International Journal of Control, Automation and Systems 17(X) (2019) 1-12 http://dx.doi.org/10.1007/s12555-017-0322-9 ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555 Optimal Discrete-time Integral Sliding Mode Control for Piecewise Affine Systems Olfa Jedda* and Ali Douik Abstract: This paper presents an optimal discrete-time integral sliding mode control for constrained piecewise affine systems. The proposed scheme is developed on the basis of linear quadratic regulator approach and differen- tial evolution algorithm in order to ensure the stability of the closed-loop system in discrete-time sliding mode and the optimization of response characteristics in presence of control input constraints. Moreover, the controller is de- signed such that chattering phenomenon is avoided and finite-time convergence to the sliding surface is guaranteed. The follow-up of a reference model is also ensured. The efficiency of the proposed method is illustrated with an inverted pendulum system. Keywords: Differential evolution algorithm, discrete-time integral sliding mode control, inverted pendulum system, piecewise affine systems. 1. INTRODUCTION Over the last few decades, research activities in com- puter science and control have been strongly oriented to study hybrid dynamical systems since they are used for modeling the behaviour of realistic complex systems. In fact, they involve explicitly and simultaneously con- tinuous and discrete dynamics. In other words, discrete processes are employed to select, control and supervise the behaviour of continuous processes [1]. Several sub- classes of hybrid dynamical systems, such as linear com- plementarity (LC) systems [2, 3], mixed logical dynami- cal (MLD) systems [4], piecewise affine (PWA) systems [5], and max-min-plus-scaling systems (MMPS) [6], are established in literature in order to make possible analy- sis and development of control techniques which are not available for general hybrid systems. Recently, piecewise affine systems have received much attention in research not only because they can provide a useful modeling method for a large category of hybrid dynamical systems but also because they can be used to approximate nonlinear systems, given that they are equiv- alent to interconnections of linear systems and finite au- tomata [7]. Moreover, all techniques developed for PWA systems can be extended to some other subclasses, such as MLD systems [8], since they are equivalent as demon- strated by Heemls et al. in [9]. Numerous research studies concerning PWA discrete- time systems have been carried out on stability criteria Manuscript received June 5, 2017; revised April 28, 2018; accepted January 20, 2019. Recommended by Associate Editor Tae-Hyoung Kim under the direction of Editor Euntai Kim. Olfa Jedda is with the Electrical Engineering Department, National Engineering School of Monastir, University of Monastir, Tunisia (e-mail: olfa_jedda@outlook.com). Ali Douik is with the Computer Engineering Department, National Engineering School of Sousse, University of Sousse, Tunisia (e-mail: ali.douik@enim.rnu.tn). * Corresponding author. [10, 11], identification techniques [12, 13], and control methods such as optimal control [4, 14] and model pre- dictive control (MPC) [4, 15]. In this paper, we propose a new control technique for PWA discrete-time systems based on discrete-time sliding mode theory. Discrete-time sliding mode control (DSMC) appeared in the mid 1980s with Milosavljevic [16] and then has been followed with a great interest from control commu- nity [17–21] in view of the increasing use of computers for the implementation of digital controllers. Yet, invariance and robustness properties of continuous-time sliding mode control (CSMC) against parametric uncertainties, model- ing errors and external disturbances [22–25] can not be maintained in discrete-time because of the finite sampling rate. Actually, the control input is updated at each sam- pling time so that it can not be changed when the state trajectory crosses the sliding surface during the sampling period; hence the occurrence of chattering phenomenon that may excite high-frequency dynamics, adversely af- fect system performance and damage electric power cir- cuits and mechanical parts. In [19], Gao introduced the notion of quasi-sliding mode that consists in bringing the state trajectory to cross the sliding surface in finite-time and therefore to follow a zigzag motion within a boundary layer in its vicinity so that the chatter effect will be attenuated but not totally avoided. In [26], Bartolini et al. developed a discrete-time control algorithm that consists in defining the equivalent control as the piecewise-constant control that guarantees c ⃝ICROS, KIEE and Springer 2019