Promoteur UMONS : Prof. Alain VANDE WOUWER

Promoteur UFSC – Brésil : Prof. Daniel COUTINHO

C’est une co-tuelle entre l’UMONS et l’Universidade federal de Santa Catarina (UFSC), (Brésil)

Contribution to State Estimation of Semilinear Parabolic Distributed Parameter Systems with Applications to Transport Reaction Systems

Transport–reaction systems are described by semilinear parabolic partial differential
equations (PDEs) and are fundamental in applications where diffusion processes must be
considered explicitly. The state estimation problem on the basis of some in-domain distributed
measurements is non-trivial. In this work we address this problem for a certain class of
transport-reaction systems. To achieve this task, we propose observer design strategies in the
frame of both early and late lumping approaches.
Regarding the early lumping approach for the state observer design, we use the Method of
Weighted Residuals (MWR), that encompasses the orthogonal collocation method, to derive
an approximate reduced-order model, expressed as a set of ordinary differential equations
(ODEs) subject to algebraic constraints. Then, a Lyapunov-based design method is proposed
for the reduced-order model which provides sufficient design conditions in terms of standard
linear matrix inequalities (LMIs) aiming the exponential convergence of the estimation error
with a prescribed decay rate. The observer performance is further improved through an offline
optimal sensor placement algorithm considering a parameterized reduced-order output
matrix.
Concerning the late lumping approach, firstly, we studied the operator semi-group
representation which lead us to the use of the spectrum-decomposition properties related to
parabolic differential operators. Thus we aimed at obtaining sufficient state observer
synthesis conditions based on the local lipschitz properties of the reaction rate vector
functions considering a modal output injection gain. Secondly, a Lyapunov based design
method is proposed for the stabilization of the estimation error dynamics. The approach uses
positive definite matrices to parameterize a class of Lyapunov functionals that are positive in
the Lebesgue space of integrable square functions. Thus, the stability conditions can be
expressed as a set of LMI constraints which can be solved numerically using sum of squares
(SOS) and standard semi-definite programming (SDP) tools.
Throughout the chapters of this thesis, all of these proposed techniques and methods are
applied and tested numerically to the representative cases of biochemical tubular reactor
processes. Simulation results support the effectiveness of the suggested designs.
Finally, the COVID-19 spread monitoring problem is addressed in the application part of this
thesis. In particular, we tackle the state estimation of the compartmental model based on
partial differential equations (PDEs) which describes the spread of the infectious disease in a
host population. A Lyapunov based design method with SOS and polynomial parameterization
of the decision variables is used to derive a SDP problem whose solution provides the injection
gains of the Luenberger type state observer, Numerical experiments are presented to
illustrate the method efficiency.