Stability of Lyapunov redesign trajectory tracking control with unbounded perturbations -- A tube-based stability analysis

TL;DR

This paper extends Lyapunov redesign for trajectory tracking in nonlinear systems with unbounded perturbations, using tube-based stability analysis to ensure solutions stay within a contracting region around the reference.

Contribution

It generalizes classical stability criteria to nonconstant trajectories by incorporating reference trajectory dynamics into Lyapunov redesign with a tube-based approach.

Findings

Guarantees closed-loop solutions remain in a contracting tube.

Extends stability analysis to nonconstant reference trajectories.

Incorporates transient error decrease in Lyapunov redesign.

Abstract

Considering a nonlinear system in Byrnes-Isidori form that is subject to unbounded perturbations, we apply Lyapunov redesign via feedback linearisation for trajectory tracking. Leveraging the ideas of tube-based geometric characterisation of the invariance properties of the closed loop, we generalise the classical stability criterion from the~literature from constant to nonconstant reference trajectories. The proposed analysis is tailored to the Lyapunov redesign and the tracking problem insofar as we incorporate the reference trajectory and the transient decrease of the tracking error enforced by the controller. In particular, we exploit that the Lyapunov function of the tracking error satisfies a differential inequality, thereby guaranteeing that the solution of the closed loop remains in a contracting tube along the reference trajectory.