Variable Structure Control Design for Uncertain Discrete-Time Systems

The main differences between the designs of discrete-time variable structure systems (VSS) and continuous-time VSS are the determination of the switching hyperplane and the satisfaction of the hitting/sliding condition. These differences yield many difficulties in the design of sliding mode control for the discrete-time VSS. This note tries to overcome these difficulties and design a simple sliding mode control for a single input discrete-time VSS such that not only the discrete-time uncertain system is stabilized robustly, but also the chattering along the sliding mode is reduced explicitly. Instead of the conventional switching hyperplane, in the discrete-time case a “switching region,” which includes the hyperplane So and its neighborhood, is determined and used. The determination of the switching region is to locate the equilibrium point of the nominal subsystem on each hyperplane Sp outside the switching region such that the state motions inside and outside the switching region will occur alternately until the state of the closed-loop system reaches the origin. Moreover, we prove that the robustness property is retained whenever the uncertainty satisfies a standard matching condition.