Hien T. Tran

Modeling and control of physical processes using proper orthogonal decomposition

The proper orthogonal decomposition (POD) technique (or the Karhunan Loeve procedure) has been used to obtain low-dimensional dynamical models of many applications in engineering and science. In principle, the idea is to start with an ensemble of data, called snapshots, collected from an experiment or a numerical procedure of a physical system. The POD technique is then used to produce a set of basis functions which spans the snapshot collection. When these basis functions are used in a Galerkin procedure, they yield a finite-dimensional dynamical system with the smallest possible degrees of freedom. In this context, it is assumed that the physical system has a mathematical model, which may not be available for many physical and/or industrial applications. In this paper, we consider the steady-state Rayleigh-Benard convection whose mathematical model is assumed to be unknown, but numerical data are available. The aim of the paper is to show that, using the obtained ensemble of data, POD can be used to model accurately the natural convection. Furthermore, this approach is very efficient in the sense that it uses the smallest possible number of parameters, and thus, is suited for process control. Particularly, we consider two boundary control problems 1.(a) tracking problem, and 2.(b) avoiding hot spot in a certain region of the domain.

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

Abstract State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

Feedback control methodologies for nonlinear systems

A number of computational methods have been proposed in the literature to design and synthesize feedback controls when the plant is modeled by nonlinear dynamics. However, it is not immediately clear which is the best method for a given problem; this may depend on the nature of the nonlinearities, size of the system, whether the amount of control used or time needed for the method is a concern, and other factors. In this paper, a comprehensive comparison study of five methods for the synthesis of nonlinear control systems is carried out. The performance of the methods on several test problems are studied, and some recommendations are made as to which feedback control method is best to use under various conditions.

Cardiovascular and respiratory systems : modeling, analysis, and control

Preface 1. The cardiovascular system under an ergometric workload 2. Respiratory modeling 3. Cardio-Respiratory Modeling 4. Blood volume and the venous system 5. Future directions Appendix A. Supplemental calculations B. A Nonlinear feedback law C. Retarded functional differential equations: Basic theory Bibliography Index.

Reduced order modeling and control of thin film growth in an HPCVD reactor

This paper describes the development of a reduced order model-based feedback control methodology for regulation of the growth of thin films in a high-pressure chemical vapor deposition (HPCVD) reactor. Precise control of the film thickness and composition is highly desirable, making real-time control of the deposition process very important. The source vapor species transport is modeled by the standard gas dynamics partial differential equations, with species decomposition reactions, reduced down to a small number of ordinary differential equations through use of the proper orthogonal decomposition technique. This system is coupled with a reduced order model of the surface reactions involved in the source vapor decomposition and film growth on the substrate. Also modeled is the real-time observation technique used to obtain a partial measurement of the deposition process.The utilization of reduced order models greatly simplifies the mathematical formulation of the physical process so that it can be solved...

Reduced order model compensator control of species transport in a CVD reactor

We propose the use of proper orthogonal decomposition (POD) techniques as a reduced basis method for computation of feedback controls and compensators in a high-pressure chemical vapour deposition (HPCVD) reactor. In this paper, we present a proof-of-concept computational implementation of this method with a simplified growth example for III–V layers in which we implement Dirichlet boundary control of a dilute Group III reactant transported by convection and diffusion to an absorbing substrate with no reactions. We implement the model-based feedback control using a reduced order state estimator based on observations of the flux of reactant at the substrate centre. This is precisely the type of measurements available with current sensing technology. We demonstrate that the reduced order state estimator or compensator system is capable of substantial control authority when applied to a high-order system. In principle, these ideas can be extended to more general HPCVD control situations by including multiple species with gas-phase reactions and surface reactions. Copyright

Physiologically-Based Pharmacokinetic Modeling of Benzene in Humans: A Bayesian Approach

Benzene is myelotoxic and leukemogenic in humans exposed at high doses (>1 ppm, more definitely above 10 ppm) for extended periods. However, leukemia risks at lower exposures are uncertain. Benzene occurs widely in the work environment and also indoor air, but mostly below 1 ppm, so assessing the leukemia risks at these low concentrations is important. Here, we describe a human physiologically-based pharmacokinetic (PBPK) model that quantifies tissue doses of benzene and its key metabolites, benzene oxide, phenol, and hydroquinone after inhalation and oral exposures. The model was integrated into a statistical framework that acknowledges sources of variation due to inherent intra- and interindividual variation, measurement error, and other data collection issues. A primary contribution of this work is the estimation of population distributions of key PBPK model parameters. We hypothesized that observed interindividual variability in the dosimetry of benzene and its metabolites resulted primarily from known or estimated variability in key metabolic parameters and that a statistical PBPK model that explicitly included variability in only those metabolic parameters would sufficiently describe the observed variability. We then identified parameter distributions for the PBPK model to characterize observed variability through the use of Markov chain Monte Carlo analysis applied to two data sets. The identified parameter distributions described most of the observed variability, but variability in physiological parameters such as organ weights may also be helpful to faithfully predict the observed human-population variability in benzene dosimetry.

Physiologically Based Pharmacokinetic Modeling of Benzene Metabolism in Mice Through Extrapolation from In Vitro to In Vivo

Benzene (C6H6) is a highly flammable, colorless liquid. Ubiquitous exposures result from its presence in gasoline vapors, cigarette smoke, and industrial processes. Benzene increases the incidence of leukemia in humans when they are exposed to high doses for extended periods; however, leukemia risks in humans at low exposures are uncertain. The exposure-dose-response relationship of benzene in humans is expected to be nonlinear because benzene undergoes a series of metabolic transformations, detoxifying and activating, in the liver, resulting in multiple metabolites that exert toxic effects on the bone marrow. We developed a physiologically based pharmacokinetic model for the uptake and elimination of benzene in mice to relate the concentration of inhaled and orally administered benzene to the tissue doses of benzene and its key metabolites, benzene oxide, phenol, and hydroquinone. As many parameter values as possible were taken from the literature; in particular, metabolic parameters obtained from in vitro studies with mouse liver were used since comparable parameters are also available for humans. Parameters estimated by fitting the model to published data were first-order rate constants for pathways lacking in vitro data and the concentrations of microsomal and cytosolic protein, which effectively alter overall enzyme activity. The model was constrained by using the in vitro metabolic parameters (maximum velocities, first-order rate constants, and saturation parameters), and data from multiple laboratories and experiments were used. Despite these constraints and sources of variability, the model simulations matched the data reasonably well in most cases, showing that in vitro metabolic constants can be successfully extrapolated to predict in vivo data for benzene metabolism and dosimetry. Therefore in vitro metabolic constants for humans can subsequently be extrapolated to predict the dosimetry of benzene and its metabolites in humans. This will allow us to better estimate the risks of adverse effects from low-level benzene exposures.

Modeling instability in the control system for human respiration: applications to infant non-REM sleep

Mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. The phenomena collectively referred to as periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of F.S. Grodins, J. Gray, A.I. Norins, R.W. Jones [J. Appl. Physiol. 1 (1954) 283-308] in the early 1950s. The purpose of this paper is to extend the model presented by M.C.K. Khoo, R.E. Kronauer, K.P. Strohl, A.S. Slutsky [J. Appl. Physiol. 53 (3) (1982) 644-659] in 1991 to include variable delay in the feedback control loop and to study the phenomena of periodic breathing and apnea as they occur during quiet sleep in infant sleep respiration at around 4 months of age. The nonlinear mathematical model consists of a feedback control system of five delay differential equations. Numerical simulations are performed to study instabilities in the control system and the occurrence of periodic breathing and apnea in the above case which is a time frame of high incidence of sudden infant death syndrome (SIDS).

Design of an electron gun using computer optimization

This paper considers an optimization technique in which the objective is attained via alterations to the physical geometry of the system. This optimization framework, to be considered in the context of electron guns, is known as optimal shape design. Optimal shape design has been used in a number of applications including wing design, magnetic tape design, and nozzle design, among others. In this investigation, we use the methods of shape optimization to design the cathode of an electron gun. The dynamical equations modeling the electron particle path as well as the generalized shape optimization problem will be presented. Illustrative examples of the technique on gun designs that were previously limited to spherical cathodes will be given.