Optimeringslära och systemteori
SF2832 Mathematical Systems Theory
Adress of homepage for the course:
Examiner and lecturer:
room 3532, Lindstedtsv 25, phone 790 7180.
Important! pleae read here for information on course registration and
Solution to Exam April 11, 2017 (available soon after the exam has finished)
This is an introductory course in mathematical systems theory. The
subject provides the mathematical foundation of modern control theory,
with application in aeronautics, electrical networks, signal
processing, and many other areas. The aim of the course is that you
should acquire a systematic understanding of linear dynamical systems,
which is the focus of this course. The acquirement of such knowledge
is not only very useful preparation for work on system analysis and
design problems that appear in many engineering fields, but is also
necessary for advanced studies in control and signal processing.
The overall goal of the course is to provide an understanding of the
basic ingredients of linear systems theory and how these are used in
analysis and design of control, estimation and filtering systems. In
the course we take the state-space approach, which is well suited for
efficient control and estimation design. After the course you should
be able to
- Analyze the state-space model with respect to minimality,
observability, reachability, detectability and stabilizability.
- Explain the relationship between input-output
(external) models and state-space (internal) models for linear systems
and derive such models from the basic principles.
- Derive a minimal state-space model using the Kalman decomposition.
- Use algebraic design methods for state feedback design with pole
assignment, and construct stable state observers by pole assignment
and analyze the properties of the closed loop system obtained when the
observer and the state feedback are combined to an observer based
- Apply linear quadratic techniques to derive optimal state feedback
- Solve the Riccati equations that appear in optimal control and
- Design a Kalman filter for optimal state estimation of linear
systems subject to stochastic disturbances.
- Apply the methods given in the course to solve example problems (one
should also be able to use the ``Control System Toolbox'' in Matlab to
solve the linear algebra problems that appear in the examples).
For the highest grades 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 the above results and methods relate and build on each other.
- Understand the mathematical (mainly linear algebra) foundations of
the techniques used in linear systems theory and apply those
techniques flexibly to variations of the problems studied in the
- Solve fairly simple but realistic control design problems using the
methods in the course.
The required course material consists of the following lecture
notes and exercise notes, both are available online for the students.
- Anders Lindquist & Janne Sand (revised by Xiaoming Hu), An
Mathematical Systems Theory, lecture notes, KTH,
here (password will be given at the first lecture)
- Per Enqvist, Exercises in Mathematical Systems
notes (password will be given at the first lecture), KTH.
- Supplementary material can be
The course requirements consist of an obligatory final written
examination. There are also three homework sets we strongly encourage
you to do. All these optional activities will not only give you bonus
credits in the examination, but also help you understand the course
Each homework set consists of maximally five problems. The first three are
methodology problems where you practice on the topics of the course
and apply them to examples. The last one or two problems are of more
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
Each successfully completed homework set gives you maximally 5 points for
the exam. The exact requirements will be posted on each separate
homework set. The homework sets will be posted roughly ten days before
the deadline on the course homepage.
- Homework 1:
This homework set covers the material in chapters 1-4 of the lecture notes. Solution (Due on November 23, 17:00).
- Here is the first homework set of last year [pdf][solution]
- Homework 2: This homework set covers the material in chaptes 4 -6
of the lecture notes. Solution (Due on December 6).
- Here is the second homework set of last year [pdf][solution]
3: This homework set covers the material in chapters 6 - 9 of the lecture notes. Solution (Due
on December 16).
- Here is the third homework set of last year [pdf][solution]
This is an open book exam and you may bring the lecture notes, the
exercise notes, your own classnotes and Beta
Mathematics Handbook (or any equivalent handbook). The exam will
consist of five problems that give maximally 100 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 fifteen bonus points from the homework
assignments (max credit is 115
points). These grade limits can only be modified to your advantage.
The grade FX means that you are allowed to make an appeal, see below.
| Total credit (points) || Grade |
| >90 || A |
| 76-90 || B |
| 61-75 || C |
| 50-60 || D |
| 45-49 || E |
| 41-44 || FX |
- The first exam will take place on Jan. 16, 2017 at 8:00-13:00.
If your total score (exam score + maximum 15 bonus points from the
homework assignments) is in the range 41-44
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.
All the students are encouraged to answer the questionnaire on KTH Social.
Tentative schedule for 2017
| Type ||Day || Date || Time || Hall || Instr || Topic|
| L1.|| Wed || 1/11 || 8-10 || L52 || XH
| L2.|| Thu || 2/11 || 13-15|| V31|| XH
|| Linear systems
| L3.|| Fri|| 3/11 || 13-15 || U51|| XH
|| Linear systems
| E1. || Tue || 7/11 || 10-12 || U51|| AR
|| Linear systems
| L4.|| Wed || 8/11 || 8-10 || L51|| XH
| L5. || Thu || 9/11 || 14-16 || Q33 || XH
|| Reachability cont'd
| L6.|| Fri|| 10/11 || 13-15 || U51|| XH
| L7.|| Tue || 14/11 || 10-12 || Q36|| XH
|| Observability and stability
| E2. || Wed || 15/11 || 8-10 || V22|| AR
|| Reachability and observability
| L8. || Fri || 17/11 || 13-15 || U51 || XH
|| Stability cont'd and Realization theory
| L9. || Wed || 22/11 || 8-10 || U51|| XH
|| Canonical forms and Kalman decomposition
| E3. || Thu || 23/11 || 13-15|| Q36 || AR
|| Stability and realization theory
| L10. || Fri || 24/11 || 13-15 || Q36 || XH
|| Minimal realizations
| L11. || Tue || 28/11 || 10-12 || U21 || XH
|| Minimal realizations, State feedback
| E4. || Wed || 29/11 || 8-10 || L51 || AR
|| Realization theory
| L12.|| Thu|| 30/11 || 13-15|| L51 || XH
|| Observers and observer based control
| L13. || Fri|| 1/12 || 13-15 || L52|| XH
|| Linear quadratic control
| E5.|| Tue || 5/12 || 10-12 || V34 || AR
||Pole assignment and observers |
| L14.|| Wed || 6/12 || 8-10 || L51|| XH
|| LQ and algebraic Riccati equation
| L15. || Thu || 7/12 || 13-15 || L52|| XH
|| LQ, Least squares estimation|
| E6.|| Fri || 8/12 || 13-15 || V32 || AR
|| Linear quadratic control|
| L16.|| Wed|| 13/12 || 8-10|| L52 || XH
|| Kalman filtering |
| L17.|| Thu || 14/12 || 13-15 || L52 || XH
|| Kalman filtering
| E7.|| Fri || 15/12 || 13-15 ||U51 || AR
|| Kalman filtering