One of the major tasks in electronic circuit design is the ability to predict the performance of general circuits in the presence of uncertainty in key design parameters. In the mathematical literature, such a task is referred to as uncertainty quantification. Uncertainty about the key design parameters arises mainly from the difficulty of controlling the physical or geometrical features of the underlying design, especially at the nanometer level. With the constant trend to scale down the process feature size, uncertainty quantification becomes crucial in shortening the design time. To achieve the uncertainty quantification, this thesis presents a new approach based on the concept of generalized Polynomial Chaos (gPC) to perform variability analysis of general nonlinear circuits. The proposed approach is built upon a decoupling formulation of the Galerkin projection (GP) technique, where the large matrix is transformed into a block-diagonal whose diagonal blocks can be factorized independently. The proposed methodology provides a general framework for decoupling the GP formulation based on a general system of orthogonal polynomials. Moreover, it provides a new insight into the error level that is caused by the decoupling procedure, enabling an assessment of the performance of a wide variety of orthogonal polynomials. For example, it is shown that, for the same order, the Chebyshev polynomials outperforms other commonly used gPC polynomials.

Ce travail de thèse porte sur l’analyse robuste et l’optimisation fiabiliste des problèmes vibro-acoustiques (ou en interaction fluide-structure) en tenant en compte des incertitudes des paramètres d’entrée. En phase de conception et de dimensionnement, il parait intéressant de modéliser les systèmes vibro-acoustiques ainsi que leurs variabilités qui peuvent être essentiellement liées à l’imperfection de la géométrie ainsi qu’aux caractéristiques des matériaux. Il est ainsi important, voire indispensable, de tenir compte de la dispersion des lois de ces paramètres incertains afin d’en assurer une conception robuste. Par conséquent, l’objectif est de déterminer les capacités et les limites, en termes de précision et de coûts de calcul, des méthodes basées sur les développements en chaos polynomiaux en comparaison avec la technique référentielle de Monte Carlo pour étudier le comportement mécanique des problèmes vibro-acoustique comportant des paramètres incertains. L’étude de la propagation de ces incertitudes permet leur intégration dans la phase de conception. Le but de l’optimisation fiabiliste Reliability-Based Design Optimization (RBDO) consiste à trouver un compromis entre un coût minimum et une fiabilité accrue. Par conséquent, plusieurs méthodes, telles que la méthode hybride (HM) et la méthode Optimum Safety Factor (OSF), ont été développées pour atteindre cet objectif. Pour remédier à la complexité des systèmes vibro-acoustiques comportant des paramètres incertains, nous avons développé des méthodologies spécifiques à cette problématique, via des méthodes de méta-modèlisation, qui nous ont permis de bâtir un modèle de substitution vibro-acoustique, qui satisfait en même temps l’efficacité et la précision du modèle. L’objectif de cette thèse, est de déterminer la meilleure méthodologie à suivre pour l’optimisation fiabiliste des systèmes vibro-acoustiques comportant des paramètres incertains. / This PhD thesis deals with the robust analysis and reliability optimization of vibro-acoustic problems (or fluid-structure interaction) taking into account the uncertainties of the input parameters. In the design and dimensioning phase, it seems interesting to model the vibro-acoustic systems and their variability, which can be mainly related to the imperfection of the geometry as well as the characteristics of the materials. It is therefore important, if not essential, to take into account the dispersion of the laws of these uncertain parameters in order to ensure a robust design. Therefore, the purpose is to determine the capabilities and limitations, in terms of precision and computational costs, of methods based on polynomial chaos developments in comparison with the Monte Carlo referential technique for studying the mechanical behavior of vibro-acoustic problems with uncertain parameters. The study of the propagation of these uncertainties allows their integration into the design phase. The goal of the reliability-Based Design Optimization (RBDO) is to find a compromise between minimum cost and a target reliability. As a result, several methods, such as the hybrid method (HM) and the Optimum Safety Factor (OSF) method, have been developed to achieve this goal. To overcome the complexity of vibro-acoustic systems with uncertain parameters, we have developed methodologies specific to this problem, via meta-modeling methods, which allowed us to build a vibro-acoustic surrogate model, which at the same time satisfies the efficiency and accuracy of the model. The objective of this thesis is to determine the best methodology to follow for the reliability optimization of vibro-acoustic systems with uncertain parameters.

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.

This research effort develops a comprehensive computational framework to support the parametric optimal design of uncertain dynamical systems. Uncertainty comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; not accounting for uncertainty may result in poor robustness, sub-optimal performance and higher manufacturing costs. Contemporary methods for the quantification of uncertainty in dynamical systems are computationally intensive which, so far, have made a robust design optimization methodology prohibitive. Some existing algorithms address uncertainty in sensors and actuators during an optimal design; however, a comprehensive design framework that can treat all kinds of uncertainty with diverse distribution characteristics in a unified way is currently unavailable. The computational framework uses Generalized Polynomial Chaos methodology to quantify the effects of various sources of uncertainty found in dynamical systems; a Least-Squares Collocation Method is used to solve the corresponding uncertain differential equations. This technique is significantly faster computationally than traditional sampling methods and makes the construction of a parametric optimal design framework for uncertain systems feasible. The novel framework allows to directly treat uncertainty in the parametric optimal design process. Specifically, the following design problems are addressed: motion planning of fully-actuated and under-actuated systems; multi-objective robust design optimization; and optimal uncertainty apportionment concurrently with robust design optimization. The framework advances the state-of-the-art and enables engineers to produce more robust and optimally performing designs at an optimal manufacturing cost. / Ph. D.