State/signal linear time-invariant systems theory: Passive discrete time systems

Abstract

This is a continuation of previous work where we developed a discrete time-invariant linear state/signal systems theory in a general setting. In this article, the state space is required to be a Hubert space, as earlier, but the signal space is taken to be a Kreǐn space. The notion of the adjoint of a given state/signal system is introduced and exploited throughout the paper, and in particular, in the definition and the study of passive and conservative state/signal systems, which is the main subject of this paper. It is shown that each fundamental decomposition of the Kreǐn signal space is admissible for a passive state/signal system, meaning that there is a corresponding input/state/output representation of the system, a so-called scattering representation. The connection between different scattering representations and their scattering matrices (i.e. transfer functions) is explained. We show that every passive state/signal system has a minimal conservative orthogonal dilation and minimal passive orthogonal compressions. Passive signal behaviours are defined, and their passive, conservative, and H-passive realizations are studied. It is shown that the set of all positive self-adjoint operators H (which need not be bounded or have a bounded inverse) for which a state/signal system Σ is H-passive coincides with the set of generalized positive solutions H of the Kalman-Yakubovich-Popov inequality for an arbitrary scattering representation of S, and consequently, this set does not depend on the particular representation. Under an extra minimality assumption this set contains a minimal solution which defines the available storage, and a maximal solution which defines the required supply. Copyright © 2006 John Wiley & Sons, Ltd.