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

Niclas Tietze, Kai Wulff, Johann Reger

TL;DR

The paper tackles stable trajectory tracking for nonlinear systems in Byrnes-Isidori form under unbounded perturbations by applying Lyapunov redesign with a tube-based (funnel) stability framework. It designs a feedback-linearising controller with nominal and sliding-like components, then uses a contracting Lyapunov-based tube around a known reference to guarantee that the closed-loop state remains within a computable region along the trajectory. A differential inequality for the Lyapunov function together with a pre-computed tube yields local stability, an admissible initial-state set, and practical tracking with arbitrarily small ultimate error by tuning the design parameter μ. The approach also reconciles with classical results for set-point tracking and provides a geometric interpretation via Minkowski sums, making the method amenable to offline tube construction and reachability-style analysis. An illustrative example demonstrates trajectory and set-point tracking, highlighting reduced conservatism when incorporating transient error reduction through the tube-based analysis.

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.

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

TL;DR

The paper tackles stable trajectory tracking for nonlinear systems in Byrnes-Isidori form under unbounded perturbations by applying Lyapunov redesign with a tube-based (funnel) stability framework. It designs a feedback-linearising controller with nominal and sliding-like components, then uses a contracting Lyapunov-based tube around a known reference to guarantee that the closed-loop state remains within a computable region along the trajectory. A differential inequality for the Lyapunov function together with a pre-computed tube yields local stability, an admissible initial-state set, and practical tracking with arbitrarily small ultimate error by tuning the design parameter μ. The approach also reconciles with classical results for set-point tracking and provides a geometric interpretation via Minkowski sums, making the method amenable to offline tube construction and reachability-style analysis. An illustrative example demonstrates trajectory and set-point tracking, highlighting reduced conservatism when incorporating transient error reduction through the tube-based analysis.

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.

Paper Structure

This paper contains 9 sections, 4 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Main Results
  4. Control Design and Closed Loop
  5. Tracking via Tube-Based Analysis
  6. Special Case: Set-Point Tracking
  7. Illustrative Example
  8. Trajectory Tracking
  9. Set-Point Tracking

Key Result

Theorem 4

Consider the closed loop eq:closed_loop for the trajectory $\xi_{\mathrm{d}}$ from eq:desired_state. Let $\mu > 0$ and $c_0 \geq 0$ such that eq:union_in_D is satisfied. Then, the solution $(\xi,\eta)$ satisfies eq:bound_solution for all initial states $(\xi_0,\eta_0) \in \mathcal{V}_{\xi_{\mathrm{d

Figures (4)

  • Figure 1: Solution $\nu_{\mu,c_0}$ of \ref{['eq:differential_equality_bound_lyapunov_function']} for increasing $c_0 = \nu_{\mu,c_0}(0)$.
  • Figure 2: Tube-based stability analysis for a sinusoidal reference, shown in the time domain (top and bottom) and the phase plane (middle).
  • Figure 3: Tube-based stability analysis for a transition. Incorporating the transient decrease of the tracking error enforced by the controller reduces the conservatism of the stability criterion.
  • Figure 4: Stability analysis for set-point tracking. The analysis that incorporates the transient decrease of the tracking error simplifies to the conventional local stability analysis.

Theorems & Definitions (14)

  • Definition 1: Ultimate Boundedness Kha2002
  • Remark 2
  • Remark 3
  • Theorem 4
  • proof
  • Remark 5
  • Corollary 6
  • proof
  • Remark 7: Finite-Time Analysis
  • Remark 8: Discontinuous Case
  • ...and 4 more