Studies of singular solutions in dynamic optimization: I. Theoretical aspects and methods of solution

Ko, Daniel Y. C.; Steve, W. F.

doi:10.1002/aic.690170149

Cited by 8 publications

(2 citation statements)

References 9 publications

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“…As a result, the PMP does not provide enough information to derive u s (t). 41 The points in t where the control switches between singular and non-singular arcs are referred to as switching times or switching instants. When u t ð Þ ℝ nu for n u > 1, the values of u(t) given in Equation ( 72) are element-wise.…”

Section: Singular Casementioning

confidence: 99%

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Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.

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Section: Singular Casementioning

confidence: 99%

“…In this case, u ( t ) vanishes from the necessary condition for a minimum, H u ( t ) = 0. As a result, the PMP does not provide enough information to derive u s ( t ) 41 . The points in t where the control switches between singular and non‐singular arcs are referred to as switching times or switching instants .…”

Section: Optimality Conditionsmentioning

confidence: 99%

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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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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An effective computational algorithm for suboptimal singular and/or bang‐bang control II. Applications to nonlinear lumped and distributed systems

et al. 1976

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kl kZ a pure catalyst which catalyzes the reaction A Ft B only.The value of performance index, that is, the mole fraction of the substance C present in the mixture at the reactor exit at t = 1, with the pure bang-bang policy used is 0.047918. Although this policy is only a suboptimal one, it compares quite favorably with the optimal singular solution obtained by Jackson (1968), who gives a value of performance index of only 0.3% smaller, that is, 0.048065. 1966), a two-stage continuous stirred-tank reactor (CSTR) system (Siebenthal and Aris, 1964;Edgar and Lapidus, 1972b; Luus, 1974~1, 1974b, and a six-plate gas absorber with nonlinear gas-liquid equilibrium relationship (Lapidus and Luus, 1967;Weber and Lapidus, 1971). The determination of the optimal catalyst activity distribution policy in a distributed parameter, nonisothermal tubular reactor with radial heat and mass diffusion is also studied. These four test problems represent the typical chemical engineering optimal control problems with high state dimensionality, extreme nonlinearity, and multiple controls. The practical implications of the computational results are discussed, and the comparisons of the proposed algorithm with several existing techniques are given. CONCLUSIONS AND SIGNIFICANCEFrom examination of the numerical and graphical results for the chosen test problems presented in this paper, it can be concluded that the application of proposed algorithm based on the piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables yields very acceptable, suboptimal singular and/or bang-bang control solutions with little expenditure of computer storage and computa-

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Studies of singular solutions in dynamic optimization: I. Theoretical aspects and methods of solution

Cited by 8 publications

References 9 publications

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

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

An effective computational algorithm for suboptimal singular and/or bang‐bang control II. Applications to nonlinear lumped and distributed systems

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