Quasilinearization and Invariant Imbedding
With Applications to Chemical Engineering and Adaptive Control
- 1st Edition - January 1, 1968
- Imprint: Academic Press
- Author: E. Stanley Lee
- Editor: Richard Bellman
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 5 2 7 8 - 0
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 6 7 5 - 6
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… Read more
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Request a sales quoteMathematics 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.
PrefaceChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesAppendix 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 ReferencesAppendix II. The Functional Gradient Technique 1. Introduction 2. The Recurrence Relations 3. Numerical Example 4. Discussion ReferencesAuthor IndexSubject Index
- Edition: 1
- Published: January 1, 1968
- No. of pages (eBook): 350
- Imprint: Academic Press
- Language: English
- Paperback ISBN: 9781483252780
- eBook ISBN: 9781483266756
RB
Richard Bellman
Affiliations and expertise
Departments of Mathematics,
Electrical Engineering, and Medicine
University of Southern California
Los Angeles, California