The Design of Guaranteed Cost Control for Uncertain Discrete-Time Singular Systems with Time-Delay
|Course||Operational Research and Cybernetics|
|Keywords||discrete-time time-delay singular systems cost function guaranteed cost control linear matrix inequality(LMI)|
This paper considers the problem of state feedback guaranteed cost controllerdesign for discrete-time time-delay singular systems with norm-bounded parameteruncertainties.In this paper, consider uncertain discrete-time singular time-delay systemsdescribed as follows:where k∈Z,xk∈Rn is the state vector, uk∈Rp is the control input,d is an unknownconstant integer time-delay, and （?） is the known integer,φ（s） is ainitial value at s. The matrix E∈Rn×n is singular, and rank E = r<n. The systemmatrices are assumed to be as follows:Where A,Ad,B are known constant matrices with appropriate dimensions.δA（k）,δAd（k）,δB（k） are unknown matrices, denoting the uncertainties in the system（1） .In this paper, the uncertainties are norm-bounded and are assumed to be of thefollowing form:Where E1,Fa,Fd,Fb are known constant matrices with appropriate dimensions,Δ（k）∈Ri×j are unknown time-varying matrix function satisfyingGiven positive definite symmetric matrices P1 and P2, we consider the following cost functionalThe purpose of this paper is to design a state feedback controllerK∈Rp×n is a constant matrix, such that for all admissible uncertainties, theclosed-loop system formed by system （1） and controller （6） is regular, casual andasymptotically stable, and the cost value of the cost functional J has an upper bound.First, based on the restricted system equivalent （r.s.e.） transformation and byintroducing new state vectors, the singular system （1） is transformed into a standarddiscrete-time linear time-delay system, while the cost （5） is also transformed into thecorresponding form. Hence we can discuss the problem of guaranteed cost control forthe transformed standard discrete-time linear time-delay system instead of that forsingular system （1） .Then, by quoting a Lyapunov-krasovskii function and based on the Lyapunovstability theory and linear matrix inequality（LMI） method, and a sufficient condition isgiven which guarantees that the discrete-time time-delay singular system （1） withoutuncertainties and the state feedback controller is regular, causal and asymptoticallystable, and an upper bound of the cost functional J is also given. With this condition,in terms of LMI, a sufficient condition for the discrete-time time-delay singular system（1） without uncertainties to be admissible is obtained. And a sufficient condition forthe existence of the state feedback guaranteed cost controller is obtained, and thedesign method of the controller is also given.At last, a numerical example is provided to demonstrate the effectiveness of themethod given in this paper.