Dynamic response of structures with uncertain parameters

This thesis presents a technique for obtaining the response of linear structural systems with parameter uncertainties subjected to either deterministic or random excitation. The parameter uncertainties are modeled as random variables or random fields, and are assumed to be time-independent. The new method is an extension of the deterministic finite element method to the space of random functions.

First, the general formulation of the method is developed, in the case where the excitation is deterministic in time. Next, the application of this formulation to systems satisfying the one-dimensional wave equation with uncertainty in their physical properties is described. A particular physical conceptualization of this equation is chosen for study, and some engineering applications are discussed in both an earthquake ground motion and a structural context.

Finally, the formulation of the new method is extended to include cases where the excitation is random in time. Application of this formulation to the random response of a primary-secondary system is described. It is found that parameter uncertainties can have a strong effect on the system response characteristics.

Prediction of glucose excursions under uncertain parameters and food intake in intensive insulin therapy for type 1 diabetes mellitus

Considering the difficulty in the insulin dosage selection and the problem of hyper- and hypoglycaemia episodes in type 1 diabetes, dosage-aid systems appear as tremendously helpful for these patients. A model-based approach to this problem must unavoidably consider uncertainty sources such as the large intra-patient variability and food intake. This work addresses the prediction of glycaemia for a given insulin therapy face to parametric and input uncertainty, by means of modal interval analysis. As result, a band containing all possible glucose excursions suffered by the patient for the given uncertainty is obtained. From it, a safer prediction of possible hyper- and hypoglycaemia episodes can be calculated

Application of modal interval analysis to the simulation of the behaviour of dynamic systems with uncertain parameters

Els models matemàtics quantitatius són simplificacions de la realitat i per tant el comportament obtingut per simulació d'aquests models difereix dels reals. L'ús de models quantitatius complexes no és una solució perquè en la majoria dels casos hi ha alguna incertesa en el sistema real que no pot ser representada amb aquests models. Una forma de representar aquesta incertesa és mitjançant models qualitatius o semiqualitatius. Un model d'aquest tipus de fet representa un conjunt de models. La simulació del comportament de models quantitatius genera una trajectòria en el temps per a cada variable de sortida. Aquest no pot ser el resultat de la simulació d'un conjunt de models. Una forma de representar el comportament en aquest cas és mitjançant envolupants. L'envolupant exacta és complete, és a dir, inclou tots els possibles comportaments del model, i correcta, és a dir, tots els punts dins de l'envolupant pertanyen a la sortida de, com a mínim, una instància del model. La generació d'una envolupant així normalment és una tasca molt dura que es pot abordar, per exemple, mitjançant algorismes d'optimització global o comprovació de consistència. Per aquesta raó, en molts casos s'obtenen aproximacions a l'envolupant exacta. Una aproximació completa però no correcta a l'envolupant exacta és una envolupant sobredimensionada, mentre que una envolupant correcta però no completa és subdimensionada. Aquestes propietats s'han estudiat per diferents simuladors per a sistemes incerts.

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.

A disease evolution model with uncertain parameters

In this study we consider the SIS epidemiological model (susceptible-infected-susceptible) in which the transmission and recuperation rates are considered fuzzy sets. The concepts of possibility measures and fuzzy expectancy value are used to obtain the basic reproduction value for infected groups with different viral charge.

Robust Control of Systems Subjected to Uncertain Disturbances and Actuator Dynamics

Esta tesis está enfocada al diseño y validación de controladores robustos que pueden reducir de una manera efectiva las vibraciones structurales producidas por perturbaciones externas tales como terremotos, fuertes vientos o cargas pesadas. Los controladores están diseñados basados en teorías de control tradicionalamente usadas en esta area: Teoría de estabilidad de Lyapunov, control en modo deslizante y control clipped-optimal, una técnica reciente mente introducida : Control Backstepping y una que no había sido usada antes: Quantitative Feedback Theory. La principal contribución al usar las anteriores técnicas, es la solución de problemas de control estructural abiertos tales como dinámicas de actuador, perturbaciones desconocidas, parametros inciertos y acoplamientos dinámicos. Se utilizan estructuras típicas para validar numéricamente los controladores propuestos. Especificamente las estructuras son un edificio de base aislada, una plataforma estructural puente-camión y un puente de 2 tramos, cuya configuración de control es tal que uno o mas problemas abiertos están presentes. Se utilizan tres prototipos experimentales para implementar los controladores robustos propuestos, con el fin de validar experimentalmente su efectividad y viabilidad. El principal resultado obtenido con la presente tesis es el diseño e implementación de controladores estructurales robustos que resultan efectivos para resolver problemas abiertos en control estructural tales como dinámicas de actuador, parámetros inciertos, acoplamientos dinámicos, limitación de medidas y perturbaciones desconocidas.

On Switched Regulator Design of Uncertain Nonlinear Systems Using Takagi-Sugeno Fuzzy Models

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Info-gap theory and robust design of surveillance for invasive species : the case study of Barrow Island

Surveillance for invasive non-indigenous species (NIS) is an integral part of a quarantine system. Estimating the efficiency of a surveillance strategy relies on many uncertain parameters estimated by experts, such as the efficiency of its components in face of the specific NIS, the ability of the NIS to inhabit different environments, and so on. Due to the importance of detecting an invasive NIS within a critical period of time, it is crucial that these uncertainties be accounted for in the design of the surveillance system. We formulate a detection model that takes into account, in addition to structured sampling for incursive NIS, incidental detection by untrained workers. We use info-gap theory for satisficing (not minimizing) the probability of detection, while at the same time maximizing the robustness to uncertainty. We demonstrate the trade-off between robustness to uncertainty, and an increase in the required probability of detection. An empirical example based on the detection of Pheidole megacephala on Barrow Island demonstrates the use of info-gap analysis to select a surveillance strategy.

Economic case study of ABCD cane management practices in the Burdekin River Irrigation Area (BRIA) region

A case study was undertaken to determine the economic impact of a change in management class as detailed in the A, B, C and D management class framework. This document focuses on the implications of changing from D to C, C to B and B to A class management in the Burdekin River irrigation area (BRIA) and if the change is worthwhile from an economic perspective. This report provides a guide to the economic impact that may be expected when undertaking a particular change in farming practices and will ultimately lead to more informed decisions being made by key industry stakeholders. It is recognised that these management classes have certain limitations and in many cases the grouping of practices may not be reflective of the real situation. The economic case study is based on the A, B, C and D management class framework for water quality improvement developed in 2007/2008 for the Burdekin natural resource management region. The framework for the Burdekin is currently being updated to clarify some issues and incorporate new knowledge since the earlier version of the framework. However, this updated version is not yet complete and so the Paddock to Reef project has used the most current available version of the framework for the modelling and economics. As part of the project specification, sugarcane crop production data for the BRIA was provided by the APSIM model. The information obtained from the APSIM crop modelling programme included sugarcane yields and legume grain yield (legume grain yield only applies to A class management practice). Because of the complexity involved in the economic calculations, a combination of the FEAT, PiRisk and a custom made spreadsheet was used for the economic analysis. Figures calculated in the FEAT program were transferred to the custom made spreadsheet to develop a discounted cash flow analysis. The marginal cash flow differences for each farming system were simulated over a 5-year and 10-year planning horizon to determine the net present value of changing across different management practices. PiRisk was used to test uncertain parameters in the economic analysis and the potential risk associated with a change in value.

Economic case study of ABCD cane management practices in the Burdekin Delta region.

A case study was undertaken to determine the economic impact of a change in management class as detailed in the A, B, C and D management class framework. This document focuses on the implications of changing from D to C, C to B and B to A class management in the Burdekin Delta region and if the change is worthwhile from an economic perspective. This report provides a guide to the economic impact that may be expected when undertaking a particular change in farming practices and will ultimately lead to more informed decisions being made by key industry stakeholders. It is recognised that these management classes have certain limitations and in many cases the grouping of practices may not be reflective of the real situation. The economic case study is based on the A, B, C and D management class framework for water quality improvement developed in 2007/2008 for the Burdekin natural resource management region. The framework for the Burdekin is currently being updated to clarify some issues and incorporate new knowledge since the earlier version of the framework. However, this updated version is not yet complete and so the Paddock to Reef project has used the most current available version of the framework for the modelling and economics. As part of the project specification, sugarcane crop production data for the Burdekin Delta region was provided by the APSIM model. The information obtained from the APSIM crop modelling programme included sugarcane yields and legume grain yield (legume grain yield only applies to A class management practice). Because of the complexity involved in the economic calculations, a combination of the FEAT, PiRisk and a custom made spreadsheet was used for the economic analysis. Figures calculated in the FEAT program were transferred to the custom made spreadsheet to develop a discounted cash flow analysis. The marginal cash flow differences for each farming system were simulated over a 5-year and 10-year planning horizon to determine the Net Present Value of changing across different management practices. PiRisk was used to test uncertain parameters in the economic analysis and the potential risk associated with a change in value.

Economic case study of ABCD cane management practices in the Tully region

A case study was undertaken to determine the economic impact of a change in management class as detailed in the A, B, C and D management class framework. This document focuses on the implications of changing from D to C, C to B and B to A class management in the Tully region and if the change is worthwhile from an economic perspective. This report provides a guide to the economic impact that may be expected when undertaking a particular change in farming practices and will ultimately lead to more informed decisions being made by key industry stakeholders. It is recognised that these management classes have certain limitations and in many cases the grouping of practices may not be reflective of the real situation. The economic case study is based on the A, B, C and D management class framework for water quality improvement developed in 2007/2008 by the wet tropics natural resource management region. The framework for wet tropics is currently being updated to clarify some issues and incorporate new knowledge since the earlier version of the framework. However, this updated version is not yet complete and so the Paddock to Reef project has used the most current available version of the framework for the modelling and economics. As part of the project specification, sugarcane crop production data for the Tully region was provided by the APSIM model. Because of the complexity involved in the economic calculations, a combination of the FEAT, PiRisk and a custom made spreadsheet was used for the economic analysis. Figures calculated in the FEAT program were transferred to the custom made spreadsheet to develop a discounted cash flow analysis. The marginal cash flow differences for each farming system were simulated over a 5-year and 10-year planning horizon to determine the Net Present Value of changing across different management practices. PiRisk was used to test uncertain parameters in the economic analysis and the potential risk associated with a change in value.

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.