CONDITIONING OF STATE FEEDBACK POLE ASSIGNMENT PROBLEMS

• Published in: Taiwanese journal of mathematics Vol. 16; no. 1; pp. 283 - 304
• Main Authors: Chu, Eric King-Wah, Weng, Chang-Yi, Wang, Chern-Shuh, Yen, Ching-Chang
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.02.2012
• Subjects: Analytic functions; Analyticity; Eigenvalues; Eigenvectors; Error bounds; Implicit functions; Jordan matrices; Matrices; Power series; Solvability
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/twjm/1500406541

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 numberκ≤c 0‖B †‖ [κX + (1 + ‖F‖2)1/2], where the closed-loop system matrixA+BF=XΛX −1, the closed-loop spectrum in Λ is pre-determined,κX ≡‖X‖‖X −1‖, the operatorsPc (·) ≡ (A+BF)(·) − (·)Λ and 𝒩(·) ≡ (I−BB †)Pc (·), andc 0≡‖I(·) −Pc [𝒩†(I−BB †) (·)]‖. WithcB ≡‖B‖ ‖B †‖ andc 1≡ (‖B‖‖F‖)−1, the relative condition number κ r ≤ c 0 c B [ c 1 k X ‖ Λ ‖   +   ( c 1 2 ‖ A ‖ 2 +   1 ) 1 / 2 ] . WithBwell-conditioned and Λ well chosen,κandκr can be small even when Λ (not necessary in Jordan form) possesses defective eigenvalues, depending onc 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. 2010Mathematics Subject Classification: 65F15, 65F35, 65G05, 93B05, 93B55. Key words and phrases: Condition number, Perturbation, Pole assignment, Stabilization, Jordan form, Kronecker form, State feedback.