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Necessary and sufficient conditions for zero assignment by constant squaring down. (English) Zbl 0679.93012
Summary: The problem of zero assignment by constant squaring down (CZAP) is studied for minimal systems described by a transfer function G(s)\(\in {\mathbb{R}}^{m\times l}(s)\), \(m>\ell\), using tools from exterior algebra and algebraic geometry. Conditions for its solvability are given in terms of the numbers m, \(\ell\), the Forney dynamical order \(\delta\), and the rank \(\rho_{\delta}\) of the Plücker matrix \(P_{\delta}.\)
For the cases \(\ell =l\) or \(\ell =m-1\), it is shown that G(s) is completely zero-assignable (CZA) if and only if \(\rho_{\delta}=\delta +1\). For the cases \(\ell \neq 1\), m-1, it is proved that G(s) is CZA, or generically zero-assignable (GZA), under a complex squaring down if and only if \(\ell (m-\ell)\geq \delta +1\) and \(\rho_{\delta}=\delta +1.\)
The latter conditions also provide necessary conditions for the existence of a real solution. For a generic G(s), sufficient conditions for solvabilty of CZAP by a real squaring down are that \(\ell (m-\ell)\geq \delta +1\) and that the number \[ g(a_ 0,...,a_{\ell -1})=(\delta +1)!/a_ 0!...a_{\ell -1}!\prod_{i>j}(a_ i-a_ j) \] is odd for sme set \(\{a_ 0,...,a_{\ell -1}\}\) satisfying \[ \delta +1=\sum^{\ell -1}_{i=0}a_ i-\ell (\ell -1)/2\quad and\quad 0\leq a_ 0<a_ 1<...<a_{\ell -1}\leq m-1. \] Apart from the existence results, the present approach also allows the computation of the solutions (whenever they exist). Finally, it is proved that the only fixed zeros of CZAP are the zeros of G(s), and a procedure for computing the solutions of CZAP, whenever such solutions exist, based on an optimization problem is given.

MSC:
93B55 Pole and zero placement problems
93B25 Algebraic methods
93B27 Geometric methods
93C05 Linear systems in control theory
Full Text: DOI
References:
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