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On hyperexponential stabilization of a chain of integrators in continuous and discrete time subject to unmatched perturbations

Moussa Labbadi, Denis Efimov

TL;DR

This work generalizes hyperexponential stabilization from a double integrator to an arbitrary-order chain of integrators subject to unmatched perturbations by designing a recursive, time-varying feedback with increasing gains. In continuous time, the method yields hyperexponential convergence for $x_1$ while ensuring $x_2$ remains bounded and higher-order states are ISS through gain saturation; a corresponding implicit Euler discretization preserves these properties in discrete time. The authors provide explicit constructions for low-order cases, establish conditions on the gain sequence, and prove robustness to disturbances, including a coordinate transformation that isolates matched and mismatched effects. Numerical examples and simulations illustrate rapid convergence and bounded responses, highlighting practical applicability to fast stabilization under uncertainty. Overall, the approach offers a principled way to achieve accelerated stabilization of high-order systems with unmatched perturbations in both continuous and discrete domains.

Abstract

A recursive time-varying state feedback is presented for a chain of integrators with unmatched perturbations in continuous and discrete time. In continuous time, it is shown that hyperexponential convergence is achieved for the first state variable \(x_1\), while the second state \(x_2\) remains bounded. For the other states, we establish ISS {\cb property} by saturating the growing {\cb control} gain. In discrete time, we use implicit Euler discretization to {\cb preserve} hyperexponential convergence. The main results are demonstrated through several examples of the proposed control laws, illustrating the conditions established for both continuous and discrete-time systems.

On hyperexponential stabilization of a chain of integrators in continuous and discrete time subject to unmatched perturbations

TL;DR

This work generalizes hyperexponential stabilization from a double integrator to an arbitrary-order chain of integrators subject to unmatched perturbations by designing a recursive, time-varying feedback with increasing gains. In continuous time, the method yields hyperexponential convergence for while ensuring remains bounded and higher-order states are ISS through gain saturation; a corresponding implicit Euler discretization preserves these properties in discrete time. The authors provide explicit constructions for low-order cases, establish conditions on the gain sequence, and prove robustness to disturbances, including a coordinate transformation that isolates matched and mismatched effects. Numerical examples and simulations illustrate rapid convergence and bounded responses, highlighting practical applicability to fast stabilization under uncertainty. Overall, the approach offers a principled way to achieve accelerated stabilization of high-order systems with unmatched perturbations in both continuous and discrete domains.

Abstract

A recursive time-varying state feedback is presented for a chain of integrators with unmatched perturbations in continuous and discrete time. In continuous time, it is shown that hyperexponential convergence is achieved for the first state variable , while the second state remains bounded. For the other states, we establish ISS {\cb property} by saturating the growing {\cb control} gain. In discrete time, we use implicit Euler discretization to {\cb preserve} hyperexponential convergence. The main results are demonstrated through several examples of the proposed control laws, illustrating the conditions established for both continuous and discrete-time systems.

Paper Structure

This paper contains 13 sections, 9 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform hyperexponential stability
  4. Auxiliary properties
  5. Problem statement
  6. Control in continuous time
  7. Special cases
  8. Second-order system
  9. Third-Order System
  10. Discretization of the controlled system \ref{['eq:sysm-0']}
  11. Illustrative examples
  12. Simulations
  13. Conclusions

Key Result

Lemma 1

labbadi2024hyperexponential For all $\tau \geq 0$, and $a,\alpha>0$ such that $\alpha a > 1$, the following holds: where

Figures (2)

  • Figure 1: Objective vs time.
  • Figure 2: Error norms in logarithmic scale vs time.

Theorems & Definitions (17)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • Theorem 2
  • Corollary 3
  • proof
  • ...and 7 more