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Observer-Based Sampled-Data Stabilisation of Switched Systems with Lipschitz Nonlinearities and Dwell-Time

Rami Katz, Antonio Russo, Gian Paolo Incremona, Patrizio Colaneri, Giulia Giordano

Abstract

We investigate the stabilisation of nominally linear switched systems with uncertain Lipschitz nonlinearities under dwell-time constraints, using a sampled-data switching law based on a state observer. We design the switching law based on Lyapunov-Metzler inequalities, accounting for the sampled-data output measurements, and we derive time-dependent LMI conditions for global asymptotic stability of the resulting closed-loop system. We obtain an estimate of the average quadratic cost and a bound on its maximum deviation from the actual cost. We also discuss the feasibility of the derived LMIs, provide equivalent reduced-order LMI conditions, and prove that the time dependence of the LMIs can be removed by discretising on a finite grid. Numerical examples illustrate our theoretical results.

Observer-Based Sampled-Data Stabilisation of Switched Systems with Lipschitz Nonlinearities and Dwell-Time

Abstract

We investigate the stabilisation of nominally linear switched systems with uncertain Lipschitz nonlinearities under dwell-time constraints, using a sampled-data switching law based on a state observer. We design the switching law based on Lyapunov-Metzler inequalities, accounting for the sampled-data output measurements, and we derive time-dependent LMI conditions for global asymptotic stability of the resulting closed-loop system. We obtain an estimate of the average quadratic cost and a bound on its maximum deviation from the actual cost. We also discuss the feasibility of the derived LMIs, provide equivalent reduced-order LMI conditions, and prove that the time dependence of the LMIs can be removed by discretising on a finite grid. Numerical examples illustrate our theoretical results.

Paper Structure

This paper contains 9 sections, 8 theorems, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Observer-based switching law design
  4. Observer design
  5. Proposed switching strategy
  6. Ultimate boundedness for switched affine systems
  7. LMI feasibility guarantees and discretization
  8. Numerical examples
  9. Conclusions

Key Result

Theorem 1

Consider the closed-loop system eq:ClosedLoop under Assumptions ass:lipschitz and ass:symultaneous_LF, with gains $L_i$ fixed (e.g., as in Remark rem:gains). Let $\zeta,T>0$, $\left\{X_i\right\}_{i\in [\ell]}\subseteq \mathbb{R}^{n \times n}$, $X_i\succ 0$, $i\in [\ell]$, and $\Pi \in \mathcal{M}_n$ Given $\left\{ \Psi_i\right\}_{i\in [\ell]}$ in eq:Psi, tuning parameters $h>0$, $\alpha>0$ and $\k

Figures (4)

  • Figure 1: Time evolution of the state $x$ (blue) and of the observed state $\varphi$ (black) in Example \ref{['sec:example_unstable']}.
  • Figure 2: Time evolution of the switching law $\sigma(t)$ in Example \ref{['sec:example_unstable']}.
  • Figure 3: Time evolution of the state $x$ (blue) and of the observed state $\varphi$ (black) in Example \ref{['sec:example_noHurwitz']}.
  • Figure 4: Time evolution of the switching law $\sigma(t)$ in Example \ref{['sec:example_noHurwitz']}.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • ...and 4 more