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In this thesis we develop a rigorous mathematical framework for analyzing and approximating optimal sensor placement problems for distributed parameter systems and apply these results to PDE problems defined by the convection-diffusion equations. The mathematical problem is formulated as a distributed parameter optimal control problem with integral Riccati equations as constraints. In order to prove existence of the optimal sensor network and to construct a framework in which to develop rigorous numerical integration of the Riccati equations, we develop a theory based on Bochner integrable solutions of the Riccati equations. In particular, we focus on ℐ_p-valued continuous solutions of the Bochner integral Riccati equation. We give new results concerning the smoothing effect achieved by multiplying a general strongly continuous mapping by operators in ℐ_p. These smoothing results are essential to the proofs of the existence of Bochner integrable solutions of the Riccati integral equations. We also establish that multiplication of continuous ℐ_p-valued functions improves convergence properties of strongly continuous approximating mappings and specifically approximating C₀-semigroups. We develop a Galerkin type numerical scheme for approximating the solutions of the integral Riccati equation and prove convergence of the approximating solutions in the ℐ_p-norm. Numerical examples are given to illustrate the theory. / Ph. D.

Recently there has been growing interest to characterize and reduce uncertainty in stochastic dynamical systems. This drive arises out of need to manage uncertainty in complex, high dimensional physical systems. Traditional techniques of uncertainty quantification (UQ) use local linearization of dynamics and assumes Gaussian probability evolution. But several difficulties arise when these UQ models are applied to real world problems, which, generally are nonlinear in nature. Hence, to improve performance, robust algorithms, which can work efficiently in a nonlinear non-Gaussian setting are desired. The main focus of this dissertation is to develop UQ algorithms for nonlinear systems, where uncertainty evolves in a non-Gaussian manner. The algorithms developed are then applied to state estimation of real-world systems. The first part of the dissertation focuses on using polynomial chaos (PC) for uncertainty propagation, and then achieving the estimation task by the use of higher order moment updates and Bayes rule. The second part mainly deals with Frobenius-Perron (FP) operator theory, how it can be used to propagate uncertainty in dynamical systems, and then using it to estimate states by the use of Bayesian update. Finally, a method to represent the process noise in a stochastic dynamical system using a nite term Karhunen-Loeve (KL) expansion is proposed. The uncertainty in the resulting approximated system is propagated using FP operator. The performance of the PC based estimation algorithms were compared with extended Kalman filter (EKF) and unscented Kalman filter (UKF), and the FP operator based techniques were compared with particle filters, when applied to a duffing oscillator system and hypersonic reentry of a vehicle in the atmosphere of Mars. It was found that the accuracy of the PC based estimators is higher than EKF or UKF and the FP operator based estimators were computationally superior to the particle filtering algorithms.

The PhD presents a combined approach to improving individual car efficiency. An optimal observer, the Extended Kalman Filter, is used to create an efficiency model for the car. Particular attention was paid to handling the asynchronous and redundant nature of the measurement data. A low-cost sensor suite developed to measure data is described. This sensor suite was installed on multiple vehicles to good success. It employsan accelerometer, gps, fuel injector timer, and Vss input to measure all the data necessary to reconstruct the car's state. This observer and sensor suite can be used as the base for any study which requires car efficiency maps, allowing research to proceed without manufacturer supplied data. Once the efficiency map is found, it is then curve-fitted in order to reduce model complexity. The simplified model is then used as a basis for optimal control through Pontryagin's Maximum Principle. Real-world test results are given, both for efficiency mapping, and for optimal control. Detailed discussion of the observer and controller is presented, in order to ease understanding and save implementation time

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-20T15:14:06Z No. of bitstreams: 1 bernardosottomaiorperalva.pdf: 4608167 bytes, checksum: c63c1f7fc453965f36158791fb85964e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-04-24T16:49:03Z (GMT) No. of bitstreams: 1 bernardosottomaiorperalva.pdf: 4608167 bytes, checksum: c63c1f7fc453965f36158791fb85964e (MD5) / Made available in DSpace on 2017-04-24T16:49:04Z (GMT). No. of bitstreams: 1 bernardosottomaiorperalva.pdf: 4608167 bytes, checksum: c63c1f7fc453965f36158791fb85964e (MD5) Previous issue date: 2012-05-07 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta dissertação, são apresentados métodos para detecção de sinais e estimação de energia para calorimetria de altas energias aplicados no calorímetro hadrônico (TileCal) do ATLAS. A energia depositada em cada célula do calorímetro é adquirida por dois canais eletrônicos de leitura e é estimada, separadamente, através da reconstrução da amplitude do pulso digitalizado amostrado a cada 25 ns. Este trabalho explora a aplicabilidade de uma aproximação do Filtro Casado no ambiente do TileCal para detectar sinais e estimar sua amplitude. Além disso, este trabalho explora o impacto na detecção de eventos válidos e estimação da amplitude quando somam-se os sinais referentes à mesma célula antes da aplicação do filtro. O método proposto é comparado com o Filtro Ótimo atualmente utilizado pelo TileCal para reconstrução de energia. Os resultados para dados simulados e de colisão mostram que, para condições em que a linha de base do sinal de entrada pode ser considerada estacionária, a técnica proposta apresenta uma melhor eficiência de detecção e estimação do que a alcançada pelo Filtro Ótimo empregada no TileCal. / The Tile Barrel Calorimeter (TileCal) is the central section of the hadronic calorimeter of ATLAS at LHC. The energy deposited in each cell of the calorimeter is read out by two electronic channels for redundancy and is estimated, per channel, by reconstructing the amplitude of the digitized signal pulse sampled every 25 ns. This work presents signal detection and energy estimation methods for high energy calorimetry, applied to the TileCal environment. It investigates the applicability of a Matched Filter and, furthermore, it explores the impact when summing the signals belonging to the same cell before the estimating and detecting procedures. The proposed method is compared to the Optimal Filter algorithm, that is currently been used at TileCal for energy reconstruction. The results for simulated and collision data sets showed that for conditions where the signal pedestal could be considered stationary, the proposed method achieves better detection and estimation efficiencies than the Optimal Filter technique employed in TileCal.

Ce mémoire de thèse présente une méthode permettant d'améliorer laconduite automobile. Le filtre de Kalman étendu est utilisé pour identifierun modèle de la voiture. Ce filtre est particulièrement étudié afin de prendreen compte la redondance des informations et leur mesure asynchrone.Un ensemble cohérent et bon marché de capteurs - incluant accéléromètres,GPS, temps d'ouverture des injecteurs et vitesse - a été développé et installédans plusieurs véhicules. Ces mesures sont utilisées afin de reconstituer lafonction d'efficacité du moteur. Cette méthodologie peut-être utilisée pourtoute étude requérant la connaissance de cette fonction.La fonction d'efficacité est approchée par une fonction polynomiale etle modèle obtenu est la base d'une optimisation utilisant le principe dumaximum de Pontryagin.Les résultats des tests en condition réelle sont donnés et montrent l'efficicacité de l'observateur et du contrôleur / The PhD presents a combined approach to improving individual car efficiency. An optimal observer, the Extended Kalman Filter, is used to create an efficiency model for the car. Particular attention was paid to handling the asynchronous and redundant nature of the measurement data. A low-cost sensor suite developed to measure data is described. This sensor suite was installed on multiple vehicles to good success. It employsan accelerometer, gps, fuel injector timer, and Vss input to measure all the data necessary to reconstruct the car's state. This observer and sensor suite can be used as the base for any study which requires car efficiency maps, allowing research to proceed without manufacturer supplied data. Once the efficiency map is found, it is then curve-fitted in order to reduce model complexity. The simplified model is then used as a basis for optimal control through Pontryagin's Maximum Principle. Real-world test results are given, both for efficiency mapping, and for optimal control. Detailed discussion of the observer and controller is presented, in order to ease understanding and save implementation time