Orthogonal Embedding

Abstract

In chapter 5, we saw how a state realization of a time-varying transfer operator T can be computed. The realizations which we obtained were in principle either in input normal form (A*A + B*B = I) or in output normal form (AA* + CC* = I). In chapter 6, we considered unitary systems V with unitary realizations. Such realizations are both in input normal form and in output normal form, and satisfy the additional property that both ‖V‖ = 1 and ‖V‖ = 1, while for T in either normal form, we have ‖T‖ ≥ 1, whether ‖T‖ is small or not. Since ‖T‖ tells something about the sensitivity of the realization, i.e., the transfer of errors in either the input or the current state to the output and the next state, it is interesting to know whether it is possible to have a realization of T for which ‖T‖ ≤ 1 when ‖T‖ ≤ 1. This issue can directly be phrased in terms of the problem which is the topic in this chapter: the orthogonal embedding problem. This problem is, given a transfer operator T ∈ T, to extend this system by adding more inputs and outputs to it such that the resulting system Σ, a 2 × 2 block operator with entries in T,

$$ \sum { = \left[ {\begin{array}{*{20}{c}} {\sum{_{11}}}&{\sum{_{12}}}\\ {\sum{_{21}}}&{\sum{_{22}}} \end{array}}\right]} $$

is inner and has T as its partial transfer when the extra inputs are forced to zero: T = Σ₁₁. See figure 12.1. Since the unitarity of Σ implies T*T + T _c *T _c = I, (where T _c = Σ₂₁), it will be possible to find solutions to the embedding problem only if T is contractive: I−T*T ≥ 0, so that ‖T‖ ≤ 1. Since Σ is inner, it has a unitary realization Σ, and a possible realization T of T is at each point k in time a submatrix of Σ _k (with the same A _k , and smaller dimensional B _k ,C _k ,D _k ), and hence T is a contractive realization.