Conditioning of state feedback pole assignment problems

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 P_c(·) ≡ (A + BF)(·) - (·)Λ and N(·) ≡ (I - BB^†)P_c(·),and c₀ ≡ {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 c₁ ≡ ({norm of matrix}B{norm of matrix}{norm of matrix}F{norm of matrix})^-1, the relative condition number κ_r ≤ c₀c_B [c₁κ_X{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 κ_r 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.