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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.

Předkládaná práce se zabývá návrhem modelu dynamiky paralelního manipulátoru optimálního pro účely návrhu řízení. Zvolený přístup je založen na modelování dynamiky systému v simulačním prostředí Matlab SimMechanics následovaném linearizací modelu. Výsledný stavový lineární model mimo jiné umožňuje snadné posouzení řiditelnosti a pozorovatelnosti modelu. Díky své relativní jednoduchosti je model také výpočetně nenáročný. Přístup je demonstrován na návrhu dvouvrstvého řízení SimMechanics modelu Stewartovy platformy, na kterém bylo následně navržené řízení úspěšně testováno. Podstatná část práce obsahuje přístup k modelování neurčitých parametrů dynamického modelu Stewartovy platformy a stejnosměrného motoru Maxon RE 35 a jeho výsledky. Předložený přístup je založen na modelování parametrické neurčitosti způsobem, kdy je neurčitost definována individuálně pro jednotlivé prvky stavových matic modelu. Samotná neurčitost je potom určena rozdílem mezi jednotlivými parametry příslušných matic nominálního modelu a modelu se stanovenou maximální neurčitostí parametrů. Výsledný neurčitostní model je vzhledem ke své stavové reprezentaci vhodný pro návrh regulátoru založeném na metodách návrhu robustního řízení, například minimalizaci normy H-nekonečno. Popsaná metoda byla použita pro kompenzaci posunu mezi pracovními body, okolo kterých je prováděna linearizace a pro kompenzaci nepřesnosti modelování vybraných parametrů modelů Stewartovy platformy a stejnosměrného motoru. Získané modely (v prostředí SimMechanics a neurčitostní model) byly experimentálně porovnány s chováním jednoho z lineárních pohonů Stewartovy platformy. Rozdíl v datech obdržených ze simulace v prostředí SimMechanics a naměřených na reálném stroji byl téměř kompletně pokryt neurčitostním modelem. Prezentovaná metoda neurčitostního modelování je velice univerzální a aplikovatelná na libovolný stavový model.

Fault detection and diagnosis have gained central importance in the chemical process industries over the past decade. This is due to several reasons, one of them being that copious amount of data is available from a large number of sensors in process plants. Moreover, since industrial processes operate in closed loop with appropriate output feedback to attain certain performance objectives, instrument faults have a direct effect on the overall performance of the automation system. Extracting essential information about the state of the system and processing the measurements for detecting, discriminating, and identifying abnormal readings are important tasks of a fault diagnosis system. The goal of this dissertation is to develop such fault diagnosis systems, which use limited information about the process model to robustly detect, discriminate, and reconstruct instrumentation faults. Broadly, the proposed method consists of a novel nonlinear state and parameter estimator coupled with a fault detection, discrimination, and reconstruction system. The first part of this dissertation focuses on designing fault diagnosis systems that not only perform fault detection and isolation but also estimate the shape and size of the unknown instrument faults. This notion is extended to nonlinear processes whose structure is known but the parameters of the process are a priori uncertain and bounded. Since the uncertainty in the process model and instrument fault detection interact with each other, a novel two-time scale procedure is adopted to render overall fault diagnosis. Further, some techniques to enhance the convergence properties of the proposed state and parameter estimator are presented. The remaining part of the dissertation extends the proposed model-based fault diagnosis methodology to processes for which first principles modeling is either expensive or infeasible. This is achieved by using an empirical model identification technique called subspace identification for state-space characterization of the process. Finally the proposed methodology for fault diagnosis has been applied in numerical simulations to a non-isothermal CSTR (continuous stirred tank reactor), an industrial melter process, and a debutanizer plant.

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Les travaux de cette thèse portent sur l’analyse de stabilité et la synthèse de commandes robustes pour les systèmes linéaires à commutation en temps discret avec des incertitudes polytopiques et des incertitudes sur la loi de commutation. On considère des lois de commutations arbitraires et on montre que l’utilisation des fonctions de Lyapunov commutées dépendant de paramètres permet de déterminer des critères de stabilité et de stabilisation robuste moins conservatifs. Ensuite, des conditions de stabilité robuste pour les systèmes en temps discret avec une loi de commutation incertaine sont présentées en termes de temps minimum de séjour. Les résultats obtenus s’avèrent utiles dans le contexte de la commande numérique des systèmes continus en présence d’imprécisions sur les instants d’échantillonnage et d’application des commandes. Nous montrons comment une modélisation à base d’évènements permet de ramener le problème original à un problème spécifique aux systèmes à commutation avec des incertitudes polytopiques. Les résultats sont étendus au cas des systèmes à commutation continus commandés par des correcteurs numériques / This PhD thesis is dedicated to the study of robust stability analysis and control synthesis for discrete time uncertain switching systems under arbitrary switching. Polytopic uncertainties are considered. We show that Lyapunov functions that depend on the uncertain parameter and that take into account the structure of the system may be used in order to reduce the conservatism related to uncertainty problems. Next, we consider the case of discrete time switched systems that are stabilized by a switched state feedback for which the switching signal may be temporary uncertain. Dwell time conditions for stability analysis of such systems are given. These results are usefull in the context of continuous time are stabilized via a computer when uncertainties occur on the sampling and actuation events. We present a new event based discrete-time model and we show that the stabilizability of this system can be achieved by finding a control for a switched polytopic system. The methodology is extended to the case of switched system

The engineered barrier system is a basic element in the design of repository to isolate high level radioactive waste (HLW). In this system, the clay barrier plays a prominent role in dispersing the heat generated from the waste, reduce the flow of pore water from the host rock, and maintaining the structural stability of the waste canister. The compacted expansive clay (generally bentonite blocks) is initially in unsaturated state. During the life time of the repository, the barrier will undergo different coupled thermal, hydrological and mechanical (THM) phenomena due to heating (from the heat-emitting nuclear waste) and hydration (from the saturated host rock). The design of nuclear waste disposal requires the prediction of the long term barrier behavior (i.e. hundred or thousand years), so numerical modeling is a basic component of the repository design. The numerical analyses are performed using mathematical THM formulation and the associated numerical code. Constitutive models are an essential part of the numerical simulations. Those constitutive models represent the intrinsic behavior of the material for the individual physical phenomenon (i.e. thermal, hydraulic and mechanical). Deterministic analyses have shown the potential of such mathematical formulations to describe the physical behavior of the engineered barrier system. However, the effect of the inherent uncertainties associated with the different constitutive models on the global behavior of the isolation system has not been explored yet. The first part of this thesis is related to application of recent probabilistic methods to understand and assess the impact of uncertainties on the global THM model response. Experimental data associated with the FEBEX project has been adopted for the case study presented in this thesis. CODE_BRIGHT, a fully coupled THM finite element program, is used to perform the numerical THM analysis. The second part of this thesis focuses on the complex mechanical behavior observed in a barrier material subjected (during 5 years) to heating and hydration under actual repository conditions The studied experiment is the (ongoing) full scale in-situ FEBEX test at Grimsel test site, Switzerland. A partial dismantling of this experiment has allowed the inspection of the barrier material subjected to varying stresses due to hydration and heating. The clay underwent both elastic and plastic volumetric deformations at different suction and temperature levels with changes in the pre-consolidation pressure and voids ratio that are difficult to explain with conventional models. In this thesis a double structure elasto plastic model is proposed to study the mechanical behavior of this barrier material. The numerical modeling was performed with CODE_BRIGHT. The study shows that the double structure model explains satisfactorily the observed changes in the mechanical behavior of the clay material.

Mechanical systems operate under parametric and external excitation uncertainties. The polynomial chaos approach has been shown to be more efficient than Monte Carlo approaches for quantifying the effects of such uncertainties on the system response. This work uses the polynomial chaos framework to develop new methodologies for the simulation, parameter estimation, and control of mechanical systems with uncertainty. This study has led to new computational approaches for parameter estimation in nonlinear mechanical systems. The first approach is a polynomial-chaos based Bayesian approach in which maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. The second approach is based on the Extended Kalman Filter (EKF). The error covariances needed for the EKF approach are computed from polynomial chaos expansions, and the EKF is used to update the polynomial chaos representation of the uncertain states and the uncertain parameters. The advantages and drawbacks of each method have been investigated. This study has demonstrated the effectiveness of the polynomial chaos approach for control systems analysis. For control system design the study has focused on the LQR problem when dealing with parametric uncertainties. The LQR problem was written as an optimality problem using Lagrange multipliers in an extended form associated with the polynomial chaos framework. The solution to the Hâ problem as well as the H2 problem can be seen as extensions of the LQR problem. This method might therefore have the potential of being a first step towards the development of computationally efficient numerical methods for Hâ design with parametric uncertainties. I would like to gratefully acknowledge the support provided for this work under NASA Grant NNL05AA18A. / Ph. D.