Polynomial approach to pole placement in MIMO n-D systems

A lot of dynamical properties of a linear system can be naturally expressed in terms of the positions of its poles. That is why the problem of pole placement has become so popular in control. In scalar linear systems, it is sufficient to assign just the characteristic polynomial. In multi-input multi-output systems, however, one must assign separately all the invariant polynomials (and not merely their multiple — the characteristic polynomial). The reason is that the characteristic polynomial itself does not say enough about the internal dynamics of a multi-input multi-output system. For a scalar 2-D system, the problem of pole placement has been solved by several authors recently. We shall follow here the approach of [7] which is based on 2-D polynomial equations. The progress described in the present paper is twofold: at first, the multi-input multi-output systems are considered which call for matrix (instead of scalar) polynomial equations. At second, n-D systems are involved so that general n-D equations (n ^ 2) apply.