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Accelerated Stabilization of Switched Linear MIMO Systems using Generalized Homogeneity

Moussa Labbadi, Andrey Polyakov, Denis Efimov

TL;DR

The paper addresses stabilizing switched linear MIMO systems under uncertainty by developing a generalized homogenization framework that renders the closed-loop dynamics homogeneous of a prescribed degree using a linear state feedback plus a homogeneous correction. It unifies exponential, finite-time, and nearly fixed-time convergence under both common and mode-dependent Lyapunov function settings, with stability guarantees under dwell-time conditions and LMIs that synthesize controller gains. Robustness to disturbances is analyzed, and the approach is validated through numerical simulations on representative switched MIMO systems. The work offers a systematic design procedure with practical implications for rapid stabilization in switched networks and uncertain environments.

Abstract

This paper addresses the problem of exponential and accelerated finite-time, as well as nearly fixed-time, stabilization of switched linear MIMO systems. The proposed approach relies on a generalized homogenization framework for switched linear systems and employs implicit Lyapunov functions for control design, covering both common and multiple Lyapunov function settings. Linear matrix equations and inequalities are derived to characterize the dilation generator and to synthesize the controller gains. Robustness of the resulting control laws with respect to system uncertainties and external disturbances is analyzed. The effectiveness of the proposed approach is illustrated through numerical examples.

Accelerated Stabilization of Switched Linear MIMO Systems using Generalized Homogeneity

TL;DR

The paper addresses stabilizing switched linear MIMO systems under uncertainty by developing a generalized homogenization framework that renders the closed-loop dynamics homogeneous of a prescribed degree using a linear state feedback plus a homogeneous correction. It unifies exponential, finite-time, and nearly fixed-time convergence under both common and mode-dependent Lyapunov function settings, with stability guarantees under dwell-time conditions and LMIs that synthesize controller gains. Robustness to disturbances is analyzed, and the approach is validated through numerical simulations on representative switched MIMO systems. The work offers a systematic design procedure with practical implications for rapid stabilization in switched networks and uncertain environments.

Abstract

This paper addresses the problem of exponential and accelerated finite-time, as well as nearly fixed-time, stabilization of switched linear MIMO systems. The proposed approach relies on a generalized homogenization framework for switched linear systems and employs implicit Lyapunov functions for control design, covering both common and multiple Lyapunov function settings. Linear matrix equations and inequalities are derived to characterize the dilation generator and to synthesize the controller gains. Robustness of the resulting control laws with respect to system uncertainties and external disturbances is analyzed. The effectiveness of the proposed approach is illustrated through numerical examples.
Paper Structure (10 sections, 14 theorems, 88 equations, 3 figures)

This paper contains 10 sections, 14 theorems, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Stability Notions
  5. Generalized Homogeneity
  6. On the Matrix Equation $XA - AX = X$
  7. Stabilization of Linear MIMO Systems
  8. Main result
  9. Simulations
  10. Conclusions

Key Result

Theorem 1

(see Bhat2000Laghrouche2017) Let $V :\mathbb{R}^n\to\mathbb{R}^+$ be a positive definite $C^1$ function on an open neighborhood of the origin $\mathcal{D} \subseteq \mathbb{R}^n$, and there exist real constants $\kappa > 0$ and $\alpha \geq 0$, such that for the system (eq:SN): Then, depending on $\alpha$, the origin exhibits stability with the following types of convergence: If $\mathcal{D} = \

Figures (3)

  • Figure 1: Switching signal: 0 corresponds to subsystem 1 ($A_1, B_1$), 1 corresponds to subsystem 2 ($A_2, B_2$).
  • Figure 2: System states and control input versus time for the finite-time controller.
  • Figure 3: System states and control input versus time for the nearly fixed-time controller.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Theorem 2
  • Definition 4
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Theorem 4
  • ...and 12 more