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# SF2852 Optimal Control,   2021, 7.5hp.

Course registration: If you have problems registering for the course or for exams, please contact the Student affairs office , e.g., via email: studentoffice@math.kth.se.

http://www.math.kth.se/optsyst/grundutbildning/kurser/SF2852/.

Examiner and lecturer:
Johan Karlsson, email: johan.karlsson@math.kth.se,
room 3550, Lindstedtsv 25, phone: 790 8440

Tutorial exercises:
Yibei Li, yibei@kth.se,
room 3736, Lindstedtsv 25, phone: 790 6294.

Introduction
Optimal control is the problem of determining the control function for a dynamical system to minimize a performance index. The subject has its roots in the calculus of variations but it evolved to an independent branch of applied mathematics and engineering in the 1950s. The rapid development of the subject during this period was due to two factors. The first are two key innovations, namely the maximum principle by L. S. Pontryagin and the dynamic programming principle by R. Bellman. The second was the space race and the introduction of the digital computer, which led to the development of numerical algorithms for the solution of optimal control problems. The field of optimal control is still very active and it continues to find new applications in diverse areas such as robotics, finance, economics, and biology.

Course goals
The goal of the course is to provide an understanding of the main results in optimal control and how they are used in various applications in engineering, economics, logistics, and biology. After the course you should be able to

• describe how the dynamic programming principle works (DynP) and apply it to discrete optimal control problems over finite and infinite time horizons,
• use continuous time dynamic programming and the associated Hamilton-Jacobi-Bellman equation to solve linear quadratic control problems,
• use the Pontryagin Minimum Principle (PMP) to solve optimal control problems with control and state constraints,
• use Model Predictive Control (MPC) to solve optimal control problems with control and state constraints. You should also be able understand the difference between the explicit and implicit MPC control and explain their respective advantages,
• formulate optimal control problems on standard form from specifications on dynamics, constraints and control objective. You should also be able to explain how various control objectives affect the optimal performance,
• explain the principles behind the most standard algorithms for numerical solution of optimal control problems and use Matlab to solve fairly simple but realistic problems.

For the highest grade you should be able to integrate the tools you have learnt during the course and apply them to more complex problems. In particular you should be able to

• explain how PMP and DynP relates to each other and know their respective advantages and disadvantages. In particular, you should be able to describe the difference between feedback control versus open loop control and you should use be able to compare PMP and DynP with respect to computational complexity.
• explain the mathematical methods used to derive the results and combine them to derive the solution to variations of the problems studied in the course.
Course topics
• Dynamic Programming Discrete dynamic programming, principle of optimality, Hamilton-Jacobi-Bellman equation, verification theorem.
• Pontryagin minimum principle Several versions of Pontryagin Minimum Principle (PMP) will be discussed.
• Infinite Horizon Optimal Control Optimal control over an infinite time horizon, stability, LQ optimal control.
• Model Predictive Control Explicit and implicit model predictive control.
• Applications Examples from economics, logistics, aeronautics, and robotics will be discussed.
• Computational Algorithms The most common methods for numerical solution of optimal control problems are presented.

Course material
The required course material consists of the following lecture and exercise notes on sale at Kårbokhandeln. [Lecture notes], [Exercise notes].

• Ulf Jönsson et. al. Optimal Control, Lecture notes, KTH.
• Peter Ögren et. al. Exercise Notes on Optimal Control , KTH.
• Supplementary material will be handed out during the course.

Prerequisites
The student is required to have passed the course optimization SF1841 or a course with similar content. The student should hence be familiar with concepts and theory for optimization: linear, quadratic, and nonlinear optimization; optimality conditions, lagrangian relaxation and duality theory. Familiarity with systems theory and state space is not required but recommended.

Course requirements
The course requirements consista of three mandatory homework sets and a final written examination. The homework sets may also give you bonus credits in the examination.

PhD course SF3852
It is possible to read this course as a PhD level course. For this, an extra project and at least a B on the exam is required. Email the examiner to get details regarding the project.

Homework sets
Homework set 0: This homework set provides some review of systems theory and optimization as well as a Matlab exercise that use the toolbox CVX. I recommend that everyone does problem 2. Homework set 0 is optional and does not give bonus points to the exam, however, you can get feedback on your solutions if you hand it in before the deadline.

Each of the homework sets 1-3 are mandatory and consists of three-five problems. The first two-three problems are methodology problems where you practice on the topics of the course and apply them to examples. Among the last two problems, one will focus on more theoretical nature and helps you to understand the mathematics behind the course. It can, for example, be to derive an extension of a result in the course or to provide an alternative proof of a result in the course. The other will focus on implementation and the student is required to make a Matlab program that solve a problem numerically.

You are required to get at least 10 points on each of the homeworks 1 and 3, and do the project in homework 2. Each successfully completed homework set handed in on time also gives you maximally 2 bonus points for the exam. The bonus is only valid during the year it is acquired. The exact requirements will be posted on each separate homework set. The homework sets will be posted on the homepage roughly two weeks before the deadline. The solutions to the homeworks should be uploaded on Canvas. Please prepare the solutions as a pdf in LaTeX or comparable software.

• Homework 0: This homework set covers some basic systems theory and optimization. (Due beginning of excercise session E2). Here is the first homework set: [pdf].
• Homework 1: This homework set covers problems on discrete dynamic programming and model predictive control. (Due beginning of excercise session E4). Here is the first homework set: [pdf].

### Matlab code

Here are some Matlab routines that are used in the excerise notes. You may use this for the solution of your homeworks.

Written exam
You may use Beta Mathematics Handbook and the following formula sheet (pdf) . The exam will consist of five problems that give maximally 50 points. These problems will be similar to those in the homework assignments and the tutorial exercises. The preliminary grade levels are distributed according to the following rule, where the total score is the sum of your exam score and maximally 6 bonus points from the homework assignments (max credit is 56 points). These grade limits can only be modified to your advantage.

45-56 A
39-44 B
33-38 C
28-32 D
25-27 E
23-24 FX
The grade FX means that you are allowed to make an appeal, see below.

• [info, room, etc.].
• You need to register for the exam. Information on how to register for the exam can be found here.

Appeal
If your total score (exam score + maximum 6 bonus points from the homework assignments and the computational exercises) is in the range 23-24 points then you are allowed to do a complementary examination for grade E. In the complementary examination you will be asked to solve two problems on your own. The solutions should be handed in to the examiner in written form and you must be able to defend your solutions in an oral examination. Contact the examiner no later than three weeks after the final exam if you want to do a complementary exam.

Course evaluation
At the end of the course you will be asked to complete a course evaluation form online.

The lectures and exercises are given on zoom.

Lectures
https://kth-se.zoom.us/j/61600010596

Exercises
https://kth-se.zoom.us/j/61945102589

Office hours: Fridays 14.00-15.00
https://kth-se.zoom.us/j/61945102589

Preliminary schedule for 2021
Type Date Time Topic Content (preliminary)
L1 2021-08-30 13:15 Introduction
Discrete dynamic programming
Pages 17-23
L2 2021-08-31 10:15 Discrete dynamic programming
Discrete PMP
Pages 22-24
E1 2021-09-01 13:15 Discrete dynamic programming
Linear systems

L3 2021-09-02 08:15 Discrete dynamic programming
Infinite time horizon
Pages 24-26
L4 2021-09-06 13:15 Model predictive control

E2 2021-09-07 10:15 Model predictive control

L5 2021-09-08 13:15 Dynamic programming
Pages 35-39
L6 2021-09-09 08:15 Dynamic programming
and review
Pages 5-7, 39-44
E3 2021-09-13 13:15 Dynamic programming

L7 2021-09-14 10:15 Mathematical preliminaries
(ODE theory etc)
Pages 47-54
L8 2021-09-15 13:15 Pontryagins minimum principle
(PMP) (using small variations)
Pages 59-62 and a basic example
L9 2021-09-16 08:15 PMP (control constraints)
Examples on pages 62-63, 74-75
E4 2021-09-20 13:15 PMP I

L10 2021-09-21 10:15 PMP (optimal control
to a manifold)
Pages 71-81
L11 2021-09-23 08:15 PMP (generalizations)
Pages 81-88
E5 2021-09-27 13:15 PMP II:
Time optimal control

L12 2021-09-28 10:15 PMP applications
Pages 90-96
E6 2021-09-30 08:15 PMP III

E7 2021-10-04 13:15 PMP IV

L13 2021-10-06 13:15 Computational methods Seminar
(student presentation)

L14 2021-10-07 08:15 Topics: Infinite time horizon
optimal control
Pages 97-109
L15 2021-10-11 13:15 Topics/Backup

E8 2021-10-13 13:15 Infinite time horizon optimal control and
Review: old exams

L16 2021-10-14 08:15 Review

Exam 2021-10-28 08:00 Exam

Some of last years exams can be found here:
2020
exam and solutions 20201022.
exam and solutions 20201216.
2019
exam and solutions
2018
exam and solutions
2017
exam and solutions
exam and solutions
2016
exam and solutions
exam and solutions
2015
exam and solutions