Zeros and poles of matrix transfer functions and their dynamical interpretation
C. Desoer, Jane Schulman · IEEE Transactions on Circuits and Systems · 1974
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 intoD_{l}(S)^{-1} N_{l}(s) = N_{r}(s)D_{r}(s)^{-1},whereD_{l}(\cdot),N_{l}(\cdot),N_{r}(\cdot), andD_{r}(\cdot)are polynomial matrices of appropriate size, withD_{l}(\cdot)andN_{i}(\cdot)left coprime andN_{r}(\cdot)andD_{r}(\cdot)right coprime. A zero ofH(\cdot)is defined to be a pointzwhere the local rank ofN_{l}(\cdot)drops below the normal rank. The theorems make precise the intuitive concept that a multiport blocks the transmission of signals proportional toe^{zt}if and only ifzis a zero ofH(\cdot). We show that p is a pole ofH(\cdot)if and only if some "singular" input creates a zero-state response of the formre^{pt}, fort > 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.