Event-triggered Boundary Control of a Class of Reaction-Diffusion PDEs with Time-dependent Reactivity

This paper presents an event-triggered boundary control strategy for a class of reaction-diffusion PDEs with time-varying reactivity under Robin actuation. The control approach consists of a backstepping full-state feedback boundary controller and a dynamic event-triggering condition, which determines the time instants when the control input needs to be updated. It is proved that under the proposed event-triggered boundary control approach, there is a uniform minimal dwell-time between two event times. Furthermore, the well-posedness and the global exponential convergence of the closed-loop system to zero in $L^2$-sense are established. A simulation is conducted to validate the theoretical developments

Observation and control of PDE with disturbances

In this Thesis, the problem of controlling and Observing some classes of distributed parameter systems is addressed. The particularity of this work is to consider partial differential equations (PDE) under the effect of external unknown disturbances. We consider generalized forms of two popular parabolic and hyperbolic infinite dimensional dynamics, the heat and wave equations. Sliding-mode control is used to achieve the control goals, exploiting the robustness properties of this robust control technique against persistent disturbances and parameter uncertainties

Contrôle de systèmes hyperboliques par analyse Lyapunov

In this thesis we have considered different aspects for the control of hyperbolic systems.First, we have studied switched hyperbolic systems. They contain an interaction between a continuous and a discrete dynamics. Thus, the continuous dynamics may evolve in different modes: these modes are imposed by the discrete dynamics. The change in the mode may be controlled (in case of a closed-loop system), or may be uncontrolled (in case of an open-loop system). We have focused our interest on the former case. We procedeed with a Lyapunov analysis, and construct three switching rules. We have shown how to modify them to get robustness and ISS properties. We have shown their effectiveness with numerical tests.Then, we have considered the trajectory generation problem for 2x2 linear hyperbolic systems. We have solved it with backstepping. Then, we have considered the tracking problem with a Proportionnal-Integral controller. We have shown that it stabilizes the error system around the reference trajectory with a new non-diagonal Lyapunov function. The integral action has been shown to be able to reject in-domain, as well as boundary disturbances.Finally, we have considered numerical aspects for the Lyapunov analysis. The conditions for the stability and design of controllers by quadratic Lyapunov functions involve an infinity of matrix inequalities. We have shown how to reduce this complexity by polytopic embeddings of the constraints.Many obtained results have been illustrated by academic examples and physically relevant dynamical systems (as Shallow-Water equations and Aw-Rascle-Zhang equations).Dans cette thèse nous avons étudié différents aspects pour le contrôle de systèmes hyperboliques.Tout d'abord, nous nous sommes intéressés à des systèmes hyperboliques à commutations. Cela signifie qu'il existe une interaction entre une dynamique continue et une dynamique discrète. Autrement dit, il existe différents modes dans lesquels peut évoluer la dynamique continue: ces modes sont dictés par la dynamique discrète. Ce changement de mode peut être contrôlé (dans le cas d'une boucle fermée), ou non-contrôlé (dans le cas d'une boucle ouverte). Nous nous sommes intéressés au premier cas. Par une analyse Lyapunov nous avons construit trois règles de commutations capables de stabiliser le système. Nous avons montré comment modifier deux d'entre elles pour obtenir des propriétés de robustesse et de stabilité entrée-état. Ces règles de commutations ont été testées numériquement.Ensuite, nous avons considéré la génération de trajectoire pour des systèmes hyperboliques linéaires 2x2 par backstepping. L'étape suivante a été de considérer une action Proportionnelle-Intégrale pour stabiliser la solution du système autour de la trajectoire de référence. Pour cela nous avons construit une fonction Lyapunov non-diagonale. Nous avons montré que l'action intégrale est capable de rejeter des erreurs distribuées et frontières.Enfin, nous avons considéré des aspects numériques pour l'analyse Lyapunov. Les conditions pour la stabilité et la conception de contrôleurs obtenues par des fonctions de Lyapunov quadratiques font intervenir une infinité d'inégalités matricielles. Nous avons montré que cette complexité peut être réduite en considérant une sur-approximation polytopique de ces contraintes.Les résultats obtenus ont été illustrés par des exemples académiques et des systèmes dynamiques physiques (comme les équations de Saint-Venant et les équations de Aw-Rascle-Zhang)

PDE–Based Modelling and Control Strategies for Manufacturing Processes

This work aims to design boundary control strategies to solve demand tracking and backlog problems for manufacturing systems in terms of conservation laws coupled with ODEs in different network topologies. The OCPs are investigated in the dispersing and the merging networks. The problems are optimized utilizing open-loop optimal control based on the direct and the indirect approaches. The proposed approaches enable the solution of the OCPs. All of the approaches, in general, reach a local minima with similar behaviour that leads to the steady-state. The results analysis reveals that each method has its own distinct characteristics. The indirect methodology is characterized by excellent accuracy and minimal processing burden; yet, due to the information necessary to compute the gradient, it is a sensitive method. The ease of use and flexibility to any problem distinguishes the direct method. However, this approach takes substantially longer to achieve a solution when compared to the indirect method. Also, the AMPC was introduced to investigate demand tracking and backlog problems in the context of the complex network of production systems. The addressed network includes structures that are dispersing and merging. Furthermore, the appropriate way to handle the parameters of the AMPC for both control and prediction horizons is addressed. Moreover, the proposed AMPC provides for the solutions of demand tracking and backlog problems. In general, AMPC and traditional MPC attain local minima with similar behaviour that leads to steady-state convergence. When compared to a typical MPC, the AMPC's performance shows a considerable reduction in computational time. Additionally, because it provides a mathematical insight into the method's structure, the AMPC allows for great accuracy of optimal solutions. Finally, the AMPC is characterized by its robustness according to perturbation effects

Observation and control of PDE with disturbances

In this Thesis, the problem of controlling and Observing some classes of distributed parameter systems is addressed. The particularity of this work is to consider partial differential equations (PDE) under the effect of external unknown disturbances. We consider generalized forms of two popular parabolic and hyperbolic infinite dimensional dynamics, the heat and wave equations. Sliding-mode control is used to achieve the control goals, exploiting the robustness properties of this robust control technique against persistent disturbances and parameter uncertainties