### 抄録

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H
_{∞}
control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H
_{∞}
state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.

元の言語 | 英語 |
---|---|

ページ（範囲） | 303-316 |

ページ数 | 14 |

ジャーナル | International Journal of Control |

巻 | 92 |

発行部数 | 2 |

DOI | |

出版物ステータス | 出版済み - 2 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Computer Science Applications

### これを引用

_{∞}state feedback control problem

*International Journal of Control*,

*92*(2), 303-316. https://doi.org/10.1080/00207179.2017.1351625

**
Application of facial reduction to H
_{∞}
state feedback control problem
.** / Waki, Hayato; Sebe, Noboru.

研究成果: ジャーナルへの寄稿 › 記事

_{∞}state feedback control problem ',

*International Journal of Control*, 巻. 92, 番号 2, pp. 303-316. https://doi.org/10.1080/00207179.2017.1351625

}

TY - JOUR

T1 - Application of facial reduction to H ∞ state feedback control problem

AU - Waki, Hayato

AU - Sebe, Noboru

PY - 2019/2/1

Y1 - 2019/2/1

N2 - One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H ∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H ∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.

AB - One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H ∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H ∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.

UR - http://www.scopus.com/inward/record.url?scp=85025143403&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85025143403&partnerID=8YFLogxK

U2 - 10.1080/00207179.2017.1351625

DO - 10.1080/00207179.2017.1351625

M3 - Article

AN - SCOPUS:85025143403

VL - 92

SP - 303

EP - 316

JO - International Journal of Control

JF - International Journal of Control

SN - 0020-7179

IS - 2

ER -