SF2832 Mathematical Systems Theory,   2012

Examiner and lecturer:
Xiaoming Hu, hu@kth.se , room 3712, Lindstedtsv 25, phone 790 7180.

Tutorial exercises:
Johan Thunberg , jthu02@math.kth.se, phone 790 75 07.

Solution to 2012 Exam

Introduction

Course goals

• 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).

• 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

• Anders Lindquist & Janne Sand (revised by Xiaoming Hu), An Introduction to Mathematical Systems Theory, lecture notes, KTH, 2012.
  
• Per Enqvist, Exercises in Mathematical Systems Theory, excercise notes, KTH.
• Supplementary material prepared by Ulf Jönsson and others can be downloaded here.

Course requirements

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 problems from the first three chapters in compendium. [solution] (Due on Friday February 10, 17:00).
• Here is the first homework set of 2011 [pdf][solution]

• Homework 2: This homework set covers the material in chapter 4 - chapter 5 of the lecture notes. [solution](Due on Wednesday February 29).
• Here is the second homework set of 2011 [pdf][solution]

• Homework 3: This homework set covers problems chapter 6 - chapter 9 in the lecture notes. [solution] (Due on Friday March 9).
• Here is the third homework set of 2011 [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.

Total credit (points)	Grade
>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 March 12, 2012 at 14:00-19:00.
• The second exam will take place on June 7, 2012 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
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 2012

Type	Day	Date	Time	Hall	Instr	Topic
L1.	Mon	16/1	15-17	E33	XH	Introduction
L2.	Wed	18/1	13-15	E33	XH	Linear systems
L3.	Fri	20/1	15-17	E53	XH	Linear systems
E1.	Mon	23/1	15-17	E33	JT	Linear systems
L4.	Wed	25/1	13-15	E33	XH	Reachability
L5.	Fri	27/1	15-17	L21	XH	Reachability cont'd
L6.	Mon	30/1	15-17	E33	XH	Observability
E2.	Wed	1/2	13-15	L21	JT	Reachability and observability
L7.	Fri	3/2	15-17	L21	XH	Stability
L8.	Mon	6/2	15-17	E33	XH	Stability cont'd and Realization theory
L9.	Wed	8/2	13-15	L21	XH	Canonical forms and Kalman decomposition
E3.	Fri	10/2	15-17	L21	JT	Stability and realization theory
L10.	Mon	13/2	15-17	E33	XH	Minimal realizations
L11.	Wed	15/2	13-15	L21	XH	Minimal realizations, State feedback
E4.	Fri	17/2	15-17	L21	JT	Realization theory
L12.	Mon	20/2	15-17	E33	XH	Observers and observer based control
L13.	Wed	22/2	13-15	L21	XH	Linear quadratic control
E5.	Fri	24/2	15-17	L21	JT	Pole assignment and observers
L14.	Mon	27/2	15-17	L21	XH	Riccati equation
L15.	Tue	28/2	13-15	L21	XH	Least squares estimation
E6.	Wed	29/2	13-15	L21	JT	Linear quadratic control
L16.	Fri	2/3	15-17	L21	XH	Kalman filtering and the separation theorem
E7.	Mon	5/3	15-17	L21	JT	LQ, Kalman filtering and the separation theorem
L17.	Wed	7/3	13-15	L21	XH	Kalman filtering, and the separation theorem
E8.	Fri	9/3	15-17		JT	LQ, Kalman filtering, and the separation theorem

Welcome!