We define two versions of compositions of matrix-valued rational functions of appropriate sizes and whenever analytic at infinity, offer a set of formulas for the corresponding state-space realization, in terms of the realizations of the original functions. Focusing on positive real functions, the first composition is applied to electrical circuits theory along with introducing a connection to networks of feedback loops. The second composition is applied to Stieltjes functions.
D. Alpay & I. Lewkowicz, “Composition of rational functions: State-space realization and applications”, Lin. Alg. & Appl., vol. 580, pp. 359-383, 2019.