Polynomial Matrices and Matrix Fraction Descriptions

Abstract

In §21, we discovered (c.f. Theorem 21.9) that a real m-input, p -output time-invariant continuous- or discrete-time input-output linear system with impulse response t → H(t) or k → H(k) was realizable if and only if its transfer function s → G (s) or z → G (z) agreed on its region of convergence ROC(H) with a real proper (p × m) matrix rational function of s or of z. The constant realizations for such a system was shown in Facts 21.10 to lie in one-to-one correspondence with matrix quadruples (A, B, C, D) which satisfied $$\begin{aligned} & & & G\left( s \right) = C{\left( {s{I_n} - A} \right)^{ - 1}}B + D,\quad s \in ROC\left( H \right), \\ or \\ G\left( z \right) = C{\left( {z{I_n} - A} \right)^{ - 1}}B + D,\quad z \in ROC\left( H \right). \\ \end{aligned} $$