Optimal Control for Chemical Engineers.
| Main Author: | |
|---|---|
| Format: | eBook |
| Language: | English |
| Published: |
Milton :
Taylor & Francis Group,
2012.
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| Edition: | 1st ed. |
| Subjects: | |
| Online Access: | Click to View |
Table of Contents:
- Cover
- Half Title
- Title Page
- Copyright Page
- Dedication
- Table of Contents
- Preface
- Notation
- 1 Introduction
- 1.1 Definition
- 1.2 Optimal Control versus Optimization
- 1.3 Examples of Optimal Control Problems
- 1.3.1 Batch Distillation
- 1.3.2 Plug Flow Reactor
- 1.3.3 Heat Exchanger
- 1.3.4 Gas Diffusion in a Non-Volatile Liquid
- 1.3.5 Periodic Reactor
- 1.3.6 Nuclear Reactor
- 1.3.7 Vapor Extraction of Heavy Oil
- 1.3.8 Chemotherapy
- 1.3.9 Medicinal Drug Delivery
- 1.3.10 Blood Flow and Metabolism
- 1.4 Structure of Optimal Control Problems
- Bibliography
- Exercises
- 2 Fundamental Concepts
- 2.1 From Function to Functional
- 2.1.1 Functional as a Multivariable Function
- 2.2 Domain of a Functional
- 2.2.1 Linear or Vector Spaces
- 2.2.2 Norm of a Function
- 2.3 Properties of Functionals
- 2.4 Differential of a Functional
- 2.4.1 Fréchet Differential
- 2.4.2 Gâteaux Differential
- 2.4.3 Variation
- 2.4.4 Summary of Differentials
- 2.4.5 Relations between Differentials
- 2.5 Variation of an Integral Objective Functional
- 2.5.1 Equivalence to Other Differentials
- 2.5.2 Application to Optimal Control Problems
- 2.6 Second Variation
- 2.6.1 Second Degree Homogeneity
- 2.6.2 Contribution to Functional Change
- 2.A Second-Order Taylor Expansion
- Bibliography
- Exercises
- 3 Optimality in Optimal Control Problems
- 3.1 Necessary Condition for Optimality
- 3.2 Application to Simplest Optimal Control Problem
- 3.2.1 Augmented Functional
- 3.2.2 Optimal Control Analysis
- 3.2.3 Generalization
- 3.3 Solving an Optimal Control Problem
- 3.3.1 Presence of Several Local Optima
- 3.4 Sufficient Conditions
- 3.4.1 Weak Sufficient Condition
- 3.5 Piecewise Continuous Controls
- 3.A Differentiability of λ
- 3.B Vanishing of (Fy + λGy + λ) at t=0
- 3.C Mangasarian Sufficiency Condition.
- Bibliography
- Exercises
- 4 Lagrange Multipliers
- 4.1 Motivation
- 4.2 Role of Lagrange Multipliers
- 4.3 Lagrange Multiplier Theorem
- 4.3.1 Generalization to Several Equality Constraints
- 4.3.2 Generalization to Several Functions
- 4.3.3 Application to Optimal Control Problems
- 4.4 Lagrange Multiplier and Objective Functional
- 4.4.1 General Relation
- 4.5 John Multiplier Theorem for Inequality Constraints
- 4.5.1 Generalized John Multiplier Theorem
- 4.5.2 Remark on Numerical Solutions
- 4.A Inverse Function Theorem
- Bibliography
- Exercises
- 5 Pontryagin's Minimum Principle
- 5.1 Application
- 5.2 Problem Statement
- 5.2.1 Class of Controls
- 5.2.2 New State Variable
- 5.2.3 Notation
- 5.3 Pontryagin's Minimum Principle
- 5.3.1 Assumptions
- 5.3.2 Statement
- 5.4 Derivation of Pontryagin's Minimum Principle
- 5.4.1 Pulse Perturbation of Optimal Control
- 5.4.2 Temporal Perturbation of Optimal Control
- 5.4.3 Effect on Final State
- 5.4.4 Choice of Final Costate
- 5.4.5 Minimality of the Hamiltonian
- 5.4.6 Zero Hamiltonian at Free Final Time
- 5.A Convexity of Final States
- 5.B Supporting Hyperplane of a Convex Set
- Bibliography
- 6 Different Types of Optimal Control Problems
- 6.1 Free Final Time
- 6.1.1 Free Final State
- 6.1.2 Fixed Final State
- 6.1.3 Final State on Hypersurfaces
- 6.2 Fixed Final Time
- 6.2.1 Free Final State
- 6.2.2 Fixed Final State
- 6.2.3 Final State on Hypersurfaces
- 6.3 Algebraic Constraints
- 6.3.1 Algebraic Equality Constraints
- 6.3.2 Algebraic Inequality Constraints
- 6.4 Integral Constraints
- 6.4.1 Integral Equality Constraints
- 6.4.2 Integral Inequality Constraints
- 6.5 Interior Point Constraints
- 6.6 Discontinuous Controls
- 6.7 Multiple Integral Problems
- Bibliography
- Exercises
- 7 Numerical Solution of Optimal Control Problems.
- 7.1 Gradient Method
- 7.1.1 Free Final Time and Free Final State
- 7.1.2 Iterative Procedure
- 7.1.3 Improvement Strategy
- 7.1.4 Algorithm for the Gradient Method
- 7.1.5 Fixed Final Time and Free Final State
- 7.2 Penalty Function Method
- 7.2.1 Free Final Time and Final State on Hypersurfaces
- 7.2.2 Free Final Time but Fixed Final State
- 7.2.3 Algebraic Equality Constraints
- 7.2.4 Integral Equality Constraints
- 7.2.5 Algebraic Inequality Constraints
- 7.2.6 Integral Inequality Constraints
- 7.3 Shooting Newton-Raphson Method
- 7.A Derivation of Steepest Descent Direction
- 7.A.1 Objective
- 7.A.2 Sufficiency Check
- Bibliography
- Exercises
- 8 Optimal Periodic Control
- 8.1 Optimality of Periodic Controls
- 8.1.1 Necessary Conditions
- 8.2 Solution Methods
- 8.2.1 Successive Substitution Method
- 8.2.2 Shooting Newton-Raphson Method
- 8.3 Pi Criterion
- 8.4 Pi Criterion with Control Constraints
- 8.A Necessary Conditions for Optimal Steady State
- 8.B Derivation of Equation (8.12)
- 8.C Fourier Transform
- 8.D Derivation of Equation (8.25)
- Bibliography
- Exercises
- 9 Mathematical Review
- 9.1 Limit of a Function
- 9.2 Continuity of a Function
- 9.2.1 Lower and Upper Semi-Continuity
- 9.3 Intervals and Neighborhoods
- 9.4 Bounds
- 9.5 Order of Magnitude
- 9.5.1 Big-O Notation
- 9.6 Taylor Series and Remainder
- 9.7 Autonomous Differential Equations
- 9.7.1 Non-Autonomous to Autonomous Transformation
- 9.8 Differential
- 9.9 Derivative
- 9.9.1 Directional Derivative
- 9.10 Leibniz Integral Rule
- 9.11 Newton-Raphson Method
- 9.12 Composite Simpson's 1/3 Rule
- 9.13 Fundamental Theorem of Calculus
- 9.14 Mean Value Theorem
- 9.14.1 For Derivatives
- 9.14.2 For Integrals
- 9.15 Intermediate Value Theorem
- 9.16 Implicit Function Theorem
- 9.17 Bolzano-Weierstrass Theorem.
- 9.18 Weierstrass Theorem
- 9.19 Linear or Vector Space
- 9.20 Direction of a Vector
- 9.21 Parallelogram Identity
- 9.22 Triangle Inequality for Integrals
- 9.23 Cauchy-Schwarz Inequality
- 9.24 Operator Inequality
- 9.25 Conditional Statement
- 9.26 Fundamental Matrix
- Bibliography
- Index.