### Abstract

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback $H_\infty$ control problem. It is parameterized by the $H_\infty$ norm bound $\gamma$ 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 $\gamma$. It turns out that a stabilizing solution exists for all but finitely many values of $\gamma$ larger than some a priori determined boundary $\gamma_{-}$. On the other hand for values smaller than $\gamma_{-}$ 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 $\gamma$. 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 $L_\infty$ norm strictly less than $\gamma$.
Keywords: The $H_\infty$ control problem, The Algebraic Riccati Equation, J-spectral factorization, Wiener-Hopf factorization.

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Technische Universiteit Eindhoven |

Number of pages | 15 |

Publication status | Published - 1993 |

### Publication series

Name | Memorandum COSOR |
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Volume | 9325 |

ISSN (Print) | 0926-4493 |

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## Cite this

Stoorvogel, A. A. (1993).

*Stabilizing solutions of the $H_\infty$ algebraic Riccati equation*. (Memorandum COSOR; Vol. 9325). Eindhoven: Technische Universiteit Eindhoven.