Stabilization of Control-Affine Systems by Local Approximations of Trajectories

TL;DR

This paper investigates how to stabilize control-affine systems using local trajectory approximations and output feedback, leveraging Lie brackets and convergence properties to ensure exponential stability.

Contribution

It introduces a novel approach to extremum seeking control by approximating Lie brackets through trajectory sequences, enabling stabilization with output feedback.

Findings

Establishes convergence of trajectories under specific control sequences.

Shows boundedness of control vector fields improves approximation quality.

Derives conditions for exponential stability of the proposed control law.

Abstract

We study convergence and stability properties of control-affine systems. Our considerations are motivated by the problem of stabilizing a control-affine system by means of output feedback for states in which the output function attains an extreme value. Following a recently introduced approach to extremum seeking control, we obtain access to the ascent and descent directions of the output function by approximating Lie brackets of suitably defined vector fields. This approximation is based on convergence properties of trajectories of control-affine systems under certain sequences of open-loop controls. We show that a suitable notion of convergence for the sequences of open-loop controls ensures local uniform convergence of the corresponding sequences of flows. We also show how boundedness properties of the control vector fields influence the quality of the approximation. Our convergence results are sufficiently strong to derive conditions under which the proposed output feedback control law induces exponential stability. We also apply this control law to the problem of purely distance-based formation control for nonholonomic multi-agent systems.