SF2832 Mathematical Systems Theory

SF2832 Mathematical Systems Theory,   Fall 2022 (Schedule )

Examiner and lecturer:
Xiaoming Hu, hu@kth.se , room 3533, Lindstedtsv 25, phone 790 7180.

Tutorial exercises:
Michele Mascherpa, micmas@kth.se

Course registration:
Important! please read here for information on course registration and exam application.

Information for support for students with disabilities for this course can be found here.

Introduction

This is an introductory course in mathematical systems theory. The subject provides the mathematical foundation of modern control theory, with wide engineering, financial and societal  applications. The aim of the course is that you should acquire a systematic understanding of linear dynamical systems, which is the focus of the 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.
• Supplementary material can be downloaded here . Detailed solution to a sample exam.

Course requirements

The course requirements consist of an obligatory final written examination. There are also three homework sets that are obligatory . The purpose of the homework is to 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.
The homework sets will be posted roughly ten days before the deadline under "Assignments".

• Homework 1: This homework set covers the material in chapters 1-4 of the lecture notes. 
• Here is a sample of the first homework set  [pdf][solution]

• Homework 2: This homework set covers the material in chaptes 4 -6 of the lecture notes. 
• Here is a sample of the second homework set [pdf][solution]

• Homework 3: This homework set covers the material in chapters 6 - 9 of the lecture notes. 
• Here is a sample of the third homework set [pdf][solution]

Written exam
This is an open book exam and you may bring the course compendium, the exercise notes by Per Enqvist, your own hand-written 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. These grade limits can only be modified to your advantage.

The grade FX means that you are allowed to make an appeal, see below.

• The first exam will take place on 10 January, 2023, 14:00-19:00. 

Appeal
If your total score is in the range 35-39 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.