VTechWorks Repository :: Browsing by Author "Herdman, Terry L."

Browsing by Author "Herdman, Terry L."

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Algebraic theory for discrete models in systems biology
Hinkelmann, Franziska (Virginia Tech, 2011-08-01)
This dissertation develops algebraic theory for discrete models in systems biology. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods. Formal definitions and theorems for PDS and the concept of PDS as models of biological systems are introduced in section 1.3. Constructing a model for given time-course data is a challenging problem. Several methods for reverse-engineering, the process of inferring a model solely based on experimental data, are described briefly in section 1.3. If the underlying dependencies of the model components are known in addition to experimental data, inferring a "good" model amounts to parameter estimation. Chapter 2 describes a parameter estimation algorithm that infers a special class of polynomials, so called nested canalyzing functions. Models consisting of nested canalyzing functions have been shown to exhibit desirable biological properties, namely robustness and stability. The algorithm is based on the parametrization of nested canalyzing functions. To demonstrate the feasibility of the method, it is applied to the cell-cycle network of budding yeast. Several discrete model types, such as Boolean networks, logical models, and bounded Petri nets, can be translated into the framework of PDS. Section 3 describes how to translate agent-based models into polynomial dynamical systems. Chapter 4, 5, and 6 are concerned with analysis of complex models. Section 4 proposes a new method to identify steady states and limit cycles. The method relies on the fact that attractors correspond to the solutions of a system of polynomials over a finite field, a long-studied problem in algebraic geometry which can be efficiently solved by computing Gröbner bases. Section 5 introduces a bit-wise implementation of a Gröbner basis algorithm for Boolean polynomials. This implementation has been incorporated into the core engine of Macaulay 2. Chapter 6 discusses bistability for Boolean models formulated as polynomial dynamical systems.
An alternating direction search algorithm for low dimensional optimization: an application to power flow
Burrell, Tinal R. (Virginia Tech, 1993-05-05)
Presented in this paper is a scheme for minimizing the cost function of a three-source technique to arrive at an approximation point (I,J) that is within one unit of the true minimum. The Line-Step algorithm is applied to several systems and is also compared to other minimization techniques, including the Equal Incremental Loss Algorithm. Variations are made on the Line-Step Algorithm for faster convergence and also to handle inequality constraints.
Analysis and Approximation of Viscoelastic and Thermoelastic Joint-Beam Systems
Fulton, Brian I. (Virginia Tech, 2006-07-21)
Rigidizable/Inflatable space structures have been the focus of renewed interest in recent years due to efficient packaging for transport. In this work, we examine new mathematical systems used to model small-scale joint dynamics for inflatable space truss structures. We investigate the regularity and asymptotic behavior of systems resulting from various damping models, including Kelvin-Voigt, Boltzmann, and thermoelastic damping. Approximation schemes will also be introduced. Finally, we look at optimal control for the Kelvin-Voigt model using a linear feedback regulator.
An Analysis of Stability Margins for Continuous Systems
Albanus, Julie C. (Virginia Tech, 1999-05-06)
When designing or reviewing control systems, it is important to understand the limitations of the system's design. Many systems today are designed using numerical methods. Although the numerical model may be controllable, stabilizable, or stable, small perturbations of the system parameters can result in the loss of these properties. In this thesis, we investigate these issues for finite element approximations of a thermal convection loop.
Approximation and control of a thermoviscoelastic system
Liu, Zhuangyi (Virginia Polytechnic Institute and State University, 1989)
In this paper consider the problem of controlling a thermoviscoelastic system. We present a semigroup setting for this system, and prove the well-posedness by applying a general theorem which is given in this paper. We also study the stability of the system. We give a finite element/averaging scheme to approximate the linear quadratic regulator problem governed by the system. We prove that yields faster convergence. We give a proof of convergence of the simulation problem for singular kernels and of the control problem for L2 kernels. We carry on the numerical computation to investigate the effect of heat transfer on damping and the closed-loop system.
Approximation of integro-partial differential equations of hyperbolic type
Fabiano, Richard H. (Virginia Polytechnic Institute and State University, 1986)
A state space model is developed for a class of integro-partial differential equations of hyperbolic type which arise in viscoelasticity. An approximation scheme is developed based on a spline approximation in the spatial variable and an averaging approximation in the de1ay variable. Techniques from linear semigroup theory are used to discuss the well-posedness of the state space model and the convergence properties of the approximation scheme. We give numerical results for a sample problem to illustrate some properties of the approximation scheme.
Approximation of the LQR control problem for systems governed by partial functional differential equations
Miller, Robert Edwin (Virginia Polytechnic Institute and State University, 1988)
We present an abstract framework for state space formulation and a generalized theorem on well-posedness which can be applied to a class of partial functional differential equations which arise in the modeling of viscoelastic and certain thermo-viscoelastic systems. Examples to which the theory applies include both second- and fourth-order equations with a variety of boundary conditions. The theory presented here allows for singular kernels as well as flexibility in the choice of state space. We discuss an approximation scheme using spline in the spatial variable and an averaging scheme in the delay variable. We compare a uniform mesh to a nonuniform mesh and give numerical results which indicate that the non-uniform mesh, which gives a better approximation of the kernel near the singularity, yields faster convergence. We give a proof of convergence of the simulation problem for singular kernels and of the control problem for bounded kernels. We use techniques of semigroup theory to establish the results on well-posedness and convergence.
Approximations and Object-Oriented Implementation for a Parabolic Partial Differential Equation
Camphouse, Russell C. (Virginia Tech, 1999-01-27)
This work is a numerical study of the 2-D heat equation with Dirichlet boundary conditions over a polygonal domain. The motivation for this study is a chemical vapor deposition (CVD) reactor in which a substrate is heated while being exposed to a gas containing precursor molecules. The interaction between the gas and the substrate results in the deposition of a compound thin film on the substrate. Two different numerical approximations are implemented to produce numerical solutions describing the conduction of thermal energy in the reactor. The first method used is a Crank-Nicholson finite difference technique which tranforms the 2-D heat equation into an algebraic system of equations. For the second method, a semi-discrete method is used which transforms the partial differential equation into a system of ordinary differential equations. The goal of this work is to investigate the influence of boundary conditions, domain geometry, and initial condition on thermal conduction throughout the reactor. Once insight is gained with respect to the aforementioned conditions, optimal design and control can be investigated. This work represents a first step in our long term goal of developing optimal design and control of such CVD systems. This work has been funded through Partnerships in Research Excellence and Transition (PRET) grant number F49620-96-1-0329.
Approximations for Singular Integral Equations
Herdman, Darwin T. (Virginia Tech, 1999-05-12)
This work is a numerical study of a class of weakly singular neutral equations. The motivation for this study is an aeroelastic system. Numerical techniques are developed to approximate the singular integral equation component appearing in the complete dynamical model for the elastic motions of a three degree of freedom structure, an airfoil with trailing edge flap, in a two dimensional unsteady flow. The flap can be viewed as an active control surface to dampen vibrations that contribute to flutter. The goal of this work is to provide accurate approximations for weakly singular neutral equations using finite elements as basis functions for the initial function space. Several examples are presented in order to validate the numerical scheme.
Bifurcation Analysis of a Model of the Frog Egg Cell Cycle
Borisuk, Mark T. (Virginia Tech, 1997-04-21)
Fertilized frog eggs (and cell-free extracts) undergo periodic oscillations in the activity of "M-phase promoting factor" (MPF), the crucial triggering enzyme for mitosis (nuclear division) and cell division. MPF activity is regulated by a complex network of biochemical reactions. Novak and Tyson, and their collaborators, have been studying the qualitative and quantitative properties of a large system of nonlinear ordinary differential equations that describe the molecular details of this system as currently known. Important clues to the behavior of the model are provided by bifurcation theory, especially characterization of the codimension-1 and -2 bifurcation sets of the differential equations. To illustrate this method, I have been studying a system of 9 ordinary differential equations that describe the frog egg cell cycle with some fidelity. I will describe the bifurcation diagram of this system in a parameter space spanned by the rate constants for cyclin synthesis and cycling degradation. My results suggest either that the cell cycle control system should show dynamical behavior considerably more complex than the limit cycles and steady states reported so far, or that the biochemical rate constants of the system are constrained to avoid regions of parameter space where complex bifurcation points unfold.
Cascade analysis and synthesis of transfer functions of infinite dimensional linear systems
Carpenter, Lon E. (Virginia Tech, 1992-06-13)
Problems of cascade connections (synthesis) and decomposition (analysis) are analyzed for two classes of linear systems with infinite dimensional state spaces, namely, 1) admissible systems in the sense of Bart, Gohberg and Kaashoek and 2) regular systems as recently introduced by Weiss. For the class of BGK-admissible systems, it is shown that the product of two admissible systems is again admissible and that a Wiener-Hopf factorization problem can be solved just as in the finite-dimensional case. For the class of regular systems, it is shown that the cascade connection of a rational stable and antistable system has an additive stable-antistable decomposition; this involves giving a distribution interpretation to the solution of a linear Sylvester equation involving unbounded operator coefficients. As an application, some preliminary work is presented toward obtaining a state space solution of the sensitivity minimization problem for a pure delay plant.
Compensator design for a system of two connected beams
Huang, Wei (Virginia Tech, 1994-08-05)
The goal of this paper is to study the LQG problem for a class of infinite dimensional systems. We investigate the convergence of compensator gains for such systems when standard finite element schemes are used to discretize the problem. We are particularly interested in the analysis of the uniformly exponential stability of the corresponding closed - loop systems resulting from the finite dimensional compensators. A specific multiple component flexible structure is used to focus the analysis and to test problem in numerical simulations. An abstract framework for analysis and approximation of the corresponding dynamics system is developed and used to design finite - dimensional compensators. Linear semigroup theory is used to establish that the systems are well posed and to prove the convergence of generic approximation schemes. Approximate solutions of the optimal regulator and optimal observer are constructed via Galerkin - type approximations. Convergence of the scheme is established and numerical results are presented to illustrate the method
Computational Algorithms for Face Alignment and Recognition
Bellino, Kathleen Ann (Virginia Tech, 2002-04-27)
Real-time face recognition has recently become available for the government and industry due to developments in face recognition algorithms, human head detection algorithms, and faster/low cost computers. Despite these advances, however, there are still some critical issues that affect the performance of real-time face recognition software. This paper addresses the problem of off-centered and out-of-pose faces in pictures, particularly in regard to the eigenface method for face recognition. We first demonstrate how the representation of faces by the eigenface method, and ultimately the performance of the software depend on the location of the eyes in the pictures. The eigenface method for face recognition is described: specifically, the creation of a face basis using the singular value decomposition, the reduction of dimension, and the unique representation of faces in the basis. Two different approaches for aligning the eyes in images are presented. The first considers the rotation of images using the orthogonal Procrustes Problem. The second approach looks at locating features in images using energy-minimizing active contours. We then conclude with a simple and fast algorithm for locating faces in images. Future research is also discussed.
Computational Methods for Control of Queueing Models in Bounded Domains
Menéndez Gómez, José María (Virginia Tech, 2007-06-08)
The study of stochastic queueing networks is quite important due to the many applications including transportation, telecommunication, and manufacturing industries. Since there is often no explicit solution to these types of control problems, numerical methods are needed. Following the method of Boué-Dupuis, we use a Dynamic Programming approach of optimization on a controlled Markov Chain that simulates the behavior of a fluid limit of the original process. The search for an optimal control in this case involves a Skorokhod problem to describe the dynamics on the boundary of closed, convex domain. Using relaxed stochastic controls we show that the approximating numerical solution converges to the actual solution as the size of the mesh in the discretized state space goes to zero, and illustrate with an example.
Computational Methods for Sensitivity Analysis with Applications to Elliptic Boundary Value Problems
Stanley, Lisa Gayle (Virginia Tech, 1999-07-08)
Sensitivity analysis is a useful mathematical tool for many designers, engineers and mathematicians. This work presents a study of sensitivity equation methods for elliptic boundary value problems posed on parameter dependent domains. The current focus of our efforts is the construction of a rigorous mathematical framework for sensitivity analysis and the subsequent development of efficient, accurate algorithms for sensitivity computation. In order to construct the framework, we use the classical theory of partial differential equations along with the method of mappings and the Implicit Function Theorem. Examples are given which illustrate the use of the framework, and some of the shortcomings of the theory are also identified. An overview of some computational methods which make use of the method of mappings is also included. Numerical results for a specific example show that convergence (energy norm) of the sensitivity approximations can be influenced by the specific structure of the computational scheme.
Configurable, Coordinated, Model-based Control in Electrical Distribution Systems
Hambrick, Joshua Clayton (Virginia Tech, 2010-04-28)
Utilities have been planning, building, and operating electrical distribution systems in much the same way for decades with great success. The electrical distribution system in the United States has been consistently reliable; an impressive feat considering its amazing complexity. However, in recent years, the electrical distribution system landscape has started to undergo drastic changes. Emerging applications of technologies such as distributed generation, communications, and power electronics offer both opportunities and challenges to power system operators as well as customers and developers. In this work, Graph Trace Analysis along with an integrated system model are used to develop algorithms and analysis methods necessary to facilitate the implementation of these new technologies on the electrical distribution system. A penetration limit analysis is developed to analyze the impact of distributed generation on radial distribution feeders. The analysis considers generation location, equipment rating, voltage violations, and flicker to determine the amount of generation that can be safely attached to a circuit. A real-time, hierarchical, model-based control method is developed that coordinates the operation of all control devices on electrical distribution circuits. The controller automatically compensates for changes in circuit topology as well as the addition or removal of control devices from the active circuit. Additionally, the controller allows the integration of modern, "smart" equipment with legacy control devices to facilitate incremental modernization strategies. Finally, a framework is developed to allow the testing of new analysis and control methodologies for electrical distribution systems. The framework can be used to test scenarios over multiple consecutive hourly or sub-hourly time points. The framework is used to demonstrate the effectiveness of the model-based controller versus existing operating methods for a distribution circuit test case.
Control of Burgers' Equation With Mixed Boundary Conditions
Massa, Kenneth L. (Virginia Tech, 1998-03-23)
We consider the problems of simulation and control for Burgers' equation with mixed boundary conditions. We first conduct numerical experiments to test the convergence and stability of two standard finite element schemes for various Robin boundary conditions and a variety of Reynolds numbers. These schemes are used to compute LQR feedback controllers for Burgers' equation with boundary control. Numerical studies of these feedback control laws are used to evaluate the performance and practicality of this approach to boundary control of non-linear systems.
A control problem for Burgers' equation
Kang, Sungkwon (Virginia Tech, 1990-04-05)
Burgers' equation is a one-dimensional simple model for convection-diffusion phenomena such as shock waves, supersonic flow about airfoils, traffic flows, acoustic transmission, etc. For high Reynolds number, the open-loop system (no control) produces steep gradients due to the nonlinear nature of the convection. The steep gradients are stabilized by feedback control laws. In this phase, sufficient conditions for the control input functions and the location of sensors are obtained. Also, explicit exponential decay rates for open-loop and closed-loop systems are obtained. Numerical experiments are given to illustrate some of typical results on convergence and stability.
Convergence and Boundedness of Probability-One Homotopies for Model Order Reduction
Wang, Yuan (Virginia Tech, 1997-08-13)
The optimal model reduction problem, whether formulated in the H² or H^∞ norm frameworks, is an inherently nonconvex problem and thus provides a nontrivial computational challenge. This study systematically examines the requirements of probability-one homotopy methods to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems, and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the optimal projection equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms.
A Distributed Parameter Approach to Optimal Filtering and Estimation with Mobile Sensor Networks
Rautenberg, Carlos Nicolas (Virginia Tech, 2010-03-31)
In this thesis we develop a rigorous mathematical framework for analyzing and approximating optimal sensor placement problems for distributed parameter systems and apply these results to PDE problems defined by the convection-diffusion equations. The mathematical problem is formulated as a distributed parameter optimal control problem with integral Riccati equations as constraints. In order to prove existence of the optimal sensor network and to construct a framework in which to develop rigorous numerical integration of the Riccati equations, we develop a theory based on Bochner integrable solutions of the Riccati equations. In particular, we focus on ℐ_p-valued continuous solutions of the Bochner integral Riccati equation. We give new results concerning the smoothing effect achieved by multiplying a general strongly continuous mapping by operators in ℐ_p. These smoothing results are essential to the proofs of the existence of Bochner integrable solutions of the Riccati integral equations. We also establish that multiplication of continuous ℐ_p-valued functions improves convergence properties of strongly continuous approximating mappings and specifically approximating C₀-semigroups. We develop a Galerkin type numerical scheme for approximating the solutions of the integral Riccati equation and prove convergence of the approximating solutions in the ℐ_p-norm. Numerical examples are given to illustrate the theory.

Browsing by Author "Herdman, Terry L."

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