Title

Process design using optimal control methodology

Date of Award

1990

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Biomedical and Chemical Engineering

Advisor(s)

John Heydweiller

Keywords

variational problem, reactor design

Subject Categories

Chemical Engineering

Abstract

Optimal control problems arise naturally in many fields of science and engineering. In particular, these problems appear frequently in reactor design applications. It is typically difficult, and frequently impossible, to obtain an analytical solution to a realistic optimal control problem. Therefore, a common approach is the employment of a numerical approximation strategy.

In this work, the Rayleigh-Ritz method is used by replacing the control function by a polynomial approximation with unknown coefficients. The coefficients in the approximation then become the search variables in the resulting nonlinear programming problem. It is demonstrated that, provided that the optimal control function is smooth, the Rayleigh-Ritz method can produce an accurate solution for an optimal control problem. It is also demonstrated, however, that the Rayleigh-Ritz method with a global polynomial approximation produces inaccurate and infeasible control function approximations when the optimal control function is not smooth.

With the application of the Modified Rayleigh-Ritz method, nonsmooth control functions are handled by employing piecewise polynomial approximations. The break points of the approximation are made search variables, and the control function is allowed to have a corner or discontinuity at each break point. It is demonstrated that, provided that a sufficiently large number of break points is used in the approximation, an accurate approximation for a nonsmooth control function can be obtained with this method. An adaptive version of the Modified Rayleigh-Ritz method, namely the Adaptive Approximation Strategy, is developed to handle cases where the number of corners/discontinuities is not known in advance. This strategy, which employs an heuristic criterion for the increasing of the number of break points in the approximation, is shown to provide significant savings in computational effort.

One major weakness of the Modified Rayleigh-Ritz method is that a nonlinear (i.e., piecewise cubic) approximation is guaranteed to satisfy the bounds on the control function only at the interpolation points of the approximation. It is demonstrated here that, even when the number of break points used is equal to the number of corners/discontinuities in the actual solution, the Modified Rayleigh-Ritz method may produce an infeasible and inaccurate approximation. The Bounded Control Profile method developed in this work is capable of handling nonlinear and nonsmooth control functions, and the approximation obtained using this method is guaranteed to satisfy the bounds everywhere.

Since repeated numerical integration of a set of ordinary differential equations can be prohibitively expensive, an approximate method, namely the orthogonal collocation on finite elements is used here to evaluate the state functions. An important pitfall in the application of this popular method, or any other approximate integration method, to optimal control problems is identified here: Small errors in the state functions may lead to incorrect control function approximations.

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