Thesis of Tatiana Kharkovskaia

Design of interval observers for uncertain distributed systems

This work presents new results on interval state estimation for uncertain distributed systems, the state of which has an infinite dimension and is described by partial (PDEs) or (FDEs) functional differential equations. An interval observer evaluates at each time instant a set of admissible values for the state (an interval), consistently with the measured output. The design is based on the positive systems theory. Chapters 2 and 3 focus on an interval observer design for a parabolic PDE with Dirichlet boundary conditions. The method in Chapter 2 is based on a finite-element Galerkin approximation, the interval inclusion of the state is calculated using the error estimates of the approximation. Chapter 3 presents an interval observer in the form of PDEs without Galerkin projection. In both chapters, the obtained interval estimates are applied to the design of a dynamic output feedback stabilizing controller. Chapter 4 deals with a second-order delay differential equation with uncertainties corresponding, for instance, to a mechanical system with delayed position measurements, which has form of an FDE. The proposed method contains two consecutive interval observers. The first one estimates, at each instant of time, the interval for the delay-free position using new delay-dependent conditions on positivity. Then, derived estimates of the position are used to design the second observer providing an interval for the velocity. All the obtained results are supported by numerical simulations. In particular, Chapter 2 includes experiments on the Black–Scholes model.


Directeur de thèse : Jean-Pierre RICHARD, Professeur, Centrale Lille Institut, VN d’Ascq Artem KREMLEV, Ass.Professor, ITMO State Univ., St Petersbourg, Russie Rapporteurs : Hugues MOUNIER, Professeur, Université Paris Sud, L2S, Orsay Yury ORLOV, Professeur, CICESE, Ensenada, Mexique Membres : Dorothée NORMAND-CYROT, Directeur de Recherche CNRS, LSS, Orsay Sergiy ZHUK, PhD, Ingénieur Recherche, IBM Research Dublin Alexey BOBSTOV, Professeur, ITMO State Univ., St Petersbourg, Russie Invités : Denis EFIMOV, Chargé de Recherche Inria, HDR, Inria Lille Nord-Europe Andrey POLYAKOV, Chargé de Recherche Inria, HDR, Inria Lille Nord-Europe Jean-Pierre RICHARD, Professeur, Centrale Lille Institut, VN d’Ascq

Thesis of the team VALSE defended on 02/12/2019