Quantifying chaos: practical estimation of the correlation dimension

Be it a physical object or a mathematical model, a nonlinear dynamical system can display complicated aperiodic behavior, or "chaos." In many cases, this chaos is associated with motion on a strange attractor in the system's phase space. And the dimension of the strange attractor indicates the effective number of degrees of freedom in the dynamical system.

In this thesis, we investigate numerical issues involved with estimating the dimension of a strange attractor from a finite time series of measurements on the dynamical system.

Of the various definitions of dimension, we argue that the correlation dimension is the most efficiently calculable and we remark further that it is the most commonly calculated. We are concerned with the practical problems that arise in attempting to compute the correlation dimension. We deal with geometrical effects (due to the inexact self-similarity of the attractor), dynamical effects (due to the nonindependence of points generated by the dynamical system that defines the attractor), and statistical effects (due to the finite number of points that sample the attractor). We propose a modification of the standard algorithm, which eliminates a specific effect due to autocorrelation, and a new implementation of the correlation algorithm, which is computationally efficient.

Finally, we apply the algorithm to chaotic data from the Caltech tokamak and the Texas tokamak (TEXT); we conclude that plasma turbulence is not a low- dimensional phenomenon.

Estimation and inference for grasping and manipulation tasks using vision and kinesthetic sensors

This thesis presents a novel framework for state estimation in the context of robotic grasping and manipulation. The overall estimation approach is based on fusing various visual cues for manipulator tracking, namely appearance and feature-based, shape-based, and silhouette-based visual cues. Similarly, a framework is developed to fuse the above visual cues, but also kinesthetic cues such as force-torque and tactile measurements, for in-hand object pose estimation. The cues are extracted from multiple sensor modalities and are fused in a variety of Kalman filters.

A hybrid estimator is developed to estimate both a continuous state (robot and object states) and discrete states, called contact modes, which specify how each finger contacts a particular object surface. A static multiple model estimator is used to compute and maintain this mode probability. The thesis also develops an estimation framework for estimating model parameters associated with object grasping. Dual and joint state-parameter estimation is explored for parameter estimation of a grasped object's mass and center of mass. Experimental results demonstrate simultaneous object localization and center of mass estimation.

Dual-arm estimation is developed for two arm robotic manipulation tasks. Two types of filters are explored; the first is an augmented filter that contains both arms in the state vector while the second runs two filters in parallel, one for each arm. These two frameworks and their performance is compared in a dual-arm task of removing a wheel from a hub.

This thesis also presents a new method for action selection involving touch. This next best touch method selects an available action for interacting with an object that will gain the most information. The algorithm employs information theory to compute an information gain metric that is based on a probabilistic belief suitable for the task. An estimation framework is used to maintain this belief over time. Kinesthetic measurements such as contact and tactile measurements are used to update the state belief after every interactive action. Simulation and experimental results are demonstrated using next best touch for object localization, specifically a door handle on a door. The next best touch theory is extended for model parameter determination. Since many objects within a particular object category share the same rough shape, principle component analysis may be used to parametrize the object mesh models. These parameters can be estimated using the action selection technique that selects the touching action which best both localizes and estimates these parameters. Simulation results are then presented involving localizing and determining a parameter of a screwdriver.

Lastly, the next best touch theory is further extended to model classes. Instead of estimating parameters, object class determination is incorporated into the information gain metric calculation. The best touching action is selected in order to best discern between the possible model classes. Simulation results are presented to validate the theory.

New directions in sparse sampling and estimation for underdetermined systems

A central objective in signal processing is to infer meaningful information from a set of measurements or data. While most signal models have an overdetermined structure (the number of unknowns less than the number of equations), traditionally very few statistical estimation problems have considered a data model which is underdetermined (number of unknowns more than the number of equations). However, in recent times, an explosion of theoretical and computational methods have been developed primarily to study underdetermined systems by imposing sparsity on the unknown variables. This is motivated by the observation that inspite of the huge volume of data that arises in sensor networks, genomics, imaging, particle physics, web search etc., their information content is often much smaller compared to the number of raw measurements. This has given rise to the possibility of reducing the number of measurements by down sampling the data, which automatically gives rise to underdetermined systems.

In this thesis, we provide new directions for estimation in an underdetermined system, both for a class of parameter estimation problems and also for the problem of sparse recovery in compressive sensing. There are two main contributions of the thesis: design of new sampling and statistical estimation algorithms for array processing, and development of improved guarantees for sparse reconstruction by introducing a statistical framework to the recovery problem.

We consider underdetermined observation models in array processing where the number of unknown sources simultaneously received by the array can be considerably larger than the number of physical sensors. We study new sparse spatial sampling schemes (array geometries) as well as propose new recovery algorithms that can exploit priors on the unknown signals and unambiguously identify all the sources. The proposed sampling structure is generic enough to be extended to multiple dimensions as well as to exploit different kinds of priors in the model such as correlation, higher order moments, etc.

Recognizing the role of correlation priors and suitable sampling schemes for underdetermined estimation in array processing, we introduce a correlation aware framework for recovering sparse support in compressive sensing. We show that it is possible to strictly increase the size of the recoverable sparse support using this framework provided the measurement matrix is suitably designed. The proposed nested and coprime arrays are shown to be appropriate candidates in this regard. We also provide new guarantees for convex and greedy formulations of the support recovery problem and demonstrate that it is possible to strictly improve upon existing guarantees.

This new paradigm of underdetermined estimation that explicitly establishes the fundamental interplay between sampling, statistical priors and the underlying sparsity, leads to exciting future research directions in a variety of application areas, and also gives rise to new questions that can lead to stand-alone theoretical results in their own right.

Intramolecular conflict : conformation and self-assembly of architecturally complex macromolecules in solution

The solution behavior of linear polymer chains is well understood, having been the subject of intense study throughout the previous century. As plastics have become ubiquitous in everyday life, polymer science has grown into a major field of study. The conformation of a polymer in solution depends on the molecular architecture and its interactions with the surroundings. Developments in synthetic techniques have led to the creation of precision-tailored polymeric materials with varied topologies and functionalities. In order to design materials with the desired properties, it is imperative to understand the relationships between polymer architecture and their conformation and behavior. To meet that need, this thesis investigates the conformation and self-assembly of three architecturally complex macromolecular systems with rich and varied behaviors driven by the resolution of intramolecular conflicts. First we describe the development of a robust and facile synthetic approach to reproducible bottlebrush polymers (Chapter 2). The method was used to produce homologous series of bottlebrush polymers with polynorbornene backbones, which revealed the effect of side-chain and backbone length on the overall conformation in both good and theta solvent conditions (Chapter 3). The side-chain conformation was obtained from a series of SANS experiments and determined to be indistinguishable from the behavior of free linear polymer chains. Using deuterium-labeled bottlebrushes, we were able for the first time to directly observe the backbone conformation of a bottlebrush polymer which showed self-avoiding walk behavior. Secondly, a series of SANS experiments was conducted on a homologous series of Side Group Liquid Crystalline Polymers (SGLCPs) in a perdeuterated small molecule liquid crystal (5CB). Monodomain, aligned, dilute samples of SGLCP-b-PS block copolymers were seen to self-assemble into complex micellar structures with mutually orthogonally oriented anisotropies at different length scales (Chapter 4). Finally, we present the results from the first scattering experiments on a set of fuel-soluble, associating telechelic polymers. We observed the formation of supramolecular aggregates in dilute (≤0.5wt%) solutions of telechelic polymers and determined that the choice of solvent has a significant effect on the strength of association and the size of the supramolecules (Chapter 5). A method was developed for the direct estimation of supramolecular aggregation number from SANS data. The insight into structure-property relationships obtained from this work will enable the more targeted development of these molecular architectures for their respective applications.

Value Estimation and Comparison in Multi-Attribute Choice

The following work explores the processes individuals utilize when making multi-attribute choices. With the exception of extremely simple or familiar choices, most decisions we face can be classified as multi-attribute choices. In order to evaluate and make choices in such an environment, we must be able to estimate and weight the particular attributes of an option. Hence, better understanding the mechanisms involved in this process is an important step for economists and psychologists. For example, when choosing between two meals that differ in taste and nutrition, what are the mechanisms that allow us to estimate and then weight attributes when constructing value? Furthermore, how can these mechanisms be influenced by variables such as attention or common physiological states, like hunger?

In order to investigate these and similar questions, we use a combination of choice and attentional data, where the attentional data was collected by recording eye movements as individuals made decisions. Chapter 1 designs and tests a neuroeconomic model of multi-attribute choice that makes predictions about choices, response time, and how these variables are correlated with attention. Chapter 2 applies the ideas in this model to intertemporal decision-making, and finds that attention causally affects discount rates. Chapter 3 explores how hunger, a common physiological state, alters the mechanisms we utilize as we make simple decisions about foods.

Force chains, friction, and flow: Behavior of granular media across length scales

We study the behavior of granular materials at three length scales. At the smallest length scale, the grain-scale, we study inter-particle forces and "force chains". Inter-particle forces are the natural building blocks of constitutive laws for granular materials. Force chains are a key signature of the heterogeneity of granular systems. Despite their fundamental importance for calibrating grain-scale numerical models and elucidating constitutive laws, inter-particle forces have not been fully quantified in natural granular materials. We present a numerical force inference technique for determining inter-particle forces from experimental data and apply the technique to two-dimensional and three-dimensional systems under quasi-static and dynamic load. These experiments validate the technique and provide insight into the quasi-static and dynamic behavior of granular materials.

At a larger length scale, the mesoscale, we study the emergent frictional behavior of a collection of grains. Properties of granular materials at this intermediate scale are crucial inputs for macro-scale continuum models. We derive friction laws for granular materials at the mesoscale by applying averaging techniques to grain-scale quantities. These laws portray the nature of steady-state frictional strength as a competition between steady-state dilation and grain-scale dissipation rates. The laws also directly link the rate of dilation to the non-steady-state frictional strength.

At the macro-scale, we investigate continuum modeling techniques capable of simulating the distinct solid-like, liquid-like, and gas-like behaviors exhibited by granular materials in a single computational domain. We propose a Smoothed Particle Hydrodynamics (SPH) approach for granular materials with a viscoplastic constitutive law. The constitutive law uses a rate-dependent and dilation-dependent friction law. We provide a theoretical basis for a dilation-dependent friction law using similar analysis to that performed at the mesoscale. We provide several qualitative and quantitative validations of the technique and discuss ongoing work aiming to couple the granular flow with gas and fluid flows.

Estimation of catalyst activity profiles and deactivation parameters from reactor operating data with applications to naphtha reforming

Techniques are developed for estimating activity profiles in fixed bed reactors and catalyst deactivation parameters from operating reactor data. These techniques are applicable, in general, to most industrial catalytic processes. The catalytic reforming of naphthas is taken as a broad example to illustrate the estimation schemes and to signify the physical meaning of the kinetic parameters of the estimation equations. The work is described in two parts. Part I deals with the modeling of kinetic rate expressions and the derivation of the working equations for estimation. Part II concentrates on developing various estimation techniques.

Part I: The reactions used to describe naphtha reforming are dehydrogenation and dehydroisomerization of cycloparaffins; isomerization, dehydrocyclization and hydrocracking of paraffins; and the catalyst deactivation reactions, namely coking on alumina sites and sintering of platinum crystallites. The rate expressions for the above reactions are formulated, and the effects of transport limitations on the overall reaction rates are discussed in the appendices. Moreover, various types of interaction between the metallic and acidic active centers of reforming catalysts are discussed as characterizing the different types of reforming reactions.

Part II: In catalytic reactor operation, the activity distribution along the reactor determines the kinetics of the main reaction and is needed for predicting the effect of changes in the feed state and the operating conditions on the reactor output. In the case of a monofunctional catalyst and of bifunctional catalysts in limiting conditions, the cumulative activity is sufficient for predicting steady reactor output. The estimation of this cumulative activity can be carried out easily from measurements at the reactor exit. For a general bifunctional catalytic system, the detailed activity distribution is needed for describing the reactor operation, and some approximation must be made to obtain practicable estimation schemes. This is accomplished by parametrization techniques using measurements at a few points along the reactor. Such parametrization techniques are illustrated numerically with a simplified model of naphtha reforming.

To determine long term catalyst utilization and regeneration policies, it is necessary to estimate catalyst deactivation parameters from the the current operating data. For a first order deactivation model with a monofunctional catalyst or with a bifunctional catalyst in special limiting circumstances, analytical techniques are presented to transform the partial differential equations to ordinary differential equations which admit more feasible estimation schemes. Numerical examples include the catalytic oxidation of butene to butadiene and a simplified model of naphtha reforming. For a general bifunctional system or in the case of a monofunctional catalyst subject to general power law deactivation, the estimation can only be accomplished approximately. The basic feature of an appropriate estimation scheme involves approximating the activity profile by certain polynomials and then estimating the deactivation parameters from the integrated form of the deactivation equation by regression techniques. Different bifunctional systems must be treated by different estimation algorithms, which are illustrated by several cases of naphtha reforming with different feed or catalyst composition.