A structural approach to unknown inputs observation for switching linear systems

The problem of devising an asymptotic observer for a given function of the state of a switching linear system in the presence of unknown inputs is considered. Solvability is studied both in the case of sufficiently large dwell time and in that of dwell time greater than a fixed threshold. A complete characterization of solvability in terms of necessary and sufficient conditions is given in both cases. It is shown that the necessary and sufficient conditions can be checked in practice in the first case and, under slightly more restrictive hypotheses, also in the second case by means of algorithmic procedures, which also provide a method to synthesize the observer sought for. The employed methodology makes use of geometric concepts to reveal the structural aspects of the problem and to derive its solutions. In particular, a key role is played by the novel notion of robust conditioned invariant subspace that is minimal with respect to the properties of containing a given subspace and of being externally stabilizable.