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BIRS Workshop Lecture Videos

Bezout Equations for Stable Rational Matrix Functions: The Least Squares Solution and Description of all Solutions Kaashoek, Rien


This talk concerns the corona type Bezout equation $G(z)X(z)=I_p$, $z$ in $BD$. The function $G$ is the given function and $X$ is the unknown. Both functions are stable rational matrix functions, $G$ is of size $p\times q$ and $X$ of size $q\times p$, and $I_p$ stands for the $p\times p$ identity matrix. Here \emph{stable} means that the poles of the rational matrix functions involved are all outside the closed unit disc, and hence the entries of these matrix functions are $H^\infty$- functions. We shall use state space techniques from mathematical system theory to obtain necessary and sufficient conditions for existence of solutions. Furthermore, we derive an explicit formula for the least squares solution and an explicit description of all solutions, all in terms of a state space realization of the given function $G$. The state space formula for the least squares solution is easy to use in Matlab, and it shows that the McMillan degree of this solution is less than or equal to the McMillan degree of $G$. The talk is based on joint work with Art Frazho (Purdue University) and Andre Ran (VU Amsterdam).

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