In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness

This paper addresses the set-point control problem of a heat equation with in-domain actuation. The proposed scheme is based on the framework of zero dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planing problem of a multiple-input, multiple-out system, which is solved by a Green's function-based reference trajectory decomposition. The validity of the proposed method is assessed through convergence and solvability analysis of the control algorithm. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.Comment: Preprint of an original research pape

Null controllability of the 1D heat equation using flatness

We derive in a straightforward way the null controllability of a 1-D heat equation with boundary control. We use the so-called {\em flatness approach}, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control law achieving the exact steering to zero. We also give accurate error estimates when the various series involved are replaced by their partial sums, which is paramount for an actual numerical scheme. Numerical experiments demonstrate the relevance of the approach

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

Sub-Optimality of a Dyadic Adaptive Control Architecture

The dyadic adaptive control architecture evolved as a solution to the problem of designing control laws for nonlinear systems with unmatched nonlinearities, disturbances and uncertainties. A salient feature of this framework is its ability to work with infinite as well as finite dimensional systems, and with a wide range of control and adaptive laws. In this paper, we consider the case where a control law based on the linear quadratic regulator theory is employed for designing the control law. We benchmark the closed-loop system against standard linear quadratic control laws as well as those based on the state-dependent Riccati equation. We pose the problem of designing a part of the control law as a Nehari problem. We obtain analytical expressions for the bounds on the sub-optimality of the control law

Long time control with applications

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 24-04-2020This thesis is concerned with the study of some control problems in a large time horizon. The first part of the thesis is devoted to controllability of Partial Differential Equations under state and/or control constraints. In chapter 4, we address the controllability under positivity constraints of semilinear heat equations. We firstly obtain steady state controllability, by employing a ``stair-case argument''. Then, supposing dissipativity of the free dynamics, we extend our previous result to constrained controllability to trajectories. In any case, the targets must be defined by positive controls. We prove further the positivity of the minimal controllability time under positivity constraints, by applying a new method, based on the choice of a particular test function in the definition of weak solutions to evolution equations. Hence, despite the infinite velocity of propagation for parabolic equations, a waiting time phenomenon occurs in the constrained case. In chapter 5, controllability under positivity constraints is analyzed for wave equations. In this case, the zero state is reachable, by nonnegative controls. In chapter 6, we get a global turnpike result for an optimal control problem, governed by a semilinear heat equation. The running target in the cost functional is required to be small, whereas the initial datum for the evolution equation can be chosen arbitrarily. This is done by combining the available local results [116, 137], with an estimate of the L1 norm of the optima (uniform in the time horizon) and an estimate of the time needed to get close to the turnpike. If the target is large, we produce an example, where the steady problem admits (at least) two solutions (chapter 7). In chapter 8, we present an application of stabilization/turnpike theory to a problem of rotor balancingEsta tesis concierne el estudio de algunos problemas de control en un largo horizonte temporal. La primera parte de la tesis está dedicada a la controlabilidad de Ecuaciones en Derivadas Parciales bajo restricciones de estado y/o control. En el capítulo 4, abordamos la controlabilidad bajo restricciones de positividad para la ecuación del calor semilineal. En primer lugar, obtenemos la controlabilidad entre estados estacionarios, mediante el uso de un ``stair-case argument''. Luego, suponiendo disipatividad en la dinámica libre, extendemos nuestro resultado anterior a la controlabilidad bajo restricciones hacia trayectorias. En cualquier caso, los targets deben definirse mediante controles positivos. Ademas, probamos la positividad del tiempo mínimo de controlabilidad bajo restricciones de positividad, mediante la aplicación de un nuevo método, basado en la elección de una función test particular en la definición de solucione débil para la ecuación de evolución. Por lo tanto, a pesar de la velocidad infinita de propagación para las ecuaciones parabólicas, se produce un fenómeno de tiempo de espera en el caso restringido. En el capítulo 5, la controlabilidad bajo restricciones de positividad se analiza para la ecuación de ondas. En este caso, el estado cero es alcanzable por controles positivos. En el capítulo 6, obtenemos un resultado de turnpike global para un problema de control optimo, sujeto a una ecuación del calor semilineal. En este caso, requerimos que el target en el funcional de coste sea pequeño, mientras que el dato inicial para la ecuación de evolución se puede elegir arbitrariamente. Esto se realiza combinando los resultados locales disponibles en [116, 137], con una estimación de la norma L1 para los optimos (uniforme en el horizonte temporal) y una estimación del tiempo necesario para acercarse al turnpike. Para el caso de target grande, damos un ejemplo, donde el problema estacionario admite (al menos) dos soluciones (capítulo 7). En el capítulo 8, presentamos una aplicación de la teoría de estabilización/turnpike a un problema de equilibrio para un rotorThis thesis has been mainly funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 694126-DyCon), and for an applied research secondment by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 77782

Modeling and optimal control of multiphysics problems using the finite element method

Interdisciplinary research like constrained optimization of partial differential equations (PDE) for trajectory planning or feedback algorithms is an important topic. Recent technologies in high performance computing and progressing research in modeling techniques have enabled the feasibility to investigate multiphysics systems in the context of optimization problems. In this thesis a conductive heat transfer example is developed and techniques from PDE constrained optimization are used to solve trajectory planning problems. In addition, a laboratory experiment is designed to test the algorithms on a real world application. Moreover, an extensive investigation on coupling techniques for equations arising in convective heat transfer is given to provide a basis for optimal control problems regarding heating ventilation and air conditioning systems. Furthermore a novel approach using a flatness-based method for optimal control is derived. This concept allows input and state constraints in trajectory planning problems for partial differential equations combined with an efficient computation. The stated method is also extended to a Model Predictive Control closed-loop formulation. For illustration purposes, all stated problems include numerical examples