### Abstract

In [26, 27, 35], condition numbers and perturbation bounds were produced for the state feedback pole assignment problem (SFPAP), for the single- and multi-input cases with simple closed-loop eigenvalues. In this paper, we consider the same problem in a different approach with weaker assumptions, producing simpler condition numbers and perturbation results. For the SFPAP, we shall show that the absolute condition number κ ≤ c_{0}{norm of matrix} B^{†}{norm of matrix} [κX +(1 + {norm of matrix}F{norm of matrix}^{2})^{1/2}], where the closed-loop system matrix A + BF = X Λ X^{-1}, the closed-loop spectrum in Λ is pre-determined, κX ≡ {norm of matrix}X{norm of matrix}{norm of matrix}X^{-1}{norm of matrix}, the operators P_{c}(·) ≡ (A + BF)(·) - (·)Λ and N(·) ≡ (I - BB^{†})P_{c}(·),and c_{0} ≡ {norm of matrix}I(·) - P_{c}{norm of matrix}N^{†}(I - BB^{†})(·){norm of matrix}. With c_{B} ≡ {norm of matrix}B{norm of matrix}{norm of matrix}B^{†}{norm of matrix} and c_{1} ≡ ({norm of matrix}B{norm of matrix}{norm of matrix}F{norm of matrix})^{-1}, the relative condition number κ_{r} ≤ c_{0}c_{B} [c_{1}κ_{X}{norm of matrix}Λ{norm of matrix}+(c_{2}/_{1}{norm of matrix}A{norm of matrix}^{2} +1{norm of matrix})^{1/2}]. With B well-conditioned and Λ well chosen, κ and κ_{r} can be small even when Λ (not necessary in Jordan form) possesses defective eigenvalues, depending on c_{0}. Consequently, the SFPAP is not intrinsically ill-conditioned. Similar results were obtained in [23], although differentiability was not established for its local perturbation analysis. Simple as well as general multiple closed-loop eigenvalues are treated.

Original language | English |
---|---|

Pages (from-to) | 283-304 |

Number of pages | 22 |

Journal | Taiwanese Journal of Mathematics |

Volume | 16 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Taiwanese Journal of Mathematics*,

*16*(1), 283-304. https://doi.org/10.11650/twjm/1500406541

}

*Taiwanese Journal of Mathematics*, vol. 16, no. 1, pp. 283-304. https://doi.org/10.11650/twjm/1500406541

**Conditioning of state feedback pole assignment problems.** / Chu, Eric King Wah; Weng, Chang Yi; Wang, Chern-Shuh; Yen, Ching Chang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Conditioning of state feedback pole assignment problems

AU - Chu, Eric King Wah

AU - Weng, Chang Yi

AU - Wang, Chern-Shuh

AU - Yen, Ching Chang

PY - 2012/1/1

Y1 - 2012/1/1

N2 - In [26, 27, 35], condition numbers and perturbation bounds were produced for the state feedback pole assignment problem (SFPAP), for the single- and multi-input cases with simple closed-loop eigenvalues. In this paper, we consider the same problem in a different approach with weaker assumptions, producing simpler condition numbers and perturbation results. For the SFPAP, we shall show that the absolute condition number κ ≤ c0{norm of matrix} B†{norm of matrix} [κX +(1 + {norm of matrix}F{norm of matrix}2)1/2], where the closed-loop system matrix A + BF = X Λ X-1, the closed-loop spectrum in Λ is pre-determined, κX ≡ {norm of matrix}X{norm of matrix}{norm of matrix}X-1{norm of matrix}, the operators Pc(·) ≡ (A + BF)(·) - (·)Λ and N(·) ≡ (I - BB†)Pc(·),and c0 ≡ {norm of matrix}I(·) - Pc{norm of matrix}N†(I - BB†)(·){norm of matrix}. With cB ≡ {norm of matrix}B{norm of matrix}{norm of matrix}B†{norm of matrix} and c1 ≡ ({norm of matrix}B{norm of matrix}{norm of matrix}F{norm of matrix})-1, the relative condition number κr ≤ c0cB [c1κX{norm of matrix}Λ{norm of matrix}+(c2/1{norm of matrix}A{norm of matrix}2 +1{norm of matrix})1/2]. With B well-conditioned and Λ well chosen, κ and κr can be small even when Λ (not necessary in Jordan form) possesses defective eigenvalues, depending on c0. Consequently, the SFPAP is not intrinsically ill-conditioned. Similar results were obtained in [23], although differentiability was not established for its local perturbation analysis. Simple as well as general multiple closed-loop eigenvalues are treated.

AB - In [26, 27, 35], condition numbers and perturbation bounds were produced for the state feedback pole assignment problem (SFPAP), for the single- and multi-input cases with simple closed-loop eigenvalues. In this paper, we consider the same problem in a different approach with weaker assumptions, producing simpler condition numbers and perturbation results. For the SFPAP, we shall show that the absolute condition number κ ≤ c0{norm of matrix} B†{norm of matrix} [κX +(1 + {norm of matrix}F{norm of matrix}2)1/2], where the closed-loop system matrix A + BF = X Λ X-1, the closed-loop spectrum in Λ is pre-determined, κX ≡ {norm of matrix}X{norm of matrix}{norm of matrix}X-1{norm of matrix}, the operators Pc(·) ≡ (A + BF)(·) - (·)Λ and N(·) ≡ (I - BB†)Pc(·),and c0 ≡ {norm of matrix}I(·) - Pc{norm of matrix}N†(I - BB†)(·){norm of matrix}. With cB ≡ {norm of matrix}B{norm of matrix}{norm of matrix}B†{norm of matrix} and c1 ≡ ({norm of matrix}B{norm of matrix}{norm of matrix}F{norm of matrix})-1, the relative condition number κr ≤ c0cB [c1κX{norm of matrix}Λ{norm of matrix}+(c2/1{norm of matrix}A{norm of matrix}2 +1{norm of matrix})1/2]. With B well-conditioned and Λ well chosen, κ and κr can be small even when Λ (not necessary in Jordan form) possesses defective eigenvalues, depending on c0. Consequently, the SFPAP is not intrinsically ill-conditioned. Similar results were obtained in [23], although differentiability was not established for its local perturbation analysis. Simple as well as general multiple closed-loop eigenvalues are treated.

UR - http://www.scopus.com/inward/record.url?scp=84862950793&partnerID=8YFLogxK

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U2 - 10.11650/twjm/1500406541

DO - 10.11650/twjm/1500406541

M3 - Article

VL - 16

SP - 283

EP - 304

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

SN - 1027-5487

IS - 1

ER -