Multiple Delay Case
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
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Multiple Delay Case
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.
3.1 Preliminaries We consider now a retarded type time-delay system of the form dx(t) = dt
m
∑ A j x(t − h j ),
t ≥ 0,
(3.1)
j=0
where A j , j = 0, 1, . . . , m, are given real n × n matrices and 0 = h0 < h1 < · · · < hm = h are time delays. V.L. Kharitonov, Time-Delay Systems: Lyapunov Functionals and Matrices, Control Engineering, DOI 10.1007/978-0-8176-8367-2 3, © Springer Science+Business Media, LLC 2013
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3 Multiple Delay Case
Definition 3.1 ([3]). The n × n matrix K(t) is said to be the fundamental matrix for any system of the form (3.1) there is only one fundamental matrix d K(t) = dt
m
∑ K(t − h j )A j ,
t ≥ 0,
(3.2)
j=0
and K(t) = 0n×n for t < 0, K(0) = I. Theorem 3.1 ([3]). Given an initial function ϕ ∈ PC([−h, 0], Rn ), the following equality holds: m
x(t, ϕ ) = K(t)ϕ (0) + ∑
0
K(t − θ − h j )A j ϕ (θ )dθ ,
t ≥ 0.
(3.3)
j=1 −h j
This equality is known as the Cauchy formula for system (3.1). Proof. Assume that t > 0, and compute for ξ ∈ (0,t) the partial derivative m ∂ [K(t − ξ )x(ξ , ϕ )] = − ∑ K(t − ξ − h j )A j x(ξ , ϕ ) ∂ξ j=0 +K(t − ξ )
m
∑ A j x(ξ − h j , ϕ )
j=0
=
m
∑ [K(t − ξ )A j x(ξ − h j , ϕ ) − K(t − ξ − h j )A j x(ξ , ϕ )] .
j=1
Integrating the preceding equality by ξ on the segment [0,t], we obtain that the integral of the left-hand side is equal to t 0
∂ [K(t − ξ )x(ξ , ϕ )] dξ = x(t, ϕ ) − K(t)ϕ (0). ∂ξ
Before computing the integral on the right-hand side we evaluate the integral J=
t
[K(t − ξ )A j x(ξ − h j , ϕ ) − K(t − ξ − h j )A j x(ξ , ϕ )] dξ
0 t−h j
=
K(t − θ − h j )A j x(θ , ϕ )dθ −
−h j
+
0
−
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