Lyapunov Differential Equation Hierarchy and Polynomial Lyapunov Functions for Switched Linear Systems

This work studies the problem of searching for homogeneous polynomial Lyapunov functions for stable switched linear systems. Specifically, we show an equivalence between polynomial Lyapunov functions for systems of this class and quadratic Lyapunov functions for a related hierarchy of Lyapunov differential equations. This creates an intuitive procedure for checking the stability properties of switched linear systems, and a computationally competitive algorithm is presented for generating high-order homogeneous polynomial Lyapunov functions in this manner. Additionally, we provide a comparison between polynomial Lyapunov functions generated with our proposed approach and Lyapunov functions generated with a more traditional sum-of-squares based approach.