Probabilistic robust control : theory and applications

In this work, the development of a probabilistic approach to robust control is motivated by structural control applications in civil engineering. Often in civil structural applications, a system's performance is specified in terms of its reliability. In addition, the model and input uncertainty for the system may be described most appropriately using probabilistic or "soft" bounds on the model and input sets. The probabilistic robust control methodology contrasts with existing H∞/μ robust control methodologies that do not use probability information for the model and input uncertainty sets, yielding only the guaranteed (i.e., "worst-case") system performance, and no information about the system's probable performance which would be of interest to civil engineers.

The design objective for the probabilistic robust controller is to maximize the reliability of the uncertain structure/controller system for a probabilistically-described uncertain excitation. The robust performance is computed for a set of possible models by weighting the conditional performance probability for a particular model by the probability of that model, then integrating over the set of possible models. This integration is accomplished efficiently using an asymptotic approximation. The probable performance can be optimized numerically over the class of allowable controllers to find the optimal controller. Also, if structural response data becomes available from a controlled structure, its probable performance can easily be updated using Bayes's Theorem to update the probability distribution over the set of possible models. An updated optimal controller can then be produced, if desired, by following the original procedure. Thus, the probabilistic framework integrates system identification and robust control in a natural manner.

The probabilistic robust control methodology is applied to two systems in this thesis. The first is a high-fidelity computer model of a benchmark structural control laboratory experiment. For this application, uncertainty in the input model only is considered. The probabilistic control design minimizes the failure probability of the benchmark system while remaining robust with respect to the input model uncertainty. The performance of an optimal low-order controller compares favorably with higher-order controllers for the same benchmark system which are based on other approaches. The second application is to the Caltech Flexible Structure, which is a light-weight aluminum truss structure actuated by three voice coil actuators. A controller is designed to minimize the failure probability for a nominal model of this system. Furthermore, the method for updating the model-based performance calculation given new response data from the system is illustrated.

On the solution of first excursion problems by simulation with applications to probabilistic seismic performance assessment

In a probabilistic assessment of the performance of structures subjected to uncertain environmental loads such as earthquakes, an important problem is to determine the probability that the structural response exceeds some specified limits within a given duration of interest. This problem is known as the first excursion problem, and it has been a challenging problem in the theory of stochastic dynamics and reliability analysis. In spite of the enormous amount of attention the problem has received, there is no procedure available for its general solution, especially for engineering problems of interest where the complexity of the system is large and the failure probability is small.

The application of simulation methods to solving the first excursion problem is investigated in this dissertation, with the objective of assessing the probabilistic performance of structures subjected to uncertain earthquake excitations modeled by stochastic processes. From a simulation perspective, the major difficulty in the first excursion problem comes from the large number of uncertain parameters often encountered in the stochastic description of the excitation. Existing simulation tools are examined, with special regard to their applicability in problems with a large number of uncertain parameters. Two efficient simulation methods are developed to solve the first excursion problem. The first method is developed specifically for linear dynamical systems, and it is found to be extremely efficient compared to existing techniques. The second method is more robust to the type of problem, and it is applicable to general dynamical systems. It is efficient for estimating small failure probabilities because the computational effort grows at a much slower rate with decreasing failure probability than standard Monte Carlo simulation. The simulation methods are applied to assess the probabilistic performance of structures subjected to uncertain earthquake excitation. Failure analysis is also carried out using the samples generated during simulation, which provide insight into the probable scenarios that will occur given that a structure fails.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

A bayesian probabilistic approach to structural health monitoring

A Bayesian probabilistic methodology for on-line structural health monitoring which addresses the issue of parameter uncertainty inherent in problem is presented. The method uses modal parameters for a limited number of modes identified from measurements taken at a restricted number of degrees of freedom of a structure as the measured structural data. The application presented uses a linear structural model whose stiffness matrix is parameterized to develop a class of possible models. Within the Bayesian framework, a joint probability density function (PDF) for the model stiffness parameters given the measured modal data is determined. Using this PDF, the marginal PDF of the stiffness parameter for each substructure given the data can be calculated.

Monitoring the health of a structure using these marginal PDFs involves two steps. First, the marginal PDF for each model parameter given modal data from the undamaged structure is found. The structure is then periodically monitored and updated marginal PDFs are determined. A measure of the difference between the calibrated and current marginal PDFs is used as a means to characterize the health of the structure. A procedure for interpreting the measure for use by an expert system in on-line monitoring is also introduced.

The probabilistic framework is developed in order to address the model parameter uncertainty issue inherent in the health monitoring problem. To illustrate this issue, consider a very simplified deterministic structural health monitoring method. In such an approach, the model parameters which minimize an error measure between the measured and model modal values would be used as the "best" model of the structure. Changes between the model parameters identified using modal data from the undamaged structure and subsequent modal data would be used to find the existence, location and degree of damage. Due to measurement noise, limited modal information, and model error, the "best" model parameters might vary from one modal dataset to the next without any damage present in the structure. Thus, difficulties would arise in separating normal variations in the identified model parameters based on limitations of the identification method and variations due to true change in the structure. The Bayesian framework described in this work provides a means to handle this parametric uncertainty.

The probabilistic health monitoring method is applied to simulated data and laboratory data. The results of these tests are presented.

Charts for the rapid estimation of percentages of plants and/or animals in samples

The need to estimate percentages and/or numbers occurs frequently during practical research work; accurate but rapid estimates can be useful when planning research programmes. Charts are provided that may be used as a visual aid to estimating numbers of animals/plants in a specific situation, for example, the number of fish fry in a subsample from a hatchery tank, or the percentage composition of a sample such as the percentage algal cover in a pond.

Subnotificação da Tuberculose no Concelho da Praia no período de 2006 a 2012

Segundo a OMS, Cabo Verde possui uma importante subnotificação de tuberculose, quase metade dos casos, que é muito preocupante. Ainda assim a TUberculose é um problema de saúde pública, devido à sua elevada incidência, com tendência a aumento nos últimos anos. Essa doença é a quarta causa de óbito e sua morbidade diminui a qualidade de vida. O estudo objetiva descrever a situação de subnotificação da tuberculose no concelho da Praia no período de 2006 a 2012. A subnotificação foi avaliada com base na comparação entre o número de registros das unidades de saúde e oconsolidado enviado ao PNLTL. Também foi usado o relacionamento probabilístico entre os bancos do registro dos exames laboratoriais e do registro de hospitalização dos casos de Tuberculose. Três estratégias foram adotadas para extração de dados de acordo com os documentos disponíveis nas unidades: (i) centros de saúde; (ii) laboratórios da delegacia de saúde e do HAN e (iii) hospital (HAN). Nos centros de saúde foram extraídos dados das fichas de atendimento dos pacientes diagnosticados com TB e do livro de registros dos casos de TB. Esses documentos continham dados de identificação do paciente, dados clínicos e laboratoriais. No hospital, como não havia livro de registro de casos de TB buscou-se no arquivo nosológico pacientes . Nesses prontuários buscou-se extrair os mesmo dados que dos do centro de saúde, ou seja, dados de identificação individual, dados clínicos e laboratoriais. Nos laboratórios (HAN e Delegacia) foram extraídos dados de pacientes com resultado positivo para a TB. A análise consistiu na avaliação da qualidade dos bancos e remoção de registros duplicados por intermédio do relacionamento probabilístico. Para o relacionamento dos bancos foi empregada a função reclink usando a versão 10 do programa STATA. Foram calculadas taxas de subnotificação ou sobrenotificação para cada unidade e cada ano de estudo. O cálculo considerou a diferença entre o número de casos encontrados nos registros menos o número de casos notificados dividido pelo número de casos encontrados, expresso em percentual. Valores positivos indicam a ocorrência de subnotificação enquanto valores negativos indicam sobrenotificação. Os resultados permitiram concluir que existe importante subnotificação da tuberculose em Cabo Verde, no período 2006 a 2012. A subnotificação teve maior magnitude no hospital do que nas unidades básicas de saúde. A maior parte da subnotificação detectada nesse trabalho pode ser atribuída ao desconhecimento dos resultados de exames laboratoriais pelos profissionais responsáveis pelo diagnóstico dos casos e consequentemente por sua notificação. O maior número de casos não notificados foi encontrado na listagem de resultados de exames baciloscópicos positivos no laboratório. A segunda grande fonte de casos não notificados é o registro dos pacientes internados no HAN para tratamento da TB. Com base nesse estudo recomendamos medidas de aperfeiçoamento da vigilância epidemiológica da tuberculose em Cabo Verde.