### Browsing by Author "Hannsgen, Kenneth B."

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- Adaptive Self-Tuning Neuro Wavelet Network ControllersLekutai, Gaviphat (Virginia Tech, 1997-03-31)
Show more Single layer feed forward neural networks with hidden nodes of adaptive wavelet functions (wavenets) have been successfully demonstrated to have potential in many applications. Yet applications in the process control area have not been investigated. In this paper an application to a self-tuning design method for an unknown nonlinear system is presented. Different types of frame wavelet functions are integrated for their simplicity, availability, and capability of constructing adaptive controllers. Infinite impulse response (IIR) recurrent structures are combined in cascade to the network to provide a double local structure resulting in improved speed of learning. In particular, neuro-based controllers assume a certain model structure to approximate the system dynamics of the "unknown" plant and generate the control signal. The capability of neuro-controllers to self-tuning of an unknown nonlinear plants is then illustrated through design examples. Simulation results demonstrate that the self-tuning design methods are directly applicable for a large class of nonlinear control systems.Show more - Approximation and control of a thermoviscoelastic systemLiu, Zhuangyi (Virginia Polytechnic Institute and State University, 1989)
Show more 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.Show more - Approximation of the LQR control problem for systems governed by partial functional differential equationsMiller, Robert Edwin (Virginia Polytechnic Institute and State University, 1988)
Show more 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.Show more - Bifurcation Analysis of a Model of the Frog Egg Cell CycleBorisuk, Mark T. (Virginia Tech, 1997-04-21)
Show more 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.Show more - Closability of differential operators and subjordan operatorsFanney, Thomas R. (Virginia Polytechnic Institute and State University, 1989)
Show more A (bounded linear) operator J on a Hilbert space is said to be jordan if J = S + N where S = S* and N² = 0. The operator T is subjordan if T is the restriction of a jordan operator to an invariant subspace, and pure subjordan if no nonzero restriction of T to an invariant subspace is jordan. The main operator theoretic result of the paper is that a compact subset of the real line is the spectrum of some pure subjordan operator if and only if it is the closure of its interior. The result depends on understanding when the operator D = θ + d/dx : L²(μ) —> L²(v) is closable. Here θ is an L²(μ) function, μ and v are two finite regular Borel measures with compact support on the real line, and the domain of D is taken to be the polynomials. Approximation questions more general than what is needed for the operator theory result are also discussed. Specifically, an explicit characterization of the closure of the graph of D for a large class of (θ, μ, v) is obtained, and the closure of the graph of D in other topologies is analyzed. More general results concerning spectral synthesis in a certain class of Banach algebras and extensions to the complex domain are also indicated.Show more - Compensator design for a system of two connected beamsHuang, Wei (Virginia Tech, 1994-08-05)
Show more 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 methodShow more - A control problem for Burgers' equationKang, Sungkwon (Virginia Tech, 1990-04-05)
Show more 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.Show more - Finite dimensional approximations of distributed parameter control systemsHill, David Dean (Virginia Polytechnic Institute and State University, 1989)
Show more In this paper we consider two separate approaches to the development of finite dimensional control systems for approximating distributed parameter models. One method uses the “standard finite element” approximations to construct the basic system matrices. The resulting system can then be balanced by any of several balancing algorithms. The second method is based on truncating infinite dimensional balanced realizations of the input-output map. Both approaches are applied to a control problem governed by the heat equation. We present a comparison of the resulting finite dimensional models.Show more - Idempotents in group ringsMarciniak, Zbigniew (Virginia Polytechnic Institute and State University, 1982)
Show more The von Neumann finiteness problem for k[G] is still open. Kaplansky proved it in characteristic zero. He used the nonvanishing of the trace: tr(e) = 0 implies e = 0 for any idempotent e ∊ k[G]. Assume now that char k = p > 0. Now tr can vanish on nonzero idempotents. Instead, we study the lifted trace ltr. For e = e² ∊ k[G], define ltr(e) by ê(1) where ê = Σ{x ∊ G}ê(x)x lifts e. Here ê is an infinite series with |ê(x)|_{p}→0, where each ê(x) lives in the Witt vector ring of k. We prove that ltr(e) depends on e only, it is a p-adic integer and ltr(e) = ltr(f) if f is equivalent to e. Also ltr(e) ∊ Q and ltr(e) = 0 implies e = 0 if G is polycyclic-by-finite. We conjecture that -log_{p}|ltr(e)|_{p}< |supp(e)|. We prove this for e central and for e = e² ∊ k[G] with |G| ≤ 30. In the last section, we give the example of an idempotent e such ath supp(f) is infinite for all f ~ e. Finally we estimate || for central idempotents e.Show more - Interpolation by rational matrix functions with minimal McMillan degreeKang, Jeongook Kim (Virginia Tech, 1990-12-05)
Show more Interpolation conditions on rational matrix functions expressed in terms of residues are studied. As a compact way of expressing tangential interpolation conditions of arbitrarily high multiplicity possibly from both sides simultaneously, interpolation conditions are represented in terms of residues. The minimal possible complexity, measured by the McMillan degree, of interpolants is found in terms of the controllability and the observability indices of certain pairs of matrices which are part of given data. An interpolant of such complexity is obtained in realization form. This leads to another approach to the partial realization problem. As a generalization of the well-known Lagrange interpolation problem for scalar polynomials, the problem of seeking for a matrix polynomial interpolant of low complexity is studied. The main tool is state space methods borrowed from systems theory. After adoption of state space methods, problems concerning rational matrix functions are reduced to the realm of linear algebra.Show more - Mathematical Modeling of Circadian Rhythms in Drosophila melanogasterHong, Christian I. (Virginia Tech, 1999-04-13)
Show more Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm. A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback.Show more - Mathematical modeling of pathways involved in cell cycle regulation and differentiationRavi, Janani (Virginia Tech, 2011-12-01)
Show more Cellular processes critical to sustaining physiology, including growth, division and differentiation, are carefully governed by intricate control systems. Deregulations in these systems often result in complex diseases such as cancer. Hence, it is crucial to understand the interactions between molecular players of these control systems, their emergent network dynamics, and, ultimately, the overall contribution to cellular physiology. In this dissertation, we have developed a mathematical framework to understand two such cellular systems: an early checkpoint (START) in the budding yeast cell cycle (Chapter 1), and the canonical Wnt signaling pathway involved in cell proliferation and differentiation (Chapter 2). START transition is an important decision point where the cell commits to one round DNA replication followed by cell division. Several years of experimental research have gone into uncovering molecular details of this process, but a unified understanding is yet to emerge. In chapter one, we have developed a comprehensive mathematical model of START transition that incorporates several findings including information about the phosphorylation state of key START proteins and their subcellular localization. In the second chapter, we focus on modeling the canonical Wnt signaling pathway, a cellular circuit that plays a key role in cell proliferation and differentiation. The Wnt pathway is often deregulated in colon cancers. Based on some evidence of bistability in the Wnt signaling pathway, we proposed the existence of a positive feedback loop underlying the activation and inactivation of the core protein complex of the pathway. Bistability is a common feature of biological systems that toggle between ON and OFF states because it ensures robust switching back and forth between the two states. To study and explain the behavior of this dynamical system, we developed a mathematical model. Based on experimentally determined interactions, our simple model recapitulates the observed phenomena of bimodality (bistability) and hysteresis under the effects of the physiological signal (Wnt), a Wnt-mimic (LiCl), and a stabilizer of one of the key members of core complex (IWR-1). Overall, we believe that cell biologists and molecular geneticists can benefit from our work by using our model to make novel quantitative predictions for experimental verification.Show more - Mathematical modelling, finite dimensional approximations and sensitivity analysis for phase transitions in shape memory alloysSpies, Ruben Daniel (Virginia Tech, 1992)
Show more Shape Memory Alloys (SMA’s) are intermetallic materials (chemical compounds of two or more elements) that are able to sustain a residual deformation after the application of a large stress, but they “remember” the original shape to which they creep back, without the application of any external force, after they are heated above a certain critical temperature. A general one-dimensional dynamic mathematical model is presented which accounts for thermal coupling, time-dependent distributed and boundary inputs and internal variables. Well-posedness is obtained using an abstract formulation in an appropriate Hilbert space and explicit decay rates for the associated linear semigroup are derived. Numerical experiments using finite-dimensional approximations are performed for the case in which the thermodynamic potential is given in the Landau-Devonshire form. The sensitivity of the solutions with respect to the model parameters is studied. Finally, an alternative approach to the stress-strain laws is presented which is able to capture the dependence on the strain history.Show more - Multigroup transport equations with nondiagonal cross section matricesWillis, Barton L. (Virginia Polytechnic Institute and State University, 1985)
Show more It is shown that multigroup transport equations with nondiagonal cross section matrices arise when the modal approximation is applied to energy dependent transport equations. This work is a study of such equations for the case that the cross section matrix is nondiagonalizable. For the special case of a two-group problem with a noninvertible scattering matrix, the problem is solved completely via the Wiener-Hopf method. For more general problems, generalized Chandrasekhar H equations are derived. A numerical method for their solution is proposed. Also, the exit distribution is written in terms of the H functions.Show more - A nonlinear Volterra equation of nonconvolution typeSmith, Manfred Charles (Virginia Tech, 1977-12-05)
Show more The purpose of this paper is to study the asymptotic behavior of bounded solutions x(t) of the integrodifferential equation.Show more - Numerical solutions for a class of singular integrodifferential equationsChiang, Shihchung (Virginia Tech, 1996-05-05)
Show more In this study, we consider numerical schemes for a class of singular integrodifferential equations with a nonatomic difference operator. Our interest in this particular class has been motivated by an application in aeroelasticity. By applying nonconforming finite element methods, we are able to establish convergence for a semi-discrete scheme. We use an ordinary differential equation solver for the semi-discrete scheme and then improve the result by using a fully discretized scheme. We report our numerical findings and comment on the rates of convergence.Show more - Optimal feedback control for nonlinear discrete systems and applications to optimal control of nonlinear periodic ordinary differential equationsZhang, Xiaohong (Virginia Tech, 1993)
Show more This dissertation presents a discussion of the optimal feedback control for nonliner systems (both discrete and ODE) and nonquadratic cost functions in order to achieve improved performance and larger regions of asymptotic stability in the nonlinear system context. The main work of this thesis is carried out in two parts; the first involves development of nonlinear, nonquadratic theory for nonlinear recursion equations and formulation, proof and application of the stable manifold theorem as it is required in this context in order to obtain the form of the optimal control law. The second principal part of the dissertation is the development of nonlinear, nonquadratic theory as it relates to nonautonomous systems of a particular type; specifically periodic time varying systems with a fixed, time invariant critical point.Show more - Precise energy decay rates for some viscoelastic and thermo-viscoelastic rodsInch, Scott E. (Virginia Tech, 1992-09-14)
Show more Energy dissipation in systems with linear viscoelastic damping is examined. It is shown that in such viscoelastically damped systems the use of additional dissipation mechanisms (such as boundary velocity feedback or thermal coupling) may not improve the rate of energy decay. The situation where the viscoelastic stress relaxation modulus decreases to its (positive) equilibrium modulus at a subexponential rate, e.g., like (1 + t)^{-x}+ E, where α > 0, E > 0 is examined. In this case, the nonoscillatory modes (the so-called creep modes) dominate the energy decay rate. The results are in two parts. In the first part, a linear viscoelastic wave equation with infinite memory is examined. It is shown that under appropriate conditions on the kernel and initial history, the total energy is integrable against a particular weight if the kinetic energy component of the total energy is integrable against the same weight. The proof uses energy methods in an induction argument. Precise energy decay rates have recently been obtained using boundary velocity feedback. It is shown that the same decay rates hold for history value problems with conservative boundary conditions provided that an*a priori*knowledge of the decay rate of the kinetic energy term is assumed. In the second part, a simple linear thermo-viscoelastic system, namely, a viscoelastic wave equation coupled to a heat equation, is examined. Using Laplace transform methods, an integral representation formula for*W(x,s*), the transform of the displacement*w(x, t)*, is obtained. After analyzing the location of the zeros of the appropriate characteristic equation, an asymptotic expansion for the displacement*w(O,t)*is obtained which is valid for large*t*and the specific kernel*g(t) = g*(–) + Î´tÎ·-1 [over]Î (Î·), 0 < Î· < 1. With this expansion it is shown that the coupled system tends to its equilibrium at a slower rate than that of the uncoupled system.Show more - Self-efficacy, Motivational Email, and Achievement in an Asynchronous Mathematics CourseHodges, Charles B. (Virginia Tech, 2005-11-28)
Show more This study investigated the effects of motivational email messages on learner self-efficacy and achievement in an asynchronous college algebra and trigonometry course. A pretest-posttest control group design was used. Of the 196 initial participants randomly assigned to treatment groups, 125 participants with an average age of 18.21 years completed the study. The final control and experimental groups consisted of 57 (n=17 male, n=40 female) and 68 (n=14 male, n=54 female) participants respectively. Self-efficacy to learn mathematics asynchronously (SELMA) was measured before the treatment was administered. Email messages designed to be efficacy enhancing were sent to the experimental group weekly for 4 weeks. The control group was sent email messages designed to be neutral with respect to self-efficacy weekly for 4 weeks. SELMA and math achievement were measured after the email messages were sent in week 4. Analysis of covariance was performed using the pretest SELMA measure as a covariate to detect post-treatment differences in SELMA between the control and experimental groups. No significant differences were detected at the 0.05 alpha level. Paired-sample t-Tests revealed significant increases in SELMA for both the control and experimental groups over the treatment period. Linear regression analysis revealed a weak positive relationship between SELMA and math achievement. The findings are discussed in the context of the related literature and directions for future research are suggested.Show more - The sensitivity equation method for optimal designBorggaard, Jeffrey T. (Virginia Tech, 1994-12-02)
Show more In this work, we introduce the Sensitivity Equation Method (SEM) as a method for approximately solving infinite dimensional optimal design problems. The SEM couples a trust-region/quasi-Newton optimization algorithm with gradient information provided by apprOXimately solving the sensitivity equation for (design) sensitivities. The sensitivity equation is (in the problems considered here) a partial differential equation (POE) which describes the influence of a design parameter on the state of the system. It is shown that obtaining design sensitivities from the sensitivity equation has advantages over finite difference and semi-analytical methods in that there is no need to remesh or compute mesh sensitivities (even if the domain is parameter dependent), the sensitivity equation is a linear POE for the sensitivities and can be approximated in an efficient manner using the same approximation scheme used to approximate the states. The applicability of the SEM to shape optimization problems, where the state is described by the Euler equations, is studied in detail. In particular, we prove convergence of the method for a one dimensional test problem. These results are used to speculate on the applicability of the method for more complex problems. Finally. we solve a two dimensional forebody simulator design problem (for use in wind tunnel experiments) using the SEM, which is shown to be a very efficient method for this problem.Show more