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
Novel Splicing Generates Unique mRNA Isoforms from the Ribosomal Protein L22-like Gene in Drosophila melanogaster
Alternative splicing contributes to protein diversity by creating multiple mRNAs from a single gene that may encode functionally-distinct protein variants. Little is known about alternative splicing of ribosomal protein (rp) genes as a possible mechanism to generate ribosome diversity itself. As complex organelles, eukaryotic ribosomes contain several RNAs and ~80 rp components, each represented as a single copy. Paralogous proteins exist in several rp families and their presence may indicate functional redundancy or specialized functions. Incorporation of specialized paralogs into ribosomes could generate functionally-distinct classes of ribosomes, possibly dedicated to translation of specific mRNAs or to differential regulation of protein synthesis within cells/tissues/developmental stages. Drosophila rp L23a and L22 contain a unique N-terminal extension (of unknown function) with homology to histone H1 (Koyama et al., 1999). Gene expression profiling revealed a novel transcript called “L22-like” that displays an embryonic gonad-specific expression pattern by in situ hybridization compared to a constitutive pattern for L22 mRNA (Shigenobu et al., 2006; Kai et al., 2005). Noticeably, this paralog contains a similar histone H1- like domain, and if expressed at the protein level and assembled into ribosomes, could define a unique class of ribosomes within the fly gonad. Whether or not L22-like and L22 are interchangeable within the ribosome is unknown. Further insight into L22-like expression was determined by RT-PCR analysis using RNA from embryo, larval, and adult stages, as well as from S2 cells. Full-length L22-like mRNAs were detected at all stages. Surprisingly, lower molecular weight (MW) amplicons were also present. Cloning and sequencing revealed that the lower MW amplicons (designated “L22-like short”) are mRNA variants, in which non-canonical splice sites within an exon are used for RNA splicing to remove a previously unidentified intron. Northern blot analysis confirmed the presence of multiple L22-like mRNAs. Preliminary Western blot analyses show protein products of the predicted size for both L22-like and L22-like short isoforms in S2 cells. Whether or not the L22-like isoforms are incorporated into ribosomes is yet to be determined. Collectively, these experiments provide a foundation for understanding developmental regulation of L22-like rp gene expression and how rp variants may contribute to ribosome diversity within cells.
Retention and Release of Cryptosporidium parvum Oocysts by Experimental Biofilms Composed of Natural Stream Microbial Communities
Cryptosporidium is a group of waterborne protozoan parasites that cause significant gastrointestinal disease in humans. The potential exists for these parasites to accumulate in stream or pipe surface biofilms and to be subsequently released to contaminate the water supply. Natural microbial assemblages were collected from three streams for three seasons (fall, winter, and spring) and used to grow biofilms in laboratory microcosms. These biofilms were then exposed to Cryptosporidium parvum oocysts via continuous inflow to determine daily influx, efflux, and biofilm retention and sloughing. Daily oocyst counts showed that oocysts attached to the biofilm quickly (within hours) and released slowly (within days). In these experiments, at least 40% of the oocysts added to the system attached to the biofilm in the first 72 hours, and at least 20% of the oocysts remained attached to the biofilm beyond 72 hours after the oocyst supply was removed. Although variation across site and season exists, likely as a result of variable microbial communities, all biofilms captured at least 40% of the oocysts and retained some of those oocysts for at least five days. The biofilm attachment and detachment dynamics of C. parvum oocysts have important implications for public health and suggest that biofilm monitoring should be included in routine water supply monitoring.