## 5B1742 Mathematical Systems Theory (Basic Course)

### Examiner

Ulf Jönsson (ulfj@math.kth.se), room 3711, Lindstedtsv. 25, phone. 790 8450.

### Teachers

Xiaoming Hu (hu@math.kth.se), rum 3712, Lindstedtsv. 25, tel. 790 7180.
Ulf Jönsson (ulfj@math.kth.se), room 3711, Lindstedtsv. 25, phone. 790 8450.
Ryozo Nagamune (ryozo@math.kth.se), room 3737, Lindstedtsv. 25, phone. 790 7507.

### Introduction

This is an introductory course in mathematical systems theory. The subject provides the mathematical aspects of modern control theory, with application areas in aeronautics, electrical networks, signal processing, and etc. 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 also is necessary for further studies in control and signal processing.

### Course goal

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. In the course you learn how to
• Analyze the state-space model with respect to minimality, observability, reachability, detectability and stabilizability.
• Describe and 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.
• Design a Kalman filter for optimal state estimation of linear systems subject to stochastic uncertainties.
• Solve the Riccati equations that appear in optimal control and estimation problems.
• 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 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 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 and exercise notes on sale at ``studentexpeditionen'' on Lindstedtsv 25
• Anders Lindquist & Janne Sand, An Introduction to Mathematical Systems Theory, lecture notes, KTH, 50 kr.
• Per Enqvist, Exercises in Mathematical Systems Theory, excercise notes, KTH, 30 kr
• Claes Trygger, Kopior på overheadbilder, KTH, 40 kr
• Some Linear Algebra Background for Mathematical Systems Theory (will be continuously updated), [pdf].

### Course requirements

The course requirements consist of final written examination. There are also three optional homework sets, and two optional computer exercises that we strongly encourage you to do. They will also give you bonus credits in the examination.

#### Homework sets

Each homework set consists of three problems. The first two are methodology problems where you practice on the topics of the course and apply them to examples. The third problem is 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. You may discuss the homework assignments in group (maximal three students in a group). Each group needs to hand in only one report and each student has to write at least one report during the course period (if a report is written by n (n less than 3) students, then each author is considered to have written 1/n of the report). The homework set is considered successfully completed if you have solved at least two of the three problems satisfactory. Each successfully completed homework set gives you one point. If you have done all three problems right, an additional point will be awarded. As a necessary condition to pass the course, you need to earn at least three points from the homework. 50% of the homework points are also counted as bonus points for the final exam. 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 from the first three chapters of the lecture notes (Due on Wednesday April 5).
• Here is the first homework set [pdf]
• Here is a solution sketch [pdf]
• Homework 2: This homework set covers the material in chapter 4 - chapter 5 of the lecture notes (Due on Monday April 24).
• Here is the second homework set [pdf]
• Here is a solution sketch [pdf]
• Homework 3: This homework set covers material in chapter 6 - chapter 10 (Due on Wednesday May 10).
• Here is the third homework set [pdf]
• Here is a solution sketch [pdf]
Note: The date in parentheses is the last day (before 5 pm) for handing in the written solutions. The homework questions will be made available at least two weeks before the deadline.

#### Computer exercise

The intention of the computer exercises is to exemplify how easy it is to apply the theory of the course using standard linear algebra packages for numerical computations. We will use the ``Control System Toolbox'' in MATLAB, for this purpose. The exercises serve the additional purpose of applying the results of the course to realistic control problems. The examination of each computer exercise is oral and written. Firstly the results should be presented in a written report (approximately 5 pages written using Latex or Microsoft Word) and secondly each student should be prepared to answer questions about the exercise when it is handed back. Cooperation in groups of not more than two students is allowed and only one report per group should be turned in. You should mail your report to the examiner. Each successfully completed computer exercise gives you two bonus points on the final exam. The two exercises are
• Computer exercise 1: In this exercise you investigate controllability, observability, and pole placement for two famous laboratory experiments: The cart and spring system and an inverted pendulum mounted on a cart. The deadline for the report is on April 26. Do attach your commented MATLAB code to your report!

Here is the exercise [pdf]. You may also use the following m-file [m].

• Computer exercise 2: In this exercise you investigate numerical routines for observers, linear quadratic control, and Kalman filtering. Hand in the report at the latest on May 12. Do attach your commented MATLAB code to your report!

Here is the exercise [pdf].

#### Written exam

This is an open book exam and you may bring the lecture notes and Beta Mathematics Handbook (or any equivalent handbook). 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 five bonus points from the homework assignments and the computer exercises. These grade limits can only be modified to your advantage.

>= 25 3
>= 35 4
>= 45 5

• The next exam will take place on June 8, 2007 at 14:00-19:00 in room E32.

A solution sketch for the exam on Tuesday May 23 can be found here [pdf].

Old exams can be obtained at the "student expedition".

#### Appeal

If your total score (exam score + maximum 5 bonus points from the homework assignments and the computational exercises) is 23 or 24 points then you are allowed to do a complementary examination for grade 3. 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 last tutorial exercise you will be asked to complete a course evaluation form. The evaluation form will also 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 2005/2006

Type Day Date Time Hall Instr Topic
L1.Wed 15/3 8-10 E34 UJ Introduction
L2.Fri 17/3 8-10 E34 UJ Linear systems
L3.Mon 20/3 8-10 E34 UJ Linear systems
E1. Tue 21/3 10-12 M22 RN Linear systems
L4. Wed 22/3 10-12 D41 UJ Reachability
L5. Thu 23/3 10-12 D31 UJ Reachability cont'd
L6. Mon 27/3 15-17 D32 UJ Observability/Stability
L7.Wed 29/3 13-15 D35 UJ Realization theory and canonical forms
E2. Fri 31/3 8-10 E53 RN Reachability and observability
L8. Mon 3/4 15-17 E34 UJ Realization theory cont'd
L9. Wed 5/4 13-15 E33 UJ Kalman decomposition and minimal realizations
E3. Fri 7/4 15-17 E33 RN Stability and realization theory
L10. Wed 19/4 13-15 E53 UJ Realization theory and state feedback
E4. Thu 20/4 15-17 D33 RN Realization theory
L11. Fri 21/4 8-10 D41 UJ Observers and observer based control.
E5. Mon 24/4 13-15 E34 RN Pole assignment and observers
L12. Wed 26/4 8-10 E34 UJ Linear quadratic control
L13. Fri 28/4 15-17 E34 UJ The Riccati equation
L14. Tue 2/5 15-17 E34 UJ The algebraic Riccati equation
E6. Wed 3/5 13-15 E36 UJ Linear quadratic control
L15. Thu 4/5 15-17 E33 XH Least squares estimation
L16. Mon 8/5 15-17 E34 XH Kalman filtering and the separation theorem
L17 . Tue 9/5 15-17 E34 XH Kalman filtering

### Welcome!

Course information on www: http://www.math.kth.se/optsyst/studinfo/5B1742/5B1742/.
5B1742 Mathematical systems theory (basic course)
Modified: May 1, 2006 by Ulf Jönsson, ulfj@math.kth.se.