Quasilinearization and Invariant Imbedding

Quasilinearization and Invariant Imbedding

With Applications to Chemical Engineering and Adaptive Control

1st Edition - January 1, 1968

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  • Author: E. Stanley Lee
  • eBook ISBN: 9781483266756

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Mathematics in Science and Engineering, Volume 41: Quasilinearization and Invariant Imbedding presents a study on the use of two concepts for obtaining numerical solutions of boundary-value problems—quasilinearization and invariant imbedding. This book emphasizes that the invariant imbedding approach reformulates the original boundary-value problem into an initial value problem by introducing new variables or parameters, while the quasilinearization technique represents an iterative approach combined with linear approximations. This volume focuses on analytical aspects that are concerned with actual convergence rates and computational requirements, considering various efficient algorithms that are suited for various types of boundary-value problems. This publication is a good reference for chemical and control engineers and scientists interested in obtaining numerical solutions of boundary-value problems in their particular fields.

Table of Contents

  • Preface

    Chapter 1. Introductory Concepts

    1. Introduction

    2. Quasilinearization

    3. Invariant Imbedding

    4. Invariant Imbedding versus the Classical Approach

    5. Numerical Solution of Ordinary Differential Equations

    6. Numerical Solution Terminologies


    Chapter 2. Quasilinearization

    1. Introduction

    2. Nonlinear Boundary-Value Problems

    3. Linear Boundary-Value Problems

    4. Finite-Difference Method for Linear Differential Equations

    5. Discussion

    6. Newton-Raphson Method

    7. Discussion

    8. Quasilinearization

    9. Discussion

    10. Existence and Convergence

    11. Existence

    12. Convergence

    13. Maximum Operation and Differential Inequalities

    14. Construction of a Monotone Sequence

    15. Approximation in Policy Space and Dynamic Programming

    16. Discussion

    17. Systems of Differential Equations


    Chapter 3. Ordinary Differential Equations

    1. Introduction

    2. A Second-Order Nonlinear Differential Equation

    3. Recurrence Relation

    4. Computational Procedure

    5. Numerical Results

    6. Stability Problem in Numerical Solution—The Fixed Bed Reactor

    7. Finite-Difference Method

    8. Systems of Algebraic Equations Involving Tridiagonal Matrices

    9. Numerical Results

    10. Stability Problem with High Peclet Number

    11. Adiabatic Tubular Reactor with Axial Mixing

    12. Numerical Results

    13. Discussion

    14. Unstable Initial-Value Problems

    15. Discussion

    16. Systems of Differential Equations

    17. Computational Considerations

    18. Simultaneous Solution of Different Iterations


    Chapter 4. Parameter Estimation

    1. Introduction

    2. Parameter Estimation and the "Black Box" Problem

    3. Parameter Estimation and the Experimental Determination of Physical Constants

    4. A Multipoint Boundary-Value Problems

    5. The Least Squares Approach

    6. Computational Procedure for a Simpler Problems

    7. Numerical Results

    8. Nonlinear Boundary Condition

    9. Random Search Technique

    10. Numerical Results

    11. Discussion

    12. Parameter Up-Dating

    13. Discussion

    14. Estimation of Chemical Reaction Rate Constants

    15. Differential Equations with Variable Coefficients

    16. An Example

    17. III-Conditioned Systems

    18. Numerical Results

    19. Discussion

    20. An Empirical Approximation

    21. Numerical Results

    22. A Second Approximation

    23. Numerical Results

    24. Differential Approximation

    25. A Second Formulation

    26. Computational Aspects

    27. Discussion


    Chapter 5. Optimization

    1. Introduction

    2. Optimum Temperature Profiles in Tubular Reactors

    3. Numerical Results

    4. Discussion

    5. Back and Forth Integration

    6. Two Consecutive Gaseous Reactions

    7. Optimum Pressure Profile in Tubular Reactor

    8. Numerical Results

    9. Optimum Temperature Profile with Pressure as Parameter

    10. Numerical Results and Procedures

    11. Calculus of Variations with Control Variable Inequality Constraint

    12. Calculus of Variations with Pressure Drop in the Reactor

    13. Pontryagin's Maximum Principle

    14. Discussion

    15. Optimum Feed Conditions

    16. Partial Derivative Evaluation

    17. Conclusions


    Chapter 6. Invariant Imbedding

    1. Introduction

    2. The Invariant Imbedding Approach

    3. An Example

    4. The Missing Final Condition

    5. Determination of x and y in Terms of r and s

    6. Discussion

    7. Alternate Formulations—I

    8. Linear and Nonlinear Systems

    9. The Riccati Equation

    10. Alternate Formulations—II

    11. The Reflection and Transmission Functions

    12. Systems of Differential Equations

    13. Large Linear Systems

    14. Computational Considerations

    15. Dynamic Programming

    16. Discussion


    Chapter 7. Quasilinearization and Invariant Imbedding

    1. Introduction

    2. The Predictor-Corrector Formula

    3. Discussion

    4. Linear Boundary-Value Problems

    5. Numerical Results

    6. Optimum Temperature Profiles in Tubular Reactors

    7. Numerical Results

    8. Discussion

    9. Dynamic Programming and Quasilinearization—I

    10. Discussion

    11. Linear Differential Equations

    12. Dynamic Programming and Quasilinearization—II

    13. Further Reduction in Dimensionality

    14. Discussion


    Chapter 8. Invariant Imbedding, Nonlinear Filtering, and the Estimation of Variables and Parameters

    1. Introduction

    2. An Estimation Problem

    3. Sequential and Nonsequential Estimates

    4. The Invariant Imbedding Approach

    5. The Optimal Estimates

    6. Equation for the Weighting Function

    7. A Numerical Example

    8. Systems of Differential Equations

    9. Estimation of State and Parameter—An Example

    10. A More General Criterion

    11. An Estimation Problem with Observational Noise and Disturbance Input

    12. The Optimal Estimate—A Two-Point Boundary-Value Problem

    13. Invariant Imbedding

    14. A Numerical Example

    15. Systems of Equations with Observational Noises and Disturbance Inputs

    16. Discussion


    Chapter 9. Parabolic Partial Differential Equations—Fixed Bed Reactors with Axial Mixing

    1. Introduction

    2. Isothermal Reactor with Axial Mixing

    3. An Implicit Difference Approximation

    4. Computational Procedure

    5. Numerical Results—Isothermal Reactor

    6. Adiabatic Reactor with Axial Mixing

    7. Numerical Results—Adiabatic Reactor

    8. Discussion

    9. Influence of the Packing Particles

    10. The Linearized Equations

    11. The Difference Equations

    12. Computational Procedure—Fixed Bed Reactor

    13. Numerical Results—Fixed Bed Reactor

    14. Conclusion


    Appendix I. Variational Problems with Parameters

    1. Introduction

    2. Variational Equations with Parameters

    3. Simpler End Conditions

    4. Calculus of Variations with Control Variable Inequality Constraint

    5. Pontryagin's Maximum Principle


    Appendix II. The Functional Gradient Technique

    1. Introduction

    2. The Recurrence Relations

    3. Numerical Example

    4. Discussion


    Author Index

    Subject Index

Product details

  • No. of pages: 350
  • Language: English
  • Copyright: © Academic Press 1968
  • Published: January 1, 1968
  • Imprint: Academic Press
  • eBook ISBN: 9781483266756

About the Author

E. Stanley Lee

About the Editor

Richard Bellman

Affiliations and Expertise

Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California

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