A guaranteed fixed-step convergence approach to discrete sliding mode control of linear disturbed systems

Abstract

Current methods for achieving fixed-time convergence in continuous sliding mode control are extensively studied, whereas there are limited studies on fixed-step convergence in discrete sliding mode control. The control objective of this paper aims to develop control schemes assuring that the number of convergence steps of the tracking error remains within the desired range, even as the initial value increases, and this paper proposes a guaranteed fixed-step convergence approach to form the reaching law. An upper bound on convergent steps, independent of the initial value, is established, thereby having the fixed-step convergence property. The performance assessment, including attractiveness, invariance, and convergence steps, is provided to meet the required specifications. The steady-state band results offer guidance in selecting parameters to achieve the control objective. To achieve fixed-step convergence of the tracking error, one can develop a dead-beat terminal sliding mode control scheme, where the tracking error is governed by the prescribed error dynamics. To illustrate the approach more concretely, specific designs are proposed for both switching and non-switching reaching laws aimed at ensuring fixed-step convergence. The simulation and experiment results validate the performance evaluation and demonstrate effectiveness of the proposed control schemes.