Mechanical systems always operate under some degree of uncertainty, which can be due to the inherent properties of the system parameters, to random inputs or external excitations, to poorly known parameters in the interface between different systems, or to inadequate knowledge of the dynamic process. Also, mechanical systems are large and highly nonlinear, while the magnitude of uncertainties may be very large. This dissertation addresses the critical need for understanding of the stochastic nature of mechanical system, especially vehicle and terramechanics systems, and need for developing efficient computational tools to model mechanical systems in the presence of parametric and external uncertainty. This dissertation investigates the influence of parametric and external uncertainties on vehicle dynamics and terramechanics. The uncertainties studied include parametric uncertainties, stochastic external excitations, and random variables between vehicle-terrain and vehicle-soil/snow interface. The methodology developed has been illustrated on a stochastic vehicle-terrain interaction model, a stochastic vehicle-soil interaction model, two stochastic tire-snow interaction models, and two stochastic tire-force relations. The uncertainties are quantified and propagated through vehicle and terramechanics systems using a polynomial chaos approach. Algorithms which can predict the geometry of the contact patch and the interfacial forces and torques on the vehicle-soil interfaces are developed. All stochastic models and algorithms are simulated for various scenarios and maneuvers. Numerical results are analyzed from the computational effort point of view, or from the angle of vehicle dynamics and terramechanics, and provide a deeper understanding of the evolution of stochastic vehicle and terramechanics systems. They can also be used in guiding vehicle design and development. This dissertation represents a pioneer study on stochastic vehicle dynamics and terramechanics. Moreover, the methodology developed is not limited to such systems. Any mechanical system with uncertainties can be treated using the polynomial chaos approach presented, considering their specific characteristics. / Ph. D.

The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs. / Ph. D.

It is a well-known fact that a power system contains many sources of uncertainties. These uncertainties coming from the loads, the renewables, the model and the measurement, etc, are influencing the steady state and dynamic response of the power system. Facing this problem, traditional methods, such as the Monte Carlo method and the Perturbation method, are either too time consuming or suffering from the strong nonlinearity in the system. To solve these, this Dissertation will mainly focus on developing the polynomial chaos based method to replace the traditional ones. Using it, the uncertainties from the model and the measurement are propagated through the polynomial chaos bases at a set of collocation points. The approximated polynomial chaos coefficients contain the statistical information. The method can greatly accelerate the calculation efficiency while not losing the accuracy, even when the system is highly stressed. In this dissertation, both the forward problem and the inverse problem of uncertainty quantification will be discussed. The forward problems will include the probabilistic power flow problem and statistical power system dynamic simulations. The generalized polynomial chaos method, the adaptive polynomial chaos-ANOVA method and the multi-element polynomial chaos method will be introduced and compared. The case studies show that the proposed methods have great performances in the statistical analysis of the large-scale power systems. The inverse problems will include the state and parameter estimation problem. A novel polynomial-chaos-based Kalman filter will be proposed. The comparison studies with other traditional Kalman filter demonstrate the good performances of the proposed Kalman filter. We further explored the area dynamic parameter estimation problem under the Bayesian inference framework. The polynomial-chaos-expansions are treated as the response surface of the full dynamic solver. Combing with hybrid Markov chain Monte Carlo method, the proposed method yields very high estimation accuracy while greatly reducing the computing time. For both the forward problem and the inverse problems, the polynomial chaos based methods haven shown great advantages over the traditional methods. These computational techniques can improve the efficiency and accuracy in power system planning, guarantee the rationality and reliability in power system operations, and, finally, speed up the power system dynamic security assessment. / PHD / It is a well-known fact that a power system state is inherently stochastic. Sources of stochasticity include load random variations, renewable energy intermittencies, and random outages of generating units, lines, and transformers, to cite a few. These stochasticities translate into uncertainties in the models that are assumed to describe the steady-sate and dynamic behavior of a power system. Now, these models are themselves approximate since they are based on some assumptions that are typically violated in practice. Therefore, it does not come as a surprise if recent research activities in power systems are focusing on how to cope with uncertainties when dealing with power system planning, monitoring and control. This Dissertation is developing polynomial-chaos-based method in quantifying, and managing these uncertainties. Three major topics, including uncertainty quantification, state estimation and parameter estimation are discussed. The developed method can improve the efficiency and accuracy in power system planning, guarantee the rationality and reliability in power system operations in dealing with the uncertainties, and, finally, enhancing the resilience of the power systems.

L’optimisation est un sujet très important dans tous les domaines. Cependant, parmi toutes les applications de l’optimisation, il est difficile de trouver des exemples de systèmes à optimiser qui ne comprennent pas un certain niveau d'incertitude sur les valeurs de quelques paramètres. Le thème central de cette thèse est donc le traitement des différents aspects de l’optimisation en présence d’incertitudes. Nous commençons par présenter un bref état de l’art des méthodes permettant de prendre en compte les incertitudes dans l’optimisation. Cette revue de la littérature a permis de constater une lacune concernant la caractérisation des propriétés probabilistes du point d’optimum de fonctions dépendant de paramètres aléatoires. Donc, la première contribution de cette thèse est le développement de deux méthodes pour approcher la fonction densité de probabilité (FDP) d’un tel point : la méthode basée sur la Simulation de Monte Carlo et la méthode de projection en dimension finie basée sur l’Approximation par polynômes de chaos. Les résultats numériques ont montré que celle-ci est adaptée à l’approximation de la FDP du point optimal du processus d'optimisation dans les situations étudiées. Il a été montré que la méthode numérique est capable d’approcher aussi des moments d'ordre élevé du point optimal, tels que l’aplatissement et l’asymétrie. Ensuite, nous passons au traitement de contraintes probabilistes en utilisant l’optimisation fiabiliste. Dans ce sujet, une nouvelle méthode basée sur des coefficients de sécurité est développée. Les exemples montrent que le principal avantage de cette méthode est son coût de calcul qui est très proche de celui de l’optimisation déterministe conventionnelle, ce qui permet son couplage avec un algorithme d’optimisation globale arbitraire. / The optimization is a very important tool in several domains. However, among its applications, it is hard to find examples of systems to be optimized that do not possess a certain uncertainty level on its parameters. The main goal of this thesis is the treatment of different aspects of the optimization under uncertainty. We present a brief review of the literature on this topic, which shows the lack of methods able to characterize the probabilistic properties of the optimum point of functions that depend on random parameters. Thus, the first main contribution of this thesis is the development of two methods to eliminate this lack: the first is based on Monte Carlo Simulation (MCS) (considered as the reference result) and the second is based on the polynomial chaos expansion (PCE). The validation of the PCE based method was pursued by comparing its results to those provided by the MCS method. The numerical analysis shows that the PCE method is able to approximate the probability density function of the optimal point in all the problems solved. It was also showed that it is able to approximate even high order statistical moments such as the kurtosis and the asymmetry. The second main contribution of this thesis is on the treatment of probabilistic constraints using the reliability based design optimization (RBDO). Here, a new RBDO method based on safety factors was developed. The numerical examples showed that the main advantage of such method is its computational cost, which is very close to the one of the standard deterministic optimization. This fact makes it possible to couple the new method with global optimization algorithms.

Advances in machine autonomy hold great promise in advancing technology, economic markets, and general societal well-being. For example, the progression of unmanned air systems (UAS) research has demonstrated the effectiveness and reliability of these autonomous systems in performing complex tasks. UAS have shown to not only outperformed human pilots in some tasks, but have also made novel applications not possible for human pilots practical. Nevertheless, human pilots are still favored when performing specific challenging tasks. For example, transportation of suspended (sometimes called slung or sling) loads requires highly skilled pilots and has only been performed by UAS in highly controlled environments. The presented work begins to bridge this autonomy gap by proposing a trajectory optimization framework for operations involving autonomous rotorcraft with suspended loads. The framework generates optimized vehicle trajectories that are used by existing guidance, navigation, and control systems and estimates the state of the non-instrumented load using a downward facing camera. Data collected from several simulation studies and a flight test demonstrates the proposed framework is able to produce effective guidance during autonomous suspended load operations. In addition, variational integrators are extensively studied in this dissertation. The derivation of a stochastic variational integrator is presented. It is shown that the presented stochastic variational integrator significantly improves the performance of the stochastic differential dynamical programming and the extended Kalman filter algorithms. A variational integrator for the propagation of polynomial chaos expansion coefficients is also presented. As a result, the expectation and variance of the trajectory of an uncertain system can be accurately predicted.

In dieser Arbeit wird für Strukturen, die im makroskopischen aufgrund unterschiedlicher Materialeigenschaften oder komplexer Geometrien eine hohe Netzfeinheit für Finite-Elemente-Berechnungen benötigen, eine neue Möglichkeit zur Berechnung effektiver Materialkennwerte vorgestellt. Durch einen modalen Ansatz, bei dem, je nach Struktur analytisch oder numerisch, mit Hilfe der modalen Kennwerte die Formänderungsenergie eines repräsentativen Volumens der Originalstruktur mit der Formänderungsenergie eines äquivalenten homogen Vergleichsvolumens verglichen wird, können effektive Materialkennwerte ermittelt und daran anschließend eine Finite-Elemente-Berechnung mit einem im Vergleich zum Originalmodell sehr viel gröberen Netz durchgeführt werden, was eine enorme Zeiteinsparung mit sich bringt. Weiterhin enthält die vorgestellte Methode die Möglichkeit, unsichere Eingabeparameter wie Geometrieabmessungen oder Materialkennwerte mit Hilfe der polynomialen Chaos Expansion zu approximieren, um Möglichkeiten zur Aussage bzgl. der daraus resultierenden Verteilungen modaler Kenngrößen auf eine schnelle und effektive Weise zu gewinnen.

Water is a scarce resource especially as the water demand is significantly increasing due to the rapid growth of population. Hydrological modelling has gained a lot of attention, as it is the key to predict water availability, optimize the use of water resources and develop risk mitigation schemes. There are still many challenges in hydrological modelling that researchers and designers are trying to solve. These challenges include, but not limited to: i) there is no single robust model that can perform well in all watersheds; ii) model parameters are often associated with uncertainty, which makes the results inconclusive; iii) the required computational power for uncertainty quantification increases with the increase in model complexity; iv) some modelling assumptions to simplify computational complexity, such as parameter independence are, are often not realistic. These challenges make it difficult to provide robust hydrological predictions and/or to quantify the uncertainties within hydrological models in an efficient and accurate way. This study aims to provide more robust hydrological predictions by developing a set of hybrid approaches. Firstly, a hybrid hydrological data-driven (HHDD) model based on the integration of a physically-based hydrological model (HYMOD) and a data-driven model (artificial neural network, ANN) is developed. The HHDD model is capable of improving prediction accuracy and generating interval flow prediction results. Secondly, a hybrid probabilistic forecasting approach is developed by linking the polynomial chaos expansion (PCE) method with ANN. The results indicate that PCE-ANN can be as reliable as but much more efficient than the traditional Monte-Carlo (MC) method for probabilistic flow forecasting. Finally, a hybrid uncertainty quantification approach that can address parameter dependence is developed through the integration of principal component analysis (PCA) with PCE. The results from this dissertation research can provide valuable technical and decision support for hydrological modeling and water resources management under uncertainty. / Thesis / Doctor of Engineering (DEng) / There is a water scarcity problem in the world, so it is vital to have reliable decision support tools for effective water resources management. Researchers and decision-makers rely on hydrological modelling to predict water availability. Hydrological model results are then used for water resources allocation and risk mitigation. Hydrological modelling is not a simple process, as there are different sources of uncertainty associated with it, such as model structure, model parameters, and data. In this study, data-driven techniques are used with process-driven models to develop hybrid uncertainty quantification approaches for hydrological modelling. The overall objectives are: i) to generate more robust probabilistic forecasts; ii) to improve the computational efficiency for uncertainty quantification without compromising accuracy; and, iii) to overcome the limitations of current uncertainty quantification methods, such as parameter interdependency. The developed hybrid approaches can be used by decision-makers in water resources management, as well as risk assessment and mitigation.

Mathematical models describing real world processes are becoming increasingly complex to better match the dynamics of the true system. While this is a positive step towards more complete knowledge of our world, numerical evaluations of these models become increasingly computationally inefficient, requiring increased resources or time to evaluate. This has led to the need for simplified surrogates to these complex mathematical models. A growing surrogate modeling solution is with the usage of neural networks. Neural networks (NN) are known to generalize an approximation across a diverse dataset and minimize the solution along complex nonlinear boundaries. Additionally, these surrogate models can be found using only incomplete knowledge of the true dynamics. However, NN surrogates often suffer from a lack of interpretability, where the decisions made in the training process are not fully understood, and the roles of individual neurons are not well defined. We present two solutions towards this lack of interpretability. The first focuses on mimicking polynomial chaos (PC) modeling techniques, modifying the structure of a NN to produce polynomial approximations of the underlying dynamics. This methodology allows for an extractable meaning from the network and results in improvement in accuracy over traditional PC methods. Secondly, we examine the construction of a reduced order modeling scheme using NN autoencoders, guiding the decisions of the training process to better match the real dynamics. This guiding process is performed via a physics-informed (PI) penalty, resulting in a speed-up in training convergence, but still results in poor performance compared to traditional schemes. / Master of Science / The world is an elaborate system of relationships between diverse processes. To accurately represent these relationships, increasingly complex models are defined to better match what is physically seen. These complex models can lead to issues when trying to use them to predict a realistic outcome, either requiring immensely powerful computers to run the simulations or long amounts of time to present a solution. To fix this, surrogates or approximations to these complex models are used. These surrogate models aim to reduce the resources needed to calculate a solution while remaining as accurate to the more complex model as possible. One way to make these surrogate models is through neural networks. Neural networks try to simulate a brain, making connections between some input and output given to the network. In the case of surrogate modeling, the input is some current state of the true process, and the output is what is seen later from the same system. But much like the human brain, the reasoning behind why choices are made when connecting the input and outputs is often largely unknown. Within this paper, we seek to add meaning to neural network surrogate models in two different ways. In the first, we change what each piece in a neural network represents to build large polynomials (e.g., $x^5 + 4x^2 + 2$) to approximate the larger complex system. We show that the building of these polynomials via neural networks performs much better than traditional ways to construct them. For the second, we guide the choices made by the neural network by enforcing restrictions in what connections it can make. We do this by using additional information from the larger system to ensure the connections made focus on the most important information first before trying to match the less important patterns. This guiding process leads to more information being captured when the surrogate model is compressed into only a few dimensions compared to traditional methods. Additionally, it allows for a faster learning time compared to similar surrogate models without the information.

A method utilizing H2 control concepts and the numerical method of Polynomial Chaos was developed in order to create a novel robust probabilistically optimal control approach. This method was created for the practical reason that uncertainty in parameters tends to be inherent in system models. As such, the development of new methods utilizing probability density functions (PDFs) was desired. From a more theoretical viewpoint, the utilization of Polynomial Chaos for studying and designing control systems has not been very thoroughly investigated. The current work looks at expanding the H2 and related Linear Quadratic Regulator (LQR) control problems for systems with parametric uncertainty. This allows solving deterministic linear equations that represent probabilistic linear differential equations. The application of common LTI (Linear Time Invariant) tools to these expanded systems are theoretically justified and investigated. Examples demonstrating the utilized optimization process for minimizing the H2 norm and parallels to LQR design are presented. The dissertation begins with a thorough background section that reviews necessary probability theory. Also, the connection between Polynomial Chaos and dynamic systems is explained. Next, an overview of related control methods, as well as an in-depth review of current Polynomial Chaos literature is given. Following, formal analysis, related to the use of Polynomial Chaos, is provided. This lays the ground for the general method of control design using Polynomial Chaos and H2. Then an experimental section is included that demonstrates controller synthesis for a constructed probabilistic system. The experimental results lend support to the method. / Ph. D.

The main goal of this study is the use of polynomial chaos expansion (PCE) to analyze the uncertainty in calculating the lateral and longitudinal center of gravity for a vehicle from static load cell measurements. A secondary goal is to use experimental testing as a source of uncertainty and as a method to confirm the results from the PCE simulation. While PCE has often been used as an alternative to Monte Carlo, PCE models have rarely been based on experimental data. The 8-post test rig at the Virginia Institute for Performance Engineering and Research facility at Virginia International Raceway is the experimental test bed used to implement the PCE model. Experimental tests are conducted to define the true distribution for the load measurement systems' uncertainty. A method that does not require a new uncertainty distribution experiment for multiple tests with different goals is presented. Moved mass tests confirm the uncertainty analysis using portable scales that provide accurate results. The polynomial chaos model used to find the uncertainty in the center of gravity calculation is derived. Karhunen-Loeve expansions, similar to Fourier series, are used to define the uncertainties to allow for the polynomial chaos expansion. PCE models are typically computed via the collocation method or the Galerkin method. The Galerkin method is chosen as the PCE method in order to formulate a more accurate analytical result. The derivation systematically increases from one uncertain load cell to all four uncertain load cells noting the differences and increased complexity as the uncertainty dimensions increase. For each derivation the PCE model is shown and the solution to the simulation is given. Results are presented comparing the polynomial chaos simulation to the Monte Carlo simulation and to the accurate scales. It is shown that the PCE simulations closely match the Monte Carlo simulations. / Master of Science