Orthogonal Embeddings of Linear Time-Varying Systems

Abstract

Given a contractive linear system T which is represented as a causal linear contractive operator on a Hilbert space H, we study the question: When does there exist an isometric or unitary causal extension of T to H l H ? Here causality is formulated as a lower triangularity condition and we assume the same causality structure for the extension. This question is often referred to as the generalized Darlington synthesis problem for linear time-varying systems. SðsÞ ¼ TðsÞ S 12 ðsÞ S 21 ðsÞ S 22 ðsÞ ! ;where the embedding has the same complexity-viewed as a dynamical systemas the original? In operator theoretic terms ''lossless'' means isometric and thus Halmos' result seems relevant. However, the context of dynamical systems requires that S 12 ; S 21 ; S 22 fulfill analyticity conditions which reflect the physical *