Robust Stabilization And H_∞ Control Of A Class Of Uncertain Discrete Singular Systems With Time-Varying Delays

This paper is focused on the problems of designing a robust stabilizing and H_∞state feedback controller for uncertain discrete singular systems with time-varying delays. Consider uncertain discrete singular systems with time-varying delays given in the followingwhere x(k) ∈ Rⁿ is the state vector, u(k) ∈ R^m is the control input, ω(k) ∈ l₂^q [0,∞) is the exogenous disturbance signal and z(k) ∈ R^p is the controlled output. The matrix E∈ R^n×n and is assumed 0 < rank E = r d,B₁,B₂,C₀,D₀ and D₁ are known real constant matrices with appropriate dimensions; and r(k) is a positive integer and satisfies 0 < Ï„₁, ≤ Ï„(k) ≤ Ï„₂ , where Ï„₁ and Ï„₂ are known positive integer; ΔA(k),ΔA_d(k),ΔB₁(k) and ΔB₂(k) are unknown matrices representing time-varying parameter uncertainties, and are assumed to be of the formwhere D,N_a,N_d,N_b1 and N_b2 are known real constant matrices and F(·)∈R^i×j is an unknown time-varying matrix function satisfyingThroughout this paper, our purpose is to design a state feedback controllersuch thatâ‘  The closed-loop system formed by system (1) and the controller (4) is admissible for all uncertainties satisfying (2) and (3) when ω(k) = 0 ; Under the zero-initial condition, the controlled output z(k) satisfiesfor any nonzero ω(k)∈l₂ and all admissible uncertainties, where γ>0 is aprescribed scalar.First, we consider the system without the exogenous disturbance signal, Theorem 1 gives sufficient LMI condition, such that the nominal unforced singular time-delay system of system (1) is admissible, and the Theorem 3 gives the LMI condition that there exists a state feedback controller (4) such that the closed system is admissible, and the method of designing the state feedback controller. Then the robust state feedback H_∞ controller is discussed based on similar method, and LMI condition is obtained.