A note on determination of zeros and zero directions by the Moore-Penrose pseudoinverse of the first nonzero Markov parameter

An unified approach to the question of determining and interpreting invariant zeros and zero directions in MIMO LTI continuous-time strictly proper and proper systems is presented. It is shown that invariant zeros of a strictly proper system appear as eigenvalues of an appropriate real matrix (formed with the aid of the Moore-Penrose pseudoinverse of the first non-zero Markov parameter and of order equals to the dimension of the state space) whereas zeros at infinity are represented by zero eigenvalues of that matrix. An analytic form of real-valued output-zeroing inputs as well as analytic form of rectilinear motions in the space of the output matrix are derived. The geometric structure of finite and infinite zeros as well as of state-zero and input zero directions are described. An analogous discussion is carried out for proper systems.