Section: New Software and Platforms

RARL2

Participant : Martine Olivi [corresponding participant] .

Status: Currently under development. A stable version is maintained.

This software is developed in collaboration with Jean-Paul Marmorat (Centre de mathématiques appliquées (CMA), École des Mines de Paris).

RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see Section  3.3.2.2 ) http://www-sop.inria.fr/apics/RARL2/rarl2.html .

The software RARL2 computes, from a given matrix-valued function in ${{\overline{H}}^2}^{m\times l}$, a local best rational approximant in the $L^2$ norm, which is stable and of prescribed McMillan degree (see Section  3.3.2.2 ). It was initially developed in the context of linear (discrete-time) system theory and makes an heavy use of the classical concepts in this field. The matrix-valued function to be approximated can be viewed as the transfer function of a multivariable discrete-time stable system. RARL2 takes as input either:

• its internal realization,

• its first $N$ Fourier coefficients,

• discretized (uniformly distributed) values on the circle. In this case, a least-square criterion is used instead of the $L^2$ norm.

It thus performs model reduction in case 1) and 2) and frequency data identification in case 3). In the case of band-limited frequency data, it could be necessary to infer the behavior of the system outside the bandwidth before performing rational approximation (see Section  3.2.2 ). An appropriate Möbius transformation allows to use the software for continuous-time systems as well.

The method is a steepest-descent algorithm. A parametrization of MIMO systems is used, which ensures that the stability constraint on the approximant is met. The implementation, in Matlab, is based on state-space representations.

The number of local minima can be large so that the choice of an initial point for the optimization may play a crucial role. In this connection, two methods can be used: 1) An initialization with a best Hankel approximant. 2) An iterative research strategy on the degree of the local minima, similar in principle to that of RARL2, increases the chance of obtaining the absolute minimum by generating, in a structured manner, several initial conditions.

RARL2 performs the rational approximation step in our applications to filter identification (see Section  4.5 ) as well as sources or cracks recovery (see Section  4.2 ). It was released to the universities of Delft, Maastricht, Cork, Brussels and Macao. The parametrization embodied in RARL2 was also used for a multi-objective control synthesis problem provided by ESTEC-ESA, The Netherlands. An extension of the software to the case of triple poles approximants is now available. It is used by FindSources3D (see Section  5.6 ).