\(H_\infty\)/LTR synthesis of discrete-time systems (Q2704519)

The present paper is devoted to the following discrete-time autonomous linear system NEWLINE\[NEWLINEx(k+1)= Ax(k)+ B_1w(k)+ B_2u(k),\;z(k)= C_1x(k)+ D_{12}u(k),\;y(k)= C_2x(k)+ D_{21}w(k),NEWLINE\]NEWLINE where \(k\in \{0,1,2,\dots\}\), and \(x\in\mathbb{R}^n\), \(w\in\mathbb{R}^{m_1}\), \(u\in \mathbb{R}^{m_2}\), \(z\in \mathbb{R}^{p_1}\), \(y\in \mathbb{R}^{p_2}\) are the state-vector, external disturbance input-vector, control input-vector, controlled output-vector and measured output-vector, respectively, and \(A\), \(B_1\), \(B_2\), \(C_1\), \(C_2\), \(D_{12}\), \(D_{21}\) are known real constant matrices of appropriate dimensions. Under the assumptions that \((A,B_2)\), \((C_2,A)\) are stabilizable and observable, respectively, and all uncertainties can be reduced equivalently to the input member of the object, the authors obtain a necessary and sufficient condition to the following:NEWLINENEWLINENEWLINEProblem 1. Given a disturbance attenuation constant \(\gamma>0\), does there exist a state feedback control \(u(k)= K_cx(k)\) such that the matrix \(A+ B_2K_c\) is stable and the condition \(\|T_{sf}(z)\|_\infty< \gamma\) holds, where \(T_{sf}(z)\) denotes the closed loop transfer function?NEWLINENEWLINENEWLINEThe above-mentioned condition is derived as a special case of those results given by \textit{V. Ionescu} and \textit{M. Weiss} [Int. J. Control 57, No. 1, 141-195 (1993; 774.93026)] and \textit{P. A. Iglesias} and \textit{K. Glover} [ibid. 54, No. 5, 1031-1073 (1991; Zbl 0741.93016)].NEWLINENEWLINENEWLINEAnother problem (Problem 2) on given recovery degrees is also solved by the duality property of these two problems.