Section: Software

RARL2

Participants : Jean-Paul Marmorat, Martine Olivi [corresponding participant] .

RARL2 (Réalisation interne et Approximation Rationnelle L2) is a software for rational approximation (see section 3.1.4 ) http://www-sop.inria.fr/apics/RARL2/rarl2-eng.html .

This software takes as input a stable transfer function of a discrete time system represented by either of

• its internal realization,

• its first $N$ Fourier coefficients,

• discretized values on the circle.

It computes a local best approximant which is stable, of prescribed McMillan degree, in the $L^2$ norm.

It is akin to the arl2 function of Endymion (see section  5.5 ) from which it differs mainly in the way systems are represented: a polynomial representation is used in Endymion, while RARL2 uses realizations. It is implemented in Matlab. This software handles multi-variable systems (with several inputs and several outputs), and uses a parametrization with the following advantages:

• it incorporates the stability requirement in a built-in manner,

• it allows the use of differential tools,

• it is well-conditioned, and computationally efficient.

An iterative research strategy on the degree of the local minima, similar in principle to that of arl2, 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 (section 4.2 ) as well as sources or cracks recovery (section 4.1 ). It was released to the universities of Delft, Maastricht, Cork and Brussels. The parametrization embodied in RARL2 was recently 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 provides satisfactory results in the source recovery problem and it is used by FindSources3D (see 5.7 ).