A necessary and sufficient LMI condition for stability of 2D mixed continuous-discrete-time systems

Abstract

This paper addresses the problem of establishing stability of 2D mixed continuous-discrete-time systems. Traditional stability analysis for 2D systems gives a sufficient condition based on 2D version of a Lyapunov equation. Here, a linear matrix inequality (LMI) condition is proposed that extends these results by introducing complex Lyapunov functions depending polynomially on a parameter and by exploiting the Gram matrix method. It is shown that this condition is sufficient for 2D exponential stability for any chosen degree of the Lyapunov function candidate, and it is also shown that this condition is also necessary for a sufficiently large degree. Moreover, an a priori bound on the degree required for achieving necessity is given. Some numerical examples illustrate the proposed methodology.