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RARL2
Some applications
RARL2 is a generalist software which efficiently performs model reduction.It is intended to be associated with other tools in order to solve more specific inverse problems. We give here some examples that have been adressed by the APICS team.
Model reduction
The function RARL2.m can be used to
reduce the 2-input, 12-states, 2-output model of
an automobile gas turbine studied in many references:
Y.S. Hung and A.G.J. MacFarlane, Multivariable feedback: a
quasi-classical approach, Lect. Notes in Control Sci. 40, Springer-
Verlag (1982).
K. Glover, All optimal Hankel-norm approximations of linear
multivariable systems and their error bounds, Int. J. Control 39, 1115-1193
(1984).
W.-Y. Yan and J. Lam, An approximate approach to H2 optimal
model reduction, IEEE Trans. Autom. Control 44, No.7, 1341-1358 (1999).
The results are shown on the Nyquist diagrams of the discrete-time associated system (in blue). According to the theory, a very good approximant is obtained at order 8.
 
 
Identification of hyperfrequency filters
The problem is to recover the coupling parameters of a microwave filter from scattering measurements.
We propose an indentification method in two steps
- a stable and causal model of high degree is first computed from the data (completion stage); In the first stage the most is made of the data taking into account the expected behavior of the filter
- A stable reduced order model is then computed by RARL2
The whole process is performed by the software PRESTO-HF. The results of our procedure are shown at hand of a real-life example provided by the CNES. It consists of measurements of a microwave filter of 8th order in 800 frequency points. An excellent agreement is obtained between the final rational model and the measurements. The latter is obtained in less than 15 seconds on a Intel Core I7 processor, which makes our approach compatible with a real-time tuning procedure of the filter.
 
The software PRESTO-HF is currently used for this purpose by several of our industrial partners.
The coupling parameters are finally extracted from the identified model. This step is performed by the software DEDALE-HF.
Source recovery
From measurements by electrodes of the electric potential on the scalp, the problem is to recover a distribution of pointwise dipolar current
sources with moments located in the brain (modeling the presence of epileptic foci). The head is modeled as a set of three spherical nested regions (brain, skull, scalp) and in each region constant conductivities are assumed. A macroscopic model and quasi-static approximation of Maxwell-equations are used to describe the spacial behavior of the potential.
The inverse problem can be approached in two steps:
- get the anti-harmonic part of the potential from the data using propagation techniques
- recover the localizations and the moments of the sources by rational approximation on planar sections (RARL2)
The singularities (green) of the potential are aligned and the sources (black) correspond to the maximum modulus. The rational approximations (pink) accumulate to these singularities:
 
One of the advantages of this method is that it does not require the a priori knowledge of the exact number of sources. If the order of the approximation is larger than the number of sources, the extra poles accumulate to the boundary of he disks. For this application, the software RARL2 has been modified to impose triple poles for the approximant. The whole process is handled by the software FindSources3d .
Wavelets Approximation
This application of our rational approximation methods to orthogonal wavelets was proposed by R. Peeters (Maastricht University).
The problem is to implement wavelets in analog circuits in view of medical signal processing applications. A dedicated method has been developed based on an L2-approximation of the wavelet by the impulse response of a stable, causal, low order filter:
Joël M.H. Karel, A wavelet approach to cardiac signal processing for low-power hardware applications, PhD Thesis (2009).
However, this method fails to find an accurate and sufficiently small order approximation in some difficult cases (Daubechies wavelets db7 and db3).
The idea was to use the software RARL2 to perform a model reduction on an accurate high order (100-200) approximation. However, an admissibility condition for wavelets is that the integral of a wavelet equals zero, which means that it has a vanishing moment of order 0. The low order approximation is still required to have an integral zero, otherwise undesired bias will show up when the wavelet is used in an application. We thus had to adapt a version of the RARL2 software to address this constraint. Since we are dealing with a quadratic optimization problem under a linear constraint, this can be solved analytically. We could thus reformulate the problem of L2-approximation subject to this constraint into an optimization problem over the class of lossless systems. This could be handled by the software with only minor changes and we were able to perform an accurate approximation of order 8 for db7.
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