https://www.math.bgu.ac.il/~paf/HNAbalancing.pdf

A polynomial approach to Hankel norm and balanced approximations

A methodology is presented for Hankel approximation and H/sup infinity /-optimization problems that is based on a new formulation of a one-step extension problem which is solved by the Sarason interpolation theorem. The parameterization of all optimal Hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an Hermitian matrix composed directly from the coefficients of a given transfer function matrix phi . Rather than starting with the state-space realization of phi , the authors use polynomial coefficients of phi as input data. In terms of these data, a natural basis is given for the finite dimensional Sarason model space and all computations involve only manipulations with finite matrices. >

One-step extension approach to optimal Hankel-norm approximation and H/sup infinity /-optimization problems

Given a transfer function matrix, it is shown that its minimal balanced realization can be obtained directly from the singular values and the singular vectors of a constant matrix constructed from the coefficients of the least common denominator polynomial. The algorithm is based on the close relationship of controllability and the observability gramians to the singular value decomposition of the associated infinite block Hankel matrix. >

A simple algorithm on minimal balanced realization for transfer function matrices

This note provides a novel methodology for Hankel approximation and H/sup infinity /-optimization problems, based on a new formulation of the one-step extension problem, which is solved by the Sarason interpolation theorem. The present method does not require an initial balanced realization, and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a Hermitian matrix composed directly from the coefficients of the given transfer function matrix. >

An efficient algorithm on optimal Hankel-norm approximation for multivariable systems

An attempt is made to characterise, qualitatively and quantitatively, the major differences between H∞ and H2-optimal model matching control. Conditions under which the effectiveness of model matching in H∞-norm space is superior to that in H2-norm space are derived. According to these conditions, designers can easily decide between H∞ and H2 approaches to meet the required performance.

Optimal Model Matching: H2 or H∞?

A unified approach is proposed here to solve three problems simultaneously, namely H ∞ -optimization, Hankel approximation and minimal balanced realization for scalar systems. It is shown that all these problems can be reduced to the same standard form wherein merely the singular-value decomposition (SVD) of an easily constructed square matrix is required to establish the final solutions. The formulation is solely in terms of the coefficients of a transfer function and the resulting algorithm can be performed in a very computationally efficient way.

Fang Bo Yeh

Papers

Unified approach to H∞-optimization, Hankel approximation and balanced realization problems