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Multiple Delay Case

  • Vladimir L. Kharitonov
Chapter
Part of the Control Engineering book series (CONTRENGIN)

Abstract

In this chapter we address the case of retarded type linear time-delay systems with multiple delays. The fundamental matrix of such a system is defined. Then this matrix is used to derive an explicit expression for the solution of an initial value problem. Applying the scheme presented in the previous chapter, a general form of quadratic functionals with prescribed time derivatives along the solutions of the time-delay systems is obtained. It is shown that the functionals are defined by special matrix valued functions known as Lyapunov matrices for the system. A special system of matrix equations that defines Lyapunov matrices is derived. It is shown that the system admits a unique solution if and only if the spectrum of the time-delay system does not contain points arranged symmetrically with respect to the origin of the complex plane. This spectrum property is known as a Lyapunov condition. Two numerical schemes for the computation of Lyapunov matrices are presented. The first one is applicable to the case where time delays are multiple to a basic one. The other one makes it possible to compute approximate Lyapunov matrices in the case of general time delays. A measure that allows one to estimate the quality of the approximation is provided as well. Quadratic functionals of the complete type are defined, and several important applications of the functionals are presented in the final part of the chapter.

Keywords

Lyapunov Matrices Lyapunov Condition Delay-free System Quadratic Functionals Thespectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Vladimir L. Kharitonov
    • 1
  1. 1.Faculty of Applied Mathematics and Processes of ControlSaint Petersburg State UniversitySaint PetersburgRussia

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