\(H_\infty\) control for discrete-time linear systems with Frobenius norm-bounded uncertainties
The present paper deals with the problem of robust stability and control for the class of uncertain discrete-time linear systems with Frobenius norm-bounded parameter uncertainties in all matrices of the system and output equations described by \[ \begin{multlined} x_{(k+1)}= (A+ H_1\Delta(k) E_1)x_k+ (B+ H_1\Delta(k) E_2)w_k+ (B_1+ H_1\Delta(k) E_3)u_k=\\ \Biggl(A+ \sum^r_{i=1} \sum^s_{j=1} H_{1i} \Delta_{ij}(k) E_{1j}\Biggr) x_k+ \Biggl(B+ \sum^r_{i=1} \sum^s_{j=1} H_{1i} \Delta_{ij}(k) E_{2j}\Biggr) w_k+\\ \Biggl(B_1+ \sum^r_{i=1} \sum^s_{j=1} H_{1i} \Delta_{ij}(k) E_{3j}\Biggr) u_k,\end{multlined} \] \[ \begin{multlined} z_k= (C+ H_2\Delta(k) E_1) x_k+ (D+ H_2\Delta(k) E_2) w_k+ (D_1+ H_2\Delta(k) E_3) u_k=\\ \Biggl(C+ \sum^r_{i=1} \sum^s_{j=1} H_{2i} \Delta_{ij}(k) E_{1j}\Biggr) x_k+ \Biggl(D+ \sum^r_{i=1} \sum^s_{j=1} H_{2i} \Delta_{ij}(k) E_{2j}\Biggr) w_k+\\ \Biggl(D_1+ \sum^r_{i=1} \sum^s_{j=1} H_{2i} \Delta_{ij}(k) E_{3j}\Biggr) u_k,\end{multlined} \] where \(A\), \(B\), \(B_1\), \(C\), \(D\), and \(D_1\) are known real constant matrices of appropriate dimensions that describe the nominal system, \(H_{ij}\), \(E_{ij}\) are known matrices and \(\Delta_{ij}(k)\) are time dependent unknown matrices which verify the following inequality \[ \sum^r_{i=1} \sum^s_{j=1} \|\Delta_{ij}(k)\|^2\leq 1,\;\forall k. \] Necessary and sufficient conditions for the above problems are proposed. A linear static state feedback control law is designed, which is in terms of a Riccati inequality. The results obtained here show that the robust control problem of the uncertain system is equivalent to the control problem for a corresponding uncertainty-free system.
scientific article; zbMATH DE number 1353734