Impulsive-smooth behavior in multimode systems

[BS-R 9431] A `switched' or `multimode' system is one that can switch between various modes of operation. We consider here switched systems in which the modes of operation are characterized as linear finite-dimensional systems, not necessarily all of the same McMillan degree. When a switch occurs from one of the modes to another of lower McMillan degree, the state space collapses and an impulse may result, followed by a smooth evolution under the new regime. This paper is concerned with the description of such impulsive-smooth behavior on a typical interval. We propose an algebraic framework, modeled on the class of impulsive-smooth distributions as defined by Hautus. Both state-space and polynomial representations are considered, and we discuss transformations between the two forms.#[BS-R 9432] This is the second part of a two-part paper on linear multimode systems. In the first part, it has been argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant, and both first-order and polynomial representations were introduced that satisfy this requirement. Here we determine the conditions under which first-order representations are minimal. We also show how two minimal representations of the same behavior are related; this leads in particular to an appropriate state space isomorphism theorem. The minimality conditions are given a dynamic interpretation.