Promoteur UMONS : Prof. Alain Vande Wouwer

Promoteur UNAL (Universidad Nacional de Colombia) : Prof. Eduardo Majica Nava

S​eules 49 personnes peuvent y assister en présentiel.

Abstract:
The main topic of the thesis is the data-driven identification of the region of attraction (ROA) of asymptotically stable equilibrium points. Although this is the main computational contribution, satisfying the underlying conditions to make this possible constitutes most of the work of the thesis. To achieve an accurate data-driven approximation of the ROA in systems with multiple fixed or equilibrium points it is necessary to properly complete a series of steps parting from some trajectories of the system, i.e., assuming there is no access to the differential or difference model equation. The main condition is an accurate approximation of the Koopman operator because it provides a set of eigenfunctions where a particular composition of them gives another non-trivial eigenfunction with an associated eigenvalue that is unitary. The main property of this eigenfunction is that it gives the stable manifold of saddle points in the boundary of the ROA, where this stable manifold is in fact, the actual boundary of the ROA. Therefore, for this whole procedure to work, it also necessary to have an approximation of the location and stability of the fixed points of the system, recalling that the only input to the algorithm is a set of trajectories of the system. Consequently, the algorithm must be an appropriate approximation of the dynamics of the system and be able to provide a difference equation able to give the location and stability of fixed points upon further traditional non-linear system analysis. The algorithm that has the potential to achieve these requisites is the extended dynamics mode decomposition (EDMD) algorithm, where most of the work of this thesis focuses in transforming the potential into actual. For the most part, the development focus is on the numerical stability of the algorithm, reducing the computational effort and necessary steps to perform the approximation. Techniques such as the p-q-quasi norm reduction of orthogonal polynomials and polynomial element elimination according to its error, ensures that smaller bases perform the approximations while guaranteeing the existence of solutions because of the orthogonality property. Improvements such as the recovery of the state via the inverse of univariate order-one polynomials reduce the number of necessary matrix inversions. Finally, a priori expansions of the state with arbitrary trigonometric functions or any other kind of elemental functions, expand the possible types of systems that the algorithm can handle. As a consequence of these improvements, the thesis achieves the original objectives of analyzing systems and controlling sets of interconnected systems in a data-driven context. Finally, the main application of the thesis is the analysis of the ROA to the anaerobic digestion process, where the analysis of multi-stability phenomena that guarantees the proper operation of the reactor is of paramount importance.