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Prodromos Daoutidis, Principal Investigator

Dynamics and Control of Chemical Plants and Feedback Control of Incompressible Flows

The design of effective controllers for complete chemical plants is an important problem and has recently attracted considerable attention. A typical chemical plant consists of a set of chemical reactors followed by a collection of separation units, to separate the reactants and the products and one or more streams from the separation units to the reactors, to recycle the unconverted reactants. Research on chemical process control has focused on addressing the design of linear/nonlinear controllers for individual reaction and separation units on the basis of appropriate dynamic models. This implicitly assumed that the combination of these individual controllers for the respective units would comprise an effective "distributed" control scheme for the overall plant. However, in the presence of recycle streams, such a control scheme is not adequate; a recycle stream introduces a feedback interaction between the process units that often leads to significantly different and highly nonlinear dynamics as opposed to simple cascades of units without recycle. This implies the need for designing controllers on the basis of dynamic models that include feedback interactions due to recycle. However, it is not possible to design a single centralized controller on the basis of a detailed dynamic model of the plant, which typically includes several hundred coupled differential and algebraic equations (DAEs).

These researchers are currently developing and testing a systematic framework for the design of controllers for complete chemical plants. To this end, it is established that chemical plants with recycle, where the recycle flowrates are typically large compared to feed/product flowrates, exhibit a non-explicit time-scale multiplicity. This implies that the control objectives should be properly formulated and addressed in the appropriate time-scales. More specifically, in the fast time-scale, which is of the order of the residence time of a unit, the dynamics of the individual units evolve independently of each other with "weak" interactions arising from the recycle. These weak interactions can be ignored in the design of the distributed controllers to achieve the requisite stabilization/tracking objectives for the individual units in the fast time-scale. However, in the long run, these weak interactions between the individual units will give rise to a significant dynamics of the plant core. This indicates that a separate "supervisory" controller should be designed to address the overall plant control objectives in the slow time-scale.

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Two key features that guide the design of the supervisory controller for the slow plant dynamics are that these slow dynamics are usually of low order, and they are highly nonlinear. Thus, given a high-order detailed model of a complex plant, it is imperative to perform a systematic model-reduction to obtain a low-order model that describes the essential slow dynamics of the plant core and can be used for nonlinear controller design. Such a model can be obtained by ignoring the fast dynamics of the individual units with the respective distributed controllers. The resulting model is given by a high-index DAE system. For such high-index DAE systems, a controller can be designed on the basis of an appropriate minimal-order state-space realization. This approach provides a systematic framework for proper model-reduction and the design of well-conditioned supervisory controllers for complex chemical plants.

The second part of this work deals with feedback control of incompressible flows. The study of dynamics and control of fluid flows is becoming increasingly important in the context of materials/chemical processing applications. Incompressible, isothermal flows are mathematically represented by the time dependent momentum equations and the continuity constraint (Navier-Stokes equations). The difficulties associated with this representation, known as the velocity-pressure formulation, are the continuity constraint (arising from the incompressibility assumption), the nonlinear convection term, and the absence of an explicit equation for pressure. When working in two spatial dimensions, some of these difficulties can be overcome by reformulating the equations into the so called stream-function formulation. In this, the pressure is eliminated, the continuity constraint is trivially satisfied, and the three partial differential equations are reduced to one. Considerable progress has been made in the mathematical analysis of both formulations. Significant research has also gone into the approximation and numerical simulation of solutions using different spatial discretization schemes. On the other hand, apart from some optimal control approaches, very few results are available on feedback control for these systems.

The objective of this work is to study the dynamics and control of incompressible flow in a two-dimensional driven cavity (this is a benchmark example for validating computational fluid dynamics algorithms, and also represents a simplified version of manufacturing devices such as short-dwell coaters and flexible blade coaters). A driven cavity is a square region with solid boundaries. The top and bottom boundaries are allowed to move and the side boundaries are always at rest.

Work-to-date has been focussed on spatial discretization of velocity-pressure formulation via finite differences, the Galerkin Finite Element Method (GFEM), and preliminary simulation studies at low Reynolds numbers. This work is studying flows at high Reynolds numbers. To this end, these researchers are addressing the derivation of a reduced-order ordinary differential equation (ODE) model that captures the dynamic behavior of the process using the Fourier-Galerkin method with the stream-function formulation. Accuracy of this reduced-order model is evaluated through comparisons with the model obtained via a finite difference discretization of the velocity-pressure formulation. This reduced-order model is used as the basis for a detailed computational study of the dynamics of the system.

Finally, these researchers are addressing the problem of controlling the flow pattern in cavity. For this, they consider the total kinetic energy of the system as the controlled output and the acceleration of the lid as the manipulated input. Both state-feedback and output-feedback controllers are designed. Their performance in output regulation, tracking, suppressing flow instabilities, and superiority to linear control methods are illustrated by simulations.

This information is available in alternative formats upon request by individuals with disabilities. Please send email to alt-format@msi.umn.edu or call 612-624-0528.

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