Exact LMI Conditions for Stability and $\mathcal {L}_2$ Gain Analysis of 2-D Mixed Continuous–Discrete Time Systems via Quadratically Frequency-Dependent Lyapunov Functions

This article addresses the problems of establishing structural stability and the $\mathcal {L}_2$ gain of 2-D mixed continuous–discrete time systems. The first contribution is to show that Lyapunov functions quadratically dependent on the frequency are exact for establishing structural stability. This is particularly important since the existing works that exploit Lyapunov functions provide a much larger upper bound on the dependence on the frequency or other parameters. The second contribution is to propose a novel linear matrix inequality (LMI) necessary and sufficient condition for establishing the existence of such Lyapunov functions. It is shown, analytically and through several examples, for both best and worst cases, that the numerical complexity of this novel condition is significantly smaller than that of the existing methods. The third contribution is to show that the proposed methodology can be used to establish upper bounds on the $\mathcal {L}_2$ gain, in particular, deriving a novel necessary and sufficient LMI condition based on Lyapunov functions quadratically dependent on the frequency. Finally, the article presents the generalization of the proposed methodology to nonmixed 2-D systems.