Matrix RationalH₂Approximation: A Gradient Algorithm Based on Schur Analysis

Abstract

This report deals with rational approximation of any speci ed order n of transfer functions. Transfer functions are assumed to be matrices whose entries belong to the Hardy space for the complement of the closed unit disk endowed with the L 2-norm. A new approach is developed leading to an original algorithm, the rst one to our knowledge which concerns matrix transfer functions. This approach generalizes the ideas developed in the scalar case, but involves substantial new di culties. The inner-unstable factorization of transfer functions allows to express the criterion in terms of inner matrices of Mac-Millan degree n. These matrices form a di erential manifold. Based on a tangential Schur algorithm, an atlas of this manifold is given for which the coordinates vary in n copies of the unit ball. Then a gradient algorithm can be used to solve this problem. The di erent cases which can arise while processing the algorithm are studied : how to switch to another chart of the atlas, what has to be done when a boundary point is reached. In the neighbourhood of a boundary point, the criterion can be smoothly extended. Moreover, such a point can be considered as an initial point for the research of a lower degree approximant. It is explained how to cope with the decrease and the increase of the degree. The convergence of the algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example.

Hankel-type model reduction based on frequency response matching

“…Indeed, recall that (4) is general and (5) (6) is a relaxation of (5) γ 6 ≤ γ 5 and the inequality follows. Now, prove the upper bound (10). Note that G−b/a ∞ ≤ γ 6 , where a = qϕ ∼ and ϕ ∼ has only zeros outside the unit circle.…”

“…modeling of electro-magnetic structures) can be even cheaper, than inverting the state-space matrix A, as shown in [6], [7], [8]. The approximation from the frequency data is related to the celebrated Nevanlinna-Pick interpolation problem, an extension of which to Hardy spaces can be found in [9], [10] with a recent progress in [11], [12]. In [13] another approach was developed to obtain an approximation.…”

Model reduction using a frequency-limited -cost

“…The problem of finding a reduced order model that, in H 2 sense, resembles the original model well has been a goal in many investigations. Especially since the work of Meier and Luenberger (1967), and especially Wilson (1970), in which they derive first order optimality conditions for minimization of the H 2 -norm, see for example Lepschy et al (1991), Beattie and Gugercin (2007), Gugercin et al (2008), Fulcheri and Olivi (1998), Yan and Lam (1999) and references therein. One reason for this could be the fact that H 2 criterion provides a meaningful characterization of the error, both in deterministic and stochastic contexts.…”

“…The different algorithms use different techniques to assure that the reduced model is stable, to speed up the process and to guarantee convergence. Examples of these algorithms are Yan and Lam (1999), Fulcheri and Olivi (1998) and Huang et al (2001).…”

Semidefinite Hankel-Type Model Reduction Based on Frequency Response Matching

“…Computing a frequency response for particular applications (e.g., modeling of electro-magnetic structures) can be even cheaper, than inverting a state-space matrix A, as shown in [7], [8], [9]. One of the main tools for frequency domain approximation is the interpolation techniques ( [10], [11], [12]). Another tool is convex optimization as in the method proposed in [13], [14].…”