In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the spatial variable over an N-point mesh, thereby converting each partial differential equation into N ordinary differential equations. These equations can then be solved using numerical routines defined for ordinary differential equations.

System Lupus Erythematosus (SLE) is a chronic inflammatory autoimmune disorder that affects many parts of the body including skin, joints, kidneys, brains and other organs. Lupus Nephritis (LN) is a disease caused by SLE. Given the complexity of LN, we establish an optimal treatment strategy based on a previously developed mathematical model.For our thesis work, the model variables are: Immune Complexes (I), Pro-inflammatory mediators (P), Damaged tissue (D), and Anti-inflammatory mediators (A). The analysis in this research project focuses on analyzing therapeutic strategies to control damage using both parameter estimation techniques (integration of data to quantify any uncertainties associated with parameters) and optimal control with the goal of minimizing time spent on therapy for treating damaged tissue by LN.

Cette thèse s'intéresse à une étude des EDSR ergodiques, avec pour principal objectif leur application à l'étude du comportement en temps long de certaines EDP. Dans un premier temps, nous démontrons des résultats (qui sont déjà connus dans le cadre où l'EDS sous-jacente est à bruit additif) dans un cadre de bruit sous-jacent multiplicatif. Par la suite, l'introduction d'un nouvel aléa via un processus de Poisson nous permet de nous intéresser non plus au comportement en temps long d'une seule EDP, mais au comportement en temps long d'un système d'EDP couplées. Enfin, lorsque l'EDS sous-jacente est bruitée par un processus de Lévy, le lien est fait avec des équations intégro-différentielles partielles. L'application de ces équations à la résolution de problèmes de contrôle optimal est également présentée. / In this thesis, we are interested in studying ergodic BSDEs, and our main goal is to apply our results to the large time behaviour of some PDEs. First, we prove some results (already known in the case where the underlying SDE has an additive noise) in the case of an underlying multiplicative noise. Then, we introduce a Poisson process and it leads us to the large time behaviour of a system of coupled PDEs. Finally, when the underlying SDE has a Lévy noise, we make a link with partial integro-differential equations. We also apply these equations to solve some optimal control problems.

This thesis centers around an electric arc furnace (EAF) at a steel mini-mill in Wilton, IA. First, the thesis replicates previous optimization attempts. Next, the modeling is greatly altered to produce a much improved steel-melting model. Then, a new optimal control system is created and used to reduce energy and fuel costs over the melting process. Finally, results are presented. This thesis shows that when the new optimal control is simulated, the system shows significant energy and fuel savings.

L’objectif principal de ce travail est l’étude de la planification de trajectoires pour des drones de type HALE ou MALE. Les modèles cinématiques de ces drones sont étudiés. Les drones HALE sont modélisés par le système de Dubins. Pour les drones MALE, le modèle est construit en étudiant le repère cinématique du drone. Nous considérons les problèmes de planification de trajectoires point-point et point-pattern. Il s’agit, à partir de la position courante du drone, de rejoindre un point ou une figure prédéfinie dans l’espace. La planification point-point est abordée sous forme d’un problème de contrôle optimal. Deux méthodes sont proposées pour résoudre le problème point-pattern. D’abord nous présentons la synthèse en temps minimal pour le système de Dubins. Ensuite, nous développons une méthode basée sur le principe de LaSalle. La première méthode est utilisée au sein d’un algorithme de planification pour des drones HALE. La deuxième permet de stabiliser les deux types de drones considérés vers un pattern. Nous proposons une extension des algorithmes de planification développés, basée sur une discrétisation del’espace grâce aux graphes de Voronoï et une méthode de planification discrète, pour construire des trajectoiresdans des milieux encombrés. Nous étudions également le problème de couplage drone/capteur. Il s’agit de calculer une trajectoire permettant de satisfaire les objectifs du drone et de son capteur (une caméra). L’algorithme proposé est construit à partir de la résolution d’un problème quadratique sous contraintes.Dans une seconde partie, nous analysons un problème de contrôle optimal inverse. Celui-ci permet d’améliorer les résultats des méthodes de planification en s’inspirant du comportement des pilotes. Après avoir posé le problème, les résultats théoriques sont exposés et le cas particulier du système de Dubins est étudié en pratique. / This thesis is about path planning for HALE or MALE UAVs (Unmanned Aircraft Vehicles), possibly under mission constraints. As such, the study is performed at the kinematic level : HALE UAVs are represented as Dubins systems, and a model for MALE UAVs is constructed by studying their kinematic frame. In the first part, we tackle the path planning problem for a UAV that must join a target (a point or a pattern), starting from any position. The point to point path planning problem is addressed as an optimal control problem. Regarding the point to pattern path planning problem, two different methods are proposed. The former consists in solving the minimum time synthesis for the Dubins system, in order to obtain a basis for a HALE UAVs planning algorithm. The latter method relies on the LaSalle principle ; it permits to stabilize a HALE or MALE UAV to a pattern.In addition, extensions of the previously developed algorithms to cluttered environnement are provided. This extension is achieved thanks to a space discretization using Voronoi diagrams and a discrete planning method. Finally, the mission constraints are dealt with as a coupling problem between the UAV and its sensors. The proposed algorithm is presented in the form of a constrained quadratic problem.In the second part of this thesis, we want to refine the planning algorithm to get a result closer to trajectories of pilots. In order to do that, we solve an inverse optimal control problem where the cost to find is computed from the experience of pilots. Theoretical results are presented and applied to the particular case of the Dubins system.

Exploiting high altitude wind energy using power kites is an emerging topic in the field of renewable energy. The claimed advantages of power kites over traditional wind power technologies are the lower construction costs, less land occupation and more importantly, the possibility of efficiently harvesting wind energy at high altitudes, where more dense and steady wind power exists. One of the most challenging issues to bring the power kite concept to real industrialization is the controller design. While traditional wind turbines can be inherently stabilized, the airborne nature of kites causes a strong instability of the systems. This thesis aims to develop a novel economic model predictive path-following control (EMPFC) framework to tackle the path-following control of power kites, as well as provide insightful stability analysis of the proposed control scheme. Chapter 3 is focused on the stability analysis of EMPFC. We proceed with a sampled-data EMPC scheme for set-point stabilization problems. An extended definition of dissipativity is introduced for continuous-time systems, followed by giving sufficient stability conditions. Then, the EMPFC scheme for output path-following problems is proposed. Sufficient conditions that guarantee the convergence of the system to the optimal operation on the reference path are derived. Finally, an example of a 2-DoF robot is given. The simulation results show that under the proposed EMPFC scheme, the robot can follow along the reference path in forward direction with enhanced economic performance, and finally converges to its optimal steady state. In Chapter 4, the proposed EMPFC scheme is applied to a challenging nonlinear kite model. By introducing additional degrees of freedom in the zero-error manifold (i.e., the space where the output error is zero), a relaxation of the optimal operation is achieved. The effectiveness of the proposed control scheme is shown in two aspects. For a static reference path, the generated power is increased while the kite is stabilized in the neighborhood of the reference path. For a dynamic reference path, the economic performance can be further enhanced since parameters for the reference path are treated as additional optimization variables. The proposed EMPFC achieves the integration of path optimization and path-following, resulting in a better economic performance for the closed-loop system. Simulation results are given to show the effectiveness of the proposed control scheme. Finally, Chapter 5 concludes the thesis and future research topics are discussed. / Graduate / 2020-05-14

The scheduling and Control Group at the University of South Australia has been studying the optimal control of trains for many years, and has developed in-cab devices that help drivers stay on time and minimise energy use. In this thesis, we re-examine the optimal control theory for the train control problem. In particular, we study the optimal control around steep sections of track. To calculate an optimal driving strategy we need a realistic model of train performance. In particular, we need to know a coefficient of rolling resistance and a coefficient of aerodynamic drag. In practice, these coefficients are different for every train and difficult to predict. In the thesis, we study the use of mathematical filters to estimate model parameters from observations of actual train performance.

Many potentially useful applications that broadly fall under the umbrella of payload transportation operations have been proposed for aerial towed-cable (ATC) systems, namely the precise capture and delivery of payloads. There remain outstanding issues concerning the dynamics and control of ATC systems that are inhibiting the near-term demonstration of these applications. The development of simplified representations of ATC systems that retain the important dynamics, yet are simple enough for use in control system development is limited. Likewise, little research exists into the development of controllers for ATC systems, especially the development of towing strategies and cable-based control techniques for rendezvous and payload transportation. Thus, this thesis presents novel research into the development of control strategies and simulation facilities that redress these two major anomalies, thereby overcoming a number of hitherto unresolved issues. The primary objective of this thesis is to develop innovative non-linear optimal control systems to manoeuvre a cable towed beneath an aircraft to transport payloads both to and from surface locations. To appropriately satisfy this objective, accurate and efficient modelling capabilities are proposed, yielding the equations of motion for numerous models of the ATC system. A series of techniques for improving the representativeness of simple dynamical models were developed. The benefits of using these procedures were shown to be significant and possible without undue complexity or computational expense. Use of such techniques result in accurate simulations and allow representative control systems to be designed. A series of single and multi-phase non-linear optimal control problems for ATC systems are then formally proposed, which were converted into non-linear programming problems using direct transcription for expedient solution. The possibility of achieving accurate, numerous instantaneous rendezvous of the cable tip with desired surface locations on the ground, in two and three-dimensions, is successfully demonstrated. This was achieved through the use of deployment and retrieval control of the cable and/or aircraft manoeuvring. The capability of the system to safely and accurately transport payloads to and from the surface via control of the cable and/or aircraft manoeuvring is also established. A series of parametric studies were conducted to establish the impact that various parameters have on the ability of the system to perform various rendezvous and payload transportation operations. This allowed important insights into to the nature of the system to be examined. In order for the system to perform rendezvous and payload transportation operations in the presence of wind gusts, a number of simple closed loop optimal feedback controllers were developed. These feedback controllers are based on the linear quadratic regulator control methodology. A preliminary indication of the robustness of ATC systems to wind gusts is provided for through a succession of parametric investigations. The performance of the closed-loop system demonstrates that precise and robust control of the ATC system can be achieved for a wide variety of operating conditions. The research presented in this thesis will provide a solid foundation for further advancing the development of aerial tether payload transportation technology.

<p>The main contributions in this thesis are advances in parametric programming. The thesis is divided into three parts; theoretical advances, application areas and constrained control allocation. The first part deals with continuity properties and the structure of solutions to convex parametric quadratic and linear programs. The second part focuses on applications of parametric quadratic and linear programming in control theory. The third part deals with constrained control allocation and how parametric programming can be used to obtain explicit solutions to this problem.</p>

Time-optimal solutions provide us with the fastest means to regulate a system in presence of input constraints. This advantage of time-optimal control solutions is offset by the fact that their real-time implementation involves computationally intensive iterative techniques. Moreover, time-optimal controls depend on the initial state and have to be recalculated for even the slightest perturbation. Clearly time-optimal controls are not good candidates for online regulation. Consequently, the search for alternatives to time-optimal solutions is a very active area of research. The work described here is inspired by the simplicity of optimal-aim concept. The "optimal-aim strategies" provide online regulation in presence of bounded inputs with minimal computational effort. These are based purely on state-space geometry of the plant and are inherently adaptive in nature. Optimal-aim techniques involve aiming of trajectory derivative (or the state velocity vector) so as to approach the equilibrium state in the best possible manner. This thesis documents the efforts to develop an online regulation algorithm for systems with input constraints. Through a number of hypotheses focussed on trying to reproduce the exact time-optimal solution, the diffculty associated with this task is demonstrated. A modification of optimal-aim concept is employed to develop a novel regulation algorithm. In this algorithm, aim directions are chosen in a special manner to generate the time-optimal control approximately. The control scheme thus developed is shown to be globally stabilizing for systems having eigenvalues in the CLHP (closed left half-plane). It is expected that this method or its modifications can be extended to higher dimensional systems as a part of future research. An alternative control algorithm involving a simple state-space aiming concept is also developed and discussed.