According to the model free approach assumption four protein dynamic related parameters, the slow and fast local motion of the NH vector, the generalized order parameter, and the 15N shielding anisotropy can be estimated at each residue by the spectral density functions at the resonant frequencies of N (omega_N) and H (omega_H). In this work, we study the linear and nonlinear estimations of the aforementioned parameters of the two proteins C12A-p8^MTCPI and Pilin from strain K122-4. The principal components of the four parameters of C12A-p8^MTCPI are used to cluster the residues. The results show that the principle components provide useful information about the secondary structure of the protein. Finally, we propose a practical method to examine the model free assumption by characterizing the distribution of the transverse rate R_2 in multifield.

Decisions for asset allocation and protection are predicated upon accurate knowledge of the current operating environment as well as correctly characterizing the evolution of the environment over time. The desired kinematic and kinetic states of objects in question cannot be measured directly in most cases and instead are inferred or estimated from available measurements using a filtering process. Often, nonlinear transformations between the measurement domain and desired state domain distort the state domain probability density function yielding a form which does not necessarily resemble the form assumed in the filtering algorithm. The distortion effect must be understood in greater detail and appropriately accounted for so that even if sensors, state estimation algorithms, and state propagation algorithms operate in different domains, they can all be effectively utilized without any information loss due to domain transformations. This research presents an analytical investigation into understanding how non-linear transformations of stochastic, but characterizable, processes affect state and uncertainty estimation with direct application to space object surveillance and space- craft attitude determination. Analysis is performed with attention to construction of the state domain probability density function since state uncertainty and correlation are derived from the statistical moments of the probability density function. Analytical characterization of the effect nonlinear transformations impart on the structure of state probability density functions has direct application to conventional non- linear filtering and propagation algorithms in three areas: (1) understanding how smoothing algorithms used to estimate indirectly observed states impact state uncertainty, (2) justification or refutation of assumed state uncertainty distribution for more realistic uncertainty quantification, and (3) analytic automation of initial state estimate and covariance in lieu of user tuning. A nonlinear filtering algorithm based upon Bayes’ Theorem is presented to ac- count for the impact nonlinear domain transformations impart on probability density functions during the measurement update and propagation phases. The algorithm is able to accommodate different combinations of sensors for state estimation which can also be used to hypothesize system parameters or unknown states from available measurements because information is able to appropriately accounted for.

O filtro de Kalman estendido tem sido a mais popular ferramenta de filtragem não linear das últimas quatro décadas. É de fácil implementação e apresenta baixo custo computacional. Nos casos nos quais as não linearidades do sistema dinâmico são significativas, porém, o filtro de Kalman estendido pode apresentar resultados insatisfatórios. Nessas situações, o filtro de Kalman unscented substitui com vantagens o filtro de Kalman estendido, pois pode apresentar melhores estimativas de estado, embora ambos os filtros exibam complexidade computacional de mesma ordem. A qualidade das estimativas de estado do filtro unscented está intimamente ligada à sintonia dos parâmetros que controlam a transformada unscented. A versão escalada dessa transformada exibe três parâmetros escalares que determinam o posicionamento dos pontos sigma e, consequentemente, afetam diretamente a qualidade das estimativas produzidas pelo filtro. Apesar da importância do filtro de Kalman unscented, a sintonia ótima desses parâmetros é um problema para o qual ainda não há solução definitiva. Não há nem mesmo recomendações heurísticas que garantam o bom funcionamento do filtro unscented na maior parte dos problemas tratáveis por meio de filtros Gaussianos. Essa carência e a importância desse filtro para a área de filtragem não linear fazem da busca por mecanismos de sintonia automática do filtro unscented área de pesquisa ativa. Assim, este trabalho propõe técnicas para sintonia automática dos parâmetros da transformada unscented escalada. Além da sintonia desses parâmetros, também é abordado o problema de sintonizar as matrizes de covariância dos ruídos de processo e de medida demandadas pelo modelo do sistema dinâmico usado pelo filtro unscented. As técnicas propostas cobrem então a sintonia automática de todos os parâmetros do filtro. / The extended Kalman filter has been the most popular nonlinear filter of the last four decades. It is easy to implement and exhibits low computational cost. When nonlinearities are significant, though, the extended Kalman filter can display poor state estimation performance. In such situations, the unscented Kalman filter can yield better state estimates, while displaying the same order of computational complexity as the extended Kalman filter. The quality of the state estimates produced by the unscented Kalman filter is directly influenced by the tuning of the scalar parameters that govern the unscented transform. The scaled version of the unscented transform features three scalar parameters that determine the positioning of the sigma points, thus directly affecting the filter state estimation performance. Despite the importance of the unscented Kalman filter, the optimal tuning of the scaled unscented transform parameters is still an open problem. This work hence discusses algorithms for the automatic tuning of the unscented transform parameters. The discussion includes the tuning of the needed noise covariance matrices, thus covering the automatic tuning of all parameters of the unscented Kalman filter.

A new algorithm is proposed that smoothly integrates the nonlinear estimation of the attitude quaternion using Davenport's q-method and the estimation of non-attitude states within the framework of an extended Kalman filter. A modification to the q-method and associated covariance analysis is derived with the inclusion of an a priori attitude estimate. The non-attitude states are updated from the nonlinear attitude estimate based on linear optimal Kalman filter techniques. The proposed filter is compared to existing methods and is shown to be equivalent to second-order in the attitude update and exactly equivalent in the non-attitude state update with the Sequential Optimal Attitude Recursion filter. Monte Carlo analysis is used in numerical simulations to demonstrate the validity of the proposed approach. This filter successfully estimates the nonlinear attitude and non-attitude states in a single Kalman filter without the need for iterations.

Nonlinear estimation techniques play an important role for process monitoring since some states and most of the parameters cannot be directly measured. There are many techniques available for nonlinear state and parameter estimation, i.e., extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filtering (PF) and moving horizon estimation (MHE) etc. However, many issues related to the available techniques are to be solved. This dissertation discusses three important techniques in nonlinear estimation, which are the application of unscented Kalman filters, improvement of moving horizon estimation via computation of the arrival cost and different implementations of extended Kalman filters. First the use of several estimation algorithms such as linearized Kalman filter (LKF), extended Kalman filter (EKF), unscented Kalman filter (UKF) and moving horizon estimation (MHE) are investigated for nonlinear systems with special emphasis on UKF as it is a relatively new technique. Detailed case studies show that UKF has advantages over EKF for highly nonlinear unconstrained estimation problems while MHE performs better for systems with constraints. Moving horizon estimation alleviates the computational burden of solving a full information estimation problem by considering a finite horizon of the measurement data; however, it is non-trivial to determine the arrival cost. A commonly used approach for computing the arrival cost is to use a first order Taylor series approximation of the nonlinear model and then apply an extended Kalman filter. The second contribution of this dissertation is that an approach to compute the arrival cost for moving horizon estimation based on an unscented Kalman filter is proposed. It is found that such a moving horizon estimator performs better in some cases than if one based on an extended Kalman filter. It is a promising alternative for approximating the arrival cost for MHE. Many comparative studies, often based upon simulation results, between extended Kalman filters (EKF) and other estimation methodologies such as moving horizon estimation, unscented Kalman filter, or particle filtering have been published over the last few years. However, the results returned by the extended Kalman filter are affected by the algorithm used for its implementation and some implementations of EKF may lead to inaccurate results. In order to address this point, this dissertation investigates several different algorithms for implementing extended Kalman filters. Advantages and drawbacks of different EKF implementations are discussed in detail and illustrated in some comparative simulation studies. Continuously predicting covariance matrix for EKF results in an accurate implementation. Evaluating covariance matrix at discrete times can also be applied. Good performance can be expected if covariance matrix is obtained from integrating the continuous-time equation or if the sensitivity equation is used for computing the Jacobian matrix.

O filtro de Kalman estendido tem sido a mais popular ferramenta de filtragem não linear das últimas quatro décadas. É de fácil implementação e apresenta baixo custo computacional. Nos casos nos quais as não linearidades do sistema dinâmico são significativas, porém, o filtro de Kalman estendido pode apresentar resultados insatisfatórios. Nessas situações, o filtro de Kalman unscented substitui com vantagens o filtro de Kalman estendido, pois pode apresentar melhores estimativas de estado, embora ambos os filtros exibam complexidade computacional de mesma ordem. A qualidade das estimativas de estado do filtro unscented está intimamente ligada à sintonia dos parâmetros que controlam a transformada unscented. A versão escalada dessa transformada exibe três parâmetros escalares que determinam o posicionamento dos pontos sigma e, consequentemente, afetam diretamente a qualidade das estimativas produzidas pelo filtro. Apesar da importância do filtro de Kalman unscented, a sintonia ótima desses parâmetros é um problema para o qual ainda não há solução definitiva. Não há nem mesmo recomendações heurísticas que garantam o bom funcionamento do filtro unscented na maior parte dos problemas tratáveis por meio de filtros Gaussianos. Essa carência e a importância desse filtro para a área de filtragem não linear fazem da busca por mecanismos de sintonia automática do filtro unscented área de pesquisa ativa. Assim, este trabalho propõe técnicas para sintonia automática dos parâmetros da transformada unscented escalada. Além da sintonia desses parâmetros, também é abordado o problema de sintonizar as matrizes de covariância dos ruídos de processo e de medida demandadas pelo modelo do sistema dinâmico usado pelo filtro unscented. As técnicas propostas cobrem então a sintonia automática de todos os parâmetros do filtro. / The extended Kalman filter has been the most popular nonlinear filter of the last four decades. It is easy to implement and exhibits low computational cost. When nonlinearities are significant, though, the extended Kalman filter can display poor state estimation performance. In such situations, the unscented Kalman filter can yield better state estimates, while displaying the same order of computational complexity as the extended Kalman filter. The quality of the state estimates produced by the unscented Kalman filter is directly influenced by the tuning of the scalar parameters that govern the unscented transform. The scaled version of the unscented transform features three scalar parameters that determine the positioning of the sigma points, thus directly affecting the filter state estimation performance. Despite the importance of the unscented Kalman filter, the optimal tuning of the scaled unscented transform parameters is still an open problem. This work hence discusses algorithms for the automatic tuning of the unscented transform parameters. The discussion includes the tuning of the needed noise covariance matrices, thus covering the automatic tuning of all parameters of the unscented Kalman filter.

Este trabalho investigou o uso de tomografia por impedância elétrica (TIE) na discriminação de fases em sistemas bifásicos água-ar. A TIE é uma técnica não-intrusiva em que são estimados parâmetros de condutividade elétrica de um sistema de interesse a partir de correntes elétricas impostas e potenciais elétricos medidos na fronteira desse meio. Esta técnica se traduz em um problema desafiador, por ser inverso, não-linear e mal-posto. Adicionalmente, na aplicação em análise, a dinâmica do sistema é rápida a ponto de influir nas estimativas procuradas. Foi sistematizada uma abordagem para integrar informações de medições a de outras fontes, como um regularizador generalizado de Tikhonov (filtro gaussiano), parametrização de estado e modelos de evolução, construindo um modelo adaptativo de estimação. Tal combinação de métodos é inédita na literatura. Parametrização do estado (vetor de condutividades do sistema de interesse, após discretização espacial) em condutividade logarítmica foi implementada para assegurar a obtenção de valores condizentes com a física, i.e., as estimativas em condutividade são mantidas estritamente positivas, com benefícios adicionais de aumento da região de convergência monotônica e melhoria na uniformidade da taxa de convergência das estimativas. O estudo de um sistema numérico evidenciou que a parametrização do estado permitiu o aumento do fator de sub-relaxação no método de Gauss-Newton, de 4~ para 15~, o que torna o algoritmo mais rápido. Dois modelos de evolução para escoamentos foram propostos e, comparativamente com o modelo de passeio aleatório, proporcionaram convergência mais rápida, melhor distinção das fases e melhoria do grau de observabilidade do problema de TIE. Esses modelos descrevem uma velocidade representativa para o escoamento, avaliada experimentalmente em 0; 47 m_s. Ensaios experimentais estáticos sugerem que os métodos aplicados diferenciam a presença das fases em um duto. No caso em que a dinâmica é relevante (passagem de bolhas ao longo do duto), o algoritmo desenvolvido permite o devido acompanhamento de não homogeneidades. Portanto, os resultados dessa pesquisa têm o potencial de apoiar a estimação de vazões bifásicas em trabalhos futuros, uma vez que a avaliação da fração de ocupação das fases é um passo crucial para o desenvolvimento de um medidor real de vazão multifásica. / This work investigated the use of electrical impedance tomography (EIT) in phase discrimination in two-phase air-water systems. EIT is a non-intrusive technique in which electric currents are imposed and electric potentials are measured at the boundary of a system. This method is mathematically challenging, as it is non-linear, inverse, and ill-posed. Also, for the application at hand, the system dynamics is fast enough to influence the sought estimates. A systematic approach was created to combine information from measurements and other sources, including a generalized Tikhonov regularization term (Gaussian filter), state parametrization and evolution models. This adaptive estimation approach is a contribution to the literature. State parametrization (vector of conductivities of the system of interest after spatial discretization) in logarithmic conductivity was implemented to ensure that the estimates remain in physical bounds, i.e., only positive values are achieved. Additional benefits are the increase of the region that leads to monotone convergence and a more uniform convergence rate of the estimates. The comparative analysis of a numerical system showed that state parametrization allowed an increase for the under-relaxation factor in the Gauss-Newton method, from 4% to 15%, increasing the algorithm\'s speed. Two evolution models for flows were proposed and, when compared to the random walk model, provided faster convergence, better phase distinction and an improved degree of observability for the EIT problem. These models describe a representative velocity for the flow, estimated experimentally as 0:47 m/s. Experimental tests of static setups suggest that the applied methods are able to differentiate the phases in a duct. In the case where the dynamics is relevant (flow of bubbles along the duct), the algorithm developed allows for monitoring inhomogeneities. Therefore, the results of this thesis are able to support the estimation of two-phase flow rates in future work, given that evaluating void fraction is a crucial step for an online multiphase flow rate meter.

A novel application is presented for a fault-tolerant adaptive model predictive control system for small-scale helicopters. The use of a joint Extended Kalman Filter, (EKF), for the estimation of the states and parameters of the UAV, provided the advantage of implementation simplicity and accuracy. A linear model of a small-scale helicopter was utilized for testing the proposed control system. The results, obtained through the simulation of different fault scenarios, demonstrated that the proposed scheme was able to handle different types of actuator and system faults effectively. The types of faults considered were represented in the parameters of the mathematical representation of the helicopter. Benefits provided by the proposed fault-tolerant adaptive model predictive control systems include: The use of the joint Kalman filter provided a straightforward approach to detect and handle different types of actuator and system faults, which were represented as changes of the system and input matrices. The built-in adaptability provided for the handling of slow time-varying faults, which are difficult to detect using the standard residual approach. The successful inclusion of fault tolerance yielded a significant increase in the reliability of the UAV under study. A byproduct of this research is an original comparison between the EKF and the Unscented Kalman Filter, (UKF). This comparison attempted to settle the conflicting claims found in the research literature concerning the performance improvements provided by the UKF. The results of the comparison indicated that the performance of the filters depends on the approximation used for the nonlinear model of the system. Noise sensitivity was found to be higher for the UKF, than the EKF. An advantage of the UKF appears to be a slightly faster convergence.

This dissertation explores both the estimation of various parameters from a multiple antenna GPS receiver, which is used as an attitude sensor, and attitude determination using GPS-like Pseudolite signals. To use a multiple antenna GPS receiver as an attitude sensor, parameters such as baselines, integer ambiguities, line biases, and attitude, should be resolved beforehand. Also, due to a cycle slip problem a subsystem to correct this problem should be implemented. All of these tasks are called a self survey. A new algorithm to estimate these parameters from a GPS receiver is developed usingnonlinear batch filteringmethods.For convergence issues, both the nolinear least squares (NLS) and Levenberg-Marquardt (LM) methods are applied in the estimation.Acomparison ofthe NLSand LMmethods shows that the convergence of the LM method for the large initial errors is more robust than that of the NLS. In the proximity of the International Space Station (ISS), Pseudolite signals replace the GPSsignals since almostallsignals are blocked.Since the Pseudolite signals have spherical wavefronts, a new observation model should be applied. A nonlinear predictive filter, an extended Kalman filter (EKF), and an unscented filter (UF) are developed and compared using Pseudolite signals. A nonlinear predictive filter can provide a deterministic solution; however, it cannot be used for the moving case. Instead, the EKF or the UF can be used with the angular rate measurements. A comparison of EKF and UF shows that the convergence of the UF for the large initial errors is more robust than that of the EKF. Also, an alternative global navigation constellation is presented by using the Flower Constellation (FC) scheme. A comparison of FC global navigation constellation and other GPS constellations, U.S. GPS, Galileo, and GLONASS, shows that position and attitude errors of the FC constellation are smaller that those of the others.

In order to overcome the various disadvantages of standalone INS and GPS, these systems are integrated using nonlinear estimation techniques and benefits of the two complementary systems are obtained at the same time. The standard and most widely used estimation algorithm in the INS/GPS integrated systems is Extended Kalman Filter (EKF). Linearization step involved in the EKF algorithm can lead to second order errors in the mean and covariance of the state estimate. Another nonlinear estimator, Unscented Kalman Filter (UKF) approaches this problem by carefully selecting deterministic sigma points from the Gaussian distribution and propagating these points through the nonlinear function itself leading third order errors for any nonlinearity. Scaled Unscented Transformation (SUT) is one of the sigma point selection methods which gives the opportunity to adjust the spread of sigma points and control the higher order errors by some design parameters. Determination of these parameters is problem specific. In this thesis, effects of the SUT parameters on integrated navigation solution are investigated and an &ldquo / Adaptive UKF&rdquo / is designed for a tightly-coupled INS/GPS integrated system. Besides adapting process and v measurement noises, SUT parameters are adaptively tuned. A realistic fighter flight trajectory is used to simulate IMU and GPS data within Monte Carlo analysis. Results of the proposed method are compared with standard EKF and UKF integration. It is observed that the adaptive scheme used in the sigma point selection improves the performance of the integrated navigation system especially at the end of GPS outage periods.