Discrete Time Sliding Mode Control Scheme for Nonlinear Systems With Bounded Uncertainties

Discrete Time Sliding Mode Control Scheme for Nonlinear Systems With Bounded Uncertainties

Jalel Ghabi (Higher Institute of Applied Sciences and Technology of Kairouan, University of Kairouan, Kairouan, Tunisia), Ahmed Rhif (Higher Institute of Applied Sciences and Technology of Kairouan, University of Kairouan, Kairouan, Tunisia) and Sundarapandian Vaidyanathan (R & D Centre, Vel Tech University, Chennai, India)
Copyright: © 2018 |Pages: 19
DOI: 10.4018/IJSDA.2018040102

Abstract

This article introduces a sliding mode controller to stabilize a discrete-time nonlinear system in the presence of uncertainties and external disturbances. The proposed controller is derived to guarantee the existence of a quasi-sliding mode, taking into account the upper bound of uncertainties. With this method, a recursive switching function is used, which allows for recovering lost invariance and robustness properties of a discrete sliding mode control. As for the system stability, it is found that the system is stabilized and finally restricted to a known region. This control scheme ensures robustness against parametric uncertainties and external disturbances as well as the elimination of chattering. In this article, after a detailed formalization of the proposed control design, a numerical example for an inverted pendulum is considered, proving the effectiveness of the control methodology.
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1. Introduction

Sliding Mode Control (SMC) has become one of the most popular research areas in automatic control and has found a number of industrial and control applications due its high performance (Azar & Zhu, 2015; Zhu & Azar, 2015; Azar and Serrano, 2015; Mekki et al., 2015; Azar & Vaidyanathan, 2015a, 2015b, 2015c, 2016; Vaidyanathan et al., 2015; Vaidyanathan & Azar, 2015a, 2015b; Djouima et al., 2017; Meghni et al., 2017; Singh et al., 2017). The main merit of continuous sliding mode control (CSMC) is its robustness and invariance properties against parametric uncertainties, modelling errors and external disturbances (Young et al., 1999; Utkin & Chang, 2002; Plestan et al., 2013; Yang et al., 2013; Shtessel et al., 2014; Shi et al., 2017). It is well known that these properties can be obtained under the assumption that infinite switching on the sliding surface between two different control structures is possible, and that the uncertainties are bounded and matched. Sliding mode control consists of two phases: the first phase is the design of a sliding surface along which the process can slide to find its desired final value. The second is the synthesis of the control law in such a way that any state outside the sliding surface is forced to reach the desired sliding manifold in finite time and stay on it. However, one of the major drawback of CSMC is the so-called chattering phenomenon. Such a phenomenon consists of the oscillation of the control signal, tied to the discontinuous nature of the control strategy, at a frequency and with an amplitude capable of disrupting, damaging or, at least, wearing the controlled physical system. Yet, chattering excite high frequency unmodeled dynamics, which degrades the performance of the system, and even destabilize it. A number of direct ways to face the chattering phenomenon from a theoretical viewpoint have been developed such as a saturation function used in the control law instead of the sign function (Slotine & Li, 1991) or a higher order sliding mode control introduced in (Levant, 2003; Cavallo & Natale, 2004; Han & Liu, 2016). The last technique is not only able to resolve the chattering problems but also to ensure the conservation of robustness properties and system performances. However, the second-order sliding mode control is relatively simple to implement and it gives good robustness to external disturbances (Bartolini et al. 2003; Das & Mahanta, 2014; Jedda et al., 2017; Ding & Li, 2017).

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