Fuhrmann, P. A. and Ober, R.
(1993)
A functional approach to LQG balancing.
*International Journal of Control*, 57 (3), 627-741.
(doi:10.1080/00207179308934410).

## Abstract

The paper has as its theme a circle of problems related to LQG balancing, with a special emphasis on the related problems of model reduction and robust stabilization. The aim is to present a unified point of view to both previously described problems. The unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions. With an arbitrary transfer function G one can associate, canonically, two objects—the conjugate inner function which is based on the NRCF, and the R, which is associated with the LQG controller of the function G. The approach that is taken is functional emphasizing operators. A balanced realization of a stable g arises as a matrix representation of the shift realization, with respect to a basis made out of suitably normalized Hankel singular vectors. A similar result holds for LQG balanced realizations. Here, the underlying Hankel operator we study is H_{R}, where _R= U*M + V*N and U, V solve the H_{∞}-Bezout equation M¯V − N¯U = I. This Hankel operator has the same singular vectors, though different singular values and Schmidt pairs, as H. The basis of singular vectors of H_{R} determines canonically a basis for the polynomial model shift realization of g corresponding to which the matrix representation is LQG balanced. One of the central results is that the optimal Hankel norm approximant of is up to a scaling factor also conjugate inner Denoting by R the symbol associated with the LQG controller of g_{n}, we show that -R* is the strictly proper part of the best n - 1 order Hankel norm approximant of R^{*}. We will also obtain state-space representations for R* and g_{n} in terms of the parameters in the LQG balanced state space representation of g. Similar results hold for the case of Nehari complements. These are applied to robust control. As a result of this study the problems of model reduction and robust stabilization can be viewed as dual problems.

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