[Paper Review] Stability analysis for a class of linear systems governed by difference equations

Sérine Damak, Michaël Di Loreto+2 more|arXiv (Cornell University)|Dec 27, 2013

Stability and Control of Uncertain Systems8 references3 citations

TL;DR

This paper presents a Lyapunov-Krasovskii approach to analyze exponential stability for linear systems governed by continuous-time difference equations with multiple delays. It establishes sufficient LMI-based conditions for exponential stability, with necessary and sufficient conditions derived for commensurate delays, and provides estimations of the decay rate and robustness analysis under parametric uncertainties and time-varying delays.

ABSTRACT

Linear systems governed by continuous-time difference equations cover a wide class of linear systems. From the Lyapunov-Krasovskii approach, we investigate stability for such a class of systems. Sufficient conditions, and in some particular cases, necessary and sufficient conditions for exponential stability are established, for multivariable systems with commensurate or rationally independent delays. A discussion on robust stability is proposed, for parametric uncertainties and time-varying delays.

Motivation & Objective

• To develop sufficient conditions for exponential stability of linear systems governed by continuous-time difference equations with multiple delays.
• To derive necessary and sufficient conditions for exponential stability in the case of commensurate delays.
• To extend stability analysis to systems with parametric uncertainties and time-varying delays.
• To provide estimations of the exponential decay rate using Lyapunov-Krasovskii functionals.
• To ensure robustness of stability under norm-bounded parametric uncertainties and time-varying delays.

Proposed method

• Utilizes the Lyapunov-Krasovskii functional approach to analyze stability of systems described by $ x(t) = \sum_{k=1}^{N} A_k x(t - r_k) $.
• Derives sufficient LMI conditions for exponential stability using a Lyapunov functional with matrix variables $ Q_k $ and $ \eta $.
• Establishes necessary and sufficient conditions for exponential stability when delays are commensurate, based on spectral radius analysis.
• Applies spectral analysis and Banach algebra techniques to prove boundedness and invertibility of the system distribution $ g(t) = \delta(t) - \sum A_k \delta(t - r_k) $.
• Uses Laplace transform and convolution to express the solution as $ x(t) = h(t) \ast \psi(t) $, where $ h \in \ell_1^{n \times n} $, ensuring boundedness.
• Introduces a transformed variable $ z(t, \varphi) = e^{\gamma t} x(t, \varphi) $ to prove exponential decay with rate $ \gamma < \mu $, where $ \mu $ is derived from the LMI conditions.

Experimental results

Research questions

• RQ1 What are sufficient LMI-based conditions for exponential stability in linear systems with multiple delays governed by difference equations?
• RQ2 Under what conditions are the stability criteria both necessary and sufficient, particularly for commensurate delays?
• RQ3 How can the exponential decay rate be estimated and improved compared to prior LMI-based results?
• RQ4 What is the robustness of the stability conditions under parametric uncertainties and time-varying delays?
• RQ5 Can the solution be proven bounded and exponentially decaying using functional analysis and Laplace transform techniques?

Key findings

• Sufficient LMI conditions are established for exponential stability of multivariable linear systems with commensurate or rationally independent delays.
• For commensurate delays, necessary and sufficient conditions for exponential stability are derived, expressed as spectral radius inequalities.
• The exponential decay rate $ \gamma $ is estimated via the LMI conditions, with $ \|x_t(\varphi)\|_{L_2} \leq 4.9125 \|\varphi\|_c e^{-\gamma t} $ for $ \|\varphi\|_c = 2 $, as shown in Example 7.
• Robustness to parametric norm-bounded uncertainties is analyzed, with stability preserved under small perturbations.
• Sufficient conditions for exponential stability are extended to time-varying delays, using a transformed variable approach.
• The solution $ x(t, \varphi) $ is proven bounded and exponentially decaying via the invertibility of the system distribution $ g $ in the Banach algebra $ \ell_1^{n \times n} $.