Dissertation > Industrial Technology > Automation technology,computer technology > Automated basic theory > Automatic control theory

The Design of Guaranteed Cost Control for Uncertain Discrete-Time Singular Systems with Time-Delay

Author ZhangCaiYan
Tutor MaShuPing
School Shandong University
Course Operational Research and Cybernetics
Keywords discrete-time time-delay singular systems cost function guaranteed cost control linear matrix inequality(LMI)
Type Master's thesis
Year 2009
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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.

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