An Review on Realization Theory for Infinite-Dimensional Systems∗

Already since the beginning of infinite-dimensional systems theory, there has been interest in the state-space realization problem. The state-space realization problem is the problem of finding for a given function a system in state-space form whose transfer function equals the given function. If we start with a rational function, then it is well-known, that one can always find a system with finite-dimensional state space whose transfer function is the given rational function. Moreover, it is always possible to find a controllable and observable realization and all controllable and observable realizations are equivalent, i.e., let (A1, B1, C1, D1) and (A2, B2, C2, D2) be two realizations that are controllable and observable, then there exists an invertible matrix S such that A1 = SA2S, B1 = SB2, C1 = C2S, and D1 = D2. Since every finite-dimensional system has a rational transfer function, for a non-rational function we only can find a (state-space) realization with an infinite-dimensional state space. For functions that are analytic and bounded in some right-half plane, the realization problem was investigated by a number of people, e.g. Baras and Brockett [BB73], Fuhrmann [Fuh81], Helton [Hel76], Yamamoto [Yam81, Yam82], Salamon [Sal89] and Weiss [Wei89c, Wei97]. Here we present the realization theory in the language of well-posed linear systems. ∗This paper was supported by the Volkswagen Stiftung (RiP program at Oberwolfach) and by the Deutsche Forschungsgemeinschaft.