Este trabalho aborda o problema de controle ótimo por modos deslizantes via função penalidade para sistemas de tempo discreto. Para resolver este problema será desenvolvido uma estrutura matricial alternativa baseada no problema de mínimos quadrados ponderados e funções penalidade. A partir desta nova formulação é possível obter a lei de controle ótimo por modos deslizantes, as equações de Riccati e a matriz do ganho de realimentação através desta estrutura matricial alternativa. A motivação para propormos essa nova abordagem é mostrar que é possível obter uma solução alternativa para o problema clássico de controle ótimo por modos deslizantes. / This work introduces a penalty function approach to deal with the optimal sliding mode control problem for discrete-time systems. To solve this problem an alternative array structure based on the problem of weighted least squares penalty function will be developed. Using this alternative matrix structure, the optimal sliding mode control law of, the matrix Riccati equations and feedback gain were obtained. The motivation of this new approach is to show that it is possible to obtain an alternative solution to the classic problem of optimal sliding mode control.

Orientador: Prof. Dr. André Fenili / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia Mecânica, Santo André, 2018.

CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / A teoria dos jogos à um ramo da matemÃtica dedicado ao estudo de situaÃÃes que surgem quando mÃltiplos agentes de decisÃo buscam atingir seus objetivos individuais, possivelmente conflitantes entre si. Em sua formulaÃÃo dinÃmica linear quadrÃtica (LQ), as soluÃÃes de equilÃbrio de Nash dos jogadores podem ser obtidas em termos das equaÃÃes algÃbricas de Riccati acopladas, que, a depender do mÃtodo numÃrico utilizado para seu cÃlculo, podem gerar resultados insatisfatÃrios sob o ponto de vista da estabilidade e precisÃo numÃrica. Neste sentido, esta dissertaÃÃo propÃe um novo algoritmo para uma soluÃÃo alternativa das equaÃÃes algÃbricas de Riccati acopladas associadas aos jogos dinÃmicos (LQ), com estrutura de informaÃÃo em malha aberta, utilizando, para isso, conceitos da teoria da dualidade e otimizaÃÃo estÃtica convexa. Em adiÃÃo, obtÃm-se uma nova metodologia para a sÃntese de uma famÃlia de controladores Ãtimos. A teoria dos jogos tambÃm revela um enorme potencial de aplicaÃÃo em problemas de controle multiobjetivo, no qual està incluÃdo o controle Hinf, que pode ser formulado como um jogo dinÃmico de soma-zero. Considerando essa formulaÃÃo, as novas metodologias propostas neste trabalho sÃo estendidas aos problemas de controle Hinf com rejeiÃÃo de perturbaÃÃo, gerando resultados com melhores propriedades de desempenho e estabilidade que os obtidos via equaÃÃo algÃbrica de Riccati modificada. Por fim, atravÃs de exemplos numÃricos e simulaÃÃes computacionais, as novas metodologias sÃo confrontadas com as metodologias tradicionais, evidenciando-se os aspectos mais relevantes de cada abordagem.

Made available in DSpace on 2016-08-17T14:53:01Z (GMT). No. of bitstreams: 1 Fabio Nogueira da Silva.pdf: 1098466 bytes, checksum: a72dcced91748fe6c54f3cab86c19849 (MD5) Previous issue date: 2008-06-20 / FUNDAÇÃO DE AMPARO À PESQUISA E AO DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO DO MARANHÃO / We present in this work the results about two neural networks methods to solve the algebraic Riccati(ARE), what are used in many applications, mainly in the Linear Quadratic Regulator (LQR), H2 and H1 controls. First is showed the real symmetric form of the ARE and two methods based on neural computation. One feedforward neural network (FNN), that de¯nes an error as function of the ARE and a recurrent neural network (RNN), which converts a constrain optimization problem, restricted to the state space model, into an unconstrained convex optimization problem de¯ning an energy as function of the ARE and Cholesky factor. A proposal to chose the learning parameters of the RNN used to solve the ARE, by making a surface of the parameters variations, thus we can tune the neural network for a better performance. Computational experiments related with the plant matrices perturbations of the tested systems in order to perform an analysis of the behavior of the presented methodologies, that are based on homotopies methods, where we chose a good initial condition and compare the results to the Schur method. Two 6th order systems were used, a Doubly Fed Induction Generator(DFIG) and an aircraft plant. The results showed the RNN a good alternative compared with the FNN and Schur methods. / Apresenta-se nesta dissertação os resultados a respeito de dois métodos neuronais para a resolução da equação algébrica de Riccati(EAR), que tem varias aplicações, sendo principalmente usada pelos Regulador Linear Quadrático(LQR), controle H2 e controle H1. É apresentado a EAR real e simétrica e dois métodos baseados em uma rede neuronal direta (RND) que tem a função de erro associada a EAR e uma rede neuronal recorrente (RNR) que converte um problema de otimização restrita ao modelo de espaço de estados em outro de otimização convexa em função da EAR e do fator de Cholesky de modo a usufruir das propriedades de convexidade e condições de otimalidade. Uma proposta para a escolha dos parâmetros da RNR usada para solucionar a EAR por meio da geração de superfícies com a variação paramétrica da RNR, podendo assim melhor sintonizar a rede neuronal para um melhor desempenho. Experimentos computacionais relacionados a perturbações nos sistemas foram realizados para analisar o comportamento das metodologias apresentadas, tendo como base o princípio dos métodos homotópicos, com uma boa condição inicial, a partir de uma ponto de operação estável e comparamos os resultados com o método de Schur. Foram usadas as plantas de dois sistemas: uma representando a dinâmica de uma aeronave e outra de um motor de indução eólico duplamente alimentado(DFIG), ambos sistemas de 6a ordem. Os resultados mostram que a RNR é uma boa alternativa se comparado com a RND e com o método de Schur.

This thesis addresses several aerospace guidance and decision making problems using both no cooperative and cooperative game theoretical solution concepts in the differential games framework. In the first part of the thesis, state dependent Riccati equation (SDRE) method has been extended to a zero-sum nonlinear differential games setting. This framework is used to study problems of intercepting a manoeuvring target, with and without terminal impact angle constraints, in the zero-sum differential games theory perspective. The guidance laws derived according to the proposed method are in closed from and online implementable. In the second part of the thesis, cooperative game theoretic concepts are applied to make a group of unmanned aerial vehicles (UAV) achieve rendezvous, in a given finite time horizon. An algorithm has been proposed that enables the UAVs to realize Nash bargaining solution. In this context, criteria for time consistency of a cooperative solution of nonzero-sum linear quadratic differential games have been studied. The problems where the UAVs try to achieve rendezvous by implementing cooperative game theoretic strategies, based on local information structure only, is also addressed.

We present a new method for the computation of low rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low rank property of the right hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Hence, it provides a possibility to tackle large scale control problems. Besides the efficient solution of the matrix equation itself, a thorough integration of the method into several control algorithms can improve their performance to a high degree. This is demonstrated for algorithms for model reduction and optimal control. Furthermore, we propose a heuristic for determining a set of suboptimal ADI shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters.

Consider a time-dependent optimal control problem, where the state evolution is described by an initial value problem. There are a variety of numerical methods to solve these problems. The so-called indirect approach is considered detailed in this thesis. The indirect methods solve decoupled boundary value problems resulting from the necessary conditions for the optimal control problem. The so-called Pantoja method describes a computationally efficient stage-wise construction of the Newton direction for the discrete-time optimal control problem. There are many relationships between multiple shooting techniques and Pantoja method, which are investigated in this thesis. In this context, the equivalence of Pantoja method and multiple shooting method of Riccati type is shown. Moreover, Pantoja method is extended to the case where the state equations are discretised using one of implicit numerical methods. Furthermore, the concept of symplecticness and Hamiltonian systems is introduced. In this regard, a suitable numerical method is presented, which can be applied to unconstrained optimal control problems. It is proved that this method is a symplectic one. The iterative solution of optimal control problems in ordinary differential equations by Pantoja or Riccati equivalent methods leads to a succession of triple sweeps through the discretised time interval. The second (adjoint) sweep relies on information from the first (original) sweep, and the third (final) sweep depends on both of them. Typically, the steps on the adjoint sweep involve more operations and require more storage than the other two. The key difficulty is given by the enormous amount of memory required for the implementation of these methods if all states throughout forward and adjoint sweeps are stored. One of goals of this thesis is to present checkpointing techniques for memory reduced implementation of these methods. For this purpose, the well known aspect of checkpointing has to be extended to a `nested checkpointing` for multiple transversals. The proposed nested reversal schedules drastically reduce the required spatial complexity. The schedules are designed to minimise the overall execution time given a certain total amount of storage for the checkpoints. The proposed scheduling schemes are applied to the memory reduced implementation of the optimal control problem of laser surface hardening and other optimal control problems. / Es wird ein Problem der optimalen Steuerung betrachtet. Die dazugehoerigen Zustandsgleichungen sind mit einer Anfangswertaufgabe definiert. Es existieren zahlreiche numerische Methoden, um Probleme der optimalen Steuerung zu loesen. Der so genannte indirekte Ansatz wird in diesen Thesen detailliert betrachtet. Die indirekten Methoden loesen das aus den Notwendigkeitsbedingungen resultierende Randwertproblem. Das so genannte Pantoja Verfahren beschreibt eine zeiteffiziente schrittweise Berechnung der Newton Richtung fuer diskrete Probleme der optimalen Steuerung. Es gibt mehrere Beziehungen zwischen den unterschiedlichen Mehrzielmethoden und dem Pantoja Verfahren, die in diesen Thesen detailliert zu untersuchen sind. In diesem Zusammenhang wird die aequivalence zwischen dem Pantoja Verfahren und der Mehrzielmethode vom Riccati Typ gezeigt. Ausserdem wird das herkoemlige Pantoja Verfahren dahingehend erweitert, dass die Zustandsgleichungen mit Hilfe einer impliziten numerischen Methode diskretisiert sind. Weiterhin wird das Symplektische Konzept eingefuehrt. In diesem Zusammenhang wird eine geeignete numerische Methode praesentiert, die fuer ein unrestringiertes Problem der optimalen Steuerung angewendet werden kann. In diesen Thesen wird bewiesen, dass diese Methode symplectisch ist. Das iterative Loesen eines Problems der optimalen Steuerung in gewoenlichen Differentialgleichungen mit Hilfe von Pantoja oder Riccati aequivalenten Verfahren fuehrt auf eine Aufeinanderfolge der Durchlaeufetripeln in einem diskretisierten Zeitintervall. Der zweite (adjungierte) Lauf haengt von der Information des ersten (primalen) Laufes, und der dritte (finale) Lauf haeng von den beiden vorherigen ab. Ueblicherweise beinhalten Schritte und Zustaende des adjungierten Laufes wesentlich mehr Operationen und benoetigen auch wesentlich mehr Speicherplatzkapazitaet als Schritte und Zustaende der anderen zwei Durchlaeufe. Das Grundproblem besteht in einer enormen Speicherplatzkapazitaet, die fuer die Implementierung dieser Methoden benutzt wird, falls alle Zustaende des primalen und des adjungierten Durchlaufes zu speichern sind. Ein Ziel dieser Thesen besteht darin, Checkpointing Strategien zu praesentieren, um diese Methoden speichereffizient zu implementieren. Diese geschachtelten Umkehrschemata sind so konstruiert, dass fuer einen gegebenen Speicherplatz die gesamte Laufzeit zur Abarbeitung des Umkehrschemas minimiert wird. Die aufgestellten Umkehrschemata wurden fuer eine speichereffiziente Implementierung von Problemen der optimalen Steuerung angewendet. Insbesondere betrifft dies das Problem einer Oberflaechenabhaertung mit Laserbehandlung.

Le travail porte sur la problématique de l’observation des systèmes — la reconstruction de l’état complet d’un système dynamique à partir d'une mesure partielle de cet état. Nous considérons spécifiquement les systèmes non linéaires. Le filtre de Kalman étendu (EKF) est l’un des observateurs les plus utilisés à cette fin. Il souffre cependant d’une performance moindre lorsque l'état estimé n’est pas dans un voisinage de l'état réel. La convergence de l’observateur dans ce cas n’est pas prouvée. Nous proposons une solution à ce problème : l’EKF à grand gain adaptatif. La théorie de l’observabilité fait apparaître l’existence de représentations caractérisant les systèmes dit observables. C’est la forme normale d’observabilité. L’EKF à grand gain est une variante de l’EKF que l’on construit à base d’un paramètre scalaire. La convergence de cet observateur pour un système sous sa forme normale d’observabilité est démontrée pour toute erreur d’estimation initiale. Cependant, contrairement à l’EKF, cet algorithme est très sensible au bruit de mesure. Notre objectif est de combiner l’efficacit´e de l’EKF en termes de lissage du bruit, et la r´eactivit´e de l’EKF grand-gain face aux erreurs d’estimation. Afin de parvenir à ce résultat nous rendons adaptatif le paramètre central de la méthode grand gain. Ainsi est constitué l’EKF à grand gain adaptatif. Le processus d’adaptation doit être guidé par une mesure de la qualité de l’estimation. Nous proposons un tel indice et prouvons sa pertinence. Nous établissons une preuve de la convergence de notre observateur, puis nous l’illustrons à l’aide d’une série de simulations ainsi qu’une implémentation en temps réel dur. Enfin nous proposons des extensions au résultat initial : dans le cas de systèmes multi-sorties et dans le cas continu-discret. / The work concerns the “observability problem”—the reconstruction of a dynamic process’s full state from a partially measured state— for nonlinear dynamic systems. The Extended Kalman Filter (EKF) is a widely-used observer for such nonlinear systems. However it suffers from a lack of theoretical justifications and displays poor performance when the estimated state is far from the real state, e.g. due to large perturbations, a poor initial state estimate, etc. . . We propose a solution to these problems, the Adaptive High-Gain (EKF). Observability theory reveals the existence of special representations characterizing nonlinear systems having the observability property. Such representations are called observability normal forms. A EKF variant based on the usage of a single scalar parameter, combined with an observability normal form, leads to an observer, the High-Gain EKF, with improved performance when the estimated state is far from the actual state. Its convergence for any initial estimated state is proven. Unfortunately, and contrary to the EKF, this latter observer is very sensitive to measurement noise. Our observer combines the behaviors of the EKF and of the high-gain EKF. Our aim is to take advantage of both efficiency with respect to noise smoothing and reactivity to large estimation errors. In order to achieve this, the parameter that is the heart of the high-gain technique is made adaptive. Voila, the Adaptive High-Gain EKF. A measure of the quality of the estimation is needed in order to drive the adaptation. We propose such an index and prove the relevance of its usage. We provide a proof of convergence for the resulting observer, and the final algorithm is demonstrated via both simulations and a real-time implementation. Finally, extensions to multiple output and to continuous-discrete systems are given.

Dans cette thèse sont montrés des développements de l'approximation de la phase aléatoire (RPA) dans le contexte de théories à séparation de portée. On présente des travaux sur le formalisme de la RPA en général, et en particulier sur le formalisme "matrice diélectrique" qui est exploré de manière systématique. On montre un résumé d'un travail sur les équations RPA dans le contexte d'orbitales localisées, notamment des développements des orbitales virtuelles localisées que sont les "orbitales oscillantes projetées" (POO). Un programme a été écrit pour calculer des fonctions telles que le trou de d'échange, la fonction de réponse, etc... sur des grilles de l'espace réel (grilles parallélépipédiques ou de type "DFT"). On montre certaines de ces visualisations. Dans l'espace réel, on expose une adaptation de l'approximation du dénominateur effectif (EED), développée originellement dans l'espace réciproque en physique du solide. Également, les gradients analytiques des énergies de corrélation RPA dans le contexte de la séparation de portée sont dérivés. Le formalisme développé ici à l'aide d'un lagrangien permet une dérivation tout-en-un des termes courte- et longue-portée qui émergent dans les expressions du gradient, et qui montrent un parallèle intéressant. Des applications sont montrées, telles que des optimisations de géométries aux niveaux RSH-dRPA-I et RSH-SOSEX d'un ensemble de 16 petites molécules, ou encore le calcul et la visualisation des densités corrélées au niveau RSH-dRPA-I / In this thesis are shown developments in the random phase approximation (RPA) in the context of range-separated theories. We present advances in the formalism of the RPA in general, and particularly in the "dielectric matrix" formulation of RPA, which is explored in details. We show a summary of a work on the RPA equations with localized orbitals, especially developments of the virtual localized orbitals that are the "projected oscillatory orbitals" (POO). A program has been written to calculate functions such as the exchange hole, the response function, etc... on real space grid (parallelepipedic or of the "DFT" type) ; some of those visualizations are shown here. In the real space, we offer an adaptation of the effective energy denominator approximation (EED), originally developed in the reciprocal space in solid physics. The analytical gradients of the RPA correlation energies in the context of range separation has been derived. The formalism developed here with a Lagrangian allows an all-in-one derivation of the short- and long-range terms that emerge in the expressions of the gradient. These terms show interesting parallels. Geometry optimizations at the RSH-dRPA-I and RSH-SOSEX levels on a set of 16 molecules are shown, as well as calculations and visualizations of correlated densities at the RSH-dRPA-I level