On a conjugation method applied to discrete \(H^{\infty}\)-control (Q2719617)

The present paper is devoted to discrete-time \(H^\infty\) control systems. Let Ric be the mapping defined in \textit{P. A. Iglesias} and \textit{K. Glover} [Int. J. Control 54, 1031-1073 (1991; Zbl 0741.93016)]. Using the conjugation method and some results given by \textit{D. Alpay} and \textit{I. Gohberg} [Unitary rational matrix functions, Operator Theory, Adv. Appl. 33, Basel, Birkhäuser Verlag, 175-222 (1988; Zbl 0651.47008)], the following result is proved.NEWLINENEWLINENEWLINETheorem 1. Let \(G(z):= D+ C(zI- A)^{-1} B\equiv \{A, B, C, D\}\in RL^\infty_{(m+ r)(p+ q)}\) be stabilizable and detectable, where \(RL^\infty_{(m+ r)(p+ q)}\) is a rational \(p\times q\) matrix with poles on the set \(\partial\delta= \{z\in C,|z|= 1\}\), and \(A\), \(B\), \(C\), \(D\) are real matrices. Let \(\sigma(A)\cap \partial\delta= \Phi\) and \(S_1= \left(\left[\begin{smallmatrix} A & 0\\ 0 & I\end{smallmatrix}\right], \left[\begin{smallmatrix} I & BJB^T\\ 0 & A^T\end{smallmatrix}\right]\right)\). Then a necessary and sufficient condition for the existence of a stable \((J, J')\)-lossless (right) conjugate factor \(\theta(z)\) of \(G(z)\) is: \(S_1\in \text{dom(Ric)}\), \(X:= \text{Ric}(S_1)\geq 0\), \(\widehat A= (I+ BJB^TX)^{-1}A\) is stable and the matrix \(J+ B^TXB\) is invertible and has the same inertial moment as \(J\). In addition the factor \(\theta(z)\) can be derived from the corresponding discrete Riccati equation as NEWLINE\[NEWLINE\theta(z)= \left[\begin{matrix} \widehat A & BD_d\\ -JB^T X\widehat A & D_d\end{matrix}\right],NEWLINE\]NEWLINE where \(D_d\) is a matrix satisfying the equation \(D^T_d(J+ B^T XB)D_d= J\).NEWLINENEWLINENEWLINEA similar result for a stable discrete \((J, J')\)-gainless (left) conjugate factor is also obtained.