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KTH / Teknikvetenskap / Matematik / Optimeringslära och systemteori

# SF2832 Mathematical Systems Theory

Adress of homepage for the course:

### The homepage has been moved to Canvas!

Examiner and lecturer:
Xiaoming Hu, hu@kth.se , room 3532, Lindstedtsv 25, phone 790 7180.

Tutorial exercises:
Axel Ringh, aringh@math.kth.se, phone 790 6659.

Course registration:
Important! pleae read here for information on course registration and exam application.
Solution to Exam April 11, 2017 (available soon after the exam has finished)

## Introduction

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.

## Course goals

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 controller.
• Apply linear quadratic techniques to derive optimal state feedback controllers.
• Solve the Riccati equations that appear in optimal control and estimation problems.
• 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 course.
• Solve fairly simple but realistic control design problems using the methods in the course.

## Course material

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 Introduction to Mathematical Systems Theory, lecture notes, KTH, 2012. download here (password will be given at the first lecture)
• Per Enqvist, Exercises in Mathematical Systems Theory, excercise notes (password will be given at the first lecture), KTH.

## Course requirements

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 material better.

Homework
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 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 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]
• Homework 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]

Written exam
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.

>90 A
76-90 B
61-75 C
50-60 D
45-49 E
41-44 FX
The grade FX means that you are allowed to make an appeal, see below.

• The first exam will take place on Jan. 16, 2017 at 8:00-13:00.

Appeal
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.

Course evaluation
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 Introduction
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 Reachability
L5. Thu 9/11 14-16 Q33 XH Reachability cont'd
L6. Fri 10/11 13-15 U51 XH Observability
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