SF2842 Geometric Control Theory

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.

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

Welcome!