Global Stability Analysis of Nonlinear Sampled-Data Systems using Convex Methods

TL;DR

This paper develops convex optimization-based stability conditions for nonlinear sampled-data systems using Lyapunov functions that accommodate variable sampling periods, demonstrated through numerical examples.

Contribution

It introduces a novel convex optimization framework employing Sum-of-Squares methods for stability analysis of nonlinear sampled-data systems with fixed and variable sampling intervals.

Findings

Stability conditions derived using Lyapunov functions with slack variables.

Applicable to both fixed and time-varying sampling periods.

Validated through numerical examples demonstrating effectiveness.

Abstract

We consider the problem of global stability of nonlinear sampled-data systems. Sampled-data systems are a form of hybrid model which arises when discrete measurements and updates are used to control continuous-time plants. In this paper, we use a recently introduced Lyapunov approach to derive stability conditions for both the case of fixed sampling period (synchronous) and the case of a time-varying sampling period (asynchronous). This approach requires the existence of a Lyapunov function which decreases over each sampling interval. To enforce this constraint, we use a form of slack variable which exists over the sampling period, may depend on the sampling period, and allows the Lyapunov function to be temporarily increasing. The resulting conditions are enforced using a new method of convex optimization of polynomial variables known as Sum-of-Squares.We use several numerical examples to illustrate this approach.