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Polynomial families with multilinear uncertainties
Abstract
In this dissertation we study polynomials $p(s,q)$ whose coefficients depend multiaffinely on parameters $q$ which lie in a hyperrectangle $Q \in R\sp{n}$. Such a collection, or family, of polynomials arise in the study of uncertain linear systems and establishing their Hurwitzness is at the heart of many robust control problems. Our objective is to give some conditions under which this Hurwitzness can be easily deduced. Central to this dissertation is the so-called value set of a polynomial family. At a given frequency $\omega$ the value set is $p(j\omega,Q)$ and the Hurwitzness of the entire family of polynomials is ensured provided that 0 $\not\in p(j\omega,Q$) for all $\omega \in R.$ In this way, the shape of the value set becomes crucial in deducing whether a polynomial family is Hurwitz or not. This dissertation gives three results to help describe the value set $p(j\omega,Q)$. The first result completely describes the value set when $Q$ is a two-dimensional box. The second result gives a necessary and sufficient condition for $p(j\omega,Q)$ to be a convex set and provides a pleasing and elegant counterpart to Zadeh and Desoer's Mapping Theorem. Finally, we identify a nontrivial subclass of multiaffine polynomial families whose value sets can be completely described as in the two-dimensional case.
Subject Area
Mathematics
Recommended Citation
Xu, Zhong-Ling, "Polynomial families with multilinear uncertainties" (1991). Doctoral Dissertations Available from Proquest. AAI9120959.
https://scholarworks.umass.edu/dissertations/AAI9120959