Faculty of Science, School of Statistics & Actuarial Science, MSC Dissertation / This thesis follows a direction of research that deals with the theoretical foundations of stochastic differential equations on manifolds and a geometric analysis of the fundamental equations in nonlinear filtering theory. We examine the importance of modern differential geometry in developing an invariant theory of stochastic processes on manifolds, which allow us to extend current filtering techniques to an important class of manifolds. Furthermore, these tools provide us with greater insight to the infinite-dimensional nonlinear filtering problem. In particular, we apply our geometric analysis to the so called unnormalized conditional density approach expounded by M. Zakai. We exploit the geometric setting to study the geometric and algebraic properties of the Zakai equation, which is a linear stochastic partial differential equation. In particular, we investigate the use of Lie algebras and group invariance techniques for dimension analysis and for the reduction of the Zakai equation. Finally, we utilize simulation to demonstrate the superiority of the Zakai equation over the extended Kalman filter for a passive radar tracking problem.

Doctor of Philosophy (PhD). / The term `Lax pair' refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous continuous systems such as the Korteweg de Vries equation and the Painleve equations all have integrable discrete analogues, which retrieve the original systems in the continuous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations. Infinite hierarchies of integrable equations can be constructed where each equation is related to adjacent members of the hierarchy and the order of the equations can be increased arbitrarily. After a literature review, the original material in this thesis is instigated by a completeness study that finds all possible Lax pairs of a certain type, including one for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg de Vries equation is subsequently reduced to several q-discrete Painleve equations, and the reductions are used to form Lax pairs for those equations. The series of reductions suggests the presence of a hierarchy of equations, where each equation is obtained by applying a recursion relation to an earlier member of the hierarchy, this is confirmed using expansions within the Lax pairs for the q-Painleve equations. Lastly, some explorations are included into fake Lax pairs, as well as sets of equivalent nonlinear equations with similar Lax pairs.

This thesis solves the stabilizing control of an autonomous motorcycle. The control of an autonomous motorcycle is a challenging and interesting problem in the field because the plant is under-actuated, unstable and nonlinear. Two major problems that have not been considered in the literature are explicitly solved in our work: (i) the robust control problem of the plant subject to uncertainty and exogenous disturbance; (ii) the non-local stabilization of the nonlinear plant. To achieve the first goal, we propose a robust H_infty controller based on the linearized system, which provides a significant improvement in dealing model uncertainty and disturbance attenuation in comparison with those controllers given by classical linear design tools. To achieve the second goal, we propose a nonlinear controller based on the combination of a nonlinear forwarding method with several other methods for the nonlinear plant through identifying an appropriate upper triangular s tructure of the nonlinear system. This yields a stability region, the whole upper space above the level ground, such that the trajectory starting from any position in the upper hemi-sphere with arbitrary initial velocities converges to the upright position. Both results are novel and first results of their kinds in control of an autonomous motorcycle. Computer simulations verify the effectiveness of the proposed controllers.

Response behavior of a nonlinear structural system subject to environmental loadings is investigated in this study. The system contains a nonlinear restoring force due to large geometric displacement. The external excitation is modeled as a narrowband stochastic process possessing dynamic characteristics of typical environmental loadings. A semi-analytical method is developed to predict the stochastic nonlinear response behavior under narrowband excitations in both the primary and the subharmonic resonance regions. Preservation of deterministic response characteristics under the narrowband random field is assumed. The stochastic system response induced by variations in the narrowband excitations is considered as a sequence of successive transient states. Due to the system nonlinearity, under a combination of excitation conditions, several response attraction domains may co-exist. Presence of co-existence of attraction domains and variations in the excitation amplitude often induce complex response inter-domain transitions. The response characteristics are found to be attraction domain dependent. Among different response attraction domains, their corresponding response amplitude domains overlap. In addition, within an individual attraction domain, response amplitude domains corresponding to different excitation amplitudes also overlap. Overlapping of response amplitude domains and the time-dependent variations in the excitation parameters induce response intra-domain transitions. Stationary Markovian assumption is employed to characterize the stochastic behavior of the response amplitude process and the excitation parameter processes. Based on the stochastic excitation properties and the deterministic response characteristics, governing equations of the response amplitude probability inter- and intra-domain transitions are formulated. Numerical techniques and an iteration procedure are employed to evaluate the stationary response amplitude probability distribution. The proposed semi-analytical method is validated by extensive numerical simulations. The capability of the method is demonstrated by good agreements among the predicted response amplitude distributions and the simulation results in both the primary and the subharmonic resonance regions. Variations in the stochastic response behavior under varying excitation bandwidth and variance are also predicted accurately. Repeated occurrences of various subharmonic responses observed in the numerical simulations are taken into account in the proposed analysis. Comparisons of prediction results with those obtained by existing analytical methods and simulation histograms show that a significant improvement in the prediction accuracy is achieved. / Graduation date: 1999

Stochastic analysis procedures have been recently applied to analyze nonlinear dynamical systems. In this study, nonlinear responses, stochastic and/or chaotic, are examined and interpreted from a probabilistic perspective. A multi-point-moored ocean structural system under regular and irregular wave excitations is analytically examined via a generalized stochastic Melnikov function and Markov process approach. Time domain simulations and associated experimental observations are employed to assist in the interpretation of the analytical predictions. Taking into account the presence of random noise, a generalized stochastic Melnikov function associated with the corresponding averaged system, where a homoclinic connection exists near the primary resonance, is derived. The effects of random noise on the boundary of regions of possible existence of chaotic response is demonstrated via a mean-squared Melnikov criterion. The random wave field is approximated as random perturbations on regular and nearly regular (with very narrow-band spectrum) waves by adding a white noise component, or using a filtered white noise process to fit the JONSWAP spectrum. A Markov process approach is then applied explicitly to analyze the response. The evolution of the probability density function (PDF) of nonlinear stochastic response under the Markov process approach is characterized by a deterministic partial differential equation called the Fokker-Planck equation, which in this study is solved by a path integral solution procedure. Numerical evaluation of the path integral solution is based on path sum, and the short-time propagator is discretized accordingly. Short-time propagation is performed by using a fourth order Runge-Kutta scheme to calculate the most probable (i.e. mean) position in the phase space and to establish the fact that discrete contributions to the random response are locally Gaussian. Transient and steady-state PDF's can be obtained by repeat application of the short-time propagation. Based on depictions of the joint probability density functions and time domain simulations, it is observed that the presence of random noise may expedite the occurrence of "noisy" chaotic response. The noise intensity governs the transition among various types of stochastic nonlinear responses and the relative strengths of coexisting response attractors. Experimental observations confirm the general behavior depicted by the analytical predictions. / Graduation date: 1995

Thesis (Ph. D.)--Washington State University, May 2007. / Includes bibliographical references.

Analysis of the dynamic behavior on structures is one vital aspect of designing structures such as buildings and bridges. Determination of the correct damping factor is of critical importance as it is the governing factor of dynamic design. Damping on structures exhibits a very complex behavior. Different models are suggested in literature to explain damping behavior. The usefulness of a valid damping model depends on how easily it can be adopted to analyze the dynamic behavior. Ease of mathematically representing the model and ease of analyzing the dynamic behavior by using the mathematical representation are the two determining aspects of the utility of the selected model. This thesis presents a parametric representation of non-linear damping models of the form presented by [Jea86] and the mathematical techniques to use the parametrically represented damping model in dynamic behavior analysis. In the damping model used in this thesis, the damping factor is proportional to the amplitude of vibration of the structure. However, determination of the amplitude again depends on the damping of the structure for a given excitation. Also, the equations which explain the behavior of motion are differential equations in a matrix form that is generally linearly inseparable. This thesis addresses these challenges and presents a numerical method to solve the motion equations by using Runge-Kutta techniques. This enables one to use a given non-linear model of the form proposed by [Jea86] to analyze the actual response of the structure to a given excitation from wind, seismic or any other source. Several experiments were conducted for reinforced concrete and steel framed buildings to evaluate the proposed framework. The non-linear damping model proposed by [Sat03], which conforms to [Jea86] is used to demonstrate the use of the proposed techniques. Finally, a new damping model is proposed based on the actual behavior and the serviceability criteria, which better explains the damping behavior of structures.

We study the geometric classification of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as defined by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (Ω,B,??).

This work presents a control algorithm developed from the mammalian emotional processing network. Emotions are processed by the limbic system in the mammalian brain. This system consists of several components that carry out different tasks. The system level understanding of the limbic system has been previously captured in a discrete event computational model. This computational model was modified suitably to be used as a feedback mechanism to regulate the output of a continuous-time first order plant. An extension to a class of nonlinear plants is also discussed. The combined system of the modified model and the linear plant are represented as a set of bilinear differential equations valid in a half space of the 3-dimensional real space. The bounding plane of this half space is the zero level of the square of the plant output. This system of equations possesses a continuous set of equilibrium points which lies on the bounding plane of the half space. The occurrence of a connected equilibrium set is uncommon in control engineering, and to prove stability for such cases one needs an extended Lyapunov-like theorem, namely LaSalle's Invariance Principle. In the process of using this Principle, it is shown that this set of equations possesses a first integral as well. A first integral is identified using the compatibility method, and this first integral is utilized to prove asymptotic stability for a region of the connect equilibrium set.

Sensorless control of the synchronous reluctance motor has been a topic of research for more than a decade, producing several successful methods to accomplish this goal. However, a technique that has been overlooked is the full-order nonlinear observer, which is essentially a software model of the motor driven by measurements from the actual motor. Presented in this thesis is the design, implementation, and experimental testing of a full-order observer-based sensorless control technique which requires only the phase current and voltage measurements that are typically available in standard three-phase inverters. A technique is also presented for calculating a table of observer feedback gains parameterized only by the steady-state motor speed. This allows a gain-scheduling observer to be implemented which, as shown using experiments, improves the transient response of the observer over a wide speed range. The sensorless controller consists of a full-order nonlinear observer coupled with an input-output linearization speed controller. The resulting controller was implemented in Simulink and executed on a dSPACE DS1103 real-time DSP board using the Real-Time Workshop extension to Simulink. A custom built three-phase IGBT inverter was used to interface the DSP to a 100 watt synchronous reluctance motor for laboratory testing. The resulting sensorless controller was able to successfully track a varying speed reference from 150 rpm to 1800 rpm with a tracking error under 5% for most of the speed range. At the lowest speeds, the tracking error begins to increase but the observer remains stable down to 150 rpm.