### 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 |
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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