Introduction to Control Theory and Dynamic Programming
Authors
Description
The book introduces various aspects of control theory. The range of applications of this theory is very broad, allowing it to address current problems from different areas of knowledge, such as mathematics, physics, economics, engineering and other applied sciences. One of the main features of the text is the formal, mathematical treatment of the results presented. Alongside the introduction of concepts and discussion of theoretical results, emphasis is placed on the analysis of corresponding examples, motivated by practical problems. The choice of this type of approach is intended not only to allow a better understanding of theoretical results, but also to illustrate in detail the wide range of applications of the theory dealt with in the book.
Target audience
Higher education
Name: Introduction to Control Theory and Dynamic Programming
Author(s): Johann Baumaister e Antonio Leitão
Pages: 399
Publication: IMPA, 2014
ISBN: 978-85-244-0271-5
Edition: 1
1 Introduction
1.1 Introduction to Control Systems
1.2 Examples
1.2.1 Heating a Room
1.2.2 A Discrete Control Problem
1.2.3 A Minimum Time Problem
1.2.4 Launching a Rocket
1.2.5 Balancing a Stick
1.2.6 Lead in the Human Body
1.2.7 The Brachistochrone
1.2.8 An Optimal Control Problem
Exercises
2 Observability
2.1 Linear systems
2.2 Non-observable subspace
2.3 State reconstruction
Exercises
3 Controllability
3.1 Linear Systems
3.2 Controllability and Observability
3.3 Autonomous Control Systems
3.4 Normal Form of Autonomous Systems
3.5 Controllability and Optimal Strategies
3.6 Attainability of States with Control Constraints
3.7 Bang-Bang Principle and Minimum Time Problems
3.8 Controllability of Discrete Linear Systems
Exercises
4 Stability
4.1 Concept and Examples
4.2 Stability of Linear Systems
4.3 Routh-Hurwitz Criterion
4.4 Perturbation of Linear Systems
4.5 Lyapunov’s Method
4.6 Lyapunow’s Matrix Equation
4.7 Stability of Discrete Linear Systems
Exercises
5 Stabilization
5.1 Linear Systems
5.2 Pole Placement
5.3 Dynamic Observer
5.4 Output Feedback Stabilization
5.5 Operating Points
Exercises
6 Parameter Identification
6.1 Automatic Control
6.2 Identifiability
6.3 Adaptive Identification: Introduction
6.4 Adaptive Identification: Stability
Exercises
7 Variational Calculus
7.1 Variational Problems and Convexity
7.2 du Bois-Reymond and Lagrange Lemmas
7.3 Euler-Lagrange Equation
7.4 Partially Differentiable Extremals
7.5 Vector Problems
Exercises
8 Variational Principles in Mechanics
8.1 Newtonian Mechanics
8.2 Conservative Theorems in Closed Systems
8.3 Lagrangean Mechanics
8.4 Hamiltonian Mechanics
Exercises
9 Variational Calculus and Optimal Control
9.1 What is Optimal Control
9.2 Variational Problems with Constraints
9.3 Singular Extremes and Optimal Trajectories
9.4 Optimal Control and Convexity: sufficient conditions
9.5 Optimal Control and Convexity: necessary conditions
Exercises
10 Maximum Principle
10.1 Finite Horizon Problems
10.2 Infinite Horizon Problems
10.3 Applications of the Maximum Principle
Exercises
11 Discrete Dynamic Programming
11.1 Introduction
11.2 Simple Path Problem
11.3 Equipment Replacement Problem
11.4 Traveling Salesman Problem
11.5 Discrete Linear Quadratic Problem
11.6 Consecutive Decisions Problem
Exercises
12 Continuous Dynamic Programming
12.1 Optimal Value Function
12.2 Bellman Principle
12.3 Hamilton-Jakobi-Bellman Equation
12.4 Linear Quadratic Control Problem
Exercises
13 Viscous Solutions and Value Function
13.1 Viscous Solutions of PDEs
13.2 Hopf-Lax Formula
13.3 Value Function as Viscous Solution of HJB
13.4 Maximum Principle
13.5 Uniqueness of Viscous Solutions
Exercises
Appendix A: Ordinary Differential Equations
A.1 Exponential of a Matrix
A.2 Autonomous Linear Systems
A.3 Non-Autonomous Linear Systems
A.4 Non-Linear Systems: existence and uniqueness
A.5 Non-Linear Systems: continuous dependence
A.6 Shooting Method
Exercises
Appendix B: Demonstration of the Maximum Principle
B.1 Infinite Optimization
B.1.1 An Abstract Optimization Problem
B.1.2 Linearization of the Optimization Problem
B.1.3 Necessary Conditions for the Abstract Problem
B.2 An Auxiliary Problem
B.3 Necessary Conditions for Optimality
List of symbols
Bibliography
Index