Affine Parameter-Dependent Lyapunov Functions for LPV Systems with Affine Dependence

TL;DR

This paper introduces a unified approach using affine parameter-dependent Lyapunov functions to certify robust stability, convergence, and performance of LPV systems in both continuous and discrete time, highlighting differences in treatment.

Contribution

It provides a novel, uniform method for LPV system certification that accounts for differences between CT and DT cases, improving over previous approaches.

Findings

The proposed method is less conservative than existing slack variable approaches.

It successfully certifies robust stability and performance in both CT and DT LPV systems.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

This paper deals with the certification problem for robust quadratic stability, robust state convergence, and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters. We consider both continuous-time (CT) and discrete-time (DT) parameter-varying systems. In this paper, we provide a uniform method for this certification problem in both cases and we show that, contrary to what was claimed previously, the DT case requires a significantly different treatment compared to the existing CT results. In the established uniform approach, quadratic Lyapunov functions, that are affine in the parameter, are used to certify robust stability, robust convergence rates, and robust performance in terms of linear matrix inequality feasibility tests. To exemplify the procedure, we solve the certification problem for $\mathscr{L}_2$-gain performance both in the CT and the DT cases. A numerical example is given to show that the proposed approach is less conservative than a method with slack variables.