# Stabilizing solutions of the $H_\infty$ algebraic Riccati equation

A.A. Stoorvogel

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## Abstract

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback H/sub /spl infin// control problem. It is parameterized by the H/sub /spl infin// norm bound /spl gamma/ we want to achieve. The objective of this paper is to study the behaviour of the solution to the Riccati equation as a function of /spl gamma/. It turns out that a stabilizing solution exists for all but finitely many values of /spl gamma/ larger than some a priori determined boundary /spl gamma/*. On the other hand for values smaller than /spl gamma/* there does not exist a stabilizing solution. The finite number of exception points turn out to be switching points where eigenvalues of the stabilizing solution can switch from negative to positive with increasing /spl gamma/. After the final switching point the solution will be positive semi-definite. We obtain the following interpretation: the Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exist a static feedback such that the closed loop transfer matrix has no more than k unstable poles and an L/sub /spl infin// norm strictly less than /spl gamma/.
Original language English Proceedings 1994 American Control Conference (Baltimore MD, USA, June 29-July 1, 1994) Institute of Electrical and Electronics Engineers 2772-2776 5 0-7803-1783-1 https://doi.org/10.1109/ACC.1994.735073 Published - 1994

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