Synthesis of Limit Cycles and Reference Tracking via Switching Affine Systems

TL;DR

This paper presents a new method for approximating nonlinear limit cycles using switching affine systems in higher-dimensional spaces, enabling stable modeling and effective reference tracking.

Contribution

It introduces a general partition-based synthesis scheme with stability guarantees and a multi-Lyapunov approach for improved reference tracking in switching affine systems.

Findings

The method guarantees globally stable limit cycles.

It minimizes data-model error through constrained optimization.

The approach improves convergence in reference tracking.

Abstract

This paper introduces a novel method to approximate limit cycles of nonlinear ODEs by use of switching affine dynamics in order to ease data-based modeling and analysis. Previous approaches to approximating limit cycles by switching systems have been largely confined to simple partitions into two-regions or low-dimensional (often planar) settings. In contrast, this study utilizes more general partitions in higher-dimensional state spaces, augmented by external signals, to develop a synthesis scheme that guarantees a globally stable limit cycle. The synthesis task is formulated and solved based on constrained numerical optimization. Starting from sampled data of the nonlinear dynamics, the method minimizes the error between the data and the limit cycle generated by the switching affine model, while employing stability constraints to ensure global stability. Based on the obtained model, the paper tackles the problem of reference tracking for switching affine systems with periodic behavior. While the approximation scheme is based on a common Lyapunov function, the reference tracking approach uses multiple Lyapunov functions to achieve less conservative convergence results. The principle and effectiveness of the proposed methods are illustrated through a set of examples.