Numerical solution of an optimal temperature problem

Walder, T.J.; Storey, C.

doi:10.1016/0300-9467(70)85005-5

Cited by 5 publications

(2 citation statements)

References 17 publications

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“…However, they employed numerical algorithms which are rather difficult to implement and which produce values of the objective functional rather far from the true optimum. More recent studies (Seinfeld and Lapidus, 1968a, b;Storey, 1970;Nauman and Mallikarjun, 1984, Buzzi-Ferraris el al., 1984) have emphasized direct search techniques and simple functional forms for the control functions, u(x). Such a technique of using approximating function forms is also known as the Rayleigh-Ritz method.…”

Section: Introducfionmentioning

confidence: 92%

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On Optimization in Tubular Reactor Systems

Mallikarjun

Nauman

1987

Chemical Engineering Communications

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Section: Introducfionmentioning

confidence: 92%

“…1 1 7 . xl(9) Here & is the ratio of pre-exponential rate constants for the backward andDownloaded by [University of Toronto Libraries] at 11:32 20 December 2014…”

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confidence: 99%

On Optimization in Tubular Reactor Systems

Mallikarjun

Nauman

1987

Chemical Engineering Communications

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The computation of optimal singular bang‐bang control II. Nonlinear systems

Edgar

Lapidus

1972

AIChE Journal

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A general algorithm for the computation of singular/bang-bang control, previously applied to linear systems, is extended to nonlinear systems. The minimum time control of a two-stage CSTR is demonstrated. SCOPEIn the preceding paper by Edgar and Lapidus (1972) it is shown how optimal singular/bang-bang control can be computed for linear systems with fixed or nonfixed final times. The computation utilizes the conversion of the given control problem into a linear-quadratic problem (LQP), which is then solved via discrete dynamic programming with discrete penalty functions to handle end point and control constraints. Inherent in the problem solution is a limiting process, which solves the singular/bang-bang problem as the limit of a series of nonsingular/non-bangbang problems.In this work it will be shown how a singular/bang-bang control problem with a nonlinear system equation can be converted into an LQP. The method of solution for the LQP is the same as that mentioned earlier; however, the calculated control is suboptimal, as opposed to the optimal control which results for linear systems. The calculated control is suboptimal because the nonlinear state equations are approximated as time-varying linear state equations. The suboptimal control estimate is then refined by first-order and second-order descent methods in conjunction with the limiting process to yield the optimal singular/bang-bang control.A number of nonlinear system reactor engineering examples have been computed to test the effectiveness of the proposed algorithm. The optimal startup control of a four-state variable system, a probIem which cannot be solved by the method of phase plane analysis, has been computed by the new algorithm and is presented here. The results demonstrate that the new algorithm is technically not limited by system dimensionality, although this factor does affect the amount of computation time. CONCLUSIONS AND SIGNIFICANCEThe application of a general algorithm for solving singular/bang-bang problems with nonlinear systems is presented. The approach utilizes Pearson apparent linearization of the nonlinear state equations and a limiting process which solves the original problem as a series of nonsingular/nonbang-bang problems. In this work the algorithm is applied to the startup of two CSTR's in series, a problem originally described by Siebenthal and Aris (1964). The Pearson method obtains a good suboptimal estimate of the minimum time and optimal control, but for extremely nonlinear equations, the process can be quite time-consuming. The conjugate gradient (first order) and direct second variation (second order) methods readily refine the suboptimal control to an optimal control, but some care must be exercised in the use of penalty functions for the constraints.One can conclude, however, that the general algorithm effectively obtains singular/bang-bang control for systems of large state dimension, extreme nonlinearity, and multiple controls. The algorithm handles linear or nonlinear control problems and solves both the fixed ...

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Approximation methods for optimal control synthesis

1971

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A number of approximate methods for optimal control synthesis are reviewed and tested on a highly non‐linear continuous‐stirred‐tank‐reactor example. It was found that parameterizing a feed‐back controller and searching for the optimal coefficients was computationally as fast as the standard conjugate gradient approach. Ancilliary advantages of this approach make it an attractive means of optimal control synthesis.

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Numerical solution of an optimal temperature problem

Cited by 5 publications

References 17 publications

On Optimization in Tubular Reactor Systems

On Optimization in Tubular Reactor Systems

The computation of optimal singular bang‐bang control II. Nonlinear systems

Approximation methods for optimal control synthesis

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