State Feedback Fuzzy of Discrete-Time Nonlinear Systems

This chapter focuses on the state feedback SMC problem for discrete-time uncertain nonlinear 1-D systems subject to mismatched parameter uncertainties and external nonlinear disturbance based on T-S fuzzy affine models. Via sufficiently exploring the dynamical properties of the original system and combining with the sliding surface function, a descriptor system is established to characterize the holonomic sliding mode dynamics. Then based on piecewise quadratic Lyapunov functions (PQLFs) and some convexification techniques, the asymptotical stability conditions for the sliding motion are derived under a convex optimization framework, and the stability analysis results on the sliding motion tend to be less conservative. It is worth mentioning that different from the continuous-time situation, the discrete-time PQLFs will jump from one region at the time k to another region at the next time k + 1, which is known as region transition feature and will be elaborated in this chapter in detail. Considering the local models sharing one common control input matrix or not, two state feedback SMC design methods will be proposed, which can both drive the system states into a neighbourhood of the sliding surface. Simulation examples are presented to validate the effectiveness of the proposed methods.