Topological Aspects of Transfer Matrices with Entries in the Quotient Field of H∞
M. L. J. Hautus, S.Q. Zhu · Birkhäuser Boston eBooks · 1990
Let F nxm be the set of all n by m transfer matrices with entries in the quotient field F of H ∞ . This article investigates the properties of F nxm with respect to the gap topology. First, we identify the subset B n,m of F nxm consisting of all transfer matrices possessing right and left Bezout fractions, and we show that B n,m is an open subset of F nxm in the gap topology. Moreover, a bound is given in terms of the gap metric which guarantees that if the distance of a transfer matrix P 1 ∈ F nxm from P 2 ∈ B n,m is smaller than this bound, then P 1 is also in B n,m . In addition, P 1 and P 2 can be stabilized by a same controller. Furthermore, a relation between the gap of two transfer matrices and the gap of their domains is given. Using this relation, we show that, if the gap of two scalar transfer functions P 1 and P 2 is smaller than a specified number, then P 1 and P 2 and must have the same number of poles in the open right half plane. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.