June 2020

2 juin 2020 – 15h, réunion en visioconférence

Nota : Cette session, organisée par Alexandre Seuret, est dédiée à la préparation des exposés pour la conférence IFAC2020. Le format est donc spécifique : 20 min de présentation et 10 min d’interaction avec le public (questions & conseils).

Speaker: Mathieu Bajodek
Title: Insight into stability analysis of time-delay systems using Legendre polynomials
Abstract: In this paper, a numerical analysis to assess stability of time-delay systems is investigated. The proposed approach is based on the design of a finite-dimensional approximation of the infinite-dimensional space of solutions of the system. Indeed, based on the dynamical coefficients on the sequence made of the first Legendre polynomials, the original time-delay system is modelled by a finite-dimensional model interconnected to a modelling error. Putting aside the interconnection, the resulting finite-dimensional system turns out to be a nice approximation of the time-delay system. Using Pade arguments, the eigenvalues of this finite-dimensional system are proven to converge towards a set of characteristic roots of the original time-delay system. Furthermore, considering now the whole interconnected system and having a deeper look at the interconnection, an enriched Lyapunov-Krasovskii functional is proposed to develop a sufficient condition expressed in terms of linear matrix inequalities for the stability of the time-delay system. Both results are illustrated on a toy example.
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Speaker: Mathias Serieye
Title: Stabilization of switched affine systems via multiple shifted Lyapunov functions
Abstract: This paper deals with the stabilization of switched affine systems. The particularities of this class of nonlinear systems are first related to the fact that the control action is performed through the selection of the switching mode to be activated and, second, to the problem of providing an accurate characterization of the set where the solutions to the system converge to. In this paper, we propose a new method based on a control Lyapunov function, that provides a more accurate invariant set for the closed-loop systems, which is composed by the union of potentially several disjoint subsets. The main contribution is presented as a non convex optimization problem, which refers to a Lyapunov-Metzler condition. Nevertheless a gridding technique applied on some parameters allows obtaining a reasonable solution through an LMI optimization. The method is then illustrated on two numerical examples that demonstrate the potential of the method.
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Speaker: Marianne Souaiby
Title: Computation of Lyapunov functions under state constraints using semidefinite programming hierarchies
Abstract: In this article, we provide an algorithm for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, despite being convex, is computationally difficult. For conic constraints, we provide a discretization algorithm based on simplical partitioning of a simplex, so that the search of desired function is a addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. The second algorithms that we propose is tailored for semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares polynomial.
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Speaker: Matteo Tacchi
Title: Approximating regions of attraction of a sparse polynomial differential system
Abstract: Motivated by stability analysis of large scale power systems, we describe how the Lasserre (moment – sums of squares, SOS) hierarchy can be used to generate outer approximations of the region of attraction (ROA) of sparse polynomial differential systems, at the price of solving linear matrix inequalities (LMI) of increasing size. We identify specific parsity structures for which we can provide numerically certified outer approximations of the region of attraction in high dimension. For this purpose, we combine previous results on non-sparse ROA approximations with sparse semi-algebraic set volume computation.
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