A New Approach To Dynamic Feedback Linearization Control Of An

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 3, MARCH 1998

III. CONCLUSION We have presented a new type of approximate solution for nonlinear optimal control problems by making use of the concept of pseudolinearization. This approximation is expected to be less local than the standard linear-quadratic approximation about a single equilibrium, as is indicated by the relationship between the corresponding control laws as well as a comparison of the corresponding closedloop stability properties. Certainly this expectation is illustrated by our examples. Also, the available computational tools for Riccati equations and the method for computing 8(x) via splines in [1] indicate that in computational difficulty our approach is somewhere between solving the linear-quadratic problem at a single equilibrium and a direct attack on the Hamilton–Jacobi equation. For simplicity we have considered only single-input systems. Extensions to the multi-input case can be pursued without conceptual difficulty, though the details of assumptions and notations become unwieldy. Similarly, we have considered only the smooth (C ) case, though it is not difficult to relax assumptions at least to a setting consistent with C 2 solutions of the Hamilton–Jacobi equation. Finally, we note that the (exact) feedback linearization transformation can be used in a similar way. The associated linear-quadratic problem leads to an exact representation for the solution of the Hamilton–Jacobi equation that reads much like Theorem 2.3 with an even more constrained performance index. Efforts to obtain results for more useful forms of J lead to a mild generalization of [5] involving a specific and somewhat simple form of Hamilton–Jacobi equation. These issues are discussed in [6]. Also, transformations (feedback and coordinate change) yielding exact input–output linearization and input–output pseudolinearization can be considered. The results involve algebraic Riccati equations of dimension equal to the plant relative degree. However, the form of the performance indexes is, again, highly constrained by plant data. Furthermore, stability hypotheses on the plant zero dynamics are required both to ensure meaningful performance indexes and to conclude closed-loop stability properties.

1

REFERENCES [1] S. A. Bortoff, “Approximate state-feedback linearization using spline functions,” in Proc. American Control Conf., Baltimore, MD, 1994, pp. 1702–1706. [2] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992. [3] D. L. Lukes, “Optimal regulation of nonlinear dynamical systems,” SIAM J. Contr., vol. 7, pp. 75–101, 1969. [4] C. Reboulet and C. Champetier, “A new method for linearizing nonlinear systems: The pseudo-linearization,” Int. J. Contr., vol. 40, pp. 631–638, 1984. [5] W. J. Rugh, “System equivalence in a class of nonlinear optimal control problems,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 189–194, 1971. [6] H. Tan and W. J. Rugh, “Pseudolinearization and exact linearization transformations in nonlinear optimal control,” Dept. Electrical and Computer Eng., Johns Hopkins Univ., Tech. Rep. JHU-95-26,1995. Available URL http://www.ece.jhu.edu/reports/techreps/95-26.ps. [7] J. Wang and W. J. Rugh, “On the pseudo-linearization problem for nonlinear systems,” Syst. Contr. Lett., vol. 12, pp. 161–167, 1989. ) optimal control of [8] A. J. van der Schaft, “Relations between ( a nonlinear system and its linearization,” in Proc. 30th IEEE Conf. Decision and Control, Brighton, U.K., 1991, pp. 1807–1808.

H1

391

A New Approach to Dynamic Feedback Linearization Control of an Induction Motor John Chiasson

Abstract—A new approach to dynamic feedback linearization control of an induction motor is given. Previously, it has been shown that the dynamic model of an induction motor, consisting of speed, stator currents, and rotor fluxes, is dynamically feedback linearizable [1]. However, the controller and transformation were valid only as long as the motor torque was nonzero and the methodology required switching between two computationally complex transformations. Here it is shown that by considering the direct-quadrature model of the induction motor, a single dynamic feedback linearizing transformation exists and is valid (essentially) as long as the (magnitude of the) rotor flux is nonzero. Furthermore, the resulting control computations are well within the capabilities of contemporary microprocessor technology. Index Terms—Dynamic feedback linearization, induction motor.

I. INTRODUCTION The work here is to consider the use of the differential-geometric approach of dynamic feedback linearization to a nontrivial physical system, the induction motor. It is well known that the induction motor is not (static) feedback linearizable [6]. The interest here is to show that input-to-state linearization can be achieved for speed control of an induction motor, where the singularity condition is essentially that the flux be nonzero, which is not unlike the requirements for other methods such as input–output linearization. In a series of papers by Charlet et al. [22]–[24] it was shown that multi-input systems could become (static) feedback linearizable from input to state after the addition of an integrator to one of the inputs. That such an approach was possible with the induction motor was considered in [1]. There, a fifth-order state-space model of the induction motor in the so-called a-b coordinate system [16] was considered in which the inputs consisted of the two stator voltages uSa ; uSb , and the state variables consisted of the two stator currents iSa ; iSb , the two rotor fluxes Ra ; Rb , and the speed !. Using this model, it was shown that the induction motor is dynamically feedback linearizable, that is by adding an integrator to one of the two inputs of the induction motor, the resulting sixth-order system was then (static) feedback linearizable [1]. However, this control structure had drawbacks: 1) a nonsingular feedback linearizing transformation was shown to exist only as long as the electromagnetic torque produced by the motor was nonzero; 2) the control structure required switching between two different transformations to avoid the singularities in the transformations; and 3) the transformations and the required feedback cancellations were computationally prohibitive. The drawbacks enumerated above make the dynamic feedback linearization approach in [1] infeasible. In this work, a single dynamic feedback linearization transformation and controller are constructed whose singularity is easily avoided and whose required computations are well within the limits of current microprocessor technology. The key to this alternative approach is to start with a model of the induction motor in the direct-quadrature (d-q ) coordinate system rather than in the a-b coordinates. Of particular significance is that Manuscript received December 5, 1995; revised April 29, 1996. This work was supported in part by the University of Pittsburgh. The author is with ABB Daimler-Benz Transportation, Pittsburgh PA 15236 USA. Publisher Item Identifier S 0018-9286(98)01381-6.

0018–9286/98$10.00  1998 IEEE

392

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 3, MARCH 1998

the input voltages vd ; vq are obtained from the stator voltages uSa ; uSb of the a-b coordinate system by a rotation through the angle  = Tan01 ( Rb = Ra ). It is then shown that the addition of an integrator added to the direct axis input results in a single feedback linearizing transformation for speed control with the controller and transformation being nonsingular as long as (essentially) the flux 2 + 2 is nonzero. This singularity condition magnitude d = Ra Rb is simple to avoid and is common to the standard field-oriented controller [3], the input–output linearization controller [5]–[7], as well as being required for controllability in the passivity-based controllers [15]–[18]. The feedback cancellations required by the dynamic feedback linearization controller are computable in real-time using contemporary microprocessor technology. It is also shown that if an integrator is added to the quadrature axis input (rather than the direct axis), then the position  may be included in the model, and the resulting seventh-order model is feedback linearizable. However, this transformation and the corresponding controller are nonsingular only if the electromagnetic torque  = J d iq is nonzero which, in particular, would require there be a nonzero load-torque on the motor in order to hold the position  at any fixed value. Interestingly, the feedback cancellations required are significantly less than that required when the integrator is in the direct axis integrator. The crucial difference in this approach from that of [1] is that the inputs here are first put through the (nonlinear) d-q transformation, and then an integrator is added to d axis input. It is this input transformation that allows a simplification of the (dynamic) feedback linearizing transformation and controller. Before proceeding with the results, it is interesting to point out that the use of dynamic state feedback for the induction motor was first considered by De Luca, where full linearization of the induction motor electrical dynamics was achievable under the assumption that the motor speed was constant [11]. In [12], again with the speed considered to be a slowly varying parameter, De Luca showed that dynamic nonlinear state feedback could be used to input–output linearize the induction motor. In [12] the outputs were chosen as the flux magnitude and torque, and simulation indicated that the zero dynamics were stable. Further interesting work along these lines is given in [13] and [14].

II. THE DIRECT-QUADRATURE MODEL OF THE INDUCTION MOTOR The a-b model (which is a two-phase equivalent model of threephase motor) of the induction motor is given by [3], [6], [7]

d!=dt = ( Ra iSb 0 Rb iSa ) 0 L =J diSa =dt = 0 iSa +  Ra + np ! Rb + uSa =(LS ) diSb =dt = 0 iSb +  Rb 0 np ! Ra + uSb =(LS ) d Ra =dt = 0 Ra 0 np ! Rb + MiSa d Rb =dt = 0 Rb + np ! Ra + MiSb (1) where uSa ; uSb are the stator phase voltages, iSa ; iSb are the stator phase currents, Ra ; Rb are the rotor fluxes (rotated to the stator frame), and ! is the rotor speed. RS is the stator phase resistance, LS is the stator phase inductance, RR is the rotor phase resistance, LR is the rotor phase inductance, M is the coefficient of mutual inductance, J is the rotor moment of inertia, L is the load-torque,  = 1 0 M 2 =LS LR is the leakage factor, TR = LR =RR is the rotor time constant,  = 1=TR ; np is the number of pole-pairs,  = 2 LS ) + np M=(JLR ); = M=(LS LR ); and = M 2 RR =(LR

RS =(LS ):

As pointed out in [6], the direct-quadrature (d-q ) model can be viewed as applying the nonlinear coordinate transformation

!=!  = Tan01 ( Rb = Ra ) id = cos()iSa + sin()iSb iq = 0 sin()iSa + cos()iSb

d=

2 + 2 Ra Rb

(2)

to (1) along with defining a new set of inputs

vd = + cos()uSa + sin()uSb vq = 0 sin()uSa + cos()uSb :

(3)

This transformation of the voltages (as well as the transformation of the currents) is referred to as the direct-quadrature or d-q transformation. In this new coordinate system, (1) becomes

d!=dt =  d iq 0 L =J d d =dt = 0 d + Mid did =dt = 0 id + M d =(LR LS ) + np !iq 2 + Miq d + vd =(LS ) diq =dt = 0 iq 0 Mnp ! d =(LR LS ) 0 np !id 0 Miq id = d + vq =(LS ) d=dt = np ! + Miq = d :

(4)

A standard approach for simplifying (4) for the design of the speed control loop is to use inner proportional plus integral (PI) current loops to achieve current-command. That is, choose the inputs as

vd = KDP (idr 0 id ) + KDI vq

=

KQP (iqr 0 iq ) + KQI

t

0

t

0

dr 0 id ) dt

(i

qr 0 iq ) dt

(i

(5)

where idr and iqr are the reference (desired) currents for id and iq , respectively. Proper choice of the gains (specifically, high-gain feedback) results in id ! idr and iq ! iqr fast enough that the current dynamics can be ignored [4], [7]. That is, (4) can then be replaced (approximated) by the following reduced-order model:

d!=dt =  d iqr 0 L =J d d =dt = 0 d + Midr d=dt = np ! + Miqr = d

(6)

where idr ; iqr are now the inputs. III. DYNAMIC FEEDBACK LINEARIZING TRANSFORMATIONS It has been pointed out in [6] that (1) is not feedback linearizable, and therefore neither is the model (4). It is also easy to check that the reduced-order model (6) with inputs idr ; iqr and state variables !; d ;  is also not feedback linearizable. The concept of dynamic feedback linearization has been proposed and developed in [22]–[24], [29]. Specifically, it has been shown that for a dynamic system which is not feedback linearizable, the addition of integrators in the inputs can result in the higher order system being feedback linearizable. This phenomenon occurs only in multi-input systems as a singleinput system is dynamically feedback linearizable, if and only if it is (statically) feedback linearizable [23]. As the induction motor model is a multi-input system which is not feedback linearizable, it is natural to ask if it is dynamically feedback linearizable. As previously noted, the a-b model was used in [1] where it was shown that a dynamic feedback linearizing transformation

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 3, MARCH 1998

exists as long at the torque  = J( Ra iSb 0 Rb iSa ) was nonzero. The feedback structure in [1] required switching between two computationally complex controllers to avoid singularities in the controller. On the other hand, the torque expression in the d-q system is simply J d iq leading one to conjecture that the dynamic feedback linearizing transformation and controller might also be simpler in form if developed in this coordinate system. This is indeed shown to be the case as the resulting controller no longer requires switching between two different transformations and the computational requirements are significantly reduced. In addition, if the integrator is added to the d axis, the singularity condition no longer requires that the torque be nonzero. In the rest of this work, it is assumed that all of the state variables are available for feedback and all of the motor parameters including the load-torque L are constant and known. Realistically, this is not true since the fluxes Ra ; Rb are not usually measured nor is the load-torque typically known. However, there are now well-known reliable methods to estimate the fluxes [3], [19]. Furthermore, using the estimates of the fluxes, an estimate of the torque is available which can then be used to provide an accurate estimate of the loadtorque using observer theory (see, e.g., [8]). Finally, there are several techniques to estimate the motor parameters. For a classical approach (locked-rotor test, no-load test, etc.) see [20], while for some modern approaches using least-squares methods see [9], [21]. The issue of showing a priori stability when the state variables are not assumed known but replaced by their estimates is not considered here. That is, it is assumed that if the fluxes are estimated, then the estimation error is sufficiently small to be neglected. One advantage of the passivity-based controllers proposed by Ortega and his coworkers [15]–[18] is that global closed-loop stability can be guaranteed a priori where the desired flux is used instead of the estimated flux. Notation: Let h(x) : 0. Note from (8) that none of the state variables zi are linearly related to the speed ! . Consequently, d=dt = ! is not a linear function of the zi so that the position cannot be appended to (9) without losing linearity of the system. That is, if d=dt = ! = x1 is appended to (7), then it is straightforward to show that the resulting system is not feedback linearizable. A similar situation arises in the case of a shunt-connected dc motor [2].

0 0

y1

As d!=dt < rated =J , (12) is satisfied as long as

d>

0

=

0

M

Again using CONDENS [25], it is found that (14) is feedback linearizable with Kronecker indexes 1 = 2; 2 = 2. In this case, a solution set is easily found to be

3

(12)

g1

0 0 2 0 is straightforward to maintain and is common to other controllers [7], [3], [5], [6]). Consequently, to use this controller would require switching to another control strategy when it is desired to change the sign of the torque. By appending d=dt = ! = x1 to (14), it is straightforward to check that the fifth-order system is feedback linearizable. In fact, adding the state variable z0 =  to the transformation (15) gives a feedback linearizing controller for position control. However, to avoid the singularity of the controller, it is still required that iq 6= 0.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 3, MARCH 1998

C. Dynamic Feedback Linearization of the Full-Order d-q Model The dynamic feedback linearization of the full-order d-q model is similar to the reduced-order cases presented. Consequently, the results are just summarized. 1) Integrator in the d Axis: Letting x1 = !; x2 = d ; x3 = id ; x4 = iq ; x5 = ; x6 = vd ; dx6 =dt = u1 ; u2 = vq in (4) results in a sixth-order system which is feedback linearizable. The Kronecker indexes are 1 = 3; 2 = 3 and the output maps are the same as in the reduced-order case, that is

y1

=

y2

=

h1 (x) = x2

h2 (x) = x1 0 x22 x5 =M:

The feedback linearizing transformation is then z = T (x) = 2 T :

(6)