A multiscale approach to state estimation with applications in process operability analysis and model predictive control

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000.Includes bibliographical references.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.This thesis explores the application of multiscale ideas to the areas of state estimation and control. The work represents a significant departure from the traditional representations in the time and frequency domains, and provides a novel framework that leads to fast, efficient, and modular estimation algorithms. Multiscale methods were rediscovered through wavelet theory in the mid-eighties, as a tool for the geophysics community. Like Fourier theory, it provides a more instructive representation of data than time series alone, by decomposition into a different set of orthonormal basis functions. Multiscale models and data sets exist on multiscale trees of nodes. Each node represents a place holder corresponding to a time point in a time series. The nodes of a tree form a structure which may contain measurements, states, inputs, outputs, and uncertainties. Each level of the tree represents the set of data at a given level of resolution. This dual localization in time and frequency has benefits in the storage of information, since irrelevant data and pure noise can be identified and discarded. It also preserves time and frequency information in a way that Fourier theory cannot. Grouping and condensing of important information follows naturally, which facilitates the making of decisions at a level of detail relevant to the question being asked. Multiscale systems theory is a general approach for multiscale model construction on a tree. This thesis derives the multiscale models corresponding to the Haar transform, which produces a modified hat transform for input data. Autoregressive models, commonly used in time series analysis, give rise to multiscale models on the tree. These allow us to construct numerical algorithms that are effcient and parallelizable, and scale logarithmically with the number of data points, rather than the linear performance typical for similar time-series algorithms. This multiscale systems theory generalizes easily to other wavelet bases. Multiscale models of the underlying physics and the measurement model can be combined to construct a cost function which estimates the underlying physical states from a set of measurements. The resulting set of normal equations is sparse and contains a specialized structure, leading to a highly efficient solution strategy. A modified multiscale state estimation algorithm incorporates prior estimates, consistent with the Kalman filter, with which it is linked. A constrained multiscale state estimator incorporates constraints in the states, and in linear combinations of the states. All incarnations of the multiscale state estimator provide a framework for the optimal fusion of multiple sets of measurements, including those taken at different levels of resolution. This is particularly useful in estimation and control problems where measurement data and control strategies occur at multiple rates. The arbitrary size of the state allows for the use of higher order underlying physical models, without modification of the estimation algorithm. Finally, the algorithm accommodates an arbitrary specification of the uncertainty estimates at any combination of time points or level of resolution. The structure of the solution algorithm is sufficiently flexible to use the same intermediate variables for all of these modifications, leading to considerable reusability, both of code, and of prior calculations. Thus, the multiscale state estimation algorithm is modular and parallelizable. An uncertainty analysis of the algorithm represents state estimation error in terms of the underlying model and measurement uncertainties. Depending on the size of the problem, different techniques should be used to construct the probability distribution functions of the error estimates. This thesis demonstrates direct integration, propagation of the moments of the measurement and model errors, polynomial chaos expansions, and an approximation using Gaussian quadrature and Monte Carlo simulation. A sample of smaller case studies shows the range of uses of the algorithm. Three larger case studies demonstrate the multiscale state estimator in realistic chemical engineering examples. The terephthalic acid plant case study successfully incorporates a non-linear model of the first continuously stirred tank reactor into the multiscale state estimator. The paper-rolling case study compares the multiscale state estimator to the Karhunen-Loeve transform as a means of state estimation. Finally, the heavy oil fractionator of the Shell Control Problem demonstrates the multiscale state estimator in a control setting.by Matthew Simon Dyer.Ph.D