Linear discrete-time systems with Markovian jumps and mode dependent time-delay: stability and stabilizability
The authors consider the problem of stochastic stability and stabilizability of a class of linear discrete-time systems with Markovian jumps and mode-dependent time delays described by \[ x_{k+1}= A(r_k) x_k+ A_d(r_k) x_{k-d_{r_k}}+ B(r_k) u_k,\;x_l= \phi_l,\;l= -d_{r_0},\dots, -1,0,\tag{1} \] where \(x_k\) and \(u_k\) are the \(n\)- and \(m\)-dimensional state and control vectors, respectively, at instant \(k> 0\); \(\{r_k, k\geq 0\}\) is a homogeneous Markov chain that takes values in a finite state space \(\{S= 1,2,\dots, N\}\) with transition probability matrix function \(p_{ij}= P\{r_{k+1}= j\mid r_k= i\}\), where \(p_{ij}\geq 0\) and \(\sum^N_{j=1} p_{ij}= 1\) for all \(i,j\in S\). The matrices \(A(r_k)\), \(A_d(r_k)\) and \(B(r_k)\) are constant matrices of appropriate sizes for any fixed values of \(r_k\) in \(S\), and \(d_{r_k}\) is a positive integer representing the time-delay of the system. For system (1) the authors used the linear control law \(u_k= K(r_k)x_k\), where the control gain \(K(r_k)\) was determined for each system mode \(r_k\in S\). To obtain sufficient conditions of stability and stabilizability the authors have used linear matrix inequality techniques. They also propose an algorithm for the design of a stabilizing controller. An example is given to illustrate the obtained results.
scientific article; zbMATH DE number 2097943