Generalized quadratic Lyapunov functions for nonlinear/uncertain systems analysis

We consider the class of discrete-lime nonlinear uncertain systems described by the feedback connection of a linear time-invariant system and a "troublesome component", i.e., either a static nonlinearity or a time-varying parametric uncertainty. We propose a generalized quadratic Lyapunov function for stability analysis of such systems. In particular, the Lyapunov function is given by a quadratic form of a vector that depends on the state in a specific nonlinear manner. Introducing a quadratic-form model of the troublesome component in the spirit of integral quadratic constraints, we obtain sufficient conditions for the existence of such Lyapunov functions that proves global/regional stability. The conditions are given in terms of linear matrix inequalities that can be numerically verified in polynomial time.