Applied and Computational Optimal Control: A Control Parametrization Approach

✍ Scribed by Kok Lay Teo, Bin Li, Changjun Yu, Volker Rehbock

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 581
• Series: Springer Optimization and Its Applications, 171
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

The aim of this book is to furnish the reader with a rigorous and detailed exposition of the concept of control parametrization and time scaling transformation. It presents computational solution techniques for a special class of constrained optimal control problems as well as applications to some practical examples.  The book may be considered an extension of the 1991 monograph A Unified Computational Approach Optimal Control Problems, by K.L. Teo, C.J. Goh, and K.H. Wong. This publication discusses the development of new theory and computational methods for solving various optimal control problems numerically and in a unified fashion. To keep the book accessible and uniform, it includes those results developed by the authors, their students, and their past and present collaborators.  A brief review of methods that are not covered in this exposition, is also included.

Knowledge gained from this book may inspire advancement of new techniques to solve complex problems that arise in the future. This book is intended as reference for researchers in mathematics, engineering, and other sciences, graduate students and practitioners who apply optimal control methods in their work. It may be appropriate reading material for a graduate level seminar or as a text for a course in optimal control.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 Optimal Control Problems
1.2 Illustrative Examples
1.3 Computational Algorithms
1.3.1 Dynamic Programming and Iterative Dynamic Programming
1.3.2 Leapfrog Algorithm and STC algorithm
1.3.3 Control Parametrization
1.3.4 Collocation Methods
1.3.5 Full Parametrization
1.4 Optimal Control Software Packages
2 Unconstrained Optimization Techniques
2.1 Introduction
2.2 Basic Concepts
2.3 Gradient Methods
2.4 Steepest Descent Method
2.5 Newton's Method
2.6 Modifications to Newton's Method
2.7 Line Search
2.8 Conjugate Gradient Methods
2.8.1 Convergence of the Conjugate Gradient Methods
2.9 Quasi-Newton Methods
2.9.1 Approximation of the Inverse G-1
2.9.2 Rank Two Correction
2.9.3 BFGS Update Formula
3 Constrained Mathematical Programming
3.1 Introduction
3.2 Quadratic Programming with Linear Equality Constraints
3.3 Quadratic programming via Active Set Strategy
3.4 Constrained Quasi-Newton Method
3.5 Sequential Quadratic Programming Algorithm
4 Optimization Problems Subject to Continuous Inequality Constraints
4.1 Introduction
4.2 Constraint Transcription Technique
4.2.1 Inequality Constraints
4.2.2 Continuous Inequality Constraints
4.3 Continuous Inequality Constraint Transcription Approach
4.3.1 The First Method
4.3.2 The Second Method
4.3.2.1 Solution Method
4.4 Exact Penalty Function Method
4.4.1 Convergence Analysis
4.4.2 Algorithm and Numerical Results
4.5 Exercises
5 Discrete Time Optimal Control Problems
5.1 Introduction
5.2 Dynamic Programming Approach
5.2.1 Application to Portfolio Optimization
5.2.1.1 Problem Formulation
5.2.1.2 Analytical Solution
5.2.1.3 Numerical Simulations
5.3 Discrete Time Optimal Control Problems with Canonical Constraints
5.3.1 Gradient Formulae
5.3.2 A Unified Computational Approach
5.4 Problems with Terminal and All-Time-Step Inequality Constraints
5.4.1 Constraint Approximation
5.4.2 Convergence Analysis
5.4.3 Illustrative Examples
5.5 Discrete Time Time-Delayed Optimal Control Problem
5.5.1 Approximation
5.5.2 Gradients
5.5.3 A Tactical Logistic Decision Analysis Problem
5.6 Exercises
6 Elements of Optimal Control Theory
6.1 Introduction
6.2 First Order Necessary Condition: Euler-LagrangeEquations
6.3 The Linear Quadratic Theory
6.4 Pontryagin Maximum Principle
6.5 Singular Control
6.6 Time Optimal Control
6.7 Continuous State Constraints
6.8 The Bellman Dynamic Programming
6.9 Exercises
7 Gradient Formulae for Optimal Parameter Selection Problems
7.1 Introduction
7.2 Optimal Parameter Selection Problems
7.2.1 Gradient Formulae
7.2.2 A Unified Computational Approach
7.3 Control Parametrization
7.4 Switching Times as Decision Parameters
7.4.1 Gradient Computation
7.4.2 Time Scaling Transformation
7.4.3 Combined Piecewise Constant Control and Variable System Parameters
7.4.4 Discrete Valued Optimal Control Problems and Optimal Control of Switched Systems
7.5 Time-Lag System
7.5.1 Gradient Formulae
7.6 Multiple Characteristic Time Points
7.7 Exercises
8 Control Parametrization for Canonical Optimal Control Problems
8.1 Introduction
8.2 Problem Statement
8.3 Control Parametrization
8.4 Four Preliminary Lemmas
8.5 Some Convergence Results
8.6 A Unified Computational Approach
8.7 Illustrative Examples
8.8 Combined Optimal Control and Optimal Parameter Selection Problems
8.8.1 Model Transformation
8.8.2 Smoothness of Optimal Control
8.8.3 Illustrative Examples
8.9 Control Parametrization Time Scaling Transform
8.9.1 Control Parametrization Time Scaling Transform
8.9.2 Convergence Analysis
8.10 Examples
8.11 Exercises
9 Optimal Control Problems with State and Control Constraints
9.1 Introduction
9.2 Optimal Control with Continuous State Inequality Constraints
9.2.1 Time Scaling Transform
9.2.2 Constraint Approximation
9.2.3 A Computational Algorithm
9.2.4 Solving Problem (P ,γ (p))
9.2.5 Some Convergence Results
9.2.6 Illustrative Examples
9.3 Exact Penalty Function Approach
9.3.1 Control Parametrization and Time Scaling Transformation
9.3.2 Some Convergence Results
9.3.3 Computational Algorithm
9.3.4 Examples
9.4 Exercises
10 Time-Lag Optimal Control Problems
10.1 Time-Lag Optimal Control
10.1.1 Introduction
10.1.2 Problem Formulation
10.1.3 Control Parametrization
10.1.4 The Time-Scaling Transformation
10.1.5 Gradient Computation
10.1.6 Numerical Examples
10.2 Time-Lag Optimal Control with State-Dependent Switched System
10.2.1 Introduction
10.2.2 Problem Statement
10.2.3 Preliminaries
10.2.4 Main Results
10.2.5 Numerical Example
10.3 Min-Max Optimal Control
10.3.1 Problem Statement
10.3.2 Some Preliminary Results
10.3.3 Problem Approximation
10.3.4 Illustrative Example
10.4 Exercises
11 Feedback Control
11.1 Introduction
11.2 Neighbouring Extremals
11.2.1 Problem Formulation
11.2.2 Construction of Suboptimal Feedback Control Law
11.2.3 Numerical Examples
11.3 PID Control
11.3.1 Problem Statement
11.3.2 Constraint Approximation
11.3.3 Computational Method
11.3.4 Application to a Ship Steering Control Problem
11.4 Exercises
12 On Some Special Classes of Stochastic Optimal Control Problems
12.1 Introduction
12.2 A Combined Optimal Parameter and Optimal Control Problem
12.2.1 Deterministic Transformation
12.2.2 A Numerical Example
12.3 Optimal Feedback Control for Linear Systems Subject to Poisson Processes
12.3.1 Two Stochastic Optimal Feedback ControlProblems
12.3.2 Deterministic Model Transformation
12.3.3 An Example
12.4 Exercises
A.1 Elements of Mathematical Analysis
A.1.1 Introduction
A.1.2 Sequences
A.1.3 Linear Vector Spaces
A.1.4 Metric Spaces
A.1.5 Continuous Functions
A.1.6 Normed Spaces
A.1.7 Linear Functionals and Dual Spaces
A.1.8 Elements in Measure Theory
A.1.9 The Lp Spaces
A.1.10 Multivalued Functions
A.1.11 Bounded Variation
A.2 Global Optimization via Filled Function Approach
A.3 Elements of Probability Theory
References