### Abstract

In this paper, we consider the problem to compute the distance to uncontrollability of a given controllable pair A ∈ C^{n×n} and B ∈ C^{n×m}. It is known that this problem is equivalent to computing the minimum of the smallest singular value of [A - zI B] over z ∈ C. With this fact, Gu proposed an algorithm that correctly estimates the distance at a computation cost ο(n^{6}). On the other hand, in the community of control theory, remarkable advances have been made on the techniques to deal with parametrized linear matrix inequalities (LMIs) as well as the analysis of positive polynomials. This motivates us to explore an alternative LMI-based algorithm and shed more insight on the problem to estimate the distance to uncontrollability. In fact, this paper shows that we can establish an effective method to compute a lower bound of the distance by simply applying the existing techniques to solve parametrized LMIs. To obtain an upper bound, on the other hand, we analyze in detail the solutions resulting from the LMI optimization carried out to compute the lower bound. This enables us to estimate the location of the local, but potentially global optimizer z* ∈ C in a reasonable fashion. We thus provide a novel technique to obtain an upper bound of the distance by evaluating the smallest singular value on the estimated optimizer z*. It turns out that the lower and upper bounds are very close in all tested numerical examples. We finally derive an algebraic condition under which the exactness of the suggested computation method of the lower bound can be ensured, based on the convex duality theory. Furthermore, we show that the suggested computation method of the upper bound is closely related to the obtained algebraic condition for the exactness verification.

Original language | English |
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Title of host publication | Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC |

Pages | 5772-5777 |

Number of pages | 6 |

Publication status | Published - Dec 1 2006 |

Externally published | Yes |

Event | 45th IEEE Conference on Decision and Control 2006, CDC - San Diego, CA, United States Duration: Dec 13 2006 → Dec 15 2006 |

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
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ISSN (Print) | 0191-2216 |

### Conference

Conference | 45th IEEE Conference on Decision and Control 2006, CDC |
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Country | United States |

City | San Diego, CA |

Period | 12/13/06 → 12/15/06 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Modelling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC*(pp. 5772-5777). [4177363] (Proceedings of the IEEE Conference on Decision and Control).

**Computing the distance to uncontrollability via LMIs : Lower and upper bounds computation and exactness verification.** / Ebihara, Yoshio; Hagiwara, Tomomichi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC.*, 4177363, Proceedings of the IEEE Conference on Decision and Control, pp. 5772-5777, 45th IEEE Conference on Decision and Control 2006, CDC, San Diego, CA, United States, 12/13/06.

}

TY - GEN

T1 - Computing the distance to uncontrollability via LMIs

T2 - Lower and upper bounds computation and exactness verification

AU - Ebihara, Yoshio

AU - Hagiwara, Tomomichi

PY - 2006/12/1

Y1 - 2006/12/1

N2 - In this paper, we consider the problem to compute the distance to uncontrollability of a given controllable pair A ∈ Cn×n and B ∈ Cn×m. It is known that this problem is equivalent to computing the minimum of the smallest singular value of [A - zI B] over z ∈ C. With this fact, Gu proposed an algorithm that correctly estimates the distance at a computation cost ο(n6). On the other hand, in the community of control theory, remarkable advances have been made on the techniques to deal with parametrized linear matrix inequalities (LMIs) as well as the analysis of positive polynomials. This motivates us to explore an alternative LMI-based algorithm and shed more insight on the problem to estimate the distance to uncontrollability. In fact, this paper shows that we can establish an effective method to compute a lower bound of the distance by simply applying the existing techniques to solve parametrized LMIs. To obtain an upper bound, on the other hand, we analyze in detail the solutions resulting from the LMI optimization carried out to compute the lower bound. This enables us to estimate the location of the local, but potentially global optimizer z* ∈ C in a reasonable fashion. We thus provide a novel technique to obtain an upper bound of the distance by evaluating the smallest singular value on the estimated optimizer z*. It turns out that the lower and upper bounds are very close in all tested numerical examples. We finally derive an algebraic condition under which the exactness of the suggested computation method of the lower bound can be ensured, based on the convex duality theory. Furthermore, we show that the suggested computation method of the upper bound is closely related to the obtained algebraic condition for the exactness verification.

AB - In this paper, we consider the problem to compute the distance to uncontrollability of a given controllable pair A ∈ Cn×n and B ∈ Cn×m. It is known that this problem is equivalent to computing the minimum of the smallest singular value of [A - zI B] over z ∈ C. With this fact, Gu proposed an algorithm that correctly estimates the distance at a computation cost ο(n6). On the other hand, in the community of control theory, remarkable advances have been made on the techniques to deal with parametrized linear matrix inequalities (LMIs) as well as the analysis of positive polynomials. This motivates us to explore an alternative LMI-based algorithm and shed more insight on the problem to estimate the distance to uncontrollability. In fact, this paper shows that we can establish an effective method to compute a lower bound of the distance by simply applying the existing techniques to solve parametrized LMIs. To obtain an upper bound, on the other hand, we analyze in detail the solutions resulting from the LMI optimization carried out to compute the lower bound. This enables us to estimate the location of the local, but potentially global optimizer z* ∈ C in a reasonable fashion. We thus provide a novel technique to obtain an upper bound of the distance by evaluating the smallest singular value on the estimated optimizer z*. It turns out that the lower and upper bounds are very close in all tested numerical examples. We finally derive an algebraic condition under which the exactness of the suggested computation method of the lower bound can be ensured, based on the convex duality theory. Furthermore, we show that the suggested computation method of the upper bound is closely related to the obtained algebraic condition for the exactness verification.

UR - http://www.scopus.com/inward/record.url?scp=39649120075&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39649120075&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:39649120075

SN - 1424401712

SN - 9781424401710

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 5772

EP - 5777

BT - Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC

ER -