A note on physical interpretation of zeros in discrete-time MIMO LTI systems

A geometric interpretation of invariant zeros of MIMO LTI discrete-time systems is provided in the paper. The zeros are defined as the triples: complex number, state-zero direction, input-zero direction. This definition 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 each complex number constitutes an invariant zero). Simply, in the degenerate case, to each complex number we can assign an appropriate real initial condition and an appropriate real input sequence which produces a nontrivial solution of the state equation and identical zero system response. Naturally, when the zeros are treated merely as complex numbers such interpretation is impossible. The presented definition of invariant zeros is compared with other commonly known definitions. It is shown that each Smith zero of the system matrix (i.e., invariant zero in the sense of the Rosenbrock, Wolovich and MacFarlane-Karcanias definitions) is also an invariant zero in the sense of the definition of zeros adopted in the paper. On the other hand, simple numerical examples show that the considered definition of invariant zeros and the Davison-Wang definition are not comparable