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2 International Journal of Applied Mathematics and Computer Science

2 2001

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Time-variant Darlington synthesis and induced realizations

Pik D.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 6

1331-1360

For a block lower triangular contraction T, necessary and sufficient conditions are given in order that there exist block lower triangular contractions T1,1, T2,1 and T2,2 such that Ut= [ T1,1 T ] T2,1 T2,2 is unitary. For the case when T1,1* and T2,2 have dense ranges, all such embeddings are described. Each unitary embedding of UT induces a contractive realization of T, and various properties of this realization are characterized in terms of the unitary embedding.

The geometry of Darlington synthesis

Dewilde P.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 6

1379-1386

We revisit the classical problem of `Darlington synthesis', or Darlington embedding. Although traditionally it is solved using analytic means, a more natural way to approach it is to use the geometric properties of a well-chosen Hankel map. The method yields surprising results. In the first place, it allows us to formulate necessary and sufficient conditions for the existence of the embedding in terms of systems properties of the transfer operation to be embedded. In addition, the approach allows us to extend the solution to situations where no analytical transform is available. The paper has a high review content, as all the results presented have been obtained during the last twenty years and have been published. However, we make a systematic attempt at formulating them in a geometric way, independent of an accidental parametrization. The benefit is clarity and generality.