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 S₀ 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 S_p 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