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 thesis applies digital repetitive control to a single-phase DC-to-AC converter, with some proposed designs to improve stability and enhance performance of the converter under various load variations. A practical DC-to-AC converter is required to convert DC power to stable AC power with low harmonic distortion when attached to various linear or nonlinear loads. This thesis combines repetitive control with feedback dithering modulation and optimal state feedback to control the converter. The repetitive control is responsible for regulating output power and eliminating harmonics, while the feedback dithering modulation for switching the power transistors with reduced switching noise and the state feedback for stabilizing the converter under various load variations. The presented control and modulation schemes of the power converter are implemented on an FPGA (Field Programmable Gate Array). The experiments confirm the excellent performance and robustness of the converter, indicating a total harmonic distortion of less than 0.5% for the converter when attached to various linear or nonlinear loads.

The main motivation of this study is the design of a control strategy for semi-active vehicle suspension systems to improve ride comfort for road vehicles. In order to achieve this objective, firstly the damping characteristics of Magnetorheological dampers will be reviewed. Then an appropriate semi-active control strategy manipulating the inputs of the dampers to create suitable damping forces will be designed. Linear Quadratic Regulator (LQR) control strategy is the primary focus area on semi-active control throughout this study. Further, skyhook controllers are examined and compared with optimal LQR controllers. The semi-active controller is tuned using a linearized full (4 wheel) vehicle ride model with seven degrees of freedom. Some selected simulations are carried out by using a nonlinear model to tune LQR controller in an effort to optimize bounce, pitch, and roll motion of the vehicle. Time domain simulations and frequency response analysis are used to justify the effectiveness of the proposed LQR control strategy.

Made available in DSpace on 2014-06-11T19:27:14Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-05Bitstream added on 2014-06-13T20:55:57Z : No. of bitstreams: 1 alegre_dm_me_ilha.pdf: 954109 bytes, checksum: 3e5701b2e8747b004da37b4ca12ca513 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O presente trabalho apresenta a metodologia dos observadores de estado para a detecção e localização de falhas em sistemas contendo elementos finitos de placas de Kirchoff. Tal metodologia consiste na montagem de um banco de observadores de estado, o qual é capaz de detectar falhas presentes no sistema, além de localizar o componente danificado e a porcentagem de falha. As matrizes de ganho dos observadores de estado foram determinadas por dois métodos distintos: o método Regulador Quadrático Linear e das Desigualdades Matriciais Lineares. Nesse trabalho, foi utilizada uma placa plana fina montada sobre um sistema de suspensão similar ao de uma plataforma veicular, representando um veiculo simplificadamente. A modelagem da plataforma utilizada foi realizada mediante a utilização do método dos elementos finitos, considerando-se diferentes números de elemento no modelo. O tipo do elemento finito utilizado foi o elemento de placa de Kirchoff. Adicionalmente foi analisada a influência de elementos de controle junto à suspensão da plataforma no movimento da mesma. O modelo considerado, juntamente com os programas computacionais desenvolvidos, foram utilizados para a simulação do movimento da plataforma. Na literatura, normalmente são apresentadas simulações para o movimento de apenas ¼ do veículo, neste trabalho, no entanto, os programas desenvolvidos podem simular o movimento do veiculo inteiro. Foram realizadas simulações computacionais para o movimento da plataforma com a finalidade de se detectar e localizar falhas introduzidas nos elementos da suspensão e também foram realizados testes experimentais, com os mesmos fins. Mediante tais testes (teóricos e experimentais) verificou-se a eficácia da metodologia desenvolvida e a sua principal limitação: o número de elementos finitos considerado no modelo relacionado com o número de medidas efetuadas e a observabilidade do sistema. / This work presents the state observers methodology for the detection and location of faults in systems containing finite elements of plate of Kirchoff. This methodology consists in the assembly of a bank of state observers, which is capable of detecting faults in the system, and also to locate the damaged component and the percentage of failure. The gain matrices of the state observers were determined by two different methods, these are the method Linear Quadratic Regulator and the Linear Matrix Inequalities. In this work was considered a thin plate mounted on a suspension system that is similar to a vehicle platform, representing a vehicle in a simplified way. The modeling of the platform used was performed by using the finite element method, considering different numbers of element in the model. The finite element used was the Kirchoff’s plate element. It was also studied the influence of elements of control, together with the suspension of the platform, into its movement. The model considered, together with the developed computational programs, were used to simulate the movement of the platform. In the literature, usually are presented simulations for the movement of only ¼ of the vehicle, in this work, the developed programs can simulate the movement of the entire vehicle. It was realized computational simulations for the movement of the platform in order to detect and locate faults introduced in the elements of the suspension and experimental tests were also conducted with the same purpose. Through such tests (theoretical and experimental) it was verified the effectiveness of the developed methodology and its major limitation: the number of finite elements considered in the model related with the number of outputs and the observability of the system.

O estudo da estabilidade e desempenho em sistemas de controle é um tópico relevante na teoria de sistemas. Quando são assumidas incertezas no modelo da planta, existe uma maior dificuldade para garantir um nível de desempenho adequado do sistema dinâmico e a estabilidade pode ser comprometida. Neste trabalho são utilizados um regulador linear quadrático robusto e um filtro de Kalman robusto combinados em uma única formulação para tratar de sistemas dinâmicos incertos em tempo real. O caso de estudo selecionado é o pêndulo invertido. Seus principais desafios de controle encontrados na literatura: estabilização, seguimento e levantamento-captura, serão considerados. Os algoritmos utilizados são motivados pelo fato de que problemas estocásticos podem ser resolvidos por meio de argumentos determinísticos, baseados nos conceitos de função penalidade e mínimos quadrados regularizados. Desta forma, é possível a obtenção do melhor desempenho em contrapartida à máxima influência de incerteza admissível. A análise de desempenho do controlador robusto é realizada por meio de ensaios práticos incluindo incertezas na planta, ruído nos sensores e distúrbios no sinal de controle do pêndulo. / The study of stability and performance in control systems is a relevant topic in systems theory. When uncertainties are considered in the model of the plant, there is a greater difficulty in ensuring an appropriate performance level of the dynamic system, plus, the stability could be compromised as well. In this dissertation a robust linear quadratic regulator and a robust Kalman filter are used in a unified manner to deal with uncertain dynamic systems in real time. The selected case study is the inverted pendulum. Its main control challenges found in the literature will be considered: stabilization, tracking and catching swing-up. The used algorithms are motivated by the fact that stochastic problems can be solved through deterministic arguments based on the concepts of penalty function and regularized least-squares. Thus, it is possible to obtain an optimal performance for the maximum acceptable uncertainty. The performance analysis of the robust control is carried out by practical experiments including uncertainties in the plant, noise in the sensors and disturbance in the pendulum control signal.

Orientador: Prof. Dr. Alfeu Joãzinho Sguarezi Filho / Coorientador: Prof. Dr. José Alberto Torrico Altuna / Tese (doutorado) - Universidade Federal do ABC. Programa de Pós-Graduação em Energia, Santo André, 2018.

This thesis presents the design, construction, and control of a two-wheeled inverted pendulum (TWIP) robotic walker prototype for assisting mobility-impaired users with balance and fall prevention. A conceptual model of the robotic walker is developed and used to illustrate the purpose of this study. A linearized mathematical model of the two-wheeled system is derived using Newtonian mechanics. A control strategy consisting of a decoupled LQR controller and three state variable controllers is developed to stabilize the platform and regulate its behavior with robust disturbance rejection performance. Simulation results reveal that the LQR controller is capable of stabilizing the platform and rejecting external disturbances while the state variable controllers simultaneously regulate the system’s position with smooth and minimum jerk control. A prototype for the two-wheeled system is fabricated and assembled followed by the implementation and tuning of the control algorithms responsible for stabilizing the prototype and regulating its position with optimal performance. Several experiments are conducted, confirming the ability of the decoupled LQR controller to robustly balance the platform while the state variable controllers regulate the platform’s position with smooth and minimum jerk control.

This diploma thesis deals with control the Ball on the wheel system in real time. In the first part there is made mathematical model of system, which is described by state equations. Then the laboratory model of Ball on the wheel system and its parts are described. The laboratory model consists servo drive, servo amplifier, microcontroller with real-time operating system and distance sensors. In the practical part of the thesis there is designed control of system by linear quadratic regulator and its implementation with control peripherals into microcontroller with real-time operating system. Ball on the wheel system is controlled by microcontroller, which sends a commands to servo amplifier by CAN bus.

[pt] Quadricópteros vêm sendo o objeto de estudo de inúmeras pesquisas ao redor do mundo. Diversas técnicas de controle já foram desenvolvidas, cada uma com seus prós e contras, objetivando o aprimoramento do desempenho destes veículos aéreos na consecução de tarefas específicas. Este trabalho foca na comparação de características de desempenho de técnicas de controle aplicadas a quadricópteros. Este trabalho mostra o projeto de controles modernos aplicados à quadricópteros apresentando tais técnicas, iniciando com a realimentação de estados com polos dominantes, passando pelo controle Linear Quadratic Regulator (LQR). Por sua vez, visando otimizar o desempenho das técnicas de controle aqui estudadas, foram aplicadas técnicas de inteligência computacional para resolver um problema de otimização do LQR e para auxiliar no controle de forças dos rotores. Apresenta-se o projeto de um controle PID, que será usado como referência para as demais técnicas analisadas. O controle por realimentação de estados citado anteriormente obteve bons resultados. O tempo de assentamento foi o menor para o eixo Z, overshoots e o erro em regime permanente, os menores para os eixos X e Y. O controlador Fuzzy conseguiu fazer seu papel auxiliando a movimentação do quadricóptero. O GA otimizou o tempo de assentamento do LQR. Esse controle conseguiu alcançar os menores tempos de assentamento para os eixos X e Y, sendo mais rápido que a configuração original do LQR, escolhida por heurísticas. Com esse trabalho foi possível notar que os controladores modernos, realimentação de estado e LQR, tem um desempenho melhor que o controle PID de referência. / [en] Quadcopters are researched all over the world. A lot of techniques had been developed and many others a blistering, each one with their pros and cons. The focus of this work is the comparison between the performance of some techniques most used in quadcopters, qualifying these techniques. This work shows the designing process of a quadcopter controllers, starting with state feedback with dominant poles and going to the Linear Quadratic Regulator controller. Focusing on optimizing the performance of those control strategies, computational techniques were used to solve an LQR optimization problem and to help choose the best inputs for the rotors. This work presents a PID controller that will be used as a reference for comparison. The state feedback controller with the Fuzzy position control performed very well, being the fastest one to settle on the Z axis, having the least overshoots and the lowest steadystate errors for the X and Y axes. The Fuzzy controller did what was supposed to do, smoothing and enabling a precise movement for the quadcopter. The GA also did what was supposed to do and improved the settling time for the LQR controller. It showed that it was a nice way to tune the Q and R matrixes, allowing the controller to be the fastest one to settle in the X and Y axes. As a result of this work, the modern control techniques, state feedback and LQR, performance better than the classic PID controller, used as reference.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-08-14T18:28:45Z No. of bitstreams: 1 FabioSilva.pdf: 1098466 bytes, checksum: a72dcced91748fe6c54f3cab86c19849 (MD5) / Made available in DSpace on 2017-08-14T18:28:45Z (GMT). No. of bitstreams: 1 FabioSilva.pdf: 1098466 bytes, checksum: a72dcced91748fe6c54f3cab86c19849 (MD5) Previous issue date: 2008-06-20 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPQ) / Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e Tecnológico do Maranhão (FAPEMA) / 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.