Antonios Armaou

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

Abstract This article presents computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of spatially distributed processes governed by highly dissipative nonlinear partial differential equations (PDEs). The methods are based on spatial discretization using the method of weighted residuals with spatially global basis functions (i.e., functions that cover the entire domain of definition of the process and satisfy the boundary conditions). More specifically, we perform spatial discretization of the optimization problems using the method of weighted residuals with analytical or empirical (obtained via Karhunen–Loeve expansion) eigenfunctions as basis functions, and combination of the method of weighted residuals with approximate inertial manifolds. The proposed methods account for the fact that the dominant dynamics of highly dissipative PDE systems are low dimensional in nature and lead to approximate optimization problems that are of significantly lower order compared to the ones obtained from spatial discretization using finite-difference and finite-element techniques, and thus, they can be solved with significantly smaller computational demand. The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques. We employ backward finite differences (implicit Euler) to perform temporal discretization and solve the nonlinear programs resulting from temporal and spatial discretization using reduced gradient techniques (MINOS). We use two representative examples of dissipative PDEs, a diffusion-reaction process with constant and spatially varying coefficients, and the Kuramoto–Sivashinsky equation, a model that describes incipient instabilities in a variety of physical and chemical systems, to demonstrate the implementation and evaluate the effectiveness of the proposed optimization algorithms.

Brief Analysis and control of parabolic PDE systems with input constraints

This paper develops a general framework for the analysis and control of parabolic partial differential equations (PDE) systems with input constraints. Initially, Galerkins method is used for the derivation of ordinary differential equation (ODE) system that capture the dominant dynamics of the PDE system. This ODE systems are then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed-loop stability in the presence of constraints. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator. The theoretical results are used to stabilize an unstable steady-state of a diffusion-reaction process using constrained control action.

Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control

This work addresses the problem of global exponential stabilization of the Kuramoto{Sivashinsky equation (KSE) subject to periodic boundary conditions via distributed static output feedback control. Under the assumption that the number of measurements is equal to the total number of unstable and critically stable eigenvalues of the KSE and a necessary and sucient stability condition is satised, linear static output feedback controllers are designed that globally exponentially stabilize the zero solution of the KSE. The controllers are designed on the basis of nite-dimensional approximations of the KSE which are obtained through Galerkin’s method. The theoretical results are conrmed by computer simulations of the closed-loop system. c 2000 Elsevier Science B.V. All rights reserved.

Feedback control of the Kuramoto-Sivashinsky equation

Abstract This work focuses on linear finite-dimensional output feedback control of the Kuramoto–Sivashinsky equation (KSE) with periodic boundary conditions. Under the assumption that the linearization of the KSE around the zero solution is controllable and observable, linear finite-dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution, for any value of the instability parameter. The controllers are synthesized on the basis of finite-dimensional approximations of the KSE which are obtained through Galerkin’s method. The performance of the controllers is successfully tested through computer simulations.

Brief Robust control of parabolic PDE systems with time-dependent spatial domains

We synthesize robust nonlinear static output feedback controllers for systems of quasi-linear parabolic partial differential equations with time-dependent spatial domains and uncertain variables. The controllers are successfully applied to a typical diffusion-reaction process with moving boundary and uncertainty.

Plasma enhanced chemical vapor deposition: Modeling and control

This paper focuses on modeling and control of a single-wafer parallel electrode plasma-enhanced chemical vapor deposition process with showerhead arrangement used to deposit a 500 A amorphous silicon thin film on an 8 cm wafer. Initially, a two-dimensional unsteady-state model is developed for the process that accounts for diffusive and convective mass transfer, bulk and deposition reactions, and nonuniform fluid flow and plasma electron density profiles. The model is solved using finite-difference techniques and the radial nonuniformity of the final film thickness is computed to be almost 19%. Then, a feedback control system is designed and implemented on the process to reduce the film thickness nonuniformity. The control system consists of three spatially distributed proportional integral controllers that use measurements of the deposition rate at several locations across the wafer, to manipulate the inlet concentration of silane in the showerhead and achieve a uniform deposition rate across the wafer. The implementation of the proposed control system is shown to reduce the film thickness radial nonuniformity to 3.8%.

A Computational Procedure for Optimal Engineering Interventions Using Kinetic Models of Metabolism

The identification of optimal intervention strategies is a key step in designing microbial strains with enhanced capabilities. In this paper, we propose a general computational procedure to determine which genes/enzymes should be eliminated, repressed or overexpressed to maximize the flux through a product of interest for general kinetic models. The procedure relies on the generalized linearization of a kinetic description of the investigated metabolic system and the iterative application of mixed‐integer linear programming (MILP) optimization to hierarchically identify all engineering interventions allowing for reaction eliminations and/or enzyme level modulations. The effect of the magnitude of the allowed changes in concentrations and enzyme levels is investigated, and a variant of the method to explore high‐fold changes in enzyme levels is also analyzed. The proposed framework is demonstrated using a kinetic model modeling part of the central carbon metabolism of E. coli for serine overproduction. The proposed computational procedure is a general approach that can be applied to any metabolic system for which a kinetic description is provided.

Equation-free gaptooth-based controller design for distributed complex/multiscale processes

Abstract We present and illustrate a systematic computational methodology for the design of linear coarse-grained controllers for a class of spatially distributed processes. The approach targets systems described by micro- or mesoscopic evolution rules, for which coarse-grained, macroscopic evolution equations are not explicitly available. In particular, we exploit the smoothness in space of the process “coarse” variables (“observables”) to estimate the unknown macroscopic system dynamics. This is accomplished through appropriately initialized and connected ensembles of micro/mesoscopic simulations realizing a relatively small portion of the macroscopic spatial domain (the so-called gaptooth scheme). Our illustrative example consists of designing discrete-time, coarse linear controllers for a Lattice-Boltzmann model of a reaction-diffusion process (a kinetic-theory based realization of the FitzHugh-Nagumo equation in one spatial dimension).

Nonlinear feedback control of parabolic PDE systems with time-dependent spatial domains

Proposes a methodology for the synthesis of nonlinear finite-dimensional time-varying output feedback controllers for systems of quasi-linear parabolic partial differential equations with time-dependent spatial domains. The method is successfully applied to a typical diffusion reaction process whose spatial domain changes with time and is shown to outperform a controller design method which does not account for the variation of the spatial domain.

Coupled Enzyme Reactions Performed in Heterogeneous Reaction Media: Experiments and Modeling for Glucose Oxidase and Horseradish Peroxidase in a PEG/Citrate Aqueous Two-Phase System

The intracellular environment in which biological reactions occur is crowded with macromolecules and subdivided into microenvironments that differ in both physical properties and chemical composition. The work described here combines experimental and computational model systems to help understand the consequences of this heterogeneous reaction media on the outcome of coupled enzyme reactions. Our experimental model system for solution heterogeneity is a biphasic polyethylene glycol (PEG)/sodium citrate aqueous mixture that provides coexisting PEG-rich and citrate-rich phases. Reaction kinetics for the coupled enzyme reaction between glucose oxidase (GOX) and horseradish peroxidase (HRP) were measured in the PEG/citrate aqueous two-phase system (ATPS). Enzyme kinetics differed between the two phases, particularly for the HRP. Both enzymes, as well as the substrates glucose and H2O2, partitioned to the citrate-rich phase; however, the Amplex Red substrate necessary to complete the sequential reaction partitioned strongly to the PEG-rich phase. Reactions in ATPS were quantitatively described by a mathematical model that incorporated measured partitioning and kinetic parameters. The model was then extended to new reaction conditions, i.e., higher enzyme concentration. Both experimental and computational results suggest mass transfer across the interface is vital to maintain the observed rate of product formation, which may be a means of metabolic regulation in vivo. Although outcomes for a specific system will depend on the particulars of the enzyme reactions and the microenvironments, this work demonstrates how coupled enzymatic reactions in complex, heterogeneous media can be understood in terms of a mathematical model.