A Chain-Scattering Approach to LMI Multi-objective Control

International audienceThis paper revisits, from a chain-scattering perspective, the LMI solution based on Youla-Kucera parametrisation of the general multi-objective control problem. The conceptual and computational advantages of the chain-scattering formalism are demonstrated by allowing a more direct derivation of some known results as well as by hinting to some new research directions

Identification and matrix rational H2H^2 approximation : a gradient algorithm based on Schur analysis

This report deals with rational approximation of any specified 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 first one to our knowledge which concerns matrix transfer functions. This approach generalizes the ideas developed in the scalar case, but involves substantial new difficulties. 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 differential 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 different 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

Bilan carbone d'une équipe de recherche à l'Inria

Nous avons réalisé le bilan des émissions de gaz à effet de serre (GES) d’une équipe du centre de recherche Inria de Sophia Antipolis pour l’année 2019. L’équipe comprenait cette année-là 5 permanents, 5 doctorants, 1 post-doc, une assistante d’équipe à mi-temps et a accueilli pendant quelques mois deux stagiaires de master

Canonical lossless state-space systems: Staircase forms and the Schur algorithm

A new finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chart can be left out of the atlas without losing the property that the atlas covers the manifold

Rational Approximation of Transfer Functions for Non-Negative EPT Densities

International audienceAn Exponential-Polynomial-Trigonometric (EPT) function is defined on [0,∞) by a minimal realization (A, b, c). A stable non-negative EPT function of a fixed degree is fitted to the histogram of a large set of data using an L2 criterion. If we neglect the non-negativity constraint this is shown to be equivalent to a rational approximation problem which is approached using the RARL2 software. We show how, under the additional assumption of the existence of a strictly dominant real pole of the rational function, the non-negativity constraint on the EPT function can be imposed by performing a constraint convex optimization on b at each stage at which an (A, c) pair is determined. In this convex optimization step a recent generalized Budan-Fourier sequence approach to determine non-negativity of an EPT function on a finite interval plays a major role

A unified approach to Nevanlinna-Pick interpolation problems

International audienceThis work deals with Complex-valued interpolation by a Schur rational function of given degree at a set of nodes located in the closed lower half-plane, with prescribed maximum points for the modulus (i.e. points where it is equal to 1) on the real axis. The motivation comes from broadband matching, for which the technique we develop offers a new tool

Boundary Nevanlinna-Pick interpolation with prescribed peak points. Application to impedance matching

International audienceWe study a generalized Nevanlinna Pick interpolation problem on the half-plane for rational functions of prescribed degree, where peak points are imposed and interpolation conditions may lie on the real axis. This generalizes previous work by T. Giorgiou, C. Byrnes, A Lindquist and A. Megretski. The problem is motivated by the issue of broadband matching in electronics and microwave system design. We prove existence and uniqueness of a solution by differential-topological techniques. The approach is put to work numerically on a real example, using a continuation method

Nudelman interpolation, parametrizations of lossless functions and balanced realizations

We investigate the parametrization issue for discrete-time stable all-pass multivariable systems by means of a Schur algorithm involving a Nudelman interpolation condition. A recursive construction of balanced realizations is associated with it, that possesses a very good numerical behavior. Several atlases of charts or families of local parametrizations are presented and for each atlas a chart selection strategy is proposed. The last one can be viewed as a nice mutual encoding property of lossless functions and turns out to be very efficient. These parametrizations allow for solving optimization problems within the fields of system identification and optimal control

A unified approach to Nevanlinna-Pick interpolation problems

International audienceThis work deals with Complex-valued interpolation by a Schur rational function of given degree at a set of nodes located in the closed lower half-plane, with prescribed maximum points for the modulus (i.e. points where it is equal to 1) on the real axis. The motivation comes from broadband matching, for which the technique we develop offers a new tool

Polynomial structure of 3 x 3 reciprocal inner matrices

International audienceThe objective of our work is the derivation of efficient algorithms for the synthesis of microwave multiplexers. In our opinion, an efficient frequency design process calls for the understanding of the structure of n x n inner (or lossless) reciprocal rational functions for n > 2. Whereas the case n = 2 is completely understood and a keystone of filter synthesis very little seems to be known about the polynomial structure of such matrices when they involve more than 2 ports. We therefore start with the analysis of the 3 x 3 case typically of practical use in the manufacturing of diplexers. Based on recent results obtained on minimal degree reciprocal Darlington synthesis, we derive a polynomial model for 3 x 3 reciprocal inner rational matrices with given McMillan degree