Zeros and poles of matrix transfer functions and their dynamical interpretation

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{l}(S)^{-1} N_{l}(s) = N_{r}(s)D_{r}(s)^{-1}</tex> ,where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{l}(\cdot),N_{l}(\cdot),N_{r}(\cdot)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{r}(\cdot)</tex> are polynomial matrices of appropriate size, with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{l}(\cdot)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{i}(\cdot)</tex> left coprime and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{r}(\cdot)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{r}(\cdot)</tex> right coprime. A zero of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(\cdot)</tex> is defined to be a point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> where the local rank of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{l}(\cdot)</tex> drops below the normal rank. The theorems make precise the intuitive concept that a multiport blocks the transmission of signals proportional to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e^{zt}</tex> if and only if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> is a zero of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(\cdot)</tex> . We show that p is a pole of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(\cdot)</tex> if and only if some "singular" input creates a zero-state response of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">re^{pt}</tex> , for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t &gt; 0</tex> . 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.