A Note on Zeros, Output-Nulling Subspaces and Zero-Dynamics in MIMO LTI Systems

In standard MIMO LTI continuous-time systems S(A,B,C) the classical notion of the Smith zeros does not characterize fully the output-zeroing problem nor the zero dynamics. The question how this notion can be extended and related to the state-space methods is discussed. Nothing is assumed about the relationship of the number of inputs to the number of outputs nor about the normal rank of the underlying system matrix. The aforementioned extension used in this paper treats multivariable zeros (called further the invariant zeros) as the triples (complex number, nonzero state-zero direction, input-zero direction). Such treatment is strictly connected with the output zeroing problem and in that spirit the zeros can be easily interpreted even in the degenerate case (i.e., when any complex number is such zero). In this paper a simple sufficient and necessary condition of nondegeneracy is presented. The condition decomposes the class of all systems S(A,B,C) such that 0 B ≠ and 0 C ≠ into two disjoint subclasses: of nondegenerate and degenerate systems. In nondegenerate systems the Smith zeros and the invariant zeros are exactly the same objects which are determined as the roots of the so-called zero polynomial. The degree of this polynomial equals the dimension of the maximal (A,B)-invariant subspace contained in Ker C, while the zero dynamics are independent upon control vector. In degenerate systems the zero polynomial determines merely the Smith zeros, while the set of the invariant zeros equals the whole complex plane. The dimension of the maximal (A,B)-invariant subspace contained in Ker C is strictly larger than the degree of the zero polynomial, whereas the zero dynamics essentially depend upon control vector.