Computing Bounds on $L_{\infty}$-induced Norm for Linear Time-Invariant Systems Using Homogeneous Lyapunov Functions

TL;DR

This paper introduces a method using homogeneous Lyapunov functions, derived from lifted higher-order systems, to compute tighter bounds on the $L_{ abla}$-induced norm of LTI systems, surpassing traditional quadratic approaches.

Contribution

It proposes a novel approach employing homogeneous Lyapunov functions for less conservative bounds on the $L_{ abla}$-induced norm of LTI systems, improving over quadratic Lyapunov methods.

Findings

Homogeneous Lyapunov functions yield tighter bounds than quadratic functions.

Lifting LTI systems via Kronecker product enables the use of quadratic Lyapunov functions for higher-order systems.

Examples demonstrate significant improvements in bounds using the proposed method.

Abstract

Quadratic Lyapunov function has been widely used in the analysis of linear time invariant (LTI) systems ever since it has shown that the existence of such quadratic Lyapunov function certifies the stability of the LTI system. In this work, the problem of finding upper and lower bounds for the $L_{\infty}$-induced norm of the LTI system is considered. Quadratic Lyapunov functions are used to find the star norm, the best upper on the $L_{\infty}$-induced norm, by bounding the unit peak input reachable sets by inescapable ellipsoids. Instead, a more general class of homogeneous Lyapunov functions is used to get less conservative upper bounds on the $L_{\infty}$-induced norm and better conservative approximations for the reachable sets than those obtained using standard quadratic Lyapunov functions. The homogeneous Lyapunov function for the LTI system is considered to be a quadratic Lyapunov function for a higher-order system obtained by Lifting the LTI system via Kronecker product. Different examples are provided to show the significant improvements on the bounds obtained by using Homogeneous Lyapunov functions.