State/Signal Linear Time-Invariant Systems Theory, Part IV: Affine and Generalized Transfer Functions of Discrete Time Systems

Abstract.

In the present article we continue to develop the linear discrete time-invariant state/signal systems theory that we have introduced and studied in three earlier articles (Parts I—III). The trajectories of a state/signal system ∑ with a Hilbert state space χ and a Hilbert or Kreĭn signal space ${\mathcal{W}}$ consists of a pair of sequences (x(·),w(·)) which after an admissible input/output decomposition ${\mathcal{W}} = {\mathcal{Y}}\,\dot{+}\,{\mathcal{U}}$ of the signal space can be obtained from the set of trajectories (x(·), u(·), y(·)) of a standard input/state/output system by taking w(·) = y(·) + u(·). In Part I we studied the families of all admissible decompositions of W for a given state/signal system ∑ and the corresponding input/state/output representations and their transfer functions. Here we extend that theory to the the non-admissible case, and obtain generalized input/output transfer functions, as well as right and left affine transfer functions. As opposed to a standard transfer function, a generalized transfer function not need to be holomorphic at the origin. This makes it possible to realize a much larger class of transfer functions than what is possible by using the standard input/state/output theory. For example, every rational matrixvalued function (including those that have a pole at the origin) can be realized as the generalized transfer function of a state/signal system whose state space has a finite dimension equal to the McMillan degree of the given function. Likewise, every meromorphic J-contractive matrix-valued function can be realized as the generalized transfer function of a simple conservative state/signal system, or of a minimal passive state/signal system. Similar operator-valued results are also presented. As is well-known, a rational matrix-valued function with a pole at the origin can also be realized as the transfer function of a descriptor system, but this descriptor realization has the drawback that the dimension of its state space is bigger than the dimension of the state space of our s/s realization (i.e., bigger than the McMillan degree of the function). Our s/s realizations also differ significantly from other known realizations which depend on the choice of an auxiliary point in the complex plane where the given function is holomorphic.