The Design of Guaranteed Cost Control for Uncertain DiscreteTime Singular Systems with TimeDelay 

Author  ZhangCaiYan 
Tutor  MaShuPing 
School  Shandong University 
Course  Operational Research and Cybernetics 
Keywords  discretetime timedelay singular systems cost function guaranteed cost control linear matrix inequality(LMI) 
CLC  TP13 
Type  Master's thesis 
Year  2009 
Downloads  33 
Quotes  0 
This paper considers the problem of state feedback guaranteed cost controllerdesign for discretetime timedelay singular systems with normbounded parameteruncertainties.In this paper, consider uncertain discretetime singular timedelay systemsdescribed as follows:where k∈Z,x_{k}∈R^{n} is the state vector, u_{k}∈R^{p} is the control input,d is an unknownconstant integer timedelay, and （?） is the known integer,φ（s） is ainitial value at s. The matrix E∈R^{n×n} is singular, and rank E = r<n. The systemmatrices are assumed to be as follows:Where A,A_{d},B are known constant matrices with appropriate dimensions.δA（k）,δA_{d}（k）,δB（k） are unknown matrices, denoting the uncertainties in the system（1） .In this paper, the uncertainties are normbounded and are assumed to be of thefollowing form:Where E_{1},F_{a},F_{d},F_{b} are known constant matrices with appropriate dimensions,Δ（k）∈R^{i×j} are unknown timevarying matrix function satisfyingGiven positive definite symmetric matrices P_{1} and P_{2}, we consider the following cost functionalThe purpose of this paper is to design a state feedback controllerK∈R^{p×n} is a constant matrix, such that for all admissible uncertainties, theclosedloop 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 standarddiscretetime linear timedelay system, while the cost （5） is also transformed into thecorresponding form. Hence we can discuss the problem of guaranteed cost control forthe transformed standard discretetime linear timedelay system instead of that forsingular system （1） .Then, by quoting a Lyapunovkrasovskii function and based on the Lyapunovstability theory and linear matrix inequality（LMI） method, and a sufficient condition isgiven which guarantees that the discretetime timedelay 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 discretetime timedelay 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.