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Theses

Lyapunov Functions and Ensemble Approximations for Constrained Systems using Semidefinite Programming

Marianne Souaiby 1 
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : This thesis deals with analysis of constrained dynamical systems, supported by some numerical methods. The systems that we consider can be broadly seen as a class of nonsmooth systems, where the state trajectory is constrained to evolve within a prespecified (and possibly time-varying) set. The possible discontinuities in these systems arise due to sudden change in the vector field at the boundary of the constraint set. The general framework that we adopt has been linked to different classes of nonsmooth systems in the literature, and it can be described by an interconnection of an ordinary differential equation with a static relation (such as variational inequality, or a normal cone inclusion, or complementarity relations). Such systems have found applications in modeling of several engineering and physical systems, and the results of this dissertation make some contributions to the analysis and numerical methods being developed for such system class.The first problem that we consider is related to the stability of an equilibrium point for the aforementioned class of nonsmooth systems. We provide appropriate definitions for stability of an equilibrium, and the Lyapunov functions, which take into consideration the presence of constraints in the system. In the presence of conic constraints, it seems natural to work with cone-copositive Lyapunov functions. To confirm this intuition, and as the first main result, we prove that, for a certain class of cone-constrained systems with an exponentially stable equilibrium, there always exists a smooth cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials.This later class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms precisely for that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. For conic constraints, we provide a discretization algorithm based on simplicial partitioning of a simplex, so that the search of desired function is addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. Our second algorithm is tailored to semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares. Some polynomials examples are given to illustrate our approach.Continuing with our study of state-constrained systems, we next consider the time evolution of a probability measure which describes the distribution of the state over a set. In contrast with smooth ordinary differential equations, where the evolution of this probability measure is described by the Liouville equation, the flow map associated with the nonsmooth differential inclusion is not necessarily invertible and one cannot directly derive a continuity equation to describe the evolution of the distribution of states. Instead, we consider Lipschitz approximation of our original nonsmooth system and construct a sequence of measures obtained from Liouville equations corresponding to these approximations. This sequence of measures converges in weak-star topology to the measure describing the evolution of the distribution of states for the original nonsmooth system. This allows us to approximate numerically the evolution of moments (up to some finite order) for our original nonsmooth system, using a hierarchy of semidefinite programs. Using similar methodology, we study approximation of the support of the solution (described by a measure at each time) using polynomial approximations.
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Marianne Souaiby. Lyapunov Functions and Ensemble Approximations for Constrained Systems using Semidefinite Programming. Automatic Control Engineering. INSA de Toulouse, 2021. English. ⟨NNT : 2021ISAT0009⟩. ⟨tel-03408417v2⟩

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