Section: New Results

New Techniques for Robust Control of Linear Infinite-Dimensional Systems

Participants : Yacine Bouzidi [Disco] , Petteri Laakkonen [Univ. Tampere] , Adrien Poteaux [Lille 1] , Alban Quadrat [Disco] , Arnaud Quadrat [SAGEM] , Guillaume Rance [SAGEM] , Fabrice Rouillier [Ouragan] .

Computer algebra methods for testing the structural stability of multidimensional systems

We present new computer algebra based methods for testing the structural stability of $n$-D discrete linear systems (with $n\ge 2$). More precisely, starting from the usual stability conditions which resumes to deciding if an hypersurface has points in the unit polydisc, we show that the problem is equivalent to deciding if an algebraic set has real points and use state-of-the-art algorithms for this purpose. Our strategy has been implemented in Maple and its relevance demonstrated through numerous experimentations.

Moreover, we also consider the specific case of two-dimensional systems and focus on the practical efficiency aspect. For such systems, the problem of testing the stability is reduced to that of deciding if a bivariate algebraic system with finitely many solutions has real ones. Our first contribution is an algorithm that answers this question while achieving practical efficiency. Our second contribution concerns the stability of two dimensional systems with parameters. More precisely, given a two-dimensional system depending on a set of parameters, we present a new algorithm that computes regions of the parameters space in which the considered system is structurally stable.

Computer algebra methods for the stability analysis of differential systems with commensurate time-delays

Within the frequency-domain approach, the asymptotic stability of of linear differential systems with commensurate delays is ensured by the condition that all the roots of the corresponding quasipolynomial have negative real parts. A classical approach for checking this condition consists in computing the set of critical zeros of the quasipolynomial, i.e., the roots (and the corresponding delays) of the quasipolynomial that lie on the imaginary axis, and then analyzing the variation of these roots with respect to the variation of the delay. Based on solving algebraic systems techniques, we propose a certified and efficient symbolic-numeric algorithm for computing the set of critical roots of a quasipolynomial. Moreover, using recent algorithmic results developed by the computer algebra community, we present an efficient algorithm for the computation of Puiseux series at a critical zero which allows us to finely analyze the stability of the system with respect to the variation of the delay

A fractional ideal approach to the robust regulation problem

We show how fractional ideal techniques developed in [8] can be used to obtain a general formulation of the internal model principle for stabilizable infinite-dimensional SISO plants which do not necessarily admit coprime factorization. This result is then used to obtain necessary and sufficient conditions for the robust regulation problem. In particular, we find again all the standard results obtained in the literature.

Robust control as an application to the homological perturbation lemma:

Within the lattice approach to transfer matrices developed in [8] , we have recently shown how standard results on robust control can be obtained in a unified way and generalized when interpreted as a particular case of the so-called Homological Perturbation Lemma. This lemma plays a significant role in algebraic topology, homological algebra, algebraic and differential geometry, computer algebra .... Our results show that it is also central to robust control theory for infinite-dimensional linear systems.

A symbolic-numeric method for the parametric $H_{\infty }$ loop-shaping design problem

We develop a symbolic-numeric method for solving the $H_{\infty }$ loop-shaping design problem for a low order single-input single-output system with parameters. Due to the system parameters, no purely numerical algorithm can indeed solve the problem. Using Gröbner basis techniques and the rational univariate representation of zero-dimensional algebraic varieties, we first give a parametrization of all the solutions of the two algebraic Riccati equations associated with the $H_{\infty }$ control problem. Then, using results on the spectral factorization problem, a certified symbolic-numeric algorithm is obtained for the computations of the positive definite solutions of these two algebraic Riccati equations. Finally, we present a certified symbolic-numeric algorithm which solves the $H_{\infty }$ loop-shaping design problem for the above class of systems.