CONDITIONING OF STATE FEEDBACK POLE ASSIGNMENT PROBLEMS

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 $\kappa \leq c_0 \|B^{\dagger}\| $ $ \left[ \kappa_X + \left(1 + \|F\|^2 \right)^{1/2} \right]$, where the closed-loop system matrix $A+BF = X \Lambda X^{-1}$, the closed-loop spectrum in $\Lambda$ is pre-determined, $\kappa_X \equiv\|X\| \|X^{-1}\|$, the operators $P_c (\cdot) \equiv (A+BF)(\cdot) - (\cdot) \Lambda$ and $\mathcal{N}(\cdot) \equiv (I - BB^{\dagger}) P_c(\cdot)$, and $c_0 \equiv \|I(\cdot) -P_c\left[ \mathcal{N}^{\dagger}(I - BB^{\dagger}) (\cdot) \right]\|$. With $c_B \!\equiv\!\|B\| \|B^{\dagger}\|$ and $c_1 \!\equiv\! (\|B\| \|F\|)^{-1}$, the relative condition number $\kappa_r \!\leq c_0 c_B \left[ c_1 \kappa_X \|\Lambda\| + \right. \left. \left( c_1^2 \|A\|^2 + 1 \right)^{1/2} \right]$. With $B$ well-conditioned and $\Lambda$ well chosen, $\kappa$ and $\kappa_r$ can be small even when $\Lambda$ (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.