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Matrix-valued Laurent polynomials, parametric linear systems and integrable systems

In this paper, we study transfer functions corresponding to parametric linear systems whose coefficients are block matrices. Thus, these transfer functions constitute Laurent polynomials whose coefficients are square matrices. We assume that block matrices defining the parametric linear systems are solutions of an integrable hierarchy called for us, the block matrices version of the finite discrete KP hierarchy, which is introduced and studied with certain detail in this paper. We see that the linear system defined by means of the simplest solution of the integrable system is controllable and observable. Then, as a consequence of this fact, it is possible to verify that any solution of the integrable hierarchy, obtained by the dressing method of the simplest solution, defines a parametric linear system which is also controllable and observable.