Robust orbital stabilization: A Floquet theory–based approach

• Published in: International journal of robust and nonlinear control Vol. 31; no. 16; pp. 8075 - 8108
• Main Authors: Sætre, Christian Fredrik, Shiriaev, Anton S., Freidovich, Leonid B., Gusev, Sergei V., Fridman, Leonid M.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 10.11.2021
• Subjects: Disturbances; Feedback; Invariants; Manifolds; Orbital stability; orbital stabilization; Orbits; robust nonlinear control; Robustness; Sliding; sliding mode control; Uncertainty; underactuated mechanical systems
• ISSN: 1049-8923; 1099-1239; 1099-1239
• DOI: 10.1002/rnc.5738

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 article 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.