Output Feedback Control For Uncertain Discrete-Time Markovian Jump Singular Systems With Time-Delay

This paper considers the problem of output feedback controller disign for uncertaindiscrete markovian jump singular systems with time-delay.The system considered in this paper is assumed to be a state space model as follows:where k∈Z, x_i∈Rⁿ is the system state, u_k∈R^p is the control input,ω_k∈R^q is the disturbance input which belongs to l₂, z_k∈R^m is the controlled output, y_k∈R^l is measured output. d is an unknown constant integer time-delay, and 0 < d≤d, d > 0 is the known integer,Φ(k) is an initial value at k. {r_k, k∈Z} is a Markov chain taking values in finite space (?) = {1,2, ...,N}, with transition probability form mode i at time k to mode j at time k+1:p_ij = Pr{r_k+1= j|r_k = i} (2)with p_i,j≥0 for i,j∈(?), and sum from j=1 to N p_ij = 1. The matrix E(r_k) is singular. For eachi∈(?), we haveA(k,i)=A(i)+δA(k,i),A_d(k,i)=A_d(i)+δA_d(k,i),B(k,i)=B(i)+δB(k,i)C(k,i)=C(i)+δC(k,i),C_d(k,i)=C_d(i)+δC_d(k,i),D(k,i)=D(i)+δD(k,i) where A(i), A_d(i) , B(i) , C(i) , C_d(i)å'ŒD(i) are known constant matrices with appropriate dimensions.δA(k,i),δA_d(k,i),δB(k,i),δC(k,i),δC_d(k,i) andδD(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 formwhere E₁(i),E₂(i),F₁(i),F₂(i),F₃(i) are known constant matrices with appropriate dimensions,â–³(k,i)∈R^h×s are unknown time-varying matrix function satisfyingThe purpose of this paper is to design a output feedback controllerwhere A^c(i)∈R^r_i×r_i,r_i = rank(E(i)) such that the closed-loop system is robust stochasticallystable withγ-disturbance attenuation for all uncertainties satisfying(3) and (4).At first, based on the restricted system equivalent (r.s.e.) transformation and introducing a new state vector, the system(1) is transformed into a linear standard uncertain discrete markovian jump systems with time-delay. Hence we can discuss the problem of output feedback control for standard discrete markovain jump systemswith time-delay instead of that for system(1). Section two gives this process.Section three discusses the disign of output feedback controller for the certain system. In additon, in order to solve the problem by using the Matlab toolbox, we translate the conditions into the terms of linear matrix inequalities.Based on the former results, section four solves the problem of output feedback control for the uncertain system. For the same reason, we translate the conditions into the terms of linear matrix inequalities.At last, we illustrate the validity of the result provided in this paper with a numerical example.