Examiner and lecturer

Xiaoming Hu (hu@math.kth.se), Room 3532, Lindstedtsv. 25, tel. 790 7180.

Teaching assistant

Yuexin Cao (yuexin@kth.se), Room 3731, Lindstedtsv. 25, tel. 790 7132.

Course registration

Important! pleae read here for information on course registration and exam application.

Course contents

This course is the continuation course of Mathematical Systems Theory.

Linear geometric control theory was initiated in the beginning of the 1970's. A good summary of the subject is the book by Wonham.

The term "geometric'' suggests several things. First it suggests that the setting is linear state space and the mathematics behind is primarily linear algebra (with a geometric flavor). Secondly it suggests that the underlying methodology is geometric. It treats many important system concepts, for example controllability, as geometric properties of the state space or its subspaces. These are the properties that are preserved under coordinate changes, for example, the so-called invariant or controlled invariant subspaces. On the other hand, we know that things like distance and shape do depend on the coordinate system one chooses. Using these concepts the geometric approach captures the essence of many analysis and synthesis problems and treats them in a coordinate-free fashion. By characterizing the solvability of a control problem as a verifiable property of some constructible subspace, calculation of the control law becomes much easier. In many cases, the geometric approach can convert what is usually a difficult nonlinear problem into a straight-forward linear one.

As we can already see, "invariance" is a very important concept in the geometric control theory. It is not coincidence that invariance is also very important in machine learning since it preserves the object's identity, category and so forth across changes in the specifics of the input. In this respect, geometric methods are also important in machine learning.

The linear geometric control theory was extended to nonlinear systems in the 1970's and 1980's (see the book by Isidori). The underlying fundamental concepts are almost the same, but the mathematics is different. For nonlinear systems the tools from differential geometry are primarily used.

The course compendium is organized as follows.

Chapter 1 is introduction; In Chapter 2, invariant and controlled invariant subspaces will be discussed; In Chapter 3, the disturbance decoupling problem will be introduced; In Chapter 4, we will introduce transmission zeros and their geometric interpretations; In Chapter 5, non-interacting control and tracking will be studied as applications of the zero dynamics normal form; In Chapter 6, we will discuss some input-output behaviors from a geometric point of view; In Chapter 7, we will discuss the output regulator problem in some detail; In Chapter 8, we will extend some of the central concepts in the geometric control to nonlinear systems. Finally, in Chapter 9 some applications to mobile robots will be given.

Course material

X. Hu, A. Lindquist et al, Geometric Control Theory, lecture notes, KTH, 2012, can be downloaded here:
Table of contents, Chapter one, Chapter two, Chapter three, Chapter four, Chapter five
Chapter six, Chapter seven, Chapter eight, Chapter nine, Appendix and index
Exercise notes:
Exercise 1, Exercise 2, Exercise 3, Exercise 4, Exercise 5, Exercise 6
More information can be found here, a detailed solution to an old exam

Course requirements

The course requirements consist of an obligatory final written examination, and three homework sets.

Homework

Each homework set consists of not more than five problems. Each successfully completed homework set gives you 0.5hp. 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 under assignments.

Homework 1 (due February 7)
Homework 2 (due February 22)
Homework 3 (due March 6)

2020 Homework and solution:

Homework 1
Homework 2
Homework 3

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. These grade limits can only be modified to your advantage.

Total credit (points)	Grade
>84	A
71-84	B
56-70	C
46-55	D
40-45	E
35-39	FX

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

The first exam will take place on March 14, 2023 at 08:00-13:00.

Solutions to exam of March 2020 can be found here.

Appeal

If your 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.

Preliminary Schedule for 2023

L=Lecture, E=Exercise

Type	Day	Date	Time	Hall	Topic
L1.	Tue	17/01	15-17	B24	Introduction
L2.	Wed	18/01	15-17	B21	Invariant subspaces
L3.	Thu	19/01	13-15	B22	Invariant subspaces (cont.)
E1.	Mon	23/01	15-17	B26	Linear algebra, invariant subspaces
L4.	Tue	24/01	13-15	B21	Disturbance decoupling
L5.	Wed	25/01	15-17	D32	Disturbance decoupling, and Zeros
L6.	Thu	26/01	8-10	B21	Zeros and zero dynamics (cont.)
E2.	Tue	31/01	15-17	B22	Reachability subspaces, V*-algorithm, zero dynamics
L7.	Wed	1/02	15-17	B21	Zero dynamics and high gain control
L8.	Thu	2/02	13-15	B22	Noninteracting control and tracking
L9.	Mon	6/02	15-17	B24	Input-output behavior
E3.	Tue	7/02	15-17	B22	Applications of zero dynamics
L10.	Wed	8/02	15-17	B21	Input-output behavior and Output regulation
L11.	Thu	9/02	13-15	B23	Output regulation (cont.)
L12.	Tue	14/02	15-17	B22	Nonlinear systems: examples, math preparation
E4.	Wed	15/02	15-17	D32	Sylvester equation, Output tracking input, Output regulation
L13.	Thu	16/02	13-15	B22	Nonlinear systems: controllability
L14.	Mon	20/02	15-17	B22	Nonlinear systems: Stability, steady state response
L15.	Tue	21/02	15-17	B22	Center manifold and normal form
E5.	Wed	22/02	15-17	B26	Nonlinear systems
L16.	Thu	23/02	13-15	D41	Nonlinear systems: zero dynamics and applications
L17.	Tue	28/02	15-17	B23	Exact linearization and Consensus problem
E6.	Wed	1/03	15-17	B21	Nonlinear control and multi-agent systems

Welcome!