On the Discrete–Time Modelling and Control of Induction Motors with Sliding Modes B. Castillo–Toledo † , S. Di Gennaro ‡ , A. G. Loukianov † and J. Rivera † Abstract— In this work a discrete–time controller for an induction motor is proposed. State feedback and diffeomor- phism are applied to the plant dynamics in order to be finitely discretized. Then, on the base of the sampled dynamics, a discrete–time controller is derived, achieving speed and flux modulus tracking objectives. Finally, a reduced order observer is designed for rotor fluxes and load torque observation. Index Terms— Induction motors, sliding mode control, discrete–time systems, observer. I. I NTRODUCTION I NDUCTION motors are among the most used actuators for industrial applications due to their reliability, rugged- ness and relatively low cost. On the other hand, the control of induction motor is a challenging task since the dynamical system is multivariable, coupled, and highly nonlinear. Sev- eral control techniques have been developed for induction motors [1], [3], [13], [12], among which the sliding mode technique [14],[5]. Typically, when implemented on digital devices, the control law is approximated by using zero order holders. This approximation represents a clear disadvantage. Analogously to [2] and to what done in other applications such as in [6], [11] and [4], the alternative is to design a digital controller directly using a digital model of the motor [9]. Unfortunately, the sampled model of the induction motor is only approximated, since it is expressed as an infinite series. To bypass this difficulty, following [10] in this work we obtain an exact closed form of the sampled dynamics using a preliminary continuous feedback which ensures the finite discretizability. In the case of the induction motors such a closed form discretization can be obtained in a rather simple way. The advantage of working with a closed form discretization is clear, and in this respect the use of the sliding mode technique fits well with the design of the control law directly in the digital setting. After deriving the digital controller, we will design a reduced order observer for the estimation of the load torque and motor fluxes, in order to eliminate the need of the full state measurements. The paper is organized as follows. In Section II the continuous–time induction motor model is briefly reviewed, Work supported by CONACYT Mexico under grant N. 36960A and CNR Italy. The second author was also partially supported by MIUR Italy under PRIN 02 and by Columbus Project IST–2001–38314. † Jorge Rivera, B. Castillo–Toledo and A. G. Loukianov are with CINVESTAV, Unidad Guadalajara, Apartado Postal 31-438, Guadalajara, Jalisco, C.P. 45091, Mexico. E.mail: {toledo, louk, jorger}@gdl.cinvestav.mx ‡ S. Di Gennaro is with the Department of Electrical Engineering, University of L’Aquila, Monteluco Di Roio, 67040 L’Aquila, Italy. E. mail: digennar@ing.univaq.it. and the exact sampled dynamics of this model are derived. In Section III a discrete–time sliding mode control for rotor angular velocity and square modulus of the rotor flux vector tracking is designed. To remove the hypothesis on rotor fluxes and load torque measurability, a discrete– time observer is proposed in Section IV. Section V shows the simulation of the closed-loop induction motor control system. Final comments conclude the paper. II. SAMPLED DYNAMICS OF I NDUCTION MOTORS In the following a sampled version of the dynamics of an induction motor will be derived. Under the assumptions of equal mutual inductance and linear magnetic circuit, a fifth–order induction motor model is written as follows [8] ˙ Φ = −αΦ + pωℑΦ + αL m I ˙ I = αβΦ − pβωℑΦ − γI + 1 σ u ˙ ω = μI T ℑΦ − 1 J T L ˙ θ = ω (1) where θ and ω are the rotor angular position and velocity respectively, Φ =( φ α ,φ β ) T is the rotor flux vector, I = ( i α ,i β ) T is the stator current vector, u =( u α ,u β ) T is the control input voltage vector, T L is the load torque, J is the rotor moment of inertia, ℑ =  0 −1 1 0  , α = R r L r , β = L m σLr , γ = L 2 m R r σL 2 r + R s σ , σ = L s − L 2 m Lr , μ = 3 2 L m p JLr , with L s ,L r L m being the stator, rotor and mutual inductances respectively, R s and R r are the stator and rotor resistances, and p is the number of pole pairs. The following hypothesis will be instrumental for deriv- ing the sampled model of the motor dynamics. (H 1 ) The load torque T L can be approximated by a signal C L which is constant over the sampling period δ. Hypothesis (H 1 ) is acceptable in all cases in which T L varies slowly with respect to the system dynamics. In order to obtain a finite discretization of the system dynamics (1), in the spirit of [9], [10] let us consider first the following feedback u = σpωℑ(I + βΦ)+ e pθℑ e −pθ k ℑ v. (2) Here θ k indicates the value of θ at the time instant kδ, with δ the sampling period and k =0, 1, 2, ···. Note that the first term of (2) and the term e pθℑ have to be implemented via an analogical device, while the term e −pθ k ℑ and the new control v (designed on the basis of the discrete time representation of the system) can be implemented via a digital device. Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC ThA19.4 2598