Delay-dependent Guaranteed Cost Control For Uncertain Discrete-Time Singular Systems With Time-Delay

This paper considers the problem of state feedback guaranteed cost controller design for discrete-time singular time-delay systems with norm-bounded parameter uncertainties. Consider uncertain discrete singular time-delay systemswhere x(k) ∈ Rⁿ is the state vector, u(k) ∈ R^m is the controlled input, E, A, A_d, B are unknown real constant matrices with appropriate dimensions, 0 < rankE = r < n, d is an unknown real constant delay and satisfies 0 < d ≤ d_m, where d_m is the upper bound of d. φ(k) is a compatible initial function. ΔA, ΔA_d and ΔB are unknown matrices representing time-varying parameter uncertainties, and assumed to be of the formwhere M,N_a,N_d and N_b, are known real constant matrices, F(·) ∈ R^i×j is an unknown time-varying matrix function satisfyingGiven positive definite symmetric matrices R₁,R₂, we consider the cost functionalThe purpose of this paper is to design a state feedback controller K is a constant matrix, such that for all admissible uncertainties, the closed-loop system is regular, casual and asymptotically stable and the closed-loop value of the cost functional J satisfies a bound.At first, it is assumed that 0 < rankE = r < n. Based on the restricted system equivalent (r.s.e.) transformation and state-controlled input linear transformation, the system(1) is transformed into a linear standard discrete time-delay system, while the cost (4) is also transformed into the corresponding form. Hence we can discuss the problem of guaranteed cost control for linear standard discrete time-delay system instead of that for system (1). Section two gives this process.Section three is the main work of this paper. By quoting a Lyapunov-Krasovskii function, we establish a delay-dependent sufficient condition for the existence of the state feedback guaranteed cost controller from the Lyapunov stability theory, i.e. Theorem 1. In addition, in order to solve the problem by using the Matlab toolbox, we translate the matrix inequalities of Theorem 1 into the terms of linear matrix inequalities, and then give the design method of the controller in Theorem 2.At last, we illustrate the validity of the result provided in this paper with a numerical example.