Robust Orbital Stabilization: A Floquet Theory-based Approach

TL;DR

This paper introduces a Floquet theory-based method for designing robust orbitally stabilizing feedback controllers that ensure stability despite uncertainties and disturbances, demonstrated on an underactuated Cart-Pendulum system.

Contribution

It provides a constructive, Floquet-Lyapunov transformation-based procedure for designing sliding-mode control laws for robust orbital stabilization.

Findings

Ensures asymptotic stability under uncertainties and disturbances.

Effective control of an underactuated Cart-Pendulum system.

Demonstrates robustness through simulation results.

Abstract

The design of robust orbitally stabilizing feedback is considered. From a known orbitally stabilizing controller for a nominal, disturbance-free system, a robustifying feedback extension is designed utilizing the sliding-mode control (SMC) methodology. The main contribution of the paper is to provide a constructive procedure for designing the time-invariant switching function used in the SMC synthesis. More specifically, its zero-level set (the sliding manifold) is designed using a real Floquet-Lyapunov transformation to locally correspond to an invariant subspace of the Monodromy matrix of a transverse linearization. This ensures asymptotic stability of the periodic orbit when the system is confined to the sliding manifold, despite any system uncertainties and external disturbances satisfying a matching condition. The challenging task of oscillation control of the underactuated Cart-Pendulum system subject to both matched and unmatched disturbances/uncertainties demonstrates the efficacy of the proposed scheme.