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On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays

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  • M. De la Sen

Abstract

This paper investigates the causality properties of a class of linear time‐delay systems under constant point delays which possess a finite set of distinct linear time‐invariant parameterizations (or configurations) which, together with some switching function, conform a linear time‐varying switched dynamic system. Explicit expressions are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in‐between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.

Suggested Citation

  • M. De la Sen, 2009. "On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2009(1).
    • Handle: RePEc:wly:jnlaaa:v:2009:y:2009:i:1:n:670314
    DOI: 10.1155/2009/670314
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    References listed on IDEAS

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    1. JinRong Wang & X. Xiang & W. Wei, 2008. "Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space," Abstract and Applied Analysis, Hindawi, vol. 2008, pages 1-19, February.
    2. Bogdan Sasu, 2008. "Robust Stability and Stability Radius for Variational Control Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2008(1).
    3. Chengmin Hou & Sui Sun Cheng, 2009. "Asymptotic Dichotomy in a Class of Fourth‐Order Nonlinear Delay Differential Equations with Damping," Abstract and Applied Analysis, John Wiley & Sons, vol. 2009(1).
    4. JinRong Wang & X. Xiang & W. Wei, 2008. "Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space," Abstract and Applied Analysis, John Wiley & Sons, vol. 2008(1).
    5. Chengmin Hou & Sui Sun Cheng, 2009. "Asymptotic Dichotomy in a Class of Fourth-Order Nonlinear Delay Differential Equations with Damping," Abstract and Applied Analysis, Hindawi, vol. 2009, pages 1-7, June.
    6. Bogdan Sasu, 2008. "Robust Stability and Stability Radius for Variational Control Systems," Abstract and Applied Analysis, Hindawi, vol. 2008, pages 1-29, April.
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    Cited by:

    1. Nai-feng Gan & Yu-feng LU & Ting Gong, 2014. "Stabilization with Internal Loop for Infinite-Dimensional Discrete Time-Varying Systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-7, April.
    2. P. Niamsup & G. Rajchakit, 2013. "New Results on Robust Stability and Stabilization of Linear Discrete‐Time Stochastic Systems with Convex Polytopic Uncertainties," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    3. M. De la Sen, 2012. "Stability of Switched Feedback Time‐Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).

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