Extended zeros and model matching

• Published in: SIAM journal on control and optimization 1991-05, Vol.29 (3), p.562-593
• Main Authors: SAIN, M. K, WYMAN, B. F, PECZKOWSKI, J. L
• Format: Article
• Language: English
• Subjects: Applied sciences; Computer science; control theory; systems; Control theory; Control theory. Systems; Exact sciences and technology; System theory
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/0329032

The model matching equation $T(z) = P(z)M(z)$ induces constraints upon the multivariate zero structures of $P(z)$ and $M(z)$; the nature of the constraint is best explained by extending the usual notion of zero. In particular, the extended $\Gamma $-zero module of $P(z)$ must contain as a submodule the module $Z_\Gamma $ of matching $\Gamma $-zeros, which depends only upon $T(z)$ and $M(z)$; and the extended $\Omega $-zero module of $M(z)$ must contain as a factor module the module $Z_\Omega $ of matching $\Omega $-zeros, which depends only upon $T(z)$ and $P(z)$. Essential solutions, in which the constraint is by module isomorphism, are possible if and only if the nullity of $P(z)$ does not exceed the nullity of $T(z)$, on the one hand, or the co-nullity of $M(z)$ does not exceed the co-nullity of $T(z)$, on the other. Both the matching zero modules and their finitely generated, torsion parts--which have state-space interpretation--can be given concrete, intuitive interpretation in terms of short exact sequences, though the former is less involved than the latter. Moreover, in the case of the latter, a natural notion of essential solution is not available, in marked contrast to the situation for poles.