Observer Design for Infinite dimensional SyStEm – ODISSE

Methodologically, the ODISSE project is at the crossroads of inverse problems for partial differential equations (PDE) and observer theory. These two disciplines have a long and rich history of interactions between them and their overlap is becoming more and more important. The ODISSE project proposes fundamental/theoretical contributions in observer design to reconstitute online missing parameters in some dynamical systems described by PDE.

Indeed, to analyze, monitor, control or understand physical or biological phenomena, the first step is to provide a mathematical modeling in the form of mathematical equations that describe the evolution of the system variables. Some of these variables are accessible through measurement and others are not. One of the problems in control engineering is that of designing algorithms to provide real time estimates of the unmeasured data from other measured variables. These estimation algorithms are called state observers and are used in many devices.

The implementation of such estimators in the context of hyperbolic PDE systems, which are infinite-dimensional systems in the sense that the system's state belongs to a functional space of infinite dimension, is a topic of great interest both from the practical and theoretical points of view. Systems modeled by hyperbolic PDE, that can be of order one or two, correspond to propagation phenomena and appear in many physical contexts and industrial applications.

The ODISSE project aims at developing rigorous methodological tools for the design of estimating algorithms for infinite-dimensional systems governed by hyperbolic PDE, with a particular focus on two typical equations: transport equations (hyperbolic PDE of order one) and wave equations (order two). For this purpose, observability properties of this type of PDE systems will be investigated and novel tools for analyzing their estimations will be developed.

Based on the peculiarities of each field, we try several challenges that could help in solving some observation problems for hyperbolic PDEs:
1- Find a way to connect the notion of identifiability in inverse problems and that of observability in observer design.
2- For identifiable parameters in the sense of inverse problems, find a way to synthesize a robust and online estimation algorithm (an observer).
3- Find means to incorporate recent advances in the field of observer designs for nonlinear finite dimensional systems. Conversely, study the possibility of using tools from infinite dimensional systems for observer synthesis for finite dimensional systems.
4- Implement the proposed algorithms and perform convergence analysis of the discretized (finite dimensional) systems toward the continuous initial (infinite dimension) systems.

These challenges will be addressed in the ODISSE project through a close collaboration between researchers in applied mathematics and control theory from the community of inverse problems and observers design. Several control applications will serve as a test bed to evaluate practical relevance of the theoretical tools to be developed. More specifically, we will work out analysis and design of observers for the concrete processes : batch crystallization processes, polymerization processes and transient elastography.