Indefinite Guaranteed Cost Control For Nonlinear Uncertain Discrete-time Singular Markov Jump Systems With Time-Delay

This paper considers the problem of state feedback and static output feedback guaranteed cost controller design for nonlinear discrete-time uncer-tain singular Markov jump systems with time delay, the weighting matrix in the quadratic cost is indefinite, i.e. the weighting matrix contains negative eigenvalues.In this paper, the nonlinear discrete-time uncertain singular Markov jump systems are described as follows: where xk ∈Rn is the state vector, uk ∈Rm is the control input, yk ∈Rl is the output vector, the matrix E is singular, i.e is a Markov chain taking values in a finite space following transition probability from mode i at time k to j at time k+1: condition. dk is a positive integer representing the time-vary ing delay and re known positive and finite integers. Rn is continuous function vector on xk,xk-dk,Uk, and satisfies the following bounded condition: where a is a parameter to bound nonlinear function f, Mj1,Mj2,Mj3 are known constant matrices with appropriate dimensions.For every mode rk= i, there are constant matrices,δA(k, i),δAd(k, i),δB(k, i) are unknown matrices, denoting the uncertainties in the system.In this paper, the uncertainties are norm-bounded and are assumed to be of the following form:In this paper, the purposes are to design state feedback controller uk= K(rk)xk and static output feedback controller uk= K(rk)yk, such that for all uncertainties, the closed-loop systems formed by (0.1) and the controller are regular, casual and robust stochastically stable, have unique solution in a neighborhood of the equilibrium point (xk,xk-dk)= (0,0), the cost function J have upper and lower bound.In this paper, based on the reference [38], adding the uncertainties and the quadratic cost function. Firstly considering the two cases that the tran-sition probabilities are known and unknown, respectively using linear matrix inequality(LMI) method, the sufficient conditions to assure that the indefi-nite guaranteed cost controller exist are obtained, which make the closed-loop systems are regular, casual, have unique solution in the neighborhood of the equilibrium point(xk,Xk-dk)= (0,0), robust stochastically stable, and the quadratic cost have lower bound 0 and a certain upper bound. Then by using singular value decomposition approach, the LMI sufficient condition to assure that the time-varying delay discrete-time singular Markov jump systems static output feedback indefinite guaranteed cost controller exist is obtained. At last, based on time-varying delay discrete-time singular Markov jump systems’state feedback and output feedback indefinite guaranteed cost control, the constant delay state feedback and output feedback indefinite guaranteed cost controller existing condition and design method are given respectively.