مرکز منطقه ای اطلاع رساني علوم و فناوري - A stable state-space realization in the formulation of H<sup>∞</sup>norm computation

Abstract :

In the two block H^inftyoptimization problem, usually we are given the state-space realizations of the proper rational matrices $R_{1}(s)$ and $R_{2}(s)$ whose poles are all the open right-half plane. Two problems are studied in the note. The first is the evaluation of $\\phi(s)R_{1}(s)$ at $s = s_{k}, k = 1, 2, ..., n$ , where $\\phi(s)$ is an inner function whose zeros ${s_{k}, k = 1, 2, ..., n }$ are the poles of $R_{1}(s)$ . This evaluation is essential if Chang and Pearson\´s method is used for computing the optimal H^inftynorm. The problem is solved in state space via the solutions of Lyapunov equations. Neither polynomial matrix manipulations nor numerical pole-zero cancellations are involved in the evaluation. The second problem is to find a stable state-space realization of $S(s) = U(s)R_{2}(s)$ where $U(s)$ is an inner matrix. This problem arises in the spectral factorization of $\\gamma ^{2} - R_{2}^{\\ast }R_{2}$ . Doyle and Chu had a method for constructing stable $S(s)$ based on a minimal realization of $R_{2}(s)$ . An alternate method is proposed. The alternate method does not require a minimal realization of $R_{2}(s)$ and only a Lyapunov equation is involved.