Parameter-Dependent S-Procedure And Yakubovich Lemma

TL;DR

This paper investigates parameter-dependent LMIs related to the S-procedure and Yakubovich lemma, showing continuous solutions exist under certain conditions and proposing methods for parameter-dependent Lyapunov functions in uncertain nonlinear systems.

Contribution

It introduces a new approach to solve parameter-dependent LMIs by reducing them to higher-dimensional parameter-independent LMIs, extending Yakubovich lemma generalizations.

Findings

Feasible solutions depend continuously on parameters.

Polynomially parameter-dependent LMIs can be transformed into higher-dimensional LMIs.

Method for constructing parameter-dependent Lyapunov functions for uncertain systems.

Abstract

The paper considers a linear matrix inequality (LMI) that depends on a parameter varying in a compact topological space. It turns out that if a strict LMI continuously depends on a parameter and is feasible for any value of that parameter, then it has a solution which continuously depends on the parameter. The result holds true for LMIs that arise in S-procedure and Yakubovich lemma. It is shown that the LMI which is polynomially dependent on a vector of parameters can be reduced to a parameter-independent LMI of a higher dimension. The result is based on the recent generalization of Yakubovich lemma proposed by Iwasaki and Hara and another generalization formulated in this paper. The problem of positivity verification for a non-SOS polynomial of two variables is considered as an example. To illustrate control applications, a method of parameter-dependent Lyapunov function construction is proposed for nonlinear systems with parametric uncertainty.