Incremental Stability and Performance Analysis of Discrete-Time Nonlinear Systems using the LPV Framework

TL;DR

This paper introduces a novel incremental dissipativity framework for discrete-time nonlinear systems using LPV representations, enabling trajectory-based stability and performance analysis with convex conditions.

Contribution

It extends dissipativity analysis to trajectories of nonlinear systems via LPV embedding, addressing limitations of traditional origin-based methods.

Findings

Convex conditions for incremental dissipativity are derived.

The framework is demonstrated on a controlled unbalanced disk system.

Trajectory-based analysis improves stability guarantees.

Abstract

The dissipativity framework is widely used to analyze stability and performance of nonlinear systems. By embedding nonlinear systems in an LPV representation, the convex tools of the LPV framework can be applied to nonlinear systems for convex dissipativity based analysis and controller synthesis. However, as has been shown recently in literature, naive application of these tools to nonlinear systems for analysis and controller synthesis can fail to provide the desired guarantees. Namely, only performance and stability with respect to the origin is guaranteed. In this paper, inspired by the results for continuous-time nonlinear systems, the notion of incremental dissipativity for discrete-time nonlinear systems is proposed, whereby stability and performance analysis is done between trajectories. Furthermore, it is shown how, through the use of the LPV framework, convex conditions can be obtained for incremental dissipativity analysis of discrete-time nonlinear systems. The developed concepts and tools are demonstrated by analyzing incremental dissipativity of a controlled unbalanced disk system.