Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.

Fault detection in nonlinear systems: an observer-based approach

An un-permitted deviation of at least one characteristic property or parameter of a system from standard condition is referred as a fault. Faults result in reduced efficiency of the system, reduced quality of the product, and sometimes complete breakdown of the process. This not only causes economic losses but may also result in fatalities. An early detection of faults can assist to avert these losses. Therefore, fault detection and process monitoring is becoming an essential part of modern control systems. Fault detection in linear dynamical systems has been extensively studied and well established techniques exist in the literature. However, fault detection for nonlinear dynamical systems is yet an active field of research. This work is motivated by the fact that most of real systems are nonlinear in nature and there is a need to develop fault detection techniques for nonlinear systems. Observer-based methods for fault detection have proven to be among the most capable approaches, therefore, this research is focused towards these methods. The first step in observer-based fault detection is to generate a symptom signal, called the residual signal, which carries the information of faults. This is done by comparing the measurements from the process to their estimates generated by an observer (filter). It is desired that the residual signal is sensitive to faults and robust against disturbances. This research presents new methods for designing observer (filter) to generate residual signal which is sensitive to faults and robust against disturbances. Three types of filters are proposed in this dissertation; these include a fault sensitive filter, disturbance attenuating filter, and a filter to achieve simultaneous attenuation of disturbances and amplification of faults. Despite the disturbance attenuation property of the proposed filters, the residual signal is not completely decoupled from the effect of disturbances and uncertainties. Therefore, a threshold is needed to care for the effect of disturbances and uncertainties. Selection of threshold plays an important role in the performance of the fault detection system. If it is selected too high, some faults will not be detected. Conversely, if it is selected too low, disturbances and uncertainties will result in false alarms. This research presents a new method to determine the threshold to avoid false-alarms and to minimize missed-detections. A threshold generator is proposed which is itself a dynamic system and produces a variable threshold. This threshold changes with the effects of uncertainties and disturbances and fits more tightly to the fault-free residual signal and, hence, the performance of fault detection system is improved. In addition to the residual generation stage, the efficiency of a fault detection system can also be optimized by post-filtering. A further contribution of this research is in proposing a post-filter which operates on the residual signal to generate a modified residual signal. This modified residual signal is simultaneously sensitive to faults and robust against disturbances. Together with this post-filter, a strategy is adopted to select a threshold which maximizes the fault detectability and minimizes the number of false-alarms

H2 and H∞ Filtering for Nonlinear Singular Systems

RÉSUMÉ Dans les dernières années, les systèmes singuliers des équations différentielles ont carrément explosé puisqu’on les trouve dans plusieurs champs d’applications allant des systèmes électromécaniques en passant par des circuits électroniques, réacteurs chimiques et/ou biologiques ainsi que les systèmes d’écoulement des fluides. Dans cette thèse, deux classes des systèmes singuliers non linéaires seront considérer, en l’occurrence : (i) systèmes singuliers perturbés, (ii) systèmes généralisés ou systèmes algébro-différentielles. Les techniques H2 et H∞ pour l’estimation de l’état de ces classes seront développés ainsi que des conditions suffisantes pour la résolution des problèmes en termes des équations d’Hamilton-Jacobi seront présentés. Deux systèmes, temps-continu et discrets, seront considérés et, pour plus de viabilité des résultats, des exemples pratiques seront présentés et résolus.----------ABSTRACT Singular systems of differential equations arise in many areas of science and technology, including electro-mechanical systems, electronic circuits, chemical and biological reactors, and fluid flow systems. In this thesis, two classes of singular nonlinear systems are considered; namely, (i) singularly perturbed systems, and (ii) generalized systems, or descriptor, or differential-algebraic systems. H2 and H∞ techniques for state estimation of these classes of systems are developed, and sufficient conditions for the solvability of the problems in terms of Hamilton-Jacobi equations are presented. Both continuous-time and discrete-time systems are considered, and examples are presented to show the usefulness of the results