Robust Stability of Discrete-time Singularly Perturbed Systems with Nonlinear Perturbation

This paper is concerned with the robust stability and stabilization problems of discrete-time singularly perturbed systems (DTSPSs) with nonlinear perturbations. A proper sufficient condition via the fixed-point principle is proposed to guarantee that the given system is in a standard form. Then, based on the singular perturbation approach, a linear matrix inequality (LMI) based sufficient condition is presented such that the original system is standard and input-to-state stable (ISS) simultaneously. Thus, it can be easily verified for it only depends on the solution of an LMI. After that, for the case where the nominal system is unstable, the problem of designing a control law to make the resulting closed-loop system ISS is addressed. To achieve this, a sufficient condition is proposed via LMI techniques for the purpose of implementation. The criteria presented in this paper are independent of the small parameter and the stability bound can be derived effectively by solving an optimal problem. Finally, the effectiveness of the obtained theoretical results is illustrated by two numerical examples.