## An analysis of the pole placement problem II. The multi-input case

Volker Mehrmann and Hongguo Xu

### Abstract

For the solution of the multi-input pole placement problem we derive explicit formulas for the subspace from which the feedback gain matrix can be chosen and for the feedback gain as well as the eigenvector matrix of the closed-loop system. We discuss which Jordan structures can be assigned and also when diagonalizability can be achieved. 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, the norm of the feedback matrix and the distance to uncontrollability.

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### Key words

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

### AMS subject classifications

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

### Links to the cited ETNA articles

 [22] Vol. 4 (1996), pp. 89-105 Volker Mehrmann and Hongguo Xu: An analysis of the pole placement problem. I. The single-input case