課程資訊
課程名稱
 最佳控制
Optimal Control 
開課學期
 105-2 
授課對象
 學程  全電化都會運輸系統基礎技術學分學程  
授課教師
 張時中 
課號
 EE5051 
課程識別碼
 921 U2330 
班次
  
學分
 3.0 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期二2,3,4(9:10~12:10) 
上課地點
 電二455 
備註
 核心課程
總人數上限：20人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1052EE5051_OC 
課程簡介影片
 
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課程概述

1. Introduction and course overview
2. Unconstrained nonlinear optimization-line search, gradient and Newton methods;
3. Constrained nonlinear optimization-Lagrange multipliers.
4. Dynamic programming: principle of optimality, dynamic programming, discrete-tme LQR;
5. Dynamic programming in continuous time: Hamilton-Jacobi-Bellman equation, continuous-time LQR
6. Numerical solution in MATLAB and term project definition
7. Midterm exam
8. Calculus of variations
9. Calculus of variations applied to optimal control
10. Properties of optimal control solution
11. Constrained optimal control
12. Singular arcs
13. Estimators/Observers
14. Final exam
15. Term project presentation 

課程目標
 This course studies basic optimization and the principles of optimal control. It will be mainly on deterministic and some stochastic problems for both discrete and continuous systems. The course covers solution methods including numerical search algorithms, dynamic programming, variational calculus, and approaches based on Pontryagin's maximum principle. It will include examples and applications of multiple disciplines to motivate both theoretic study and solution algorithm design. Target students include those interested in the areas of optimization and control of various dynamic systems such as energy systems, robots, vehicles, communication networks, socioeconomic systems and biological systems. 
課程要求
 PREREQUISITE: Linear Algebra, Differential Equations, Control Systems or Linear Systems, or consent of instructor. 
預期每週課後學習時數
  
Office Hours
 備註： TBD 
指定閱讀
 1. Principles of Optimal Control - MIT OpenCourseWare
https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-323-principles-of-optimal-control-spring-2008/lecture-notes/
2. B.D.O. Anderson and J.B. Moore, Optimal Control Linear Quadratic Methods, Prentice Hall
3. F.L. Lewis and V.L. Syrmos, Optimal Control, John Wiley & Sons.
4. Prof. D. P. Bertsekas' Course Lecture Slides, 2015, http://www.athenasc.com/DP_Slides_2015.pdf 
參考書目
 1. Donald Kirk, Optimal Control Theory: An Introduction. New York, NY:
Dover, 2004.
2. Daniel Liberzon, Calculus of Variations and Optimal Control Theory, a
Concise Introduction, Princeton University Press, 2012.
Online preliminary version http://liberzon.csl.illinois.edu/teaching/cvoc.pdf
3. D. P. Bertsekas, Dynamic Programming and Optimal Control, Vol. 1 (4th
ed. 2017) and 2 (4th ed. 2012), Athena Scientific. athenasc.com/dpbook.html
4. Suresh P. Sethi and Gerald L. Thompson, Optimal Control Theory:
Applications to Management Science and Economics, 2nd ed., 2009.
5. Other assigned reading materials in the class. 
評量方式
(僅供參考)
  
No.
項目
百分比
說明
1. 
 Classroom Participation 
5% 
  
2. 
 Homework 
20% 
 Late homework will not be accepted except officially approved reasons.  
3. 
 Term project 
25% 
  
4. 
 Midterm Exam 
30% 
  
5. 
 Final Exam 
20% 
  
 
課程進度
週次
日期
單元主題
Week 1
 2/21  Review of Dynamic Systems and Control
Introduction to Optimization
Typical Problems in Optimal Control and Estimation
Course Overview and Syllabus
Prof. L. Badia’s talk on “A short introduction to utility and strategies”
Reading Assignment:
Robert Stengel, “2015 Seminar Slides for Optimal Control and Estimation, seminar 1” available: http://www.princeton.edu/~stengel/MAE546Seminars.html
H. J. Sussmann and J. C. Willems, “300 years of optimal control: From the Brachystochrone to the Maximum Principle,” IEEE Control Systems, 1997.
 
Week 2
 2/28  National Holiday. No Class. 
Week 3
 3/07  Introduction to Optimization
Typical Problems in Optimal Control and Estimation
Unconstrained Nonlinear Optimization
Convexity
Reading Assignments:
1. Robert Stengel, “2015 Seminar Slides for Optimal Control and Estimation, seminar 1” available: http://www.princeton.edu/~stengel/MAE546Seminars.html
2. J. P. How, Principles of Optimal Control - MIT OpenCourseWare
Lecture note - Unit 1
3. F.L. Lewis and V.L. Syrmos, Optimal Control Chapter 1.
 
Week 4
 3/14  Unconstrained Nonlinear Optimization
Line search
Optimality conditions N-dimension problems
Gradient method
Newton method
Constrained Optimization
Optimality conditions
Lagrange multipliers
Reading Assignments:
1. Robert Stengel, “2015 Seminar Slides for Optimal Control and Estimation, seminar 1” available: http://www.princeton.edu/~stengel/MAE546Seminars.html
2. J. P. How, Principles of Optimal Control - MIT OpenCourseWare
Lecture note - Unit 1
3. F.L. Lewis and V.L. Syrmos, Optimal Control Chapter 1.
4. D. P. Bertsekas, Nonlinear Programming, Second Edition, Athena Scientific, Belmont, MA, 1999
 
Week 5
 3/21  Unconstrained Nonlinear Optimization
Rate of convergence
Gradient method
Newton method
Constrained Optimization
Optimality conditions
Lagrange multipliers
Dynamic Programming - Introduction
Reading Assignments:
1. Robert Stengel, “2015 Seminar Slides for Optimal Control and Estimation, seminar 1” available: http://www.princeton.edu/~stengel/MAE546Seminars.html
2. J. P. How, Principles of Optimal Control - MIT OpenCourseWare
Lecture note – Lec 2
3. F.L. Lewis and V.L. Syrmos, Optimal Control Chapter 1.
4. D. P. Bertsekas, Nonlinear Programming, Second Edition, Athena Scientific, Belmont, MA, 1999
 
Week 6
 3/25  Principle of Optimality
Dynamic Programming
Discrete LQR 
Week 7
 4/04  Tomb Sweeping Day and Spring Break
No class 
Week 8
 4/11  Dynamic Programming-Discrete Time
Discrete Time LQR and DP Solution
Dynamic programming – Continuous Time
HJB Equation
Continuous Time LQR and DP

Reading Assignments:
1. 2. F.L. Lewis, D. Vrabie, and V.L. Syrmos, Optimal Control Chapter 6.
2. J. P. How, Principles of Optimal Control - MIT OpenCourseWare
Lecture note – Lec 3
3. Prof. Bertsekas' Course Lecture Slides, 2015 Lecture 2 and Lecture 3 (pages 15~36/302)
 
Week 9
 4/18  Discrete Time LQR and DP Solution (Cont.)
- steady-state LQR control
- extensions: time-varying systems, tracking problems
Dynamic programming – Continuous Time
HJB Equation
Continuous Time LQR and DP

Reading Assignments:
1. F.L. Lewis, D. Vrabie, and V.L. Syrmos, Optimal Control, Secs 2.1, 2.2 and Chapter 6.
2. J. P. How, Principles of Optimal Control - MIT OpenCourseWare
Lecture note – Lec 4
3. EE363: Lecture 4 Slides Professor Stephen Boyd, Stanford University, Winter Quarter 2008-09
4. Prof. Bertsekas' Course Lecture Slides, 2015 Lecture 2 and Lecture 3 (pages 15~36/302)
 
Week 10
 4/25  Discrete Time LQR and DP Solution (Cont.)
- steady-state LQR control
- extensions: time-varying systems, tracking problems
Dynamic programming – Continuous Time
HJB Equation
Continuous Time LQR and DP
1. F.L. Lewis, D. Vrabie, and V.L. Syrmos, Optimal Control, Secs 2.1, 2.2 and Chapter 6.
2. J. P. How, Principles of Optimal Control - MIT OpenCourseWare Lecture note – Lec 4
3. EE363: Lecture 4 Slides Professor Stephen Boyd, Stanford University, Winter Quarter 2008-09
4. Prof. Bertsekas' Course Lecture Slides, 2015 Lecture 2 and Lecture 3 (pages 15~36/302)
 
Week 11
 5/02  midterm exam
term project discussion 
Week 12
 5/09  Calculus of Variations Problems:
- Fixed final time/state
- Euler-Lagrange equation
- Free final time/state
- Some examples of CoV problems

What is the connection of Optimal Control and Calculus of Variations

References:
1. J. P. How, Principles of Optimal Control - MIT OpenCourseWare Lecture note – Lec 5
2. D. Kirk, Optimal Control Theory- An introduction, Chapter 4
3. D. Liberzon, Calculus of Variations and Optimal Control Theory, Chapter 2


 
Week 13
 5/16  CoV with Corner condition
CoV with equality constraints
CoV with general ternimal conditions
Calculus of Variations and Optimal Control

References:
1. J. P. How, Principles of Optimal Control - MIT OpenCourseWare Lecture note – Lec 5, 6
2. D. Kirk, Optimal Control Theory- An introduction, Ch 4
3. Slides from KTH_A geometric explanation of the KKT conditions.pdf
4. D. S. Naidu, Optimal Control Systems, Chapter 2
 
Week 14
 5/23  Optimal Control via variational approach (CoV)
- General terminal conditions
Hamiltonian & necessary conditions of optimality
LQR revisited using CoV approach

References:
1. J. P. How, Principles of Optimal Control - MIT OpenCourseWare Lecture note – Lec 6
2. D. Kirk, Optimal Control Theory- An introduction, Ch 4
3. D. Liberzon, Calculus of Variation and Optimal Control theory, Chapter 3
 
Week 15
 5/30 
Dragon Boat Festival (Holiday)

 
Week 16
 6/06  Optimal Control with inequality constraints
Pontriyagin Maximum/Minimum principle

References:
1. J. P. How, Principles of Optimal Control - MIT OpenCourseWare Lecture note – Lec 8 & 9
2. D. Kirk, Optimal Control Theory- An introduction, Ch 5
3. D. Liberzon, Calculus of Variation and Optimal Control theory, Chapter 4
4. F. L. Lewis, V. L. Syrmos, Optimal Control, Chapter 5 
Week 17
 6/13  Pontriyagin Maximum/Minimum principle
Optimal control examples: Minimum time problem

References:
1. J. P. How, Principles of Optimal Control - MIT OpenCourseWare Lecture note – Lec 9
2. D. Kirk, Optimal Control Theory- An introduction, Sec 5.4, 5.5
3. D. S. Naidu, Optimal Control Systems, Chapter 7
4. F. L. Lewis, V. L. Syrmos, Optimal Control, Sec 5.2 
Week 18
 6/20  Final exam

 
Week 19
 6/27  Final Project Presentation