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Stability under dwell time constraints: Discretization revisited

Thomas Mejstrik, Vladimir Yu. Protasov

Abstract

We decide the stability and compute the Lyapunov exponent of continuous-time linear switching systems with a guaranteed dwell time. The main result asserts that the discretization method with step size~$h$ approximates the Lyapunov exponent with the precision~$C\,h^2$, where~$C$ is a constant. Let us stress that without the dwell time assumption, the approximation rate is known to be linear in~$h$. Moreover, for every system, the constant~$C$ can be explicitly evaluated. In turn, the discretized system can be treated by computing the Markovian joint spectral radius of a certain system on a graph. This gives the value of the Lyapunov exponent with a high accuracy. The method is efficient for dimensions up to, approximately, ten; for positive systems, the dimensions can be much higher, up to several hundreds.

Stability under dwell time constraints: Discretization revisited

Abstract

We decide the stability and compute the Lyapunov exponent of continuous-time linear switching systems with a guaranteed dwell time. The main result asserts that the discretization method with step size~ approximates the Lyapunov exponent with the precision~, where~ is a constant. Let us stress that without the dwell time assumption, the approximation rate is known to be linear in~. Moreover, for every system, the constant~ can be explicitly evaluated. In turn, the discretized system can be treated by computing the Markovian joint spectral radius of a certain system on a graph. This gives the value of the Lyapunov exponent with a high accuracy. The method is efficient for dimensions up to, approximately, ten; for positive systems, the dimensions can be much higher, up to several hundreds.
Paper Structure (14 sections, 5 theorems, 17 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 5 theorems, 17 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Overview of the main results
  4. The structure of the paper
  5. Preliminary facts. Dynamical systems on graphs
  6. The fundamental theorem
  7. Proofs of the main results
  8. Computing the joint spectral radius and an extremal multinorm for a system on a graph
  9. Step 1.
  10. Step 2.
  11. Positive systems
  12. Numerical results
  13. Examples
  14. Markovian modified Gripenberg algorithm

Key Result

Theorem 5

Let ${\mathcal{A}}$ be an irreducible continuous-time linear switching system with the dwell time constraint $m$. Then for every discretization step $h > 0$, we have where $\sigma_h = \ln \hat{\rho}({\mathcal{A}}_h)$, $\|\mathord{\,\cdot\,} \| = \{\|\mathord{\,\cdot\,} \|_j\}_{j=1}^n$ is an extremal multinorm of ${\mathcal{A}}_h$, and

Figures (3)

  • Figure 1: The dynamical system on graph, $n=3$.
  • Figure 2: Construction in proof of Lemma \ref{['l.30']}.
  • Figure 3: Polytopes from Example \ref{['ex:1']}.

Theorems & Definitions (22)

  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Remark 6
  • Corollary 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 12 more