Embedding of time-varying contractive systems in lossless realizations

The lossless embedding problem, also known as the Darlington synthesis or unitary extension problem, considers the extension of a given contractive system to become the partial input-output operator of a lossless system. In the paper, the embedding problem is solved for discrete-time time-varying systems with finite but possibly time-varying state dimensions, for the strictly contractive as well as the boundary case. The construction is done in a state space context and gives rise to a time-varying Riccati difference equation which is shown to have a closed-form solution. As a corollary, a discrete-time Bounded Real Lemma is formulated, linking contractiveness of an input-output operator to conditions on its state realization.