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

This paper considers the problem of state feedback guaranteed cost controller design for discrete-time time-delay switched singular systems with non-bounded parameter uncertainties.In this paper, consider uncertain discrete-time switched singular time-delay systems described as follows: {Ex(k+1)=(Ai+δAi)x(k)+(Adi+δAdi)x(k-d)+(Bi+δBi)u(k), x(s)=φ(s),s=-d,…,-1,0,(1) Where k∈Z,x(k)∈Rn is the system state, u(k)∈Rm is the control input.d is an unknown constant integer time-delay and 0<d≤d,d>0 is the known integer.σ(k)is the switching law,σ(k)=i implies that the i-th subsystem is activated.σ(s)∈Rn is an initial value. E,Ai,Adi,Bi∈Rn×n are known real constant matrices with appropriate dimensions,0<rankE=r<n.δAi,δAdi andδBi are unknown matrices representing time-varying parameter uncertainties,and assumed to be of the form [δAiδAdiδBi]=EliΔi(k)[Fai Fdi Fhi], (2) where E1i,Fai,Fdi and Fhi are unknown real constant matrices,Δ,(k)is unknown time-varying matrix function satisfyingΔj(k)Δ,(k)≤I. (3) Given positive definite symmetric P1,P2,we consider the cost functional J=(?)(xT(k)P1x(k)+uT(k)P2u(k)). (4) The purpose of this paper is to design a state feedback controller u(k)=Kix(k) (5) Ki is a constant matrix, such that for all admissible uncertainties, the closed-loop system described as follows: {Ex(k+1)=(Ai+δAi)x(k)+(Adi+δAdi)x(k-d)+(Bi+δBi)u(k), x(s)=φ(s),s=-d,…,-1,0,(6) is regular, casual and asymptotically stable and the closed-loop value of the cost functional J satisfies an upper bound.At first, it is assumed 0<rankE=r<n. Based on the restricted system equivalent transformation and introducing new state vectors, the system (1) is transformed into a linear standard discrete time-delay switched systems, 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 switched system instead of that systems (1). Section two gives this process.Section three is the main work of this paper. By quoting a Lyapunov-Krasovskii function and based on the Lyapunov stability theory, and a sufficient condition is given which guarantees that the discrete-time time delay switched singular system (1) without uncertainties and the state feedback controller is regular, causal and asymptotically stale, and an upper bound of the cost functional J is also given under arbitrary switching. With this condition, in terms of LMI, a sufficient condition for the discrete-time time delay switched singular system (1) without uncertainties to be admissible is given, and the design method of the controller is also given under arbitrary switching.At last, a numerical example is provided to illustrate the validity of result provided in this paper.