Zeros and poles of matrix transfer functions and their dynamical interpretation

The given rational matrix transfer function H(\cdot) is viewed as a network function of a multiport. The no X ni matrix H(s) is factored into D_{l}(S)^{-1} N_{l}(s) = N_{r}(s)D_{r}(s)^{-1} ,where D_{l}(\cdot),N_{l}(\cdot),N_{r}(\cdot) , and D_{r}(\cdot) are polynomial matrices of appropriate size, with D_{l}(\cdot) and N_{i}(\cdot) left coprime and N_{r}(\cdot) and D_{r}(\cdot) right coprime. A zero of H(\cdot) is defined to be a point z where the local rank of N_{l}(\cdot) drops below the normal rank. The theorems make precise the intuitive concept that a multiport blocks the transmission of signals proportional to e^{zt} if and only if z is a zero of H(\cdot) . We show that p is a pole of H(\cdot) if and only if some "singular" input creates a zero-state response of the form re^{pt} , for t > 0 . The order m of the zero z is similarly characterized. Although these results have state-space interpretation, they are derived by purely algebraic techniques, independently of state-space techniques. Consequently, with appropriate modifications, these results apply to the sampled-data case.