Output-ing subspaces, zero dynamics and zeros in MIMO LTI proper systems

In linear time-invariant multi-input multi-output proper systems S(A,B,C,D) the classical notion of Smith zeros does not characterize fully the output-zeroing problem nor the zero dynamics. However, introducing the notion of invariant zeros (defined as those complex numbers in which there exists a zero direction with nonzero state-zero direction) which constitutes an extension of the concept of Smith zeros, the above disadvantage can be removed. Recall that Smith zeros of S(A,B,C,D) are the roots of the so-called zero polynomial which is obtained from the Smith canonical form of the underlying system matrix P(s), while the definition of invariant zeros admits an infinite number of these zeros (then the system is called degenerate). A simple sufficient and necessary condition of non-degeneracy (degeneracy) is presented. It decomposes the class of all systems S(A,B,C,D) such that D ≠ 0, B ≠ 0, C ≠ 0 into two disjoint subclasses: of degenerate and non-degenerate systems. In a nondegenerate system the Smith and invariant zeros are exactly the same objects which are the roots of the zero polynomial. The degree of this polynomial equals the dimension of the maximal output-ing subspace, while the zero dynamics are determined fully by the Smith zeros. In a degenerate system the zero polynomial determines merely the Smith zeros, while the set of invariant zeros equals the whole complex plane. Moreover, the dimension of the maximal output-ing subspace is strictly greater than the degree of the zero polynomial, and the zero dynamics are essentially dependent upon control vector.