Neste trabalho estudamos as noções de alcançabilidade e controlabilidade para sistemas lineares a tempo contínuo com perturbações aditivas e saltos nos parâmetros sujeitos a uma cadeia de Markov geral. Definimos conceitos de alcançabilidade e controlabilidade médios de maneira natural exigindo que os valores esperados dos gramianos correspondentes sejam definidos positivos. Visando obter uma condição testável para ambos os conceitos, introduzimos conjuntos de matrizes de alcançabilidade e de controlabilidade para esta classe de sistemas e usamos certas propriedades de invariância para mostrar que: o sistema é alcançável em média, e, analogamente, controlável em média, se e somente se as matrizes respectivas, de alcançabilidade e de controlabilidade, têm posto completo. Usamos alcançabilidade média de sistemas para mostrar que a matriz de segundo momento do estado é definida positiva com uma margem uniforme. Uma consequência deste resultado no problema de estimação linear do estado é que a matriz de covariância do erro de estimação é positiva definida em média, no sentido que existe um nível mínimo de ruído nas estimativas. Na sequência, para estimadores lineares markovianos, estudamos a limitação do valor esperado da matriz de covariância do erro para mostrar que o filtro é estável num certo sentido, sendo esta uma propriedade desejável em aplicações reais. Quanto às aplicações da controlabilidade média, usamos este conceito para estabelecer condições necessárias e suficientes que garantem a existência de um processo de controle que leva a componente contínua do estado do sistema para a origem em tempo finito e com probabilidade positiva. / In this work we study the reachability and controllability notions for continuous-time linear systems with exogenous inputs and jump parameters driven by a quite general Markov chain. We define a rather natural average reachability and controllability concepts by requiring that the associated gramians are average positive definite, respectively. Aiming at testable conditions for each concept, we introduce certain sets of matrices linked with the gramians, and employ some invariance properties to find rank-based conditions. We show for average reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, for linear markovian filters we study the average boundedness of the error covariance matrix to show that the filter is stable in an appropriate sense, which consists in a property that is desirable in real applications. Regarding the average controllability concept, we show that it is a necessary and sufficient condition for the feasibility of the following control problem: find a control process that drives the continuous component of the state to zero in finite time with positive probability.

Dans cette thèse, nous étudions un problème de contrôle et un problème inverse pour les équationsde la chaleur. Notre premier travail concerne la contrôlabilité à zéro pour une équation de la chaleur semi-linéaire. Il est à noter que sans contrôle, la solution est instable et il y aura en général explosion de la solution en un temps fini. Ici, nous proposons un résultat positif de contrôlabilité à zéro sous une hypothèse quantifiée de petitesse sur la donnée initiale. La nouveauté réside en la construction de ce contrôle pour amener la solution à l’état d’équilibre.Notre second travail aborde l’équation de la chaleur rétrograde dans un domaine borné et sous la condition de Dirichlet. Nous nous intéressons à la question suivante: peut-on reconstruire la donnée initiale à partir d’une observation de la solution restreinte à un sous-domaine et à un temps donné? Ce problème est connu pour être mal-posé. Ici, les deux principales méthodes proposées sont: une approche de filtrage des hautes fréquences et une minimisation à la Tikhonov. A chaque fois, nous reconstruisons de manière approchée la solution et quantifions l’erreur d’approximation / My thesis focuses on two main problems in studying the heat equation: Control problem and Inverseproblem.Our first concern is the null controllability of a semilinear heat equation which, if not controlled, can blow up infinite time. Roughly speaking, it consists in analyzing whether the solution of a semilinear heat equation, underthe Dirichlet boundary condition, can be driven to zero by means of a control applied on a subdomain in whichthe equation evolves. Under an assumption on the smallness of the initial data, such control function is builtup. The novelty of our method is computing the control function in a constructive way. Furthermore, anotherachievement of our method is providing a quantitative estimate for the smallness of the size of the initial datawith respect to the control time that ensures the null controllability property.Our second issue is the local backward problem for a linear heat equation. We study here the followingquestion: Can we recover the source of a linear heat equation, under the Dirichlet boundary condition, from theobservation on a subdomain at some time later? This inverse problem is well-known to be an ill-posed problem,i.e their solution (if exists) is unstable with respect to data perturbations. Here, we tackle this problem bytwo different regularization methods: The filtering method and The Tikhonov method. In both methods, thereconstruction formula of the approximate solution is explicitly given. Moreover, we also provide the errorestimate between the exact solution and the regularized one.

Les systèmes d'Equations aux Dérivées Partielles Hyperboliques Linéaires du Premier Ordre (EDPs HLPO) permettent de modéliser de nombreux systèmes de lois de conservation. Ils apparaissent, par exemple, lors de la modélisation de problèmes de trafic routier, d'échangeurs de chaleurs, ou de problèmes multiphasiques. Différentes approches ont été proposées pour stabiliser ou observer de tels systèmes. Parmi elles, la méthode de backstepping consiste à transformer le système originel en un système découplé pour lequel la synthèse de la loi de commande est plus simple. Les contrôleur obtenus par cette méthode sont explicites.Dans la première partie de cette thèse, nous présentons des résultats généraux de théorie des systèmes. Plus précisément, nous résolvons les problèmes de stabilisation en temps fini pour une classe générale d'EDPs HLPO. Le temps de convergence minimal atteignable dépend du nombre d'actionneurs disponibles. Les observateurs associés à ces contrôleurs (nécessaires pour envisager une utilisation industrielle de tels contrôleurs) sont obtenus via une approche duale. Un des avantages importants de l'approche considérée dans cette thèse est de montrer que l'espace généré par les solutions de l'EDPs HLPO considérée est isomorphe à l'espace généré par les solutions d'une système neutre à retards distribués.Dans la seconde partie de cette thèse, nous montrons la nécessité d'un changement de stratégie pour résoudre les problèmes de contrôle robuste. Ces questions surviennent nécessairement lorsque sont considérées des applications industrielles pour lesquelles les différents paramètres du système peuvent être mal connus, pour lesquelles des dynamiques peuvent avoir été négligées, de même que des retards agissant sur la commande ou sur la mesure, ou encore pour lesquels les mesures sont bruitées. Nous proposons ainsi quelques modifications sur les lois de commande précédemment développées en y incorporant plusieurs degrés de liberté permettant d'effectuer un compromis entre performance et robustesse. L'analyse de stabilité et de robustesse sous-jacente est rendue possible en utilisant l'isomorphisme précédemment introduit. / Linear First-Order Hyperbolic Partial Differential Equations (LFOH PDEs) represent systems of conservation and balance law and are predominant in modeling of traffic flow, heat exchanger, open channel flow or multiphase flow. Different control approaches have been tackled for the stabilization or observation of such systems. Among them, the backstepping method consists to map the original system to a simpler system for which the control design is easier. The resulting controllers are explicit.In the first part of this thesis, we develop some general results in control theory. More precisely, we solve the problem of finite-time stabilization of a general class of LFOH PDEs using the backstepping methodology. The minimum stabilization time reachable may change depending on the number of available actuators. The corresponding boundary observers (crucial to envision industrial applications) are obtained through a dual approach. An important by-product of the proposed approach is to derive an explicit mapping from the space generated by the solutions of the considered LFOH PDEs to the space generated by the solutions of a general class of neutral systems with distributed delays. This mapping opens new prospects in terms of stability analysis for LFOH PDEs, extending the stability analysis methods developed for neutral systems.In the second part of the thesis, we prove the necessity of a change of strategy for robust control while considering industrial applications, for which the major limitation is known to be the robustness of the resulting control law to uncertainties in the parameters, delays in the loop, neglected dynamics or disturbances and noise acting on the system. In some situations, one may have to renounce to finite-time stabilization to ensure the existence of robustness margins. We propose some adjustments in the previously designed control laws by means of several degrees of freedom enabling trade-offs between performance and robustness. The robustness analysis is fulfilled using the explicit mapping between LFOH PDEs and neutral systems previously introduced.

Modern control theory is today an interdisciplinary area of research. Just as much as this can be problematic, it also provides a rich research environment where practice and theory meet. This Thesis is conducted in the borderline between computing science (numerical analysis) and applied control theory. The design and analysis of a modern control system is a complex problem that requires high qualitative software to accomplish. Ideally, such software should be based on robust methods and numerical stable algorithms that provide quantitative as well as qualitative information. The introduction of the Thesis is dedicated to the underlying control theory and to introduce the reader to the main subjects. Throughout the Thesis, the theory is illustrated with several examples, and similarities and differences between the terminology from mathematics, systems and control theory, and numerical linear algebra are highlighted. The main contributions of the Thesis are structured in two parts, dealing with two mainly unrelated subjects. Part I is devoted to the qualitative information which is provided by the stratification of orbits and bundles of matrices, matrix pencils and system pencils. Before the theory of stratification is established the reader is introduced to different canonical forms which reveal the system characteristics of the model under investigation. A stratification reveals which canonical structures of matrix (system) pencils are near each other in the sense of small perturbations of the data. Fundamental concepts in systems and control, like controllability and observability of linear continuous-time systems, are considered and it is shown how these system characteristics can be investigated using the stratification theory. New results are presented in the form of the cover relations (nearest neighbours) for controllability and observability pairs. Moreover, the permutation matrices which take a matrix pencil in the Kronecker canonical form to the corresponding system pencil in (generalized) Brunovsky canonical form are derived. Two novel algorithms for determining the permutation matrices are provided. Part II deals with numerical methods for solving periodic Riccati differential equations (PRDE:s). The PRDE:s under investigation arise when solving the linear quadratic regulator (LQR) problem for periodic linear time-varying (LTV) systems. These types of (periodic) LQR problems turn up for example in motion planning of underactuated mechanical systems, like a humanoid robot, the Furuta pendulum, and pendulums on carts. The constructions of the nonlinear controllers are based on linear versions found by stabilizing transverse dynamics of the systems along cycles. Three different methods explicitly designed for solving the PRDE are evaluated on both artificial systems and stabilizing problems originating from experimental control systems. The methods are the one-shot generator method and two recently proposed methods: the multi-shot method (two variants) and the SDP method. As these methods use different approaches to solve the PRDE, their numerical behavior and performance are dependent on the nature of the underlying control problem. Such method characteristics are investigated and summarized with respect to different user requirements (the need for accuracy and possible restrictions on the solution time). / Modern reglerteknik är idag i högsta grad ett interdisciplinärt forskningsområde. Lika mycket som detta kan vara problematiskt, resulterar det i en stimulerande forskningsmiljö där både praktik och teori knyts samman. Denna avhandling är utförd i gränsområdet mellan datavetenskap (numerisk analys) och tillämpad reglerteknik. Att designa och analysera ett modernt styrsystem är ett komplext problem som erfordrar högkvalitativ mjukvara. Det ideala är att mjukvaran består av robusta metoder och numeriskt stabila algoritmer som kan leverera både kvantitativ och kvalitativ information.Introduktionen till avhandlingen beskriver grundläggande styr- och reglerteori samt ger en introduktion till de huvudsakliga problemställningarna. Genom hela avhandlingen illustreras teori med exempel. Vidare belyses likheter och skillnader i terminologin som används inom matematik, styr- och reglerteori samt numerisk linjär algebra. Avhandlingen är uppdelade i två delar som behandlar två i huvudsak orelaterade problemklasser. Del I ägnas åt den kvalitativa informationen som ges av stratifiering av mångfalder (orbits och bundles) av matriser, matrisknippen och systemknippen. Innan teorin för stratifiering introduceras beskrivs olika kanoniska former, vilka var och en avslöjar olika systemegenskaper hos den undersökta modellen. En stratifiering ger information om bl.a. vilka kanoniska strukturer av matrisknippen (systemknippen) som är nära varandra med avseende på små störningar i datat. Fundamentala koncept i styr- och reglerteori behandlas, så som styrbarhet och observerbarhet av linjära tidskontinuerliga system, och hur dessa systemegenskaper kan undersökas med hjälp av stratifiering. Nya resultat presenteras i form av relationerna för täckande (närmsta grannar) styrbarhets- och observerbarhets-par. Dessutom härleds permutationsmatriserna som tar ett matrisknippe i Kroneckers kanoniska form till motsvarande systemknippe i (generaliserade) Brunovskys kanoniska form. Två algoritmer för att bestämma dessa permutationsmatriser presenteras. Del II avhandlar numeriska metoder för att lösa periodiska Riccati differentialekvationer (PRDE:er). De undersökta PRDE:erna uppkommer när ett linjärt kvadratiskt regulatorproblem för periodiska linjära tidsvariabla (LTV) system löses. Dessa typer av (periodiska) regulatorproblem dyker upp till exempel när man planerar rörelser för understyrda (underactuated) mekaniska system, så som en humanoid (mänsklig) robot, Furuta-pendeln och en vagn med en inverterad (stående) pendel. Konstruktionen av det icke-linjära styrsystemet är baserat på en linjär variant som bestäms via stabilisering av systemets transversella dynamik längs med cirkulära banor. Tre metoder explicit konstruerade för att lösa PRDE:er evalueras på både artificiella system och stabiliseringsproblem av experimentella styrsystem. Metoderna är sk. en- och flerskotts metoder (one-shot, multi-shot) och SDP-metoden. Då dessa metoder använder olika tillvägagångssätt för att lösa en PRDE, beror dess numeriska egenskaper och effektivitet på det underliggande styrproblemet. Sådana metodegenskaper undersöks och sammanfattas med avseende på olika användares behov, t.ex. önskad noggrannhet och tänkbar begränsning i hur lång tid det får ta att hitta en lösning.

This dissertation addresses the development and use of novel software tools and environments for the computation and visualization of canonical information as well as stratification hierarchies for matrices and matrix pencils. The simplest standard shape to which a matrix pencil with a given set of eigenvalues can be reduced is called the Kronecker canonical form (KCF). The KCF of a matrix pencil is unique, and all pencils in the manifold of strictly equivalent pencils - collectively termed the orbit - can be reduced to the same canonical form and so have the same canonical structure. For a problem with fixed input size, all orbits are related under small perturbations. These relationships can be represented in a closure hierarchy with a corresponding graph depicting the stratification of these orbits. Since degenerate canonical structures are common in many applications, software tools to determine canonical information, especially under small perturbations, are central to understanding the behavior of these problems. The focus in this dissertation is the development of a software tool called StratiGraph. Its purpose is the computation and visualization of stratification graphs of orbits and bundles (i.e., union of orbits in which the eigenvalues may change) for matrices and matrix pencils. It also supports matrix pairs, which are common in control systems. StratiGraph is extensible by design, and a well documented plug-in feature enables it, for example, to communicate with Matlab(TM). The use and associated benefits of StratiGraph are illustrated via numerous examples. Implementation considerations such as flexible software design, suitable data representations, and good and efficient graph layout algorithms are also discussed. A way to estimate upper and lower bounds on the distance between an input S and other orbits is presented. The lower bounds are of Eckhart-Young type, based on the matrix representation of the associated tangent spaces. The upper bounds are computed as the Frobenius norm F of a perturbation such that S + F is in the manifold defining a specified orbit. Using associated plug-ins to StratiGraph this information can be computed in Matlab, while visualization alongside other canonical information remains within StratiGraph itself. Also, a proposal of functionality and structure of a framework for computation of matrix canonical structure is presented. Robust, well-known algorithms, as well algorithms improved and developed in this work, are used. The framework is implemented as a prototype Matlab toolbox. The intention is to collect software for computing canonical structures as well as for computing bounds and to integrate it with the theory of stratification into a powerful new environment called the MCS toolbox. Finally, a set of utilities for generating web computing environments related to mathematical and engineering library software is presented. The web interface can be accessed from a standard web browser with no need for additional software installation on the local machine. Integration with the control and systems library SLICOT further demonstrates the efficacy of this approach.

The Finite Element Method (FEM) based computational mechanics is applied to simulate the optimal attenuation of vibrations in actively controlled structures. The simulation results provide the forces to be generated by actuators, as well as the structures response. Vibrations can be attenuated by applying either open loop or closed loop control strategies. In open loop control, the control forces for a given initial (or disturbed) configuration of the structure are determined in terms of time, and can be preprogrammed in advance. On the other hand, the control forces in closed loop control depend only on the current state of the system, which should be continuously monitored. Optimal attenuation is obtained by solving the optimality equations for the problem derived from the Pontryagins principle. These equations together with the initial and final boundary conditions constitute the two-point-boundary-value (TPBV) problem. <p>Here the optimal solutions are obtained by applying an analogy (referred to as the beam analogy) between the optimality equation and the equation for a certain problem of static beams in bending. The problem of analogous beams is solved by the standard FEM in the spatial domain, and then the results are converted into the solution of the optimal vibration control problem in the time domain. The concept of the independent-modal-space-control (IMSC) is adopted, in which the number of independent actuators control the same number of vibrations modes. <p>The steps of the analogy are programmed into an algorithm referred to as the Beam Analogy Algorithm (BAA). As an illustration of the approach, the BAA is used to simulate the open loop vibration control of a structure with several sets of actuators. Some details, such as an efficient meshing of the analogous beams and effective solving of the target condition are discussed. <p> Next, the BAA is modified to handle closed loop vibration control problems. The algorithm determines the optimal feedback gain matrix, which is then used to calculate the actuator forces required at any current state of the system. The methods accuracy is also analyzed.

The Finite Element Method (FEM) based computational mechanics is applied to simulate the optimal attenuation of vibrations in actively controlled structures. The simulation results provide the forces to be generated by actuators, as well as the structures response. Vibrations can be attenuated by applying either open loop or closed loop control strategies. In open loop control, the control forces for a given initial (or disturbed) configuration of the structure are determined in terms of time, and can be preprogrammed in advance. On the other hand, the control forces in closed loop control depend only on the current state of the system, which should be continuously monitored. Optimal attenuation is obtained by solving the optimality equations for the problem derived from the Pontryagins principle. These equations together with the initial and final boundary conditions constitute the two-point-boundary-value (TPBV) problem. <p>Here the optimal solutions are obtained by applying an analogy (referred to as the beam analogy) between the optimality equation and the equation for a certain problem of static beams in bending. The problem of analogous beams is solved by the standard FEM in the spatial domain, and then the results are converted into the solution of the optimal vibration control problem in the time domain. The concept of the independent-modal-space-control (IMSC) is adopted, in which the number of independent actuators control the same number of vibrations modes. <p>The steps of the analogy are programmed into an algorithm referred to as the Beam Analogy Algorithm (BAA). As an illustration of the approach, the BAA is used to simulate the open loop vibration control of a structure with several sets of actuators. Some details, such as an efficient meshing of the analogous beams and effective solving of the target condition are discussed. <p> Next, the BAA is modified to handle closed loop vibration control problems. The algorithm determines the optimal feedback gain matrix, which is then used to calculate the actuator forces required at any current state of the system. The methods accuracy is also analyzed.

One of the main tasks for an industrial robot is to move the end-effector in a predefined path with a specified velocity and acceleration. Different applications have different requirements of the performance. For some applications it is essential that the tracking error is extremely small, whereas other applications require a time optimal tracking. Independent of the application, the controller is a crucial part of the robot system. The most common controller configuration uses only measurements of the motor angular positions and velocities, instead of the position and velocity of the end-effector. The development of new cost optimised robots has introduced unwanted flexibilities in the joints and the links. The consequence is that it is no longer possible to get the desired performance and robustness by only measuring the motor angular positions.  This thesis investigates if it is possible to estimate the end-effector position using Bayesian estimation methods for state estimation, here represented by the extended Kalman filter and the particle filter. The arm-side information is provided by an accelerometer mounted at the end-effector. The measurements consist of the motor angular positions and the acceleration of the end-effector. In a simulation study on a realistic flexible industrial robot, the angular position performance is shown to be close to the fundamental Cramér-Rao lower bound. The methods are also verified in experiments on an ABB IRB4600 robot, where the dynamic performance of the position for the end-effector is significantly improved. There is no significant difference in performance between the different methods. Instead, execution time, model complexities and implementation issues have to be considered when choosing the method. The estimation performance depends strongly on the tuning of the filters and the accuracy of the models that are used. Therefore, a method for estimating the process noise covariance matrix is proposed. Moreover, sampling methods are analysed and a low-complexity analytical solution for the continuous-time update in the Kalman filter, that does not involve oversampling, is proposed.  The thesis also investigates two types of control problems. First, the norm-optimal iterative learning control (ILC) algorithm for linear systems is extended to an estimation-based norm-optimal ILC algorithm where the controlled variables are not directly available as measurements. The algorithm can also be applied to non-linear systems. The objective function in the optimisation problem is modified to incorporate not only the mean value of the estimated variable, but also information about the uncertainty of the estimate. Second, H∞ controllers are designed and analysed on a linear four-mass flexible joint model. It is shown that the control performance can be increased, without adding new measurements, compared to previous controllers. Measuring the end-effector acceleration increases the control performance even more. A non-linear model has to be used to describe the behaviour of a real flexible joint. An H∞-synthesis method for control of a flexible joint, with non-linear spring characteristic, is therefore proposed. / En av de viktigaste uppgifterna för en industrirobot är att förflytta verktyget i en fördefinierad bana med en specificerad hastighet och acceleration. Exempel på användningsområden för en industrirobot är bland annat bågsvetsning eller limning. För dessa typer av applikationer är det viktigt att banföljningsfelet är extremt litet, men även hastighetsprofilen måste följas så att det till exempel inte appliceras för mycket eller för lite lim. Andra användningsområden kan vara punktsvetsning av bilkarosser och paketering av olika varor. För dess applikationer är banföljningen inte det viktiga, istället kan till exempel en tidsoptimal banföljning krävas eller att svängningarna vid en inbromsning minimeras. Oberoende av applikationen är regulatorn en avgörande del av robotsystemet. Den vanligaste regulatorkonfigurationen använder bara mätningar av motorernas vinkelpositioner och -hastigheter, istället för positionen och hastigheten för verktyget, som är det man egentligen vill styra.  En del av utvecklingsarbetet för nya generationers robotar är att reducera kostnaden men samtidigt förbättra prestandan. Ett sätt att minska kostnaden kan till exempel vara att minska dimensionerna på länkarna eller köpa in billigare växellådor. Den här utvecklingen av kostnadsoptimerade robotar har infört oönskade flexibiliteter i leder och länkar. Det är därför inte längre möjligt att få den önskade prestandan och robustheten genom att bara mäta motorernas vinkelpositioner och -hastigheter. Istället krävs det omfattande matematiska modeller som beskriver dessa oönskade flexibiliteter. Dessa modeller kräver mycket arbete att dels ta fram men även för att identifiera parametrarna. Det finns automatiska metoder för att beräkna modellparametrarna men oftast krävs det en manuell justering för att få bra prestanda.  Den här avhandlingen undersöker möjligheterna att beräkna verktygspositionen med hjälp av bayesianska metoder för tillståndsskattning. De bayesianska skattningsmetoderna beräknar tillstånden för ett system iterativt. Med hjälp av en matematisk modell över systemet predikteras vad tillståndet ska vara vid nästa tidpunkt. Efter att mätningar av systemet vid den nya tidpunkten har genomförts justeras skattningen med hjälp av dessa mätningar. De metoder som har använts i avhandlingen är det så kallade extended Kalman filtret samt partikelfiltret.  Informationen på armsidan av växellådan ges av en accelerometer som är monterad på verktyget. Med hjälp av accelerationen för verktyget och motorernas vinkelpositioner kan en skattning av verktygspositionen beräknas. I en simuleringsstudie för en realistisk vek robot har det visats att skattningsprestandan ligger nära den teoretiska undre gränsen, känd som Raooch mätstörningar som påverkar roboten. För att underlätta trimningen så har en metod för att skatta processbrusets kovariansmatris föreslagits. En annan viktig del som påverkar prestandan är modellerna som används i filtren. Modellerna för en industrirobot är vanligtvis framtagna i kontinuerlig tid medan filtren använder modeller i diskret tid. För att minska felen som uppkommer då de tidskontinuerliga modellerna överförs till diskret tid har olika samplingsmetoder studerats. Vanligtvis används enkla metoder för att diskretisera vilket innebär problem med prestanda och stabilitet. För att hantera dessa problem införs översampling vilket innebär att tidsuppdateringen sker med en mycket kortare sampeltid än vad mätuppdateringen gör. För att undvika översampling kan det motsvarande tidskontinuerliga filtret användas för att prediktera tillstånden vid nästa diskreta tidpunkt. En analytisk lösning med låg beräkningskomplexitet till detta problem har föreslagits.  Vidare innehåller avhandlingen två typer av reglerproblem relaterade till industrirobotar. För det första har den så kallade norm-optimala iterative learning control styrlagen utökats till att hantera fallet då en skattning av den önskade reglerstorheten används istället för en mätning. Med hjälp av skattningen av systemets tillståndsvektor kan metoden nu även användas till olinjära system vilket inte är fallet med standardformuleringen. Den föreslagna metoden utökar målfunktionen i optimeringsproblemet till att innehålla inte bara väntevärdet av den skattade reglerstorheten utan även skattningsfelets kovariansmatris. Det innebär att om skattningsfelet är stort vid en viss tidpunkt ska den skattade reglerstorheten vid den tidpunkten inte påverka resultatet mycket eftersom det finns en stor osäkerhet i var den sanna reglerstorheten befinner sig.  För det andra har design och analys av H∞-regulatorer för en linjär modell av en vek robotled, som beskrivs med fyra massor, genomförts. Det visar sig att reglerprestandan kan förbättras, utan att lägga till fler mätningar än motorns vinkelposition, jämfört med tidigare utvärderade regulatorer. Genom att mäta verktygets acceleration kan prestandan förbättras ännu mer. Modellen över leden är i själva verket olinjär. För att hantera detta har en H∞-syntesmetod föreslagits som kan hantera olinjäriteten i modellen. / Vinnova Excellence Center LINK-SIC

A água é uma matéria-prima estratégica na indústria química, petroquímica e de alimentos, sendo utilizada em diversas etapas dos processos. Devido à importância econômica e ao impacto ambiental resultante do consumo da água na indústria, o uso racional da água vem sendo amplamente enfatizado. Nas últimas décadas, diversas metodologias foram propostas para a minimização do consumo de água e emissão de efluentes líquidos na indústria, baseadas em conceitos de integração mássica como reuso, reciclo e regeneração das correntes que conectam os processos. As metodologias propostas na literatura para a integração mássica dividem-se em dois grandes grupos: o primeiro utiliza a abordagem termodinâmica-heurística-evolutiva, enquanto que o segundo faz uso da programação matemática e otimização. Ambas as metodologias apresentam dificuldades para tratar do caso de múltiplos poluentes, sendo que na primeira, é necessário uma série de aproximações para tratar deste tipo de problema e na segunda estão presentes limitações relativas aos métodos de otimização. Além disso, sistemas integrados massicamente podem apresentar dificuldades operacionais devido ao maior número de reciclos e reusos de correntes. O presente trabalho através do uso de ferramentas de programação matemática e, baseado no problema padrão de síntese de redes proposto por Fontana (2002), propõe métodos para a redução da complexidade do problema de síntese e obtenção de redes integradas massicamente, considerando aspectos econômicos e operacionais. Inicialmente foi proposta a eliminação de variáveis inteiras do problema de otimização, transformando o problema de programação nãolinear inteira-mista (MINLP) em um problema de programação não-linear (NLP). Em seguida, foi proposta uma metodologia para a redução da complexidade do problema baseada na utilização das restrições de igualdade, reduzindo o número de variáveis de decisão. As formulações propostas apresentaram bons resultados, diminuindo consideravelmente o tempo computacional em relação ao problema MINLP original. Baseada nas metodologias desenvolvidas para redução da complexidade do problema, foi implementado ainda um gerador de estimativas iniciais viáveis utilizando números aleatórios. Também foram propostas três implementações para a síntese de redes com considerações econômicas e operacionais, resolvendo o problema de otimização em dois níveis: um externo para avaliação de critérios econômicos e um interno de onde é obtido o índice operacional. Das três formulações com considerações operacionais propostas, duas apresentaram redes com melhor desempenho operacional quando comparadas com os problemas somente com considerações econômicas. Tais resultados foram avaliados através de simulações dinâmicas com modelos linearizados, sendo observado, nestas duas implementações, aumento da robustez do sistema (redução da ação de controle sobre as variáveis manipuladas) frente a distúrbios positivos nas cargas de poluentes e mudanças nos set-points das variáveis controladas. / Water is an important raw material in the chemical, petrochemical, and food industry, being used in several stages of a process. Based on the economic importance of the water and the environmental impact through its employment in the industry, its rational consume has been receiving special attention. In the past decades, several methodologies have been proposed for the minimization of industrial water consume and wastewater emissions based on concepts of mass integration such as reuse, recycle, and regeneration of process streams. The methodologies proposed in the literature for the mass integration can be classified in two main groups: the first uses the themodynamic-heuristic-evolutive approach while the second uses mathematical programming and optimization. Both methodologies show difficulties to deal with the case of multiple pollutants: on the first one it is necessary to make some assumptions to solve this problem while the second, limitations relative to optimization methods are present. Another problem is the operational limitations due to streams recycle and reuse, that integrated processes can present. The present study, through the use of mathematical programming tools, based on the synthesis problem proposed by Fontana (2002), proposes methods to reduce the problem complexity and also for the synthesis of mass integration networks regarding economical and operational aspects. Initially it was proposed the elimination of the integer variables from the optimization problem, changing the problem from mixed-integer nonlinear programming (MINLP) to nonlinear programming (NLP). In the following, a methodology for the reduction of the problems complexity, based on the equality constraints, was applied to reduce the number of decision variables. The proposed formulations showed good results, reducing the computational time comparing to the MINLP problem. Based on the proposed methodologies for the reduction of the problem complexity described above, it was implemented a feasible initial guess generator based on random numbers. It was also proposed three formulations for the network synthesis with economic and operational considerations, solving the synthesis problem in two levels: an external level for the evaluation of economical criteria and an internal level where the operational index is obtained. Two out of the three implementations presented networks with better operational results comparing with problems only with economical aspects. These results were evaluated by dynamic simulations with linearized models and it was observed an increase in the systems robustness (reduction of the control action over the manipulated variables) when the system was subjected to positive disturbances in the pollutants loads and set-points changes.

Dans cette thèse on s'intéresse à un modèle mathématique décrivant l'adaptation du développement des populations de moustiques face à l'usage intensif des insecticides durant la nuit (moustiquaires imprégnées, répulsifs en spray, répulsifs avec diffuseur électrique, ...).Le modèle proposé dans cette thèse est structuré en âge et dépend du temps/moment où le moustique pique pour prendre son repas. Ceci nous conduità des modèles du type ultra parabolique. Le terme de renouvellement de lapopulation de moustiques est non-local, comme pour tous les problèmes démographiques, mais comporte ici un noyau qui permet à la nouvelle générationd'adapter son temps de piqure (repas). Ceci est dû à la sélection de certainsmoustiques qui piquent plus tôt ou plus tard que les autres moustiques, suite àla pression imposée par l'usage intensif des pesticides à l'intérieur des habitats et en particulier durant la nuit. Les conditions aux bords par rapport au moment de piqure (repas) seront périodiques car selon les espèces, les moustiques prennent toujours leurs repas au même moment de la journée.Les principaux résultats peuvent être classés dans 4 parties.Dans la première partie on présente un modèle structuré en âge décrivant laplasticité du moustique dans un environnement non contrôlé. On montre quele problème est bien posé via la théorie des semi-groupes. Le comportementasymptotique est décrit grâce à l'étude du spectre de l'opérateur A générateurdu C0 semi-groupe. On prouve également l'existence ou la non existence dessolutions stationnaires (sous certaines hypothèses).Dans la deuxième partie on s'intéresse à un problème de contrôle optimalde la population de moustiques. Le contrôle correspond à la proportion demoustiques éliminée et dépend du temps, de l'âge des moustiques et du tempsoù le moustique pique pour se nourrir. On démontre d’abord l’existence desolutions grâce à un argument de point fixe puis on établit des résultats decomparaisons pour notre problème. On établit ensuite l'existence d'un contrôleoptimal puis on dérive le système d'optimalité.Dans la troisième partie on s'intéresse à la question de contrôlabilité exacte locale pour le problème décrivant la capacité des moustiques à adapter leurdynamique face à l'usage intensif des insecticides. On établit une nouvelleinégalité de type Carleman pour le modèle structuré en âge avec diffusionet une condition au bord de renouvellement non-locale et des conditions auxbords périodiques par rapport au temps de piqure des moustiques.Dans la quatrième partie on s'intéresse au comportement en temps longd'un modèle non linéaire décrivant l'adaptation de la population des moustiques à l'usage intensif des insecticides. Quand le contrôle est petit (usage limité des insecticides) alors la population mature de moustiques devient grandeavec le temps et quand le contrôle est grand (usage intensif des insecticides)la population mature de moustiques devient petite avec le temps. Dans le casintermédiaire on obtient un modèle avec retard en temps pour la populationmature de moustiques qui peut être gouvernée par une sur-équation et unesous-équation. Finalement on montre que la sous-équation admet des ondesvoyageuses et la population mature de moustiques sera donc comprise entreces ondes voyageuses et les sur-solutions. / This dissertation is concerned with an age structured problem modelling mosquito plasticity. The main results can be divided into four parts.The first part presents an age structured problem modelling mosquito plasticity in a natural environment. We first investigate the analytical asymptotic solution through studying the spectrum of an operator A which is the infinitesimal generator of a C0-semigroup. Additionally, we get the existence and nonexistence of nonnegative steady solutions under some conditions.In the second part, we study the optimal control of an age structured problem. Firstly, we prove the existence of solutions and the comparison principle for a generalized system. Then, we prove the existence of the optimal control for the best harvesting. Finally, we establish necessary optimality conditions.In the third part, we investigate the local exact controllability of an age structured problem modelling the ability of malaria vectors to shift their biting time to avoid the stressful environmental conditions generated by the use of indoor residual spraying (IRs) and insecticide-treated nets (ITNs). We establish a new Carleman's inequality for our age diffusive model with non local birth processus and periodic biting-time boundary conditions.In the fourth part, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. Firstly, we prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. Finally, we prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation.