Abstract
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 this 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.
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This research was supported in part by the commission of the EC under the ESPRITBRA program 6632 (NANA2).
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van der Veen, A.J., Dewilde, P. Embedding of time-varying contractive systems in lossless realizations. Math. Control Signal Systems 7, 306–330 (1994). https://doi.org/10.1007/BF01211522
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DOI: https://doi.org/10.1007/BF01211522