In this paper, a generalized discrete time reaching law for sliding variables with arbitrary relative degree is proposed. The reaching law is based on Gao's strategy and ensures higher order switching type sliding motion, which is a novel concept in the field of discrete time plants. The controller synthesis process is simplified by performing a specific system state transformation. It is demonstrated that the width of the layer to which the representative point of the system is driven decreases as the relative degree of the sliding variable gets bigger. It has further been proven that achieving a narrower quasi-sliding mode band width is reflected in reduced state error. Finally, the reaching phase elimination for variables with arbitrary relative degree is considered. This is obtained by implementing a time-varying sliding surface in a finite number of initial time instants. Uniform ultimate boundedness of all state variables is guaranteed.

This paper presents the first attempt to design a reaching law-based discrete-time sliding mode controller with a relative degree 2 switching variable. The current value of this variable is only affected by the control signal and disturbance generated two time instants ago. It is demonstrated that the new reaching law-based strategy offers a smaller quasi-sliding mode band width when compared to a similar control scheme with relative degree 1 switching variable. This in turn leads to reduced system output error in the control system proposed in the paper.

This paper presents a concise survey on reaching law based sliding mode control. The goal is to highlight the most important developments related to reaching laws for both continuous time systems and discrete time ones. First, origins of reaching law methodology for continuous time systems are brought up and essential properties of these control methods are discussed. Advances in continuous time reaching laws are subsequently surveyed and several examples of such strategies are given. Then, discrete time reaching laws are reviewed. Furthermore, the division between switching type and non-switching type discrete reaching laws is elaborated upon and examples from both categories are provided.

Reaching law based discrete sliding mode controllers are well known to be easy to tune and robust with respect to disturbance. In recent years it has been demonstrated that their robustness can be further enhanced with the use of arbitrary relative degree sliding variables. However, reaching laws using such variables only ensure good sliding mode performance of the system when the perturbations affecting the plant are matched, which is a very restrictive assumption. To address this issue and to further improve robustness of the plant, in this paper we propose a new model reference approach for strategies with relative degree two sliding variables. In the proposed approach, the reaching law is first used to control the evolution of a disturbance free model of the plant, and then the original system state is driven towards that of the model with a secondary controller. It will be shown that the proposed approach ensures better system robustness compared to the conventional reaching law approach and that it does not require the assumption about matched uncertainties.

Variable structure control systems are known to provide a high level of robustness to external disturbances and modeling uncertainties with comparably low computational complexity. Thanks to these features, they have found applications in various fields, such as power engineering, electronics, robotics, and aviation. In recent decades, the field of sliding mode control has developed significantly. Therefore, this study aims to discuss the basic concepts and design methodology of such strategies. Although in the 20th century, continuous-time sliding mode control has been the center of the control engineering society’s attention, it has certain major shortcomings. In particular, such control schemes result in undesirable high-frequency oscillations when applied digitally. Therefore, the more recent discrete-time approach to sliding mode control has gained recognition in the 21st century. Since the introduction of discrete-time sliding mode control strategies, the reaching law-based controller design method has been designed, within which two main paradigms may be named: the switching type and the nonswitching type quasi-sliding mode. This paper presents a broad review of the discrete-time sliding mode control strategies, starting from the definition of sliding mode through the controller design procedures and up to potential applications. The aim of this study is to provide an up-to-date state of the art and introduce readers to the newest trends and achievements in the field of sliding mode control.