### Kungl Tekniska högskolan / Optimeringslära och systemteori /

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# SF2832 Mathematical Systems Theory,   Fall 2014

Adress of homepage for the course:
http://www.math.kth.se/optsyst/grundutbildning/kurser/SF2832/.

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

Tutorial exercises:
Johan Markdahl, markdahl@math.kth.se, phone 790 62 94.

# Announcements:

• I want to clarify here that calculator is not allowed for the exam.
• Office hours before the exam: 14-16, Monday Jan. 12, and 15-17 Tuesday Jan. 13. My office is at room 3532 of the math. department.
• Quite a few of the students obtained the maximal value of 80 degrees on the inverted pendulum design, thus no extra bonus points will be given to the winners.

• ## 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 on sale at KTH Bookstore, and exercise notes.
• Anders Lindquist & Janne Sand (revised by Xiaoming Hu), An Introduction to Mathematical Systems Theory, lecture notes, KTH, 2012. First 5 chapters
• Per Enqvist, Exercises in Mathematical Systems Theory, excercise notes (password will be given at the lectures), 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 Tuesday November 25, 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 Wednesday December 10).
• 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 Friday December 19).
• 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. 15, 2015 at 14:00-19:00.

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 2014

Type Day Date Time Hall Instr Topic
L1. Mon 03/11 15-17 M32 XH Introduction
L2. Tue 04/11 15-17 Q33 XH Linear systems
L3. Wed 05/11 10-12 V23 XH Linear systems
E1. Mon 10/11 15-17 M31 JM Linear systems
L4. Wed 12/11 10-12 M32 XH Reachability
L5. Fri 14/11 13-15 M32 XH Reachability cont'd
L6. Mon 17/11 10-12 E33 XH Observability
E2*. Tue 18/11 8-10 M32 JM Reachability and observability (rescheduled to 21/11)
L7*. Wed 19/11 15-17 D32 XH Stability (rescheduled to 18/11)
L8*. Fri 21/11 13-15 M33 XH Stability cont'd and Realization theory (rescheduled to 19/11)
L9. Mon 24/11 8-10 V33 XH Canonical forms and Kalman decomposition
E3. Tue 25/11 15-17 M31 JM Stability and realization theory
L10. Wed 26/11 10-12 M31 XH Minimal realizations
L11. Thu 27/11 8-10 D42 XH Minimal realizations, State feedback
E4. Mon 1/12 15-17 M31 JM Realization theory
L12. Tue 2/12 10-12 M32 XH Observers and observer based control
L13. Wed 3/12 8-10 M31 XH Linear quadratic control
E5. Thu 4/12 15-17 M32 JM Pole assignment and observers
L14. Mon 8/12 15-17 M31 XH LQ and algebraic Riccati equation
L15. Tue 9/12 8-10 M32 XH LQ, Least squares estimation
E6. Wed 10/12 15-17 M31 JM Linear quadratic control
L16. Mon 15/12 15-17 D42 XH Kalman filtering
L17. Tue 16/12 8-10 M31 XH Kalman filtering
E7. Wed 17/12 10-12 M31 JM Kalman filtering