Conditioning of state feedback pole assignment problems

Eric King Wah Chu, Chang Yi Weng, Chern-Shuh Wang, Ching Chang Yen

Research output: Contribution to journalArticle

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 ≤ 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.

Original languageEnglish
Pages (from-to)283-304
Number of pages22
JournalTaiwanese Journal of Mathematics
Volume16
Issue number1
DOIs
Publication statusPublished - 2012 Jan 1

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Pole Assignment
Assignment Problem
State Feedback
Conditioning
Norm
Condition number

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Chu, Eric King Wah ; Weng, Chang Yi ; Wang, Chern-Shuh ; Yen, Ching Chang. / Conditioning of state feedback pole assignment problems. In: Taiwanese Journal of Mathematics. 2012 ; Vol. 16, No. 1. pp. 283-304.
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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 ≤ 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.",
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Conditioning of state feedback pole assignment problems. / Chu, Eric King Wah; Weng, Chang Yi; Wang, Chern-Shuh; Yen, Ching Chang.

In: Taiwanese Journal of Mathematics, Vol. 16, No. 1, 01.01.2012, p. 283-304.

Research output: Contribution to journalArticle

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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.

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