In-Domain Control of Partial Differential Equations

RÉSUMÉ Cette thèse porte sur la commande des systèmes à dimension infinie décrit par les équa-tions aux dérivées partielles (EDP). La commande d’EDP peut être divisée approximative-ment en deux catégories en fonction de l’emplacement des actionneurs: la commande à la frontière, où les actionnements sont appliqués à la frontière des systèmes d’EDP, et la com-mande dans le domaine, où les actionneurs pénètrent à l’intérieur du domaine des systèmes d’EDP. Dans cette thèse, nous étudierons la commande dans le domaine de l’équation d’Euler-Bernoulli, de l’équation de Fisher, l’équation de Chafee-Infante et de l’équation de Burgers. L’équation d’Euler-Bernoulli est un modèle classique d’EDP linéaire décrivant la flexion pure des structures flexibles. L’équation de Fisher et l’équation de Chafee-Infante sont des EDP paraboliques semi-linéaires, qui peuvent être utilisées pour modéliser certains phénomènes physiques, chimiques ou biologiques. L’équation de Burgers peut être considérée comme une simplification d’équations de Navier-Stokes en mécanique des fluides, en dynamique des gaz, en fluidité de la circulation, etc. Ces systèmes jouent des rôles très importants en mathéma-tiques, en physique et dans d’autres domaines. Dans cette thèse, de nouvelles méthodes qui se basent sur la dynamique des zéros et le compensateur dynamique ont été développées pour la conception et l’implémentation de lois de commande pour la commande des EDP avec des actionnements dans le domaine. Tout d’abord, nous étudions le contrôle de l’équation d’Euler-Bernoulli avec plusieurs actionneurs internes. L’inverse de la dynamique des zéros a été utilisé dans la conception de la loi de commande, ce qui permet de suivre la trajectoire prescrit souhaitée. Afin de concevoir la trajectoire souhaitée, la fonction de Green est utilisée pour déterminer la commande sta-tique. La planification de mouvement est générée par des contrôleurs dynamiques basés sur la méthode de platitude di˙érentielle. Pour les équations paraboliques non linéaires, la dy-namique des zéros est régie par une EDP non linéaire. Par conséquent, nous avons recours à la méthode de décomposition d’Adomian (ADM) pour générer la commande dynamique afin de suivre les références désirées. Dans le cas de l’équation de Burgers, un compensateur dynamique a été utilisé. Pour obtenir la stabilité globale de l’équation de Burgers contrôlée, une rétroaction non linéaire a été appliquée à la frontière. La méthode d’ADM et la platitude ont été utilisées dans l’implémentation du compensateur dynamique.----------ABSTRACT This thesis addresses in-domain control of partial di˙erential equation (PDE) systems. PDE control can in general be classified into two categories according to the location of the ac-tuators: boundary control, where the actuators are assigned to the boundary of the PDE systems, and in-domain control, where the actuation penetrates inside the domain of the PDE systems. This thesis investigates the in-domain control of some well-known PDEs, including the Euler-Bernoulli equation, the Fisher’s equation, the Chafee-Infante equation, and Burgers’ equation. Euler-Bernoulli equation is a classical linear PDE used to describe the pure bending of flexible structures. Fisher’s equation and the Chafee-Infante equation are semi-linear parabolic PDEs that can be used to model physical, chemical, and biolog-ical phenomena. Burgers’ equation can be viewed as simplified Navier-Stokes equations in lower dimensions in applied mathematics, and it has been widely adopted in fluid mechan-ics, gas dynamics, traÿc flow modeling, etc. These PDE systems play important roles in mathematics, physics, and other fields. In this work, in-domain control of linear and semi-linear parabolic equations are treated based on dynamic compensators. First, we consider the in-domain control of an Euler-Bernoulli equation with multiple internal actuators. The method of zero dynamics inverse is adopted to derive the in-domain control to allow an asymptotic tracking of the prescribed desired outputs. A linear proportional boundary feedback control is employed to stabilize the Euler-Bernoulli equation around its zero dynamics. To design the desired trajectory, the Green’s function is employed to determine the static control, and then motion planning is generated by dynamic control based on di˙erential flatness. For the semi-linear parabolic equations, zero dynamics are governed by nonlinear PDEs. Therefore, the implementation of the in-domain control of linear PDEs cannot be directly applied. We resort then to the Adomian decomposition method (ADM) to implement the dynamic control in order to track the desired set-points. Finally, the in-domain control of a Burgers’ equation is addressed based on dynamic compensator. A nonlinear boundary feedback control is used to achieve the global stability of the controlled Burgers’ equation, and the ADM as well as the flatness are used in the implementation of the proposed in-domain control scheme

Event-triggered gain scheduling of reaction-diffusion PDEs

This paper deals with the problem of boundary stabilization of 1D reaction-diffusion PDEs with a time- and space- varying reaction coefficient. The boundary control design relies on the backstepping approach. The gains of the boundary control are scheduled under two suitable event-triggered mechanisms. More precisely, gains are computed/updated on events according to two state-dependent event-triggering conditions: static-based and dynamic-based conditions, under which, the Zeno behavior is avoided and well-posedness as well as exponential stability of the closed-loop system are guaranteed. Numerical simulations are presented to illustrate the results.Comment: 20 pages, 5 figures, submitted to SICO

Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media

This paper considers a class of space fractional partial differential equations (FPDEs) that describe gas pressures in fractured media. First, the well-posedness, uniqueness, and the stability in $L_(\infty{R})$of the considered FPDEs are investigated. Then, the reference tracking problem is studied to track the pressure gradient at a downstream location of a channel. This requires manipulation of gas pressure at the downstream location and the use of pressure measurements at an upstream location. To achiever this, the backstepping approach is adapted to the space FPDEs. The key challenge in this adaptation is the non-applicability of the Lyapunov theory which is typically used to prove the stability of the target system as, the obtained target system is fractional in space. In addition, a backstepping adaptive observer is designed to jointly estimate both the system's state and the disturbance. The stability of the closed loop (reference tracking controller/observer) is also investigated. Finally, numerical simulations are given to evaluate the efficiency of the proposed method.Comment: 37 pages, 9 figure

Robust Adaptive Boundary Control of Semilinear PDE Systems Using a Dyadic Controller

In this paper, we describe a dyadic adaptive control (DAC) framework for output tracking in a class of semilinear systems of partial differential equations with boundary actuation and unknown distributed nonlinearities. The DAC framework uses the linear terms in the system to split the plant into two virtual sub-systems, one of which contains the nonlinearities, while the other contains the control input. Full-plant-state feedback is used to estimate the unmeasured, individual states of the two subsystems as well as the nonlinearities. The control signal is designed to ensure that the controlled sub-system tracks a suitably modified reference signal. We prove well-posedness of the closed-loop system rigorously, and derive conditions for closed-loop stability and robustness using finite-gain L stability theory

Sub-optimal boundary control of semilinear pdes using a dyadic perturbation observer

In this paper, we present a sub-optimal controller for semilinear partial differential equations, with partially known nonlinearities, in the dyadic perturbation observer (DPO) framework. The dyadic perturbation observer uses a two-stage perturbation observer to isolate the control input from the nonlinearities, and to predict the unknown parameters of the nonlinearities. This allows us to apply well established tools from linear optimal control theory to the controlled stage of the DPO. The small gain theorem is used to derive a condition for the robustness of the closed loop system

Distributed feedback control of a fractional diffusion process

International audienceIn this paper, a control law that enforces an output tracking of a fractional diffusion process is developed. The dynamical behavior of the process is described by a space-fractional parabolic equation. The objective is to force a spatial weighted average output to track its specified output by manipulating a control variable assumed to be distributed in the spatial domain. The state feedback is designed in the framework of geometric control using the notion of the characteristic index. Then, under the assumption that the fractional diffusion process is a minimum phase system, it is shown that the developed control law guarantees exponential stability in L2 -norm for the resulting closed loop system. Numerical simulations are performed to show the tracking and disturbance rejection capabilities of the developed controller

Distributed feedback control of a fractional diffusion process

International audienceIn this paper, a control law that enforces an output tracking of a fractional diffusion process is developed. The dynamical behavior of the process is described by a space-fractional parabolic equation. The objective is to force a spatial weighted average output to track its specified output by manipulating a control variable assumed to be distributed in the spatial domain. The state feedback is designed in the framework of geometric control using the notion of the characteristic index. Then, under the assumption that the fractional diffusion process is a minimum phase system, it is shown that the developed control law guarantees exponential stability in L 2-norm for the resulting closed loop system. Numerical simulations are performed to show the tracking and disturbance rejection capabilities of the developed controller

Observer design for a nonlinear heat equation: Application to semiconductor wafer processing

In this paper, the problem of observer design for a class of 1D nonlinear heat equations with pointwise in-domain temperature measurements is addressed. A pointwise measurement injection observer is designed and the robust convergence of its estimation error in presence of bounded distributed perturbations is established by verifying input-to-state stability. The obtained convergence conditions express the underlying interplay between heat conduction and radiation and include specific dependencies on the sensor locations which are the main degrees of freedom in the design approach. The theoretical results are experimentally validated on a semiconductor wafer processing unit