Examiner and Lecturer

Silun Zhang (silunz@kth.se), Room 3437, Lindstedsvägen 25, tel. 08790 6238.

Teaching Assistant

Zhaozhan Yao (zhaozhan@kth.se), Room 3737, Lindstedtsv. 25

Course Registration

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

Course Contents

This course is the continuation course of SF2832/SF2842 Mathematical Systems Theory.

Linear geometric control theory was initiated at 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, the 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 a 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 1970s and 1980s (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

Lecture notes and references:

- X. Hu, A. Lindquist. Geometric Control Theory, lecture notes, KTH, 2012. The free version can be downloaded in the following.

Table of contents, Chapter one, Chapter two, Chapter three, Chapter four, Chapter five
Chapter six, Chapter seven, Chapter eight, Chapter nine, Appendix and index

- W. Murray Wonham. Linear Multivariable Control: a Geometric Approach. Berlin: Springer-verlag.

Exercise notes:

Exercise 1, Exercise 2, Exercise 3, Exercise 4, Exercise 5, Exercise 6

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 4)
Homework 2 (due February 20)
Homework 3 (due March 7)

Written exam

This is an open-book exam and you may bring the lecture notes, the exercise notes, your own class notes, 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, 2025.

Solutions to the exam of March 2020 can be found here.

Appeal

If your score is in the range of 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.

Preliminary Schedule for 2024

L=Lecture, E=Exercise

Type	Day	Date	Time	Hall	Topic
L1.	Tue	14/01	15-17	Q11	Introduction
L2.	Wed	15/01	10-12	B23	Invariant subspaces
L3.	Thu	16/01	13-15	B21	Invariant subspaces (cont.)
E1.	Mon	20/01	15-17	B21	Linear algebra, invariant subspaces
L4.	Tue	21/01	15-17	B23	Disturbance decoupling
L5.	Wed	22/01	15-17	B22	Disturbance decoupling, and Zeros
L6.	Thu	23/01	10-12	B22	Zeros and zero dynamics (cont.)
E2.	Tue	28/01	15-17	B25	Reachability subspaces, V*-algorithm, zero dynamics
L7.	Wed	29/01	15-17	B22	Zero dynamics and high gain control
L8.	Thu	30/01	10-12	B22	Noninteracting control and tracking
E3.	Mon	3/02	15-17	Q13	Applications of zero dynamics
L9.	Tue	4/02	13-15	B26	Input-output behavior
L10.	Wed	5/02	15-17	B21	Input-output behavior and Output regulation
L11.	Thu	6/02	10-12	B21	Output regulation (cont.)
L12.	Tue	11/02	15-17	B25	Nonlinear systems: examples, math preparation
E4.	Wed	12/02	15-17	B22	Sylvester equation, Output tracking input, Output regulation
L13.	Thu	13/02	13-15	B21	Nonlinear systems: controllability
L14.	Tue	18/02	15-17	Q11	Nonlinear systems: Stability, steady state response
L15.	Wed	19/02	15-17	B21	Center manifold and normal form
E5.	Thu	20/02	10-12	B21	Nonlinear systems
L16.	Fri	21/02	13-15	Q13	Nonlinear systems: zero dynamics and applications
L17.	Tue	25/02	15-17	B25	Exact linearization and Consensus problem
E6.	Wed	26/02	15-17	B22	Nonlinear control and multi-agent systems

Welcome!