Archive for June, 2009
Model Predictive Control of Parabolic PDE Systems with Dirichlet Boundary Conditions via Galerkin Model Reduction
In this paper, we propose a framework to solve a closed-loop, optimal tracking control problem for a nonlinear parabolic partial differential equation (PDE) via diffusivity, interior, and boundary actuation. The approach is based on model reduction via proper orthogonal decomposition (POD) and Galerkin projection methods. A conventional integration-by-parts approach during the Galerkin projection fails to effectively incorporate the considered Dirichlet boundary control into the reduced-order model (ROM). To overcome this limitation we use a spatial discretization of the interior product during the Galerkin projection. The obtained low dimensional dynamical model is bilinear as the result of the presence of the diffusivity control term in the nonlinear parabolic PDE system. We propose a closed-loop optimal controller based on a nonlinear model predictive control (MPC) scheme aimed at bating the effect of disturbances with the ultimate goal of tracking a nominal trajectory. A quasi-linear approximation approach is used to solve on-line the quadratic optimal control problem subject to the bilinear reduced-order model associated with the MPC scheme. Based on the convergence properties of the quasi-linear approximation algorithm, the symptotical stability of the closed-loop nonlinear MPC scheme is discussed. Finally, the proposed approach is applied to the current profile control problem in tokamak plasmas and its effectiveness is demonstrated in simulations.
Understanding the interactions among polyelectrolytes by using inorganic polyoxometalate molecular clusters as model systems
Nanometer-scaled polyoxometalate (POM) molecular clusters exist as hydrophilic, highly soluble macroions in water and other polar solvents. Very interestingly, they do not stay as discrete ions even in very dilute solutions. Instead, we find that they universally tend to self-assemble into highly stable, monodispersed, hollow, spherical, single-layered shell-like structures (we call them “blackberries”), by using laser light scattering, TEM, SEM and SAXS studies.
The blackberry size can be accurately tuned by adjusting solution content and/or solution pH. The transitions between discrete macroions to blackberry structures, and between blackberries with different sizes, can be also achieved. The driving forces of the blackberry formation are not due to hydrophobic interaction, van der Waals forces or chemical interactions. Instead, we believe that the counter-ion effects and hydrogen bonds are critical. Synchrotron SAXS studies clearly show the radial distribution of small cations around large POM anions and the relation between the counter-ion association and the blackberry formation. › Continue reading
Raman and UV-Vis Spectroscopy Study of Vanadium-Containing Heteropoly Acids in Aqueous Solutions
To advance liquid phase spectroscopic techniques, we have selected two types of heteropoly acids (HPAs) in aqueous solutions to serve as our experimental catalysts: H3PW12O40, (TPA-tungstophosphoric acid) and H3PMo12O40, (MPA-molybdophosphoric acid). The cage-like structure that these HPAs assume is called the Keggin structure [1,2]. Distorted forms of the Keggin structure are also known to exist. For example, the Dawson structure is composed of two Keggin anions which have each expelled three WO3 or MoO3 units and joined together as a dimeric unit [1,3].
The initial objectives of this investigation were (1) to compare the ambient and aqueous solution spectra of the HPAs and (2) to determine differences or similarities in their structures between their ambient and aqueous states. Varying levels of vanadium were introduced into the primary and secondary structure of each HPA in order to investigate the influence of vanadium when the HPAs are in solution. TPA and MPA samples containing vanadium in the primary structure are denoted as TPAVx and MPAVx. The chemical formulas for solid TPAVx and MPAVx are H3+xPW12-xVxO40 and H3+xPMo12-xVxO40 (where x=1, 2, and 3). TPA and MPA which contain vanadia on the secondary structure are denoted as VOTPA and VOMPA. › Continue reading