

D(pc ) = 0

Measure acquisitions Computation id and iq Output computation and error correction Calculation of the voltage inputs Saturation of the currents and voltages Total:

Algorithm tracking

4.5 s

id = 0 4.5 s

4.6 s

3.6 s

9.7 s

9.7 s

18.8 s

17.8 s

current control algorithm in “ ” frame reverse Park transformation of the voltages cyclic rate generation for PWM). All these tasks have been programmed in assembly language, which explains the very low computation times.

APPENDIX Signification of the used symbols and their values for the studied motor are as follows. • and are the stator voltage components in the rotor frame in the “ ” and “ ” axes, respectively. • is the “ ”-axis current and mH is the “ ”-axis inductance. • is the “ ”-axis current and mH is the “ ”-axis inductance. We have . • is the resistance of the stator windings. Wb is the “ ” component of the flux induced by • the permanent magnets of the rotor in the stator phases. • is the number of pole pairs of the motor. • and are the instantaneous angular position and rotation speed, respectively. REFERENCES

V. CONCLUSION In this paper, we have successfully linearized the equations of the salient-pole synchronous motor with sinusoidal flux strategy. distribution without recurring to the classical Hence, we are able to exploit the reluctance torque due to saliency in order to optimize the efficiency of the machine and to achieve the maximal value of the admissible instantaneous torque. An alternative to our control method would be to directly solve the equation in order to obtain the reference and which lead to the desired current values reference torque while minimizing the copper losses. The main drawback of this approach is that an explicit solution of the above-mentioned equation is not always readily available, especially for machines which do not satisfy the Park assumptions (i.e., with a nonsinusoidal flux distribution). In this case, the equation has to be expressed in the stator natural coordinates “ .” For salient-pole machines in particular, more computationally demanding methods are needed in order to solve the equation for the required reference currents. The problem gets even more complicated if we integrate the iron losses in the model of the machine. The control strategy proposed in this paper overcomes this computational challenge by not requiring one to solve the equation . This presents the main advantage, i.e., that it can be systematically

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Ouassima Akhrif (M’93) received the Diplôme ´ ´ d’Ingénieur d’Etat degree from the Ecole Mohammadia, Rabat, Morocco, in 1984 and the M.Sc.A. and Ph.D. degrees from the University of Maryland, College Park, in 1987 and 1989, respectively, all in electrical engineering. During 1989–1990, she was a Visiting Assistant Professor in the Systems Engineering Department, Case Western Reserve University, Cleveland, ´ OH. In 1992, she joined the Ecole de Technologie Supérieure, Montréal, P.Q., Canada, where she is currently a Professor in the Electrical Engineering Department. Her research interests are nonlinear geometric control, nonlinear adaptive control, and their applications in electric drives and power systems.

Damien Grenier was born in Rouen, France, in ´ 1965. He received the Ph.D. degree from the Ecole Normale Supérieure de Cachan, Cachan, France. in 1994. From 1994 to 1996, he was involved in Post´ Doctoral research at the Ecole de Technologie Supérieure, Montreal, P.Q., Canada. In 1996, he joined the Laboratory of Electrotechnics and Instrumentation (LEI) of the Faculty of Applied Sciences, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, as a Professor. His research areas include the conception, modeling, and digital control of electromechanical converters.

Yvan Bonnassieux received the Master’s and Agrégation degrees in electrical engineering from the University of Paris XI, Paris, France, in 1990 and 1991, respectively. He is currently working toward the Ph.D. degree in electrical engineering at ´ the Ecole Normale Supérieure of Cachan, Cachan, France. He is currently a Professeur Agrégé at the University Technological Institute (IUT) of Vitry, Vitry, France. He conducts research with LESiR, a laboratory associated with the National Scientific Research Center (CNRS). His main research interests deal with discrete nonlinear control and multivariable robust control of ac motors.

L.-A. Dessaint (M’88–SM’91) was born in Paris, France, in 1953. He received the B.Ing., M.Sc.A., ´ and Ph.D. degrees from the Ecole Polytechnique de Montréal, Montréal, P.Q., Canada, in 1978, 1980, and 1985, respectively. He was a Research Assistant at the Hydro-Québec Research Institute (IREQ) from 1980 to 1985, where he worked on the simulation and control of a wind energy conversion system. He is currently a ´ Professor of Electrical Engineering at the Ecole de Technologie Supérieure, Montreal, P.Q., Canada. Since 1992, he has also been the Director of the Groupe de Recherche en ´ e´ lectronique de Puissance et Commande Industrielle (GREPCI), a research group on power electronics and digital control. His main research interest is the adaptive and nonlinear control of electric drives and power systems. He is a coauthor of the MathWorks Blockset on “Power Systems.” Dr. Dessaint is a member of the Order of Engineering of Quebec, Canada.

Bruno Le Pioufle (M’92) received the Ph.D. degree in electrical engineering in 1991 from the University of Paris XI, Paris, France. He is currently an Associate Professor at the ´ Ecole Normale Supérieure of Cachan, Cachan, France, an institute designed to train future teachers, professors, and research specialists. He also currently conducts research with LESiR, a laboratory associated with the National Scientific Research Center (CNRS). His recent research has dealt with the modeling and control of electromechanical systems, nonlinear control of electrical drives, robust control, PWM strategies, control of variable-frequency converters, modeling and control of the motor–inverter association, source–inverter association modeling, and estimation of electrochemical sources state of charge.

[14]

[15] [16] [17]