# Passive linear discrete time-invariant systems

### Olof J. Staffans

Åbo Akademi University, Finland

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## Abstract

We begin by discussing linear discrete time-invariant i/s/o (input/state/output) systems that satisfy certain ‘energy’ inequalities. These inequalities involve a quadratic storage function in the state space induced by a positive self-adjoint operator *H* that may be unbounded and have an unbounded inverse, and also a quadratic supply rate in the combined i/o (input/output) space. The three most commonly studied classes of supply rates are called scattering, impedance, and transmission. Although these three classes resemble each other, we show that there are still signiﬁcant differences. We then present a new class of s/s (state/signal) systems which have a Hilbert state space and a Kreĭn signal space. The state space is used to store relevant information ı about the past evolution of the system, and the signal space is used to describe interactions with the surrounding world. A s/s system resembles an i/s/o system apart from the fact that inputs and outputs are not separated from each other. By decomposing the signal space into a direct sum of an input space and an output space one gets a standard i/s/o system, provided the decomposition is *admissible*, and different i/o decompositions lead to different i/o supply rates (for example of scattering, impedance, or transmission type). In the case of non-admissible decompositions we obtain right and left afﬁne representations, both of the s/s system itself, and of the corresponding transfer function. In particular, in the case of a passive system we obtain right and left coprime representations of the generalized transfer functions corresponding to nonadmissible decompositions of the signal space, and we end up with transfer functions which are, e.g., generalized Potapov or Nevanlinna class functions.