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KTH / Teknikvetenskap / Matematik / Optimeringslära och systemteori

SF2852 Optimal Control,   2012, 7.5hp.


Material from the lectures can be found on this page
All the homeworks have now been corrected. You can collect your homeworks from Yuecheng at his office. Lists with the final bonus points will also be available at the exam.

The re-exam given in August and solution drafts can be found here:
exam
solutions


The last exam given in May and solution drafts can be found here:
exam
solutions



Course home page address:
http://www.math.kth.se/optsyst/grundutbildning/kurser/SF2852/.

Examiner and lecturer:
Per Enqvist, email: penqvist@math.kth.se,
room 3705, Lindstedtsv 25, phone: 790 6298

Xiaoming Hu, email: hu@math.kth.se,
room 3712, Lindstedtsv 25, phone: 790 7180

Tutorial exercises:
Hildur Æsa Oddsdóttir , email: haodd@kth.se,
room 3727, Lindstedtsv 25, phone: 790 6660.

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.

    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.

    • Homework 1: This homework set covers problems on discrete dynamic programming and model predictive control. (Due on Tuesday April 24, at 15.14).
    • Homework 2: This homework set covers computational methods for solving optimal control problems. Results should be presented in seminar form on Tuesday May 15, at 13.00, and attendance is mandatory for receiving the bonus points.
    • Homework 3: This homework set covers problems on PMP and related topics (Due on Thursday May 24, at 10.14).

    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 Wednesday May 30, 2012 at 14.00-19.00. You need to register for the exam during the period April 23 to May 13, 2012.

    Appeal
    If your total score (exam score + maximum 6 bonus points from the homework assignments and the computational exercises) is in the range 21-22 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 2012

    Type Day Date Time Hall Topic
    L1-P Tue 20/3 10-12 E31 Introduction
    Discrete dynamic programming
    L2-P Thu 22/3 10-12 E31 Discrete dynamic programming
    Discrete PMP
    E1-Y Fri 23/3 15-17 E31 Discrete dynamic programming
    Linear systems
    L3-Y Tue 27/3 10-12 E31 Discrete dynamic programming
    Infinite time horizon
    L4-H Wed 28/3 13-15 E31 Model predictive control
    E5-H Fri 30/3 15-17 E31 Model predictive control
    L5-P Tue 10/4 13-15 E31 Dynamic programming
    L6-P Wed 11/4 13-15 E31 Dynamic Programming
    L7-X Fri 13/4 15-17 E31 Mathematical preliminaries (ODE theory etc)
    E3-H Tue 17/4 13-15 E51 Dynamic Programming
    L8-X Thu 19/4 15-17 E31 Pontryagins minimum principle (PMP) (using small variations)
    L9-X Fri 20/4 15-17 E31 PMP (control constraints)
    E4-H Tue 24/4 15-17 E51 PMP I
    L10-X Wed 25/4 10-12 E51 PMP (optimal control to a manifold)
    E5-Y Fri 27/4 15-17 E31 PMP II: Time optimal control
    L11-X Tue 8/5 13-15 E51 PMP (generalizations)
    E6-Y Wed 9/5 13-15 E51 PMP III
    L12-Y Fri 11/5 15-17 E51 PMP applications
    E7-Y Mon 14/5 10-12 E31 PMP IV
    L13-P Tue 15/5 13-15 D34 Infinite time horizon optimal control
    L14-P Wed 16/5 13-15 D34 Infinite time horizon optimal control
    L15-P Tue 22/5 13-15 E51 Computational methods - Seminar (student presentation)
    E8-H Wed 23/5 15-17 E51 Infinite time horizon optimal control
    E9-H Thu 24/5 10-12 E31 Mixed Topics
    Back-up Fri 25/5 13-15 E51 Back-up time

    Welcome!


    Last years exams can be found here:
    2011
    exam
    solutions
    exam
    solutions
    2010
    exam a
    solutions a
    exam b
    solutions b