Geometric conditions for finite horizon noninteraction and fault detection based on the almost controllability subspace algorithm

This work deals with the problem of noninteraction (and the dual problem of fault detection) for discrete-time linear time-invariant finite-dimensional state space systems. In particular, the standard, infinite horizon, noninteracting control requirement is replaced by a less demanding one, where noninteracting control is only sought on finite time horizons, suitably defined in connection with some structural properties of the system subblocks. A necessary and sufficient condition for solvability of the finite horizon problem thus stated is derived in terms of the almost controllability subspaces associated with the block structure of the system. The proof of the condition is constructive in the sense that it leads to a design procedure for the feedforward compensators that guarantee noninteracting control over the given time horizons. The underlying idea of the proof, i.e. the exploitation of the almost controllability subspace algorithm in the case of finite horizon noninteraction, is also compatible with a modified procedure for designing the compensators achieving infinite horizon noninteraction, which may be admissible for some specific subblocks of the system. In fact, in this latter case, it is the effective use of the controllability subspace algorithm which plays a key role. The dual counterpart in the context of fault detection introduces a structural means to identify and treat the cases where, due to the structural properties of the monitored system, some of the residuals which can be generated are only significant in a limited time. These concepts are also illustrated with a detailed numerical example.