Control and Estimation for Partial Differential Equations and Extension to Fractional Systems

Abstract

Partial differential equations (PDEs) are used to describe multi-dimensional physical phenomena. However, some of these phenomena are described by a more general class of systems called fractional systems. Indeed, fractional calculus has emerged as a new tool for modeling complex phenomena thanks to the memory and hereditary properties of fraction derivatives.

In this thesis, we explore a class of controllers and estimators that respond to some control and estimation challenges for both PDE and FPDE. We first propose a backstepping controller for the flow control of a first-order hyperbolic PDE modeling the heat transfer in parabolic solar collectors. While backstepping is a well-established method for boundary controlled PDEs, the process is less straightforward for in-domain controllers.

One of the main contributions of this thesis is the development of a new integral transformation-based control algorithm for the study of reference tracking problems and observer designs for fractional PDEs using the extended backstepping approach. The main challenge consists of the proof of stability of the fractional target system, which utilizes either an alternative Lyapunov method for time FPDE or a fundamental solution for the error system for reference tracking, and observer design of space FPDE. Examples of applications involving reference tracking of FPDEs are gas production in fractured media and solute transport in porous media.

The designed controllers, require knowledge of some system’s parameters or the state. However, these quantities may be not measurable, especially, for space-evolving PDEs. Therefore, we propose a non-asymptotic and robust estimation algorithm based on the so-called modulating functions. Unlike the observers-based methods, the proposed algorithm has the advantage that it converges in a finite time. This algorithm is extended for the state estimation of linear and non-linear PDEs with general non-linearity. This algorithm is also used for the estimation of parameters and disturbances for FPDEs.

This thesis aims to design an integral transformation-based algorithm for the control and estimation of PDEs and FDEs. This transformation is defined through a suitably designed function that transforms the identification problem into an algebraic system for non-asymptotic estimation purposes. It also maps unstable systems to stable systems to achieve control goals.