Delay-Adaptive Control of First-order Hyperbolic PIDEs

We develop a delay-adaptive controller for a class of first-order hyperbolic partial integro-differential equations (PIDEs) with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation (PDE) with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite-dimensional backstepping technique to establish global stability results. Furthermore, the well-posedness of the closed-loop system is analyzed. Finally, the effectiveness of the proposed method was validated through numerical simulation

Robust stabilization of 2×22 \times 2 first-order hyperbolic PDEs with uncertain input delay

A backstepping-based compensator design is developed for a system of $2\times2$ first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator

Prescribed-time control for a class of semilinear hyperbolic PDE-ODE systems

A prediction-based controller is shown to achieve prescribed-time stabilization of a nonlinear infinite-dimensional system, which consists of a general boundary controlled first-order semilinear hyperbolic PDE that is bidirectionally interconnected with nonlinear ODEs at its unactuated boundary. The approach uses a coordinate transformation to map the nonlinear system into a form suitable for control. In particular, this transformation is based on predictions of system trajectories, which can be obtained by solving a general nonlinear Volterra integro-differential equation. Then, a prediction-based controller is designed to stabilize the system in prescribed-time. Numerical simulations illustrate the performance of both the prescribed-time controller and an asymptotically stabilizing one, which follows as a special case

Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems

This paper presents a delay-adaptive boundary control scheme for a $2\times 2$ coupled linear hyperbolic PDE-ODE cascade system with an unknown and arbitrarily long input delay. To construct a nominal delay-compensated control law, assuming a known input delay, a three-step backstepping design is used. Based on the certainty equivalence principle, the nominal control action is fed with the estimate of the unknown delay, which is generated from a batch least-squares identifier that is updated by an event-triggering mechanism that evaluates the growth of the norm of the system states. As a result of the closed-loop system, the actuator and plant states can be regulated exponentially while avoiding Zeno occurrences. A finite-time exact identification of the unknown delay is also achieved except for the case that all initial states of the plant are zero. As far as we know, this is the first delay-adaptive control result for systems governed by heterodirectional hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in the control application of a deep-sea construction vessel with cable-payload oscillations and subject to input delay

Neural Operators for Delay-Compensating Control of Hyperbolic PIDEs

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet

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)

Modeling and Control of the Falling Film Evaporator Process

Wegen ihres energieeffizienten Betriebs und flexiblen Designs haben Fallfilmverdampfer (FFV) eine breite Anwendung in der Industrie. Neben Fragen zur Konstruktion sind dominante Totzeiten herausfordernd bzgl. Prozessmodellierung und -regelung. Insbesondere erfordert die Automatisierung von Produktionssystemen digitale Zwillinge, d.h. Anlagenmodelle, um Betreiber zu schulen oder den Designprozess zu beschleunigen. Das Herz eines FFV besteht aus Rohren, an deren Innenseiten verdampfender Flüssigkeitsfilm hinabläuft. Daher sind die Rohre primäre Quelle für Totzeiten, welche sich vornehmlich auf den Transport wichtiger Prozessgrößen wie Liquidkonzentration und Massenstrom beziehen. Allerdings ist die Modellierung des entsprechenden dynamischen Verhaltens schwierig. Aus Sicht der Regelung erzeugen Totzeiten Schwingungen der Ausgangskonzentration - im Speziellen während der Anfahrprozesse. Zusätzlich verkomplizieren starke Kopplungen zwischen Ausgangsmassenstrom und -konzentration die in modernen Produktionen erforderliche Mehrgrößenregelung. Die vorliegende Arbeit präsentiert Lösungen für alle genannten Herausforderungen. Durch Gliederung des FFV-Prozesses in Teilsysteme sind verschiedene Designs in einfacher Weise simulierbar. In diesem Kontext erfolgt die Validierung eines bestimmtes Anlagenmodell auf Basis von Realdaten, was zum digitalen Zwilling führt. Zur Entwicklung neuer Transportmodelle verdampfender Flüssigkeitsfilme werden Bilanzgleichungen ausgewertet, sodass Systeme hyperbolischer partieller Differentialgleichungen entstehen. Mittels des Charakteristikenverfahrens gelingt eine Transformation in Totzeitgleichungen; letztere sind für Simulation und Reglerentwurf vorteilhaft. Pilotanlagenexperimente zur Identifikation und Validierung eines ausgewählten Modells unterstreichen die Eignung des Ansatzes, das gemessene Ein-/Ausgangsverhalten abzubilden. Zur Beantwortung regelungstechnischer Fragen wird ein einfaches Prozessmodell entwickelt, das Zuordnungsproblem gelöst und ein Mehrgrößenregelungskonzept entworfen

Boundary control and observation of coupled parabolic PDEs

Reaction-diffusion equations are parabolic Partial Differential Equations (PDEs) which often occur in practice, e.g., to model the concentration of one or more substances, distributed in space, under the in uence of different phenomena such as local chemical reactions, in which the substances are transformed into each other, and diffusion, which causes the substances to spread out over a surface in space. Certainly, reaction-diffusion PDEs are not confined to chemical applications but they also describe dynamical processes of non-chemical nature, with examples being found in thermodynamics, biology, geology, physics, ecology, etc. Problems such as parabolic Partial Differential Equations (PDEs) and many others require the user to have a considerable background in PDEs and functional analysis before one can study the control design methods for these systems, particularly boundary control design. Control and observation of coupled parabolic PDEs comes in roughly two settingsdepending on where the actuators and sensors are located \in domain" control, where the actuation penetrates inside the domain of the PDE system or is evenly distributed everywhere in the domain and \boundary" control, where the actuation and sensing are applied only through the boundary conditions. Boundary control is generally considered to be physically more realistic because actuation and sensing are nonintrusive but is also generally considered to be the harder problem, because the \input operator" and the "output operator" are unbounded operators. The method that this thesis develops for control of PDEs is the so-called backstepping control method. Backstepping is a particular approach to stabilization of dynamic systems and is particularly successful in the area of nonlinear control. The backstepping method achieves Lyapunov stabilization, which is often achieved by collectively shifting all the eigenvalues in a favorable direction in the complex plane, rather than by assigning individual eigenvalues. As the reader will soon learn, this task can be achieved in a rather elegant way, where the control gains are easy to compute symbolically, numerically, and in some cases even explicitly. In addition to presenting the methods for boundary control design, we present the dual methods for observer design using boundary sensing. Virtually every one of our control designs for full state stabilization has an observer counterpart. The observer gains are easy to compute symbolically or even explicitly in some cases. They are designed in such a way that the observer error system is exponentially stabilized. As in the case of finite-dimensional observer-based control, a separation principle holds in the sense that a closed-loop system remains stable after a full state stabilizing feedback is replaced by a feedback that employs the observer state instead of the plant state