State/signal linear time‐invariant systems theory: passive discrete time systems

Abstract

AbstractThis 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 Hilbert space, as earlier, but the signal space is taken to be a Kreĭn 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 Kreĭn 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 Σ, 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.