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

Abstract

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback H8 control problem. It is parametrized by the H8-norm bound ¿ we want to achieve. The objective of this paper is to study the behavior of the solution to the Riccati equation as a function of ¿. It turns out that a stabilizing solution exists for all but finitely many values of ¿ larger than some a priori determined bound ¿-. On the other hand, for values smaller than ¿- there does not exist a stabilizing solution. The finite number of exception points can be characterized as switching points where eigenvalues of the stabilizing (symmetric) solution can switch from negative to positive with increasing ¿. After the final switching point the solution will be positive semidefinite. We obtain the following interpretation: The Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exists a static feedback such that the closed-loop transfer matrix has k unstable poles and an L8 norm strictly less than ¿