Pole assignment of strictly proper and proper linear systems by constant output feedback

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Abstract
A new unifying approach for the study of pole assignment by constant output feedback (CPAP) for strictly proper and proper linear systems is presented. The multilinear nature of CPAP is reduced to a linear problem and to the standard multilinear problem of decomposability of multivectors; the solvability of CPAP thus becomes a problem of finding real intersections of a linear variety with the Grassmann variety of a projective space. An alternative proof to the ml≥n(m, I, n are the numbers of outputs, inputs, states) necessary and sufficient condition for generic pole-assignability of strictly proper systems by complex output feedback is given; the above result is extended to the ml≥ n + 1 condition for the case of proper systems. Stronger necessary conditions for generic pole assignment by a real output feedback are obtained by using the recently introduced invariant, the Plücker matrix Pn such conditions are that Pn must have full rank and ml≥n (strictly proper case), or ml≥ n + 1 (proper case). It is shown that if min{m, /} = 1, then the full rank condition of Pn is necessary and sufficient for complete pole-assignability by a real output feedback. New sufficient conditions for generic pole-assignability of generic strictly proper and generic proper systems with min {m, /} ≠ 1 are obtained; for the strictly proper case, these conditions are ml> n and that a number gfα0, α 1 α1-1) is odd. For the proper case, these conditions become ml≥n+ 1 and that a number ǵ( α0 α 1,,α1,-1) is odd. For the case of strictly proper systems with ml = n, the above result is reduced to the Brockett and Byrnes (1981) sufficient condition. The approach presented in this paper is suitable for the computation of the feedback, whenever a solution exists, and this is illustrated by a number of numerical examples.

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