##
**Singular finite horizon full information \({\mathcal H}^ \infty\) control via reduced order Riccati equations.**
*(English)*
Zbl 0863.93026

For linear time-varying systems the standard finite horizon, full information \(H_\infty\) control problem is considered. The authors study the singular case when the direct feedthrough matrix that links the control input to the controlled output is not full column rank. Combining the game-theoretical approach and decomposition methods which are based on the concept of strongly controllable subspace and which has been used in the time-invariant singular case, the authors derive a solution for time-varying systems with continuously differentiable matrices. It is shown that the original problem is equivalent to a reduced order one, which under certain assumptions is regular and its solution can be obtained via a reduced order Riccati differential equation.

Reviewer: O.I.Nikonov (Ekaterinburg)

### MSC:

93B36 | \(H^\infty\)-control |

### Keywords:

\(H_ \infty\) control; linear; time-varying systems; finite horizon; singular case; reduced order Riccati differential equation
PDF
BibTeX
XML
Cite

\textit{F. Amato} and \textit{A. Pironti}, Kybernetika 31, No. 6, 601--611 (1995; Zbl 0863.93026)

### References:

[1] | F. Amato, A. Pironti: A note on singular zero-sum linear quadratic differential games. Proceedings of the 33rd IEEE Conference or. Decision and Control, Orlando 1994. |

[2] | T. Basar, G. J. Olsder: Dynamic Noncooperative Game Theory. Academic Press, New York 1989. · Zbl 0479.90085 |

[3] | S. Butman: A method for optimizing control-free costs in systems with linear controllers. IEEE Trans. Automat. Control 13 (1968), 554-556. |

[4] | J. W. Helton M. L. Walker, W. Zhan: \({\mathcal H}^\infty\) control using compensators with access to the command signals. Proceedings of the 31st Conference on Decision and Control, Tucson 1992. |

[5] | D. J. N. Limebeer B. D. O. Anderson P. P. Khargonekar, M. Green: A game theoretic approach to \({\mathcal H}^\infty\) control for time-varying systems. SIAM J. Control Optim. 30 (1992), 262-283. · Zbl 0791.93026 |

[6] | R. Ravi K. M. Nagpal, P. P. Khargonekar: \({\mathcal H}^\infty\) control of linear time-varying systems: a state space approach. SIAM J. Control Optim. 29 (1991), 1394-1413. · Zbl 0741.93017 |

[7] | J. L. Speyer, D. H. Jacobson: Necessary and sufficient condition for optimality for singular control problem. J. Math. Anal. Appl. 33 (1971). · Zbl 0203.47101 |

[8] | A. A. Stoorvogel, H. Trentelman: The quadratic matrix inequality in singular \({\mathcal H}^\infty\) control with state feedback. SIAM J. Control Optim. 28 (1990), 1190-1208. · Zbl 0717.93016 |

[9] | G. Tadmor: Worst-case design in time domain: the maximum principle and the standard \({\mathcal H}^\infty\) problem. Math. Control Signals Systems 3 (1990), 301-324. · Zbl 0715.93027 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.