From transfer matrices to realizations: Convergence properties and parametrization of robustness analysis conditions

In various branches of systems and control theory, one is confronted with the need for approximating transfer functions by a sequence of FIR expansions in the H∞-norm. The approximating sequence grows in its McMillan degree, while the limiting transfer matrix has a finite number of poles. Considering the corresponding state-space realizations of the approximating sequence and its limit, it is of interest to understand the limiting behavior of the realization matrices. This paper investigates this behavior and, thus, provides an answer for the continuous-time counterpart of FIR expansions and exponential convergence. In a similar vein, it is well-understood how to translate frequency-domain inequalities for transfer matrices into LMIs for realizations by using the Kalman–Yakubovich–Popov (KYP) Lemma. However, it is often less clear how obviously valid manipulations of the frequency-domain inequalities lift into operations on the solutions of the corresponding LMIs. The paper provides some novel insights on such questions with applications to robustness analysis.