KTH /
Teknikvetenskap
/
Matematik
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Optimeringslära och systemteori
SF2852 Optimal Control, 2013, 7.5hp.
May exam 2013 can be found here:
exam a
solutions a
Homeworks and material from the lectures can be found on this
page
Course home page address:
http://www.math.kth.se/optsyst/grundutbildning/kurser/SF2852/.
Examiner and lecturer:
Johan Karlsson,
email: johan.karlsson@math.kth.se,
room 3710,
Lindstedtsv 25,
phone: 790 8440
Tutorial exercises:
Yuecheng Yang,
yuecheng@kth.se,
room 3738,
Lindstedtsv 25,
phone: 790 7132.
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 "institutionens
elevexpedition", Lindstedtsv.
- 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.
Course requirements
The course requirements consist of an obligatory final written
examination. There are also three optional homework sets that we
strongly encourage you to do. The homework sets give you bonus credits
in the examination.
Homework sets
Each homework set 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. The last problems are of
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.
Each successfully completed homework set 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 handed out in class
roughly two weeks before the deadline. They will also be posted on the
course homepage. You may email the solutions to the homework. If you
choose to do so, the solution should be submitted as one pdf prepared
in tex or comparable software.
- Homework 1: This homework set covers problems on discrete dynamic programming and model predictive control. (Due on Wednesday April 17, at 15.14).
- Here is the first homework set:
[pdf]
-
[ Struktur.m] (Matlab structure for problem 3)
- Homework 2: This homework set covers
computational methods for solving optimal control problems.
Results should be presented in seminar form
on Wednesday May 8, at 15.15, and attendance is mandatory
for receiving the bonus points. [pdf]
- Homework 3: This homework set covers problems on PMP and related topics
(Due on Wednesday May 15, at 15.14).
[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.
| Total credit (points) | Grade
|
|---|
| 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.
- The next exam will take place on Thursday May 23, 2013 at 08.00-13.00.
You need to register for the exam during the period April 15 to May 5, 2012.
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. The evaluation form will be posted on the course
homepage and it can be handed in anonymously in the mailbox opposite
to the entrance of "studentexpeditionen" on Lindstedtsv 25. We
appreciate your candid feedback on lectures, tutorials, course
materials, homeworks and computer exercises. This helps us to
continuously improve the course.
Tentative schedule for 2013
| Type | Day | Date | Time | Hall | Topic |
Content
|
|---|
| L1 | Tue | 19/3 | 10-12 | E36
| Introduction
Discrete dynamic programming
| Pages 1-21
| | L2 | Wed | 20/3 | 15-17 | E36
| Discrete dynamic programming
Discrete
PMP
| Theorem 1, pages 22-23
Oven example, Knapsack problem
| | E1 | Thu | 21/3 | 13-15 | E36
| Discrete dynamic programming
Linear
systems
| Economic growth problem (Ex. 1.1), Discrete DynP
Recap
Linear systems
| | L3 | Mon | 25/3 | 15-17 | E53
| Discrete dynamic programming
Infinite time
horizon
| Pages 24-28
| | L4 | Tue | 26/3 | 10-12 | E36
| Model predictive control
| Pages 28-29
MPC (See handout)
| | E2 | Thu | 28/3 | 15-17 | E53
| Model predictive control
| Infeasibility and
instability examples.
| | L5 | Tue | 9/4 | 10-12 | D41
| Dynamic programming
| Pages 35-41
| | L6 | Wed | 10/4 | 15-17 | D32
| Dynamic Programming
| Pages 41-44, 5-11
| | E3 | Thu | 11/4 | 15-17 | E53
| Dynamic Programming
| Ex. 2.1, 2.2, 2.3
| | L7 | Wed | 17/4 | 13-15 | D41
| Mathematical preliminaries (ODE theory etc)
| Pages 47-54
| | L8 | Thu | 18/4 | 15-17 | E36
| Pontryagins minimum principle (PMP) (using small
variations)
| Pages 59-65
| | E4 | Fri | 19/4 | 15-17 | E36
| PMP I
| Ex. 3.2, 3.3
| | L9 | Tue | 23/4 | 10-12 | D41
| PMP (control constraints)
| Pages 71-75, 62-63
| | L10 | Wed | 24/4 | 13-15 | D42
| PMP (optimal control to a manifold)
| Pages 76-81,
nonregular example
| | E5 | Fri | 26/4 | 15-17 | D32
| PMP II: Time optimal control
| Ex. 4.1. Exam problem.
| | L11 | Mon | 29/4 | 15-17 | D32
| PMP (generalizations)
| Pages 82-87
| | E6 | Tue | 30/4 | 8-10 | D42
| PMP III
| Ex. 5.1
| | L12 | Thu | 2/5 | 15-17 | D32
| PMP applications
| Pages 90-99
| | L13 | Mon | 6/5 | 10-12 | E33
| Infinite time horizon optimal control
| Pages 100-109
| | E7 | Tue | 7/5 | 10-12 | D41
| PMP IV
| | L14 | Wed | 8/5 | 15-17 | E33
| Computational methods - Seminar (student presentation)
| | L15 | Mon | 13/5 | 10-12 | E33
| Infinite time horizon optimal control
| | E8 | Tue | 14/5 | 15-17 | E36
| Infinite time horizon optimal control
| | E9 | Wed | 15/5 | 15-17 | E36
| Mixed Topics
| | Back-up | Thu | 16/5 | 15-17 | E36
| Back-up time
|
Welcome!
Last years exams can be found here:
2012
exam b
solutions b
exam a
solutions a
2011
exam b
solutions b
exam a
solutions a
2010
exam b
solutions b
exam a
solutions a
|
