A complete model of a finite-dimensional impedance-passive system

Abstract

We extend the classes of standard discrete- and continuous-time input/state/output matrix systems by adding reverse internal and/or external channels. The reverse internal channel permits the impulse response to contain a differentiating part, and the reverse external channel allows us to include inputs which are forced to be zero and outputs which are undetermined. The purpose of this extension is obtaining a class of state-space matrix systems that can be used to realise all right-coprime positive-real rational relations—in particular non-proper positive-real rational transfer functions can be realised. We generalise the notions of impedance and scattering passivity to extended systems. When we restrict our attention to passive systems, the new class of extended impedance-passive systems is closed under the operations of interchanging the input and the output, as well as frequency inversion and duality. We generalise two system Cayley transformations to extended systems. The first transformation that we consider is the internal Cayley transformation, which maps an impedance- or scattering-passive continuous-time system into a discrete-time approximation of the original system that is again impedance passive or scattering passive, respectively. The second transformation is the external Cayley transformation that maps a contiuous- or discrete-time impedance-passive system into a scattering-passive system with the same time axis. In our extended setting, the two Cayley transformations become bijections between the respective classes of extended passive systems.