Optimal Control for Chemical Engineers

310 Pages·2013·5.61 MB·English

by  Simant Ranjan Upreti

Optimal Control for
Chemical Engineers
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Optimal Control for
Chemical Engineers
Simant Ranjan Upreti
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Upreti, Simant Ranjan.
Optimal control for chemical engineers / Simant Ranjan Upreti.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4398-3894-5 (hardback)
1. Chemical process control. 2. Mathematical optimization. I. Title.
TP155.75.U67 2012
519.6--dc23 2012030747
Visit the Taylor & Francis Web site at
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This book is dedicated to
my parents,
Drs. Hema and Murari Lal Upreti,
Grand Master Choa Kok Sui,
Shaykh Khwaja Shamsuddin Azeemi,
and
Lord Gan.apati

Contents
Preface xiii
Notation xv
1 Introduction 1
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Optimal Control versus Optimization . . . . . . . . . . . . . 4
1.3 Examples of Optimal Control Problems . . . . . . . . . . . . 5
1.3.1 Batch Distillation . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Plug Flow Reactor . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Heat Exchanger. . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 Gas Diffusion in a Non-Volatile Liquid . . . . . . . . . 9
1.3.5 Periodic Reactor . . . . . . . . . . . . . . . . . . . . . 11
1.3.6 Nuclear Reactor . . . . . . . . . . . . . . . . . . . . . 12
1.3.7 Vapor Extraction of Heavy Oil . . . . . . . . . . . . . 13
1.3.8 Chemotherapy . . . . . . . . . . . . . . . . . . . . . . 15
1.3.9 Medicinal Drug Delivery . . . . . . . . . . . . . . . . . 16
1.3.10 Blood Flow and Metabolism . . . . . . . . . . . . . . 17
1.4 Structure of Optimal Control Problems . . . . . . . . . . . . 19
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Fundamental Concepts 23
2.1 From Function to Functional . . . . . . . . . . . . . . . . . . 23
2.1.1 Functional as a Multivariable Function . . . . . . . . . 25
2.2 Domain of a Functional . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Linear or Vector Spaces . . . . . . . . . . . . . . . . . 26
2.2.2 Norm of a Function . . . . . . . . . . . . . . . . . . . 26
2.3 Properties of Functionals . . . . . . . . . . . . . . . . . . . . 27
2.4 Differential of a Functional . . . . . . . . . . . . . . . . . . . 28
2.4.1 Fr´echet Differential . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Gˆateaux Differential . . . . . . . . . . . . . . . . . . . 30
2.4.3 Variation . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.4 Summary of Differentials . . . . . . . . . . . . . . . . 38
2.4.5 Relations between Differentials . . . . . . . . . . . . . 38
2.5 Variation of an Integral Objective Functional . . . . . . . . . 40
2.5.1 Equivalence to Other Differentials . . . . . . . . . . . 41
vii
viii
2.5.2 Application to Optimal Control Problems . . . . . . . 44
2.6 Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Second Degree Homogeneity. . . . . . . . . . . . . . . 50
2.6.2 Contribution to Functional Change . . . . . . . . . . . 51
2.A Second-Order Taylor Expansion . . . . . . . . . . . . . . . . 52
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Optimality in Optimal Control Problems 57
3.1 Necessary Condition for Optimality . . . . . . . . . . . . . . 57
3.2 Application to Simplest Optimal Control Problem . . . . . . 58
3.2.1 Augmented Functional . . . . . . . . . . . . . . . . . . 59
3.2.2 Optimal Control Analysis . . . . . . . . . . . . . . . . 59
3.2.3 Generalization . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Solving an Optimal Control Problem . . . . . . . . . . . . . 71
3.3.1 Presence of Several Local Optima. . . . . . . . . . . . 73
3.4 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Weak Sufficient Condition . . . . . . . . . . . . . . . . 75
3.5 Piecewise Continuous Controls . . . . . . . . . . . . . . . . . 76
3.A Differentiability of λ . . . . . . . . . . . . . . . . . . . . . . . 78
3.B Vanishing of (F +λG +λ˙) att =0 . . . . . . . . . . . . . 81
y y
3.C MangasarianSufficiency Condition . . . . . . . . . . . . . . . 81
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Lagrange Multipliers 87
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Role of Lagrange Multipliers . . . . . . . . . . . . . . . . . . 88
4.3 Lagrange Multiplier Theorem . . . . . . . . . . . . . . . . . . 88
4.3.1 Generalization to Several Equality Constraints . . . . 94
4.3.2 Generalization to Several Functions . . . . . . . . . . 96
4.3.3 Application to Optimal Control Problems . . . . . . . 99
4.4 Lagrange Multiplier and Objective Functional . . . . . . . . 107
4.4.1 General Relation . . . . . . . . . . . . . . . . . . . . . 109
4.5 John Multiplier Theorem for Inequality Constraints . . . . . 109
4.5.1 Generalized John Multiplier Theorem . . . . . . . . . 113
4.5.2 Remark on Numerical Solutions. . . . . . . . . . . . . 115
4.A Inverse Function Theorem . . . . . . . . . . . . . . . . . . . 115
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
ix
5 Pontryagin’s Minimum Principle 123
5.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 Class of Controls . . . . . . . . . . . . . . . . . . . . . 126
5.2.2 New State Variable . . . . . . . . . . . . . . . . . . . . 127
5.2.3 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 Pontryagin’sMinimum Principle . . . . . . . . . . . . . . . . 128
5.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.2 Statement . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4 Derivation of Pontryagin’sMinimum Principle . . . . . . . . 129
5.4.1 Pulse Perturbation of Optimal Control . . . . . . . . . 131
5.4.2 Temporal Perturbation of Optimal Control . . . . . . 134
5.4.3 Effect on Final State . . . . . . . . . . . . . . . . . . . 134
5.4.4 Choice of Final Costate . . . . . . . . . . . . . . . . . 136
5.4.5 Minimality of the Hamiltonian . . . . . . . . . . . . . 137
5.4.6 Zero Hamiltonian at Free Final Time. . . . . . . . . . 145
5.A Convexity of Final States . . . . . . . . . . . . . . . . . . . . 145
5.B Supporting Hyperplane of a Convex Set . . . . . . . . . . . . 149
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Different Types of Optimal Control Problems 153
6.1 Free Final Time . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1.1 Free Final State . . . . . . . . . . . . . . . . . . . . . 153
6.1.2 Fixed Final State . . . . . . . . . . . . . . . . . . . . . 157
6.1.3 Final State on Hypersurfaces . . . . . . . . . . . . . . 158
6.2 Fixed Final Time . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2.1 Free Final State . . . . . . . . . . . . . . . . . . . . . 161
6.2.2 Fixed Final State . . . . . . . . . . . . . . . . . . . . . 161
6.2.3 Final State on Hypersurfaces . . . . . . . . . . . . . . 162
6.3 Algebraic Constraints . . . . . . . . . . . . . . . . . . . . . . 163
6.3.1 Algebraic Equality Constraints . . . . . . . . . . . . . 163
6.3.2 Algebraic Inequality Constraints . . . . . . . . . . . . 166
6.4 Integral Constraints . . . . . . . . . . . . . . . . . . . . . . . 168
6.4.1 Integral Equality Constraints . . . . . . . . . . . . . . 168
6.4.2 Integral Inequality Constraints . . . . . . . . . . . . . 171
6.5 Interior Point Constraints . . . . . . . . . . . . . . . . . . . . 172
6.6 Discontinuous Controls . . . . . . . . . . . . . . . . . . . . . 177
6.7 Multiple Integral Problems . . . . . . . . . . . . . . . . . . . 178
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183