On the problem of model reduction in the gap metric

This paper deals with the model reduction problem where, for a given linear time-invariant dynamical system of complexity n, a simpler system of complexity r <n is desired such that the gap between their respective behaviors is minimized. We describe dynamical systems as closed, shift invariant subspaces of H_2^+, represented as kernels of rational multiplicative operators that are anti-stable rational elements of RH_\infty^-. Contrary to other approaches this enables to reduce autonomous behaviors. In this paper we will give upper- and lower bounds for the minimal gap between a rational behavior and its optimal approximation in this system class. Bounds are given in terms of its Hankel Singular Values. These bounds only depend on the given system and can be computed in advance due to the use of rational operators describing the dynamical systems. This will be illustrated by a simple example.