IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 40, NO. zyxwvutsr 3, JUNE 1993 zyxwvuts 317 A Discrete Model of Induction Motors for Real-Time Control Applications Narpat Singh Gehlot and Pablo Javier Alsina Abstract-The real-time digital control of induction motors and ac servo drives often involves estimation, identification, and adaptive control algorithms. zyxwvutsrqp An efficient and numerically stable discrete model of induction motor is required to implement these algorithms in real time. This paper deals with the develop- ment of a predictor-corrector discrete model of induction mo- tors for real-time control applications. The numerical stability of the model is analyzed by a discrete root locus technique. The digital simulation of the model is presented and compared with a rigorous solution, and satisfactory results are obtained. I. INTRODUCTlON E application of modern control theory to ac motor zyxwvu T control [ll requires a fast, stable, and accurate real- time discrete model of induction machines for analysis and design of digital control strategies. With the advent of digital signal processors and advances in microelectronics, real-time digital control of ac motors is becoming increas- ingly popular in industrial drives. The vector control of small ac servo motors to large mill motors has become a standard design practice, and many practical schemes of fully digital vector controlled ac drives have been imple- mented [1]-[8]. Generally, high-performance induction motor drives involve precise estimation of rotor flux, pa- rameter identification, and adaptation and thus demand a discrete model of induction motors for the implementa- tion of these algorithms in real time. A review of microcomputer control of high dynamic performance ac drives is presented by Leonhard [l]. Vari- ous published papers on digital control of ac motors [2]-[4] utilize discrete model of induction motors; how- ever, the details of discretization have not been discussed. Nilsen [5] utilizes the Euler forward difference approxi- mation to discretize the continuous model. Ben-Brahim and Kawamura [61, zyxwvutsr [7] calculate the discrete matrices off-line and memorize them in a look-up table for the implementation of the discrete model of the motor. An interesting real-time simulation of induction motors based on the bilinear transformation and the Euler approxima- tion methods has been reported by Vanio [9], along with the hardware implementation of the discrete models using field programmable gate arrays. However, these real-time Manuscript received December 18, 1990; revised August 16, 1991 and The authors are with the, Departamento de Engenharia EICtrica, N. S. Gehlot is the author to whom all correspondence should be IEEE Log number 9207939. October 6, 1992. Universidade Federal da Paraiba, 58100 Campina Grande, Brasil. addressed. discrete models suffer from the loss of precision and stability at low sampling frequencies (below 150 Hz). A fast recursive solution of induction motor equations has been reported for power system studies [lo], however, the error analysis and stability are not discussed. This paper deals with the development of a fast stable, and accurate predictor-corrector discrete model of induc- tion motors based on stator current/rotor flux formula- tion for real-time control applications. The discrete model consists of a predictor implemented by utilizing an ap- proximate analytical solution of the system difference equations, and a corrector part, obtained by inserting a first-order-hold (FOH) between the coupled equations of the stator and the rotor. The discretization error as a function of sampling period was studied to verify the accuracy of the discrete model. The numerical stability of the model was analyzed by the discrete root locus tech- nique. To check the validity of the discrete model, simula- tion result were compared with a rigorous solution. An alternate discrete model based on stator flux/rotor flux formulation is included in Appendix IV. 11. VSI-FEDCONTINUOUS MODEL OF INDUCTION MOTOR The field-oriented control of induction motors demands an accurate knowledge of rotor flux, for which rotor flux observers are commonly used. The rotor flux observer models are generally formulated in terms of the stator currents and rotor fluxes in a stationary reference frame [4], [6], [7]; hence, this type of formulation is retained for the predictor-corrector discrete model. The proposed discrete model of the motor deals with the electrical variables only, since the electromechanical time constant is much larger than the electrical ones, therefore it is reasonable to assume that the rotor speed remains constant during a sampling period. Generally, in the field oriented control of induction motors, only elec- trical dynamics is considered, ignoring the electromechan- ical aspects; however, measured or estimated rotor speed is incorporated in the electrical model of the motor as if it were a constant parameter at each sampling period. The continuous model of an induction motor with ref- erence frame (d,q) fixed in stator and in terms of rotor fluxes and stator currents is given by [41,[61,[71: PIS = R,Is + (R21 - W,J)Fr + (l/aLs)Vs (1) pFr = R,Is + (R,I - zyx W, J)Fr (2) 0278-0046/93$03.o0 0 1993 IEEE