Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors

Abstract

This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space $${L^2([0,\infty);\mathcal W)}$$ , where $${\mathcal W}$$ is a Kreĭn space, and the inner product in $${L^2([0,\infty);\mathcal W)}$$ is the one inherited from $${\mathcal W}$$ . A state/signal system $${\Sigma=(V;\mathcal X,\mathcal W)}$$ , with a Hilbert state space $${\mathcal X}$$ and a Kreĭn signal space $${\mathcal W}$$ , is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy $${x\in C^1([0,\infty);\mathcal X)}$$ , $${w \in C([0,\infty);\mathcal W)}$$ , and $$ (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty), $$ where the generating subspace V is a given subspace of the node space $${\mathfrak K:=\mathcal X\times\mathcal X\times\mathcal W}$$ . Passivity of this systems means that V is maximal nonnegative with respect to a certain Kreĭn space inner product on $${\mathfrak K}$$ , and that $${(z,0,0)\in V}$$ implies z = 0. We present three canonical passive shift models: (a) an observable and co-energy preserving model, (b) a controllable and energy preserving model, and (c) a simple conservative model. In order to construct these models we first introduce the notions of the input map, the output map, and the past/future map of a passive state/signal system. Our canonical passive state/signal shift realizations are analogous to the corresponding de Branges–Rovnyak type input/state/output realizations of a given Schur function.