## An analysis of the pole placement problem. I. The single-input case

Volker Mehrmann and Hongguo Xu

### Abstract

For the solution of the single-input pole placement problem we derive explicit expressions for the feedback gain matrix as well as the eigenvector matrix of the closed-loop system. Based on these formulas we study the conditioning of the pole-placement problem in terms of perturbations in the data and show how the conditioning depends on the condition number of the closed loop eigenvector matrix, which is a similar to a generalized Cauchy matrix, the norm of the feedback vector and the distance to uncontrollability.

Full Text (PDF) [247 KB]

### Key words

pole placement, condition number, perturbation theory, Jordan form, explicit formulas, Cauchy matrix, stabilization, feedback gain, distance to uncontrollability.

### AMS subject classifications

65F15, 65F35, 65G05, 93B05, 93B55.

 Vol. 5 (1997), pp. 77-97 Volker Mehrmann and Hongguo Xu: An analysis of the pole placement problem II. The multi-input case Vol. 11 (2000), pp. 25-42 M. E. Cawood and C. L. Cox: Perturbation analysis for eigenstructure assignment of linear multi-input systems Vol. 20 (2005), pp. 50-63 Daniel Kressner: On the use of larger bulges in the QR algorithm

< Back