Browsing by Author "H. T. Banks, Member"

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    A Biologically-Based Controlled Growth and Differentiation Model Using Delay Differential Equations: Development, Applications and Stability Analysis.
    (2000-11-21) Whitaker, Shree Yvonne; H. T. Tran, Chair; H. T. Banks, Member; K. Ito, Member; C. J. Portier, Member
    This work investigates the development, applications and stability analysis of a biologically-based dose-response model for developmental toxicology. The biologically-based controlled growth and differentiation model is based on a model originally developed by Leroux et al. (1996). The original model had two basic states; precursor cells and differentiated cells with both states subject to a linear birth-death process. The research discussed in this dissertation describes the development of a mathematical model that is both biologically- and statistically-based. The model is developed with a highly controlled birth and death process for precursor cells. This model limits the number of replications allowed in the development of a tissue or organ and more closely reflects the presence of a true stem cell population. The mathematical formulation of the Leroux et al. (1996) model was derived from a partial differential equation for the generating function that limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model are also discussed.The versatility of the CGD model is highlighted through a discussion of two general applications. The normal developmental process of spermatocytogenesis is investigated as the first application. Time delays are introduced into the system to more accurately mimic the development of male germ cells. As the second application, the spermatocytogenesis model is then altered to demonstrate a modeling strategy for hormesis. Asymptotic stability is investigated using the system of delay differential equations for spermatocytogenesis. The direct Lyapunov method for linear differential equations without delay is modified to establish delay-dependent stability conditions for delay differential equations with multiple delays. The stability conditions are expressed in terms of the existence of a positive definite solution to the Riccati matrix equations. Numerical simulations further verify the stability conditions.
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    Modeling and Control of Thin Film Growth in a Chemical Vapor Deposition Reactor
    (2000-10-16) Beeler, Scott Colvin; H. T. Tran, Chair; H. T. Banks, Member; P. A. Gremaud, Member; K. J. Bachmann, Member
    This work describes the development of a mathematical model of ahigh-pressure chemical vapor deposition (HPCVD) reactor and nonlinearfeedback methodologies for control of the growth of thin films in thisreactor. Precise control of the film thickness and composition is highlydesirable, making real-time control of the deposition process veryimportant. The source vapor species transport is modeled by the standardgas dynamics partial differential equations, with species decomposition reactions, reduced down to a small number of ordinary differential equationsthrough use of the proper orthogonal decomposition technique. This systemis coupled with a reduced order model of the reactions on the surfaceinvolved in the source vapor decomposition and film deposition on thesubstrate wafer. Also modeled is the real-time observation technique usedto obtain a partial measurement of the deposition process. The utilization of reduced order models greatly simplifies the mathematical formulation of the physical process so it can be solved quickly enough to beused for real-time model-based feedback control. This control problem isfairly complicated, however, because the surface reactions render the modelnonlinear. Several control methodologies for nonlinear systems are studiedin this work to determine which performs best on test examples similar tothe HPCVD problem. One chosen method is extended to a tracking control toforce certain film growth properties to follow desired trajectories. Thenonlinear control method is used also in the development of a stateestimator which uses the nonlinear partial observation of the nonlinearsystem to create an estimate of the actual state, which the feedback controlformula then can use to guide the HPCVD system. The nonlinear trackingcontrol and estimator techniques are implemented on the HPCVD model and theresults analyzed as to the effectiveness of the reduced order model andnonlinear control.