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Small Gain Theorem-Based Robustness Analysis of Discrete-Time MJLSs with the Markov Chain on a Borel Space and Its Application to NCSs

Chunjie Xiao, Ting Hou, Weihai Zhang, Feiqi Deng

TL;DR

The paper addresses robustness analysis for discrete-time Markov jump linear systems (MJLSs) where the Markov chain takes values in a Borel space. It develops a small gain theorem via a Banach-space BRL framework and derives a lower bound on the stability radius, linking robust stability to $H_{\infty}$ performance. To handle the infinite-dimensional $H_{\infty}$-inequalities when the Markov state is continuous, a gridding method converts the problem into finite LMIs, with provable feasibility and controllable conservatism. The approach is applied to networked control systems with random transmission delays modeled as Markov processes, and numerical examples illustrate the finite-LMI certificates and robustness bounds achievable in practice.

Abstract

This paper is concerned with the robustness of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space. For this general class of MJLSs, a small gain theorem is first established and subsequently applied to derive a lower bound of the stability radius. On this basis, with the aid of the extended bounded real lemma and Schur complements, the robust stability problems for the MJLSs are tackled via linear matrix inequality (LMI) techniques. The novel contribution, primarily founded on the scenario where the state space of the Markov chain is restricted in a continuous set, lies in the formulation of a griding approach. The approach converts the existence problem of solutions of an inequality related to $H_{\infty}$ analysis, which is an infinite-dimensional challenge, into a finite-dimensional LMI feasibility problem. As an application, within the framework of MJLSs, a robustness issue of the sampled-data systems is addressed by using a Markov chain, which is determined by the initial distribution and the stochastic kernel, to model transmission delays existing in networked control systems (NCSs). Finally, the feasibility of the results is verified through two examples.

Small Gain Theorem-Based Robustness Analysis of Discrete-Time MJLSs with the Markov Chain on a Borel Space and Its Application to NCSs

TL;DR

The paper addresses robustness analysis for discrete-time Markov jump linear systems (MJLSs) where the Markov chain takes values in a Borel space. It develops a small gain theorem via a Banach-space BRL framework and derives a lower bound on the stability radius, linking robust stability to performance. To handle the infinite-dimensional -inequalities when the Markov state is continuous, a gridding method converts the problem into finite LMIs, with provable feasibility and controllable conservatism. The approach is applied to networked control systems with random transmission delays modeled as Markov processes, and numerical examples illustrate the finite-LMI certificates and robustness bounds achievable in practice.

Abstract

This paper is concerned with the robustness of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space. For this general class of MJLSs, a small gain theorem is first established and subsequently applied to derive a lower bound of the stability radius. On this basis, with the aid of the extended bounded real lemma and Schur complements, the robust stability problems for the MJLSs are tackled via linear matrix inequality (LMI) techniques. The novel contribution, primarily founded on the scenario where the state space of the Markov chain is restricted in a continuous set, lies in the formulation of a griding approach. The approach converts the existence problem of solutions of an inequality related to analysis, which is an infinite-dimensional challenge, into a finite-dimensional LMI feasibility problem. As an application, within the framework of MJLSs, a robustness issue of the sampled-data systems is addressed by using a Markov chain, which is determined by the initial distribution and the stochastic kernel, to model transmission delays existing in networked control systems (NCSs). Finally, the feasibility of the results is verified through two examples.

Paper Structure

This paper contains 19 sections, 17 theorems, 104 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Notations and an Auxiliary Result
  4. Model Description and Basic Definitions
  5. Some Preliminary Robustness Results
  6. The Small Gain Theorem
  7. Stability Radii
  8. Robust Stability
  9. Finite MJLSs
  10. MJLS with Continuous-Valued Markov Jump Parameters
  11. Feasibility and Conservatism of the Gridding Method
  12. An Application to NCSs
  13. Conclusions
  14. Proof of Proposition \ref{['Schurcom']}
  15. Proof of Proposition \ref{['341proposition']}
  16. ...and 4 more sections

Key Result

Proposition 1

For arbitrary $P_{1}\in \mathcal{SH}^{n}_{\infty},\ P_{2}\in \mathcal{H}^{n\times m}_{\infty},$ and $P_{3}\in \mathcal{SH}^{m}_{\infty}$, the following are equivalent: $(i)$$P=\left[ \right]\in \mathcal{H}^{(n+m)+*}_{\infty};$$(ii)$$P_{3}\in \mathcal{H}^{m+*}_{\infty}$ and $P_{1}-P_{2}P_{3}^{-1}P_{

Figures (8)

  • Figure 1: The interconnection.
  • Figure 2: Networked control system.
  • Figure 3: $\sigma_{A,i}, \sigma_{B,i}, \sigma_{C,i}$ and $\sigma_{Q,i}$.
  • Figure 4: $\alpha_{i}, \beta_{i}$ and $\rho_{i}$.
  • Figure 5: $X_{i}$. $X_{i}^{hg}$ is the $(h,g)$th entry of the matrix $X_{i}$, $h,g\in\{1,2\}.$
  • ...and 3 more figures

Theorems & Definitions (47)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • Proposition 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 3
  • ...and 37 more