System Of Hyperbolic Partial Differential Equations Research Articles

Generalized Roe schemes for 1D two-phase, free-surface flows over a mobile bed

The problem of two-phase, free-surface flows over a mobile bed is characterized by a hyperbolic partial differential equations system that shows nonconservative terms and highly nonlinear relations between primitive and conserved variables. Weak solutions of the present problem were obtained resorting both to the distribution theory and to the integral formulation of momentum conservation: the comparison of these two approaches allowed us to give a physical insight into the meaning of the nonconservative term across a discontinuity. Starting from this result, we derived the conditions necessary to obtain generalized, well-balanced Roe solvers without using the concept of a family of paths. Two numerical schemes based on the same set of matrices have been developed, one in terms of conserved variables and one in terms of primitive variables. The friction-source term has also been included by using an upwind approach. The capabilities and limits of the proposed schemes have been analyzed by comparison with exact solutions of Riemann problems and with numerical solutions obtained with the AWB-3SRS scheme.

Discrete Fundamental Solution Preconditioning for Hyperbolic Systems of PDE

We present a new preconditioner for the iterative solution of systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electromagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple one-stage iterative method. As an example of a more involved problem, we consider the steady state solution of the non-linear Euler equations in a two-dimensional, non-axisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.

Model Predictive Control of Cocurrent First-Order Hyperbolic PDE Systems

A continuum of repetition takes place in reaction−convection processes described by first-order hyperbolic partial differential equations (PDEs). Especially, in cocurrent cases, they have finite impulse response (FIR) characteristics. Employing recently developed FIR property preserving finite difference scheme for cocurrent systems, a cocurrent first-order hyperbolic PDE system is recast as a repetitive process. Then a model predictive control scheme for repetitive processes is used for efficient control of cocurrent first-order hyperbolic PDE systems.

Receding horizon control of cocurrent first order hyperbolic partial differential equation systems

Cocurrent first order hyperbolic partial differential equations (PDE’s) have finite impulse response (FIR) characteristics. A finite difference scheme that preserve these nice dynamic characteristics is recently developed [Choi, submitted]. Employing the resulting genuine FIR model, the design of receding horizon control is easier. In this paper, a receding horizon control scheme for cocurrent first order hyperbolic PDE systems is proposed using the FIR model and is elucidated with a tubular reactor example.

Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes

3R23. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. - PD Christofides (Dept of Chem Eng, UCLA, Los Angeles CA 90095-1592). Birkhauser Boston, Cambridge MA. 2001. 248 pp. ISBN 0-8176-4156-4. $69.95.Reviewed by J Chow (Adv Tech Center, Org L9-24, Lockheed Martin, 3251 Hanover St, Palo Alto CA 94304-1191).The author has written a book that seeks to present practical, general nonlinear, and robust control methods for hyperbolic and parabolic PDE systems, and to illustrate their application to transport-reaction processes that are found in the chemical industry. There is also an attempt to compare their effectiveness with respect to traditional control methods for PDE systems. This book is written for process control engineers, researchers, and students at the graduate level.The field of chemical engineering contains many different examples of hyperbolic and parabolic PDE systems. As the author rightly states in his preface, “The interest in control of nonlinear partial differential equation (PDE) systems has been triggered by the need to achieve tight distributed control of transport-reaction processes that exhibit highly nonlinear behavior and strong spatial variations.” As a result, there is a requirement to provide the chemical process community with an overview of the most recent advances in nonlinear PDE control theory. In this book, general and practical methods for synthesizing nonlinear controllers for hyperbolic and parabolic PDE systems are systematically developed, and then extended to include robustness. Geometric and Lyapunov-based control techniques are used to synthesize nonlinear and robust controllers that use a finite number of measurement sensors and control actuators to achieve a stable closed-loop system in the face of model uncertainty. All the PDE systems considered in this text are assumed to have unique solutions that are sufficiently smooth. Readers are assumed to have a basic knowledge about PDE systems and control theories. The first two chapters focus first on quasi-linear, first-order hyperbolic PDE systems, and then on the same type of hyperbolic PDE systems, but those which include time-varying uncertain variables and unmodeled dynamics. These control methods are based on geometric control concepts and are applied to nonisothermal plug-flow reactor examples. The next two chapters focus on developing general methods for quasi-linear parabolic PDE systems using Galerkin’s method and approximate inertial manifolds, both with and without time-varying uncertain variables. Examples of the methodology’s successful application to the control of temperature profiles for catalytic rods and nonisothermal reactors are presented for illustration. The following chapter deals with the nonlinear and robust control of parabolic PDE systems with moving boundaries, ie, time-dependent spatial domains. General methods for synthesizing nonlinear and robust time-varying output feedback controllers are presented, again using a combination of Galerkin’s method and the approximate inertial manifold. The final chapter presents case studies in which the control methods derived in earlier chapters for parabolic PDE systems are applied. In particular, the control of a rapid thermal chemical vapor deposition process and a Czochralski crystal growth process are illustrated and simulated. The book ends with an extensive set of mathematical proofs to support the results developed in the preceding chapters. It also has an extensive bibliography and is sprinkled throughout with clear figures and examples. Nonlinear and Robust Control of PDE Systems is a well-written book that succeeds in its objectives and is possibly the first one to do so. The control theories and synthesis methodologies are described in exhaustive mathematical detail. This book can also be recommended to researchers and engineers in other fields who are faced with the task of developing nonlinear feedback and robust controllers for hyperbolic and parabolic PDE systems. They will find this book to be very useful because it provides the reader with the general framework for nonlinear feedback control based on detailed mathematical models.

Nonlinear control of spatially inhomogenous aerosol processes

This article proposes a nonlinear feedback control methodology for spatially inhomogeneous aerosol processes for which the manipulated inputs, the control objectives and the measurements are distributed in space. Initially, a general nonlinear partial integro-differential equation model which describes the spatio-temporal evolution of the aerosol size distribution, as well as the evolution of the concentrations of species and temperature of the continuous phase is presented. The model accounts for simultaneous chemical reaction, nucleation, condensation, coagulation and convective transport. Then, under the assumption of lognormal aerosol size distribution, the method of moments is employed to reduce the original model into a set of first-order hyperbolic partial differential equations (PDEs) which accurately describes the spatio-temporal evolution of the three leading moments needed to precisely characterize the aerosol size distribution. This hyperbolic PDE system is then used as the basis for the synthesis of nonlinear distributed output feedback controllers that enforce closed-loop stability and achieve an aerosol size distribution with desired characteristics. The proposed nonlinear control method is successfully applied through simulations to a typical aerosol process and is shown to outperform a conventional proportional integral control scheme and deal effectively with disturbances in the feed to the process.

Velocity control of hyperbolic partial differential equation systems with single characteristic variable

This work addresses the problem of controlling a flow system described by a set of first-order partial differential equations with a single characteristic variable. The manipulated input variable is the characteristic flow velocity of the system while the controlled output is any function of the state variables in the outlet stream. Extending ideas and concepts from geometric control theory for ordinary differential equation systems, the notion of input/output linearization is used as a basis for the controller design. The method of characteristics is used to establish properties of the system dynamics. Because of the presence of a deadtime, the nonlinear system behaves in a manner analogous to linear systems having quasirational transfer functions. It is shown that the zero dynamics is marginally stable. Two types of controller designs are considered. Because of the nature of the zero dynamics, a continuous-time controller is shown to induce oscillations, making it unsuitable for practical use. A novel alternative controller is proposed that will induce a discrete-time linear input/output response in the closed loop. The discrete control action is taken at variable intervals of time equal to the residence time of the fluid entering the system at the start of that interval. The control methodology is implemented on a nonisothermal plug flow reactor and its performance and robustness are evaluated through simulations.

Robust control of hyperbolic PDE systems

This work concerns systems of quasi-linear first-order hyperbolic partial differential equations (PDEs) with uncertain variables and unmodeled dynamics. For systems with uncertain tain variables, the problem of complete elimination of the effect of uncertainty on the output via distributed feedback (uncertainty decoupling) is initially considered; a necessary and sufficient condition for its solvability as well as explicit controller synthesis formulas are derived. Then, the problem of synthesizing a distributed robust controller that achieves asymptotic output tracking with arbitrary degree of attenuation of the effect of uncertain variables on the output of the closed-loop system is addressed and solved. Robustness with respect to unmodeled dynamics is studied within a singular perturbation framework. It is established that controllers which are synthesized on the basis of a reduced-order slow model, and achieve uncertainty decoupling or uncertainty attenuation, continue to enforce these objectives in the presence of unmodeled dynamics, provided that they are stable and sufficiently fast. The developed controller synthesis results are successfully implemented through simulations on a fixed-bed reactor, modeled by two quasi-linear first-order hyperbolic PDEs, where the reactant wave propagates through the bed with significantly larger speed than the heat wave, and the heat of reaction is unknown and time varying.

A method for finding exact solutions to hyperbolic systems of first-order PDEs

Differential constraints are used as a means of developing a systematic method for finding exact solutions to quasilinear nonautonomous hyperbolic systems of first-order partial differential equations (PDEs) involving two independent variables. The leading assumption of the hyperbolicity of the basic system together with a strict compatibility argument permits characterization of the most general class of quasilinear first-order constraint equations, which can be appended to the governing mathematical model under consideration. Furthermore, the exact solutions can be found by integrating the resulting overdetermined and consistent system of PDEs. When the number of auxiliary constraint equations is equal to that of the field equations, the solutions provided by the ‘nonclassical’ group method for PDEs are recovered; for a very special form of the appended constraint equations simple wave solutions to hyperbolic systems of PDEs are also obtained. Within the present theoretical framework a model describing rate-type media is considered and several classes of initial- and/or boundary-value problems are solved.

Feedback control of hyperbolic PDE systems

AbstractThis article deals with distributed parameter systems described by first‐order hyperbolic partial differential equations (PDEs), for which the manipulated input, the controlled output, and the measured output are distributed in space. For these systems, a general output‐feedback control methodology is developed employing a combination of theory of PDEs and concepts from geometric control. A concept of characteristic index is introduced and used for the synthesis of distributed state‐feedback laws that guarantee output tracking in the closed‐loop system. Analytical formulas of distributed output‐feedback controllers are derived through combination of appropriate distributed state observers with the developed state‐feedback controllers. Theoretical analogies between our approach and available results on stabilization of linear hyperbolic PDEs are also identified. The developed control methodology is implemented on a nonisothermal plug‐flow reactor and its performance is evaluated through simulations.

On stochasticity in the Von Foerster hyperbolic partial differential equation system: Further applications to the modeling of an asynchronously dividing cellular system

We investigate the problem of including stochastically varying components in the Von Foerster hyperbolic partial differential equation system. In particular, we address the incorporation of stochasticity in the birth and death rates of the model. We assume, for the purposes of this analysis, that the stochasticity is slowly varying. The results of this analysis have bearing upon the modeling of biological populations which have a describable age-time dynamics. This analysis also has bearing upon some recent results concerning the modeling of asynchronously dividing cellular systems.

Modelling and simulation of the thernomechanical pulping process

The Thermomechanical Pulping Process consists of a preheater, which heats the wood chips, and a pressurized doubledisc refiner, which produces the pulp. This paper develops a dynamical model of the process. The models of the refiner and the preheater are mixed distributed and lumped parameter models. A modified Lax-Wendroff type method was used to solve the equations of the fourth-order hyperbolic partial differential equation system describing the refiner.The simulated results were compared with experimental results measured on an industrial-scale refiner. The results were used to optimize the static operating point of the refiner and to study the effects of the control variables on operation. The effects of several disturbances were also simulated. The shape of the optimal plate gap is also a feature included in the study.

Modelling and Simulation of the Thermomechanical Pulping Process

The Thermomechanical Pulping Process consists of a preheater, which heats the wood chips, and a pressurized double-disc refiner, which produces the pulp. This paper develops a dynamical model of the process. The models of the refiner and the preheater are mixed distributed and lumped parameter models. A modified Lax-Wendroff type method was used to solve the equations of the fourth-order hyperbolic partial differential equation system describing the refiner.The simulated results were compared with experimental results measured on an industrial-scale refiner. The results were used to optimize the static operating point of the refiner and to study the effects of the control variables on operation. The effects of several disturbances were also simulated. The shape of the optimal plate gap is also a feature included in the study.