Nonlinear order reduction and control of dissipative partial differential equation systems: Methods and applications to transport-reaction processes and fluid flows.

摘要

The development of general and practical control algorithms for dissipative nonlinear partial differential equation (PDE) systems that mathematically describe fluid flow and transport-reaction processes is a fundamental problem with a variety of industrially important applications. Examples range from the feedback control of turbulence for drag reduction, to suppression of fluid mechanical instabilities in coating processes and suppression of waves exhibited by falling liquid films, and from the suppression of thermal dislocations in high-purity crystals during the Czochralski crystallization to the feedback control of chemical vapor deposition and etching of thin films for microelectronics manufacturing to achieve spatially uniform thickness.; Traditional approaches for controlling chemical distributed processes are based on the simplifying assumption that the control variables are spatially uniform; yet, many industrial control problems involve regulation of variables which are distributed in space and cannot be effectively solved with these approaches. These limitations together with the recent advances in the development of fundamental mathematical models that accurately predict the behavior of transport-reaction and fluid flow processes, provide a strong motivation for developing a general framework for nonlinear controller design based on detailed models, thus exploiting the ability of a model to predict the behavior of a process and the fundamental knowledge of the underlying physico-chemical phenomena that the model contains. The key difficulty in developing model-based control methods for transport-reaction and fluid flow processes lies in the “infinite-dimensional” nature of the distributed process models, which prohibits their direct use for control system design.; Motivated by the above, this doctoral thesis presents a general and practical methodology for the nonlinear order reduction and control of a general class of process models, described by dissipative partial differential equation systems, which arise in the modeling of diffusion-convection-reaction processes with fixed and time-varying spatial domains and fluid dynamic systems. The methodology proposes a combination of Galerkin's method with approximate inertial manifolds to derive low-dimensional approximations of the distributed process model, which are employed for the selection of the control configuration and the synthesis of high-performance nonlinear feedback controllers using geometric control methods and Lyapunov techniques. A rigorous analysis of the closed-loop system (distributed process model and controller is performed to derive precise conditions which guarantee that the desired stability and performance properties are achieved in the presence of uncertainty in the values of the parameters of the process model.; The developed methodology is applied, via computer simulations, to industrially important transport-reaction processes such as the Czochralski crystal growth, plasma-enhanced chemical vapor deposition and plasma etching and towards the suppression of instabilities exhibited by falling and shallow liquid films, described by the Kuramoto-Sivashinsky and Korteweg-de-Vries Burgers equations respectively.