Pseudorational transfer functions—A survey of a class of infinite-dimensional systems

The class of pseudorational transfer functions consists, roughly speaking, of the ratio of entire functions of exponential type that are Laplace transforms of distributions with compact support. This class gives a convenient platform for dealing with distributed systems whose state space is determined by bounded-time data. Delay-differential systems are typical examples. The following appealing features are known: Starting from a completely general input/output framework, this class allows a concrete realization procedure based on the above fractional representation of transfer functions (or impulse responses). It is then also possible to give a complete characterization of spectral properties of such realizations. Such spectral properties allows us to give a stability criterion and also an appropriate relationship between internal and external stability notions. Based on a concrete representation of the state space, it also leads to a concrete characterization of left-shift invariant subspaces of H². This result has a direct consequence on H^infin control theory. We give a concise yet comprehensive and unified overview of such results. The paper is concluded with this and also a recent result on the existence of a Bezout identity in this class.