Passive and Conservative Infinite-Dimensional Impedance and Scattering Systems (From a Personal Point of View)

Abstract

Let U be a, Hilbert space. By a L(U)-valued positive analytic function on the open right half-plane we mean an anadytic function which satisfies the condition $\hat D + \hat D^* \ge 0$. This function need not be proper, i.e., it need not be bounded on any right half-plane. We give a complete answer to the question under what conditions such a function can be realized as the transfer function of a impedance passive system. By this we mean a continuous time state space system whose control and observation operators are not more unbounded than the (main) semigroup generator of the system, and in addition, there is a certain energy inequality relating the absorbed energy and the internal energy. The system is (impedance) energy preserving if this energy inequality is an equality, and it is conservative if both the system and its dual are energy preserving. A typical example of an impedance conservative system is a system of hyperbolic type with collocated sensors and actuators. We prove that a passive realization exists if and only if a conservative realization exists, and that this is true if and only if $\lim _{\text{s} \to \text{ + }\infty } \frac{1}{s}\hat D(s)u = 0$ for every $u \in U$. The physical interpretation of this condition is that the input-output response is not allowed to contain a pure derivative action. We furthermore show that the so called diagonal transform (which is a particular rescaled feedback/feedforward transform) maps an impedance passive (or energy preserving or conservative) system into a (well-posed) scattering passive (or energy preserving or conservative) system. This implies that if we apply negative output feedback to a impedance passive system, then the resulting system is both well-posed and energy stable. Finally, we study lossless scattering systems, i.e., scattering conservative systems whose transfer functions are inner.