The general problem of pole assignment‡

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Abstract
Let G be a strictly proper, rational m × l matrix, with controllability indices λ1≥λ2≥…≥λ l and observability indices μ1≥μ2≥…≥μm. Also let ϕ1 ϕ2…, ϕ l be monic polynomials, where ϕi divides ϕi−1, i = 2, 3,…,l. Does there exist a proper rational feedback matrix K which makes the invariant polynomials of the resulting closed-loop system equal to the ϕi ? What conditions will ensure that there always exists a suitable K ? These questions define the general problem of pole assignment. A number of results are proved in relation to this problem. In particular it is shown that a suitable K always exists if the degrees of the ϕ i satisfy where equality holds when k = l. This contains earlier results of Wonham, Rosenbrock, Brasch and Pearson, and others. A dual result is available in which the roles of the λ i and μ i are interchanged.

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