Delay-dependent Guaranteed Cost Control For Uncertain Singular Time-delay Systems: The Case Of State Feedback

This paper considers the problem of state feedback guaranteed cost controller design for singular time-delay systems with norm-bounded parameter uncertainty. The whole paper is divided into five sections.Section one is the introdution.Section two are problem formulation and preliminary definitions. We consider a class of uncertain singular time-delay system represented bywhere x(t) ∈ Rⁿ, u(t) ∈ R^m are the state and control input, respectively. E,A, A_Ï„ and B are known real constant matrices with appropriate dimensions and 0 m. φ(t) ∈ C_n,Ï„ is a compatible initial function. ΔA, ΔA_Ï„ and ΔB are time-invariant matrices representing norm-bounded parametric uncertainties which are of the following formwhere D,E₁,E_Ï„,E₂ are known real constant matrices and F is an uncertain real constant matrix. Given positive definite symmetric matrices R and S, we consider the cost functionalThe problem is to design a controllerK is a constant matrix, such that for all admissible uncertainties , the closed-loop system is regular, impulse free, zero solution asymptotically stable and the closed-loop value of the cost functional J satisfies a bound J* .The nominal unforced singular time-delay system of (1) isEx{t) = Ax(t) + ATx{t - t) x{t) = lx(t) = [ y{(t) yT{there y\(t) € Rp,y2{t) 6 Z?"^p,the system (1) and its corresponding normal unforced singular time-delay system (2) can be transformated respectively into :Ey(t) =y(t) =Ey{t) = Ay{t) + Ary(t - r) y(t) = N-l(t), te[-r,0] V /(4)whereA = MAN =A A11 ^1221 T = MATN =TUAA AAT AB | = j MAAN MAArN MAB \ =AD2 =[ En E n El2, B=MB=Ex ET E2 ET = ETN =\ ETl Eobviously, system (1) and (2) are respectively r.s.e. (restricted system equivalence)[11] equivalent to (3) and (4). So there only differs a full-rank coordinate transformation matrix between the states of the two systems. Moreover, the two systems have the same transformation funtions. Hence we can discuss the problem by system (3) instead of system (1). Section two narrates this process.Section three and section four are the main work of this paper. In section three, bydsquoting a Lyapunov-Krasovskii functionalV(yt) = )dadP, t>rwe establish a delay-dependent stability criterion for the normal unforced singular time-delay system i.e. theorem 1. This result has essential advancements compared with the delay-independent stability criterion presented in [9] and [10]. At the last of section three, we illustrate the validity of the result provided in this paper with a numerical example.In section four, we give a delay-dependent sufficient condition for the existence of the state feedback guaranteed cost controller from the result in section three and the design method of the controller, i.e.theorem 2. In addition, in order to solve the problem by using the Matlab toolbox, we translate the matrix inequalities of theorem 2 into the terms of linear matrix inequalities. Theorem 3 narrates the process. At last we illustrate the validity of the design method of the robust guaranteed cost controller given in this paper.Section five is the conclusion.