A geometric approach to structural model matching by output feedback in linear impulsive systems

This paper provides a complete characterization of solvability of the problem of structural model matching by output feedback in linear impulsive systems with nonuniformly spaced state jumps. Namely, given a linear impulsive plant and a linear impulsive model, both subject to sequences of state jumps which are assumed to be simultaneous and measurable, the problem consists in finding a linear impulsive compensator that achieves exact matching between the respective forced responses of the linear impulsive plant and of the linear impulsive model, by means of a dynamic feedback of the plant output, for all the admissible input functions and for all the admissible sequences of jump times. The solution of the stated problem is achieved by reducing it to an equivalent problem of structural disturbance decoupling by dynamic feedforward. Indeed, this latter problem is formulated for the so-called extended linear impulsive system, which consists of a suitable connection between the given plant and a modified model. A necessary and sufficient condition for the solution of the structural disturbance decoupling problem is first shown. The proof of sufficiency is constructive, since it is based on the synthesis of the compensator that solves the problem. The proof of necessity is based on the definition and the geometric properties of the unobservable subspace of a linear impulsive system subject to unequally spaced state jumps. Finally, the equivalence between the two structural problems is formally established and proven.