Commande Robuste et Optimisation Convexe

La COMET SCA et le GT MOSAR (Méthodes et Outils pour la Synthèse et l’Analyse en Robustesse) du GDR MACS ont co-organisé un workshop sur la Commande robuste et l’optimisation convexe.

L’objectif du séminaire était de discuter des enjeux théoriques et numériques des méthodes de commande et d’analyse robuste et d'en illustrer les intérêts et limitations par des exemples concrets d’applications dans le domaine du spatial. Ce séminaire était donc l’occasion de venir rencontrer les experts des méthodes de commande et d’analyse robuste et de partager votre expérience dans ce domaine, ou simplement de venir découvrir ou approfondir vos connaissances sur ces méthodes.

Le séminaire s'est déroulé sur 2 jours: 12 mars de 11h à 17h et le 13 mars de 9h à 15h, à Meeting business center situé à Montaudran (Toulouse).

Le programme est disponible ci-dessous, il comprend les interventions suivantes :

• Carsten Scherer (U. Stuttgart)
  Tutorial presentation about IQCs for robust analysis
• Yohei Hosoe (U. Kyoto)
  Stability of discrete-time systems with stochastic dynamics
  The influence of stochastic noise is more or less inevitable in practical control problems.  If the influence is larger than we can disregard, it would be better to take account of the presence of noise in modeling and control.  The main purpose of this talk is to provide a new option for resolving this issue such that the randomness of noise can be taken into account in control problems.  The system to be mainly dealt with in this talk is a discrete-time linear system whose dynamics itself is determined by a stochastic process.  In particular, we assume that the process is independent and identically distributed (i.i.d.) with respect to the discrete time, and discuss several results that can be obtained under the assumption.  We first talk about the equivalence of some notions for second-moment stability of the systems, and show a Lyapunov inequality characterizing the stability.  Since our Lyapunov inequality involves decision variables contained in the expectation operation, we also provide an idea for solving it as a standard linear matrix inequality.  Based on these results, a stabilization state feedback can be designed.  As a practical control approach, we further discuss an output feedback control framework using a sequential Monte Carlo method called the ensemble Kalman filter and our control theory. Then, we conclude this talk with some future prospects.
• Charles Poussot-Vassal (ONERA)
  Approximation of infinite dimensional systems and its applications
• Christelle Pittet (CNES)
  Feedback about robust control and analysis activities at CNES
• Andres Marcos (U. Bristol)
  Advances in launcher guidance and control design: from robust control to convex optimization
• Victor Magron (LAAS-CNRS)
  Two-player games between polynomial optimizers and semidefinite solvers
  We interpret some wrong results, due to numerical inaccuracies, already observed when solving semidefinite programming (SDP) relaxations for polynomial optimization, on a double precision floating point solver. It turns out that this behavior can be explained and justified satisfactorily by a relatively simple paradigm. In such a situation, the SDP solver - and not the user - performs some "robust optimization" without being told to do so. In other words, the resulting procedure can be viewed as a "max-min" robust optimization problem with two players: the solver which maximizes on a ball of arbitrary small radius, centered at the input polynomial, and the user who minimizes over the original decision variables. Next, we consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of polynomials, while relying on arbitrary-precision SDP solvers. We provide a perturbation-compensation algorithm computing exact decompositions for polynomials lying in the interior of the SOS cone. First, the user perturbs the input polynomial to obtain an approximate SOS decomposition. Then, one obtains an exact SOS decomposition after compensating the numerical error with the perturbation terms. We prove that this algorithm runs in Boolean time, which is polynomial in the degree of the input and simply exponential in the number of variables. We apply this algorithm to compute exact Polya and Putinar's representations, respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. These results, obtained through collaborations with Jean-Bernard Lasserre (LAAS CNRS) and Mohab Safey El Din (Sorbonne Univ., CNRS, INRIA, LIP6, PolSys), are related to two distinct articles, available at https://arxiv.org/abs/1811.02879 and https://arxiv.org/abs/1811.10062.
• Anthony Bourdelle & Jean-Marc Biannic (ONERA)
  From modeling to robust LPV-based observation of fuel slosh dynamics with application to small satellite attitude tracking control
• Dimitri Peaucelle (LAAS-CNRS)
  S-variables for the Positivity Check of Matrix Polynomials with Matrix Indeterminates
  Positivity check of multivariate polynomials has lived a major breakthrough in the past two decades thanks to sum-of-squares relaxations. These relaxations provide hierarchies of convex semi-definite programming formulations with decreasing and, under mild assumptions, vanishing conservatism. With this very general framework most robust stability analysis are in some sense considered as being solved, at least when leaving numerical issues apart. But when looking at the robust control results, sum-of-squares relaxations are not the unique methodology and are usually combined to other techniques such as S-procedure, KYP-lemma, DG-scalings, vertex separators, S-variables to mention just a few. Based on a joint work with Masayuki Sato, the objective of this talk is to discuss the connections between these various techniques when reformulating the problem as the positivity check of matrix valued polynomials with real and complex, scalar and matrix indeterminates.