The effect of an applied electric potential on the dynamics of gas bubble formation from a single nozzle in glycerol was studied experimentally. Dry nitrogen was bubbled into glycerol through a nozzle having an electrified tip while pressure measurements were made upstream of the nozzle. As the applied electric potential increased from zero, bubble size reduced, bubble shape became more spherical, and bubbling frequency increased. At constant gas flow, bubble-formation exhibited a classic period-doubling route to chaos with increasing potential. Although electric potential and gas flow appear to have similar effects on the bifurcation process, the relative impact of electric potential is smaller. Nevertheless, it appears feasible to use electric potential as a control parameter for bubble formation dynamics. We also define an electric Bond number assuming that both the liquid and gas phases are conducting. This is in contrast to all previous studies where one phase was considered a perfect conductor and the other one a perfect non-conductor or insulator. Based on the derived electric Bond number, it is shown that the higher electric stress forces can be achieved with the same potential difference by placing the ground electrode closer to the nozzle.

Distillation is the most common unit operation used in the chemical engineering industry. As indicated and reinforced by practical experience, distillation columns must be operated at near-flooding conditions. However, there is no reliable way of anticipating flooding, which leads to much downtime and reduced productivity.

Our previous research with gas-liquid contacting operations has shown that even with a one-hole sparger, the behavior of gas bubbles through the liquid mass can be highly nonlinear and indeed chaotic. Therefore, it is worthwhile to study distillation with nonlinear analysis tools to probe if the behavior of gas bubbles can be better understood in the context of nonlinear dynamics and chaos.

This study focuses on pressure measurements collected on a laboratory distillation column, with the goal of exploring if nonlinear time series analysis tools can increase our understanding of flooding and help avoid it by anticipating it.

Five sensors were placed in the distillation column to record the gauge pressure over time. Data was collected for two liquid flow rates and four to five gas flow rates, with the goal of anticipating flooding. We use nonlinear and symbolization analysis in computing univariate and bivariate mutual information, and find that the general redundancy increases when approaching flooding. Other findings are also discussed.

In this paper we develop the concept of Cluster-linked principal curves which is a non-parametric technique for reducing the dimensionality while preserving most of the information in it. The algorithm is an adaptation of Hastie and Stuetzle’s Principal Curve algorithm, and differs from the latter significantly as it treats all dimensions symmetrically and equally. The computational cost of the CLPC algorithm is an order of magnitude lower than that for the principal curve or other similar algorithms. The algorithm has only one hyper-parameter which can be chosen based on informational complexity. For illustration, we use simulated and real data. The experimental data consists of delay embedding formed from the measurements on a chaotic bubble column. Measures to evaluate the fit of the algorithm are also developed. Finally, we outline how the distribution of arc lengths or projections on the curve can be used for process monitoring, fault diagnosis and forecasting. Other applications are also discussed.

Most chemical plants involve gas-liquid interaction in some form, e.g., distillation, absorption, etc. Though not readily apparent, the dynamics of bubble formation from a submerged nozzle or capillary is quite nonlinear and complex, and exhibits chaotic behavior for a range of gas and liquid flows. It has been recorded that a bubble column follows the period-doubling route to chaos as the gas flow rate is increased. Our recent research has shown that applying electrostatic potential across the gas injector nozzle also has a large impact on bubble formation. This work expands on our previous work by considering different nozzle geometries, orifice sizes and larger electric fields. Return maps and bifurcation diagrams are employed to depict and characterize the transitions in bubbling dynamics. It was observed that increasing the gas flow rate and potential increases the bubbling rate much more for capillaries than it does for other nozzle geometry studied. We find that the capillaries are much more influenced by electric forces, and that for small orifice capillaries the potential of 4 to 5 kV is sufficient to transform the bubbling dynamics from a simple period-1 to chaotic. The movement of the attractor in response to electrostatic potential is substantial and thus affords the possibility of implementing bubble control using electrostatic potential as a manipulated variable, which is expected to be much faster than that achieved by manipulating flow rates or acoustic fluctuations. However, it was also found that the impact of electrostatic potential on bubbling dynamics is complex and increasing flow rate and potential do not always increase bubbling rate but sometimes decrease it and cause the chaotic dynamics to revert to periodic behavior. The sensitivity of bubbling through capillaries is discussed, and we close by outlining and recommending future work.

Two approaches to characterize global dynamics are developed in this dissertation. In particular, the concern is with nonlinear and chaotic time series obtained from physical systems. The objective is to identify the features that adequately characterize a time series, and can consequently be used for fault diagnosis and process monitoring, and for improved control.

This study has two parts. The first part is concerned with obtaining a skeletal description of the data using Cluster-linked principal curves (CLPC). A CLPC is a non-parametric hypercurve that passes through the center of the data cloud, and is obtained through the iterative Expectation-Maximization (E-M) principle. The data points are then projected on the curve to yield a distribution of arc lengths along it. It is argued that if some conditions are met, the arc length distribution uniquely characterizes the dynamics. This is demonstrated by testing for stationarity and reversibility based on the arc length distributions.

The second part explores the use of mutual information vector to characterize a system. The mutual information vector formed via symbolization is reduced in dimensionality and subjected to K-means clustering algorithm in order to examine stationarity and to compare different processes.

The computations required to implement the techniques for online monitoring and fault diagnosis are reasonable enough to be carried out in real time. For illustration purposes time series measurements from a liquid-filled column with an electrified capillary and a fluidized bed are employed.

Most chemical plants involve gas-liquid interaction in some form –e.g., distillation, absorption, etc. Though not readily apparent, the dynamics of bubble formation from a submerged nozzle or capillary is quite nonlinear and complex, and exhibits chaotic behavior for a range of gas and liquid flows. In fact, there is an unmistakable qualitative similarity between the bubble formation dynamics and the classic dripping faucet experiment. Similarities between bubble column behavior and fluidized bed dynamics have also been noted. It is useful and desirable to operate the bubble column at higher flow rates in order to obtain better mixing and lower required residence times. The bubble formation dynamics at higher flow rates though, is apparently unpredictable and complex. Our concern is to identify and characterize the bubble column, and consequently to control the dynamics. It has been recorded that a bubble column follows the period-doubling route to chaos as the gas flow rate is increased. Our recent research has shown that applying electrostatic potential across the gas injector nozzle also has a large impact on bubble formation. We discuss how electrostatic potential, nozzle design and orifice size impact bubble formation and affect period bifurcation and chaos. Elevated electrostatic potentials have shown the ability to cause bifurcations at lower gas flow rates. This movement of the attractor in response to electrostatic potential changes demonstrates the possibility of implementing bubble control using electrostatic potential as a manipulated variable. Our analysis is based on the data we have collected for various different gas flow rates, electrostatic potentials and three different capillaries and a ‘button’ nozzle. Bifurcation diagrams are detailed. Time-based Return maps (variants of Poincaré sections) are employed to depict and characterize the transitions in bubbling dynamics as parameters are changed. Occurrences of shifts in dynamics (non-stationarity) at constant operating conditions are noted. We also discuss how one could apply Ott-Grebogi-Yorke (OGY) Control scheme on the chaotic bubbling column.

Principal Component Analysis (PCA) is a powerful technique for reducing the dimensionality of and removing noise from multivariable data. However, it is not useful in the context of nonlinear systems since it exploits only linear correlation. A global model, no matter how complex, cannot adequately approximate a chaotic system. It is possible though to model the latter with locally linear models. Such a model is called principal curve, or a generalized principal component. We adapt and expand the algorithm suggested by Hastie and Stuetzle(1989) to develop a non-parametric approach to locally approximate the structure and scatter of a PDF. The iterative algorithm involves a Projection step followed by Conditional Expectation (C-E) step. The former involves projecting the observations on the principal curve, and the latter employs a variant of scatterplot smoothing. We demonstrate how our algorithm can be used to estimate the probability density of an attractor and to remove noise by reducing the effective dimension of the system or space. We also discuss testing for stationarity and reversibility in its framework. For illustration we use the (time-based) return maps and embedding space formed by the measurements collected from an electrified capillary and a fluidized bed. Our approach is unique as it bypasses the potentially costly computations associated with training a Neural network or maximizing the likelihood ratio, and yet makes no distributional assumptions.

Chaotic systems are characterized by exponential divergence of nearby trajectories, and this property of theirs excludes the use of classic time series modeling methods like ARMA for chaotic time series data. In order to better characterize the dynamics of a nonlinear system, methods relying on Takens’ embedding theorem need to be used. Takens (1981) establisdarkRowC1 the upper limit for the embedding dimension for reconstruction of attractor geometry. Often the upper limit provided by Takens’ theorem is too high. A good way to characterize the dimensionality of data is to consider the number of local principal components that adequately describe the system.

The optimum number of principal components required to represent the embedding system above should suggest the dimension of the system under study. Also, in cases of noisy time series, it has been shown that neighborhood-based smoothing as discussed by Kantz and Schreiber (1997) is quite useful.

However, usually deciding upon the optimal parameter requires computation of many correlation integrals and human decision making. It definitely is worthwhile to see if these selection processes could be, as it were, automated. It is with this goal in mind that we undertook this study. We use the informational complexity framework developed by Bozdogan (1987, 1994) to choose the optimum number of principal components to be retained as well as to decide upon an optimum radius of neighborhood or kernel radius. This method relies on the concept of informational complexity in choosing between the various models presented.

We use pressure measurements from a laboratory fluidized bed that was known to be showing chaotic dynamics. The data were described and discussed in Daw et al (1999). The measurement available is the differential pressure between two points in the fluidized bed.

This talk is based on my doctoral dissertation, which is directed towards finding ways to characterize a dynamical system based on the observations or measurements made on it over the course of time. We focus our attention on systems that are too complex to model, so that the most important source of information about them is the time series they generate. A system or process is uniquely identified by its generating function that can be seen as the (infinite dimensional) joint probability distribution of measurements. It is very desirable to characterize the probability distribution accurately in far fewer dimensions; unfortunately there are no clear-cut methods to uniquely identify nonlinear and chaotic systems in presence of healthy amounts of noise. Thus, our goal is to identify a sufficient pattern vector for dynamics (in as much as it can be reconstructed from the series). To that end we explore nonlinear dynamical techniques like Poincaré maps and return maps; information theoretical quantities like mutual information, and other quantities for their potential in characterizing system dynamics.

In this talk, we discuss how return maps or Poincaré sections, Principal curves (piecewise linear estimators or smooth splines), and mutual information, followed by hard or soft clustering fare in distinguishing between different dynamics. We use the methods to detect dynamical changes (non-stationarity) in a time series, and to distinguish different dynamic modes. The methods, besides detecting non-stationarity, can be used for Fault Diagnosis, Process Monitoring, and have useful implications for improving Control. The methods have also been successfully applied to streams of images, or movies.

We will also show a sophisticated GUI-based MATLAB software that was developed for the Measurement and Control Engineering Center (MCEC) as a part of Control and Nonlinear Dynamics for Industrial Engineering Systems (CANDIES) project. MCEC CANDIES project provided financial support for doctoral study.

Two methods based on nonlinear features (return maps) are developed and implemented to compare nonlinear and chaotic system states. In the first method, we compare the structure of data points on the return maps by comparing the means and autocovariance matrices, that capture, respectively, the shifts in mean and changes in structure. In the second method, we utilize the power of principal curves or nonlinear PCA (NLPCA) to approximate the "curve" on the return map. PDFs of Nonlinear Principal Scores (NLPS) can be compared using the classic c2 statistic. Furthermore, the NLPS reveal important information about the system dynamics (period-2 or -4, e.g.). Comparisons of NLPS prove to be very accurate in detecting structural changes and movements. The Q statistic can also be used as a measure of dissimilarity between two process states. The methods presented are very general and can be applied to almost any time series measurements.

The methods are used for testing stationarity, and extended to cover process monitoring, event detection and fault diagnosis; and have important implications for control. In this paper we use the time series data from a nonlinear bubble column for illustration.

In this work a new method for recognizing and classifying images based on concepts derived from Logical Combinatorial Pattern Recognition (LCPR) is introduced. The concept of typical segment descriptor is given with an algorithm for calculating all of these descriptors from any chain code representation: Freeman chain code, first derivative, vertex chain code, Bribiesca 3D-chain code and any other chain code representation. Also, algorithms for recognizing known images and classifying new images based on typical segment descriptors from their chain code representations are given. Several similarity measures between images are introduced. An interpretation of typical segment descriptor in terms of a mask for segmenting a known image in a scene is given. An algorithm for searching a known image in a non-segmented scene also is introduced. Several examples show the behavior of the proposed method and algorithms. All of these procedures work with not occluded and occluded images. However, the quality of the recognition is not the same in the last case.