Prerequisites
The course is aimed at students in their first year of their master in Mathematics or Applied Mathematics. Apart from a solid knowledge of linear algebra, calculus and ordinary differential equations there is no specialized knowledge necessary. Some examples and exercise require a basic understanding of electrical circuits and mass-spring-damper systems.
Note that this course consists of four bi-weekly lectures in Utrecht, an intensive project week at the U Twente (accommodation costs are covered by mastermath) a final presentation day in Groningen and the final exam in Utrecht, see detailed schedule below.
Aim of the course
The course aims at students in pure and applied mathematics with an interest in applications of (linear) algebra and ordinary differential equations. The purpose of the course is to introduce the students to basic concepts and more advanced notions of the mathematical theory of systems and control.
Course description
Mathematical systems theory is concerned with problems related to dynamic phenomena in interaction with their environment. These problems include:
* modelling. Obtaining a mathematical model that reflects the main features. A mathematical model may be represented by difference or differential equations, but also by inequalities, algebraic equations and logical constraints.
* Analysis and simulation of the mathematical model.
* Prediction and estimation.
* Control. By choosing inputs or, more general, by imposing additional constraints on some of the variables, the system may be influenced so as to obtain certain desired behavior. Feedback is an important example of control.
The main objects of study in this course are systems modeled by linear time-invariant differential equations. The content of the course closely follows the book “Introduction to Mathematical Systems Theory” by J.W. Polderman and J.C. Willems.
In particular, the following topics will be covered during the course:
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Representation of dynamical systems using polynomial matrices.
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Several representations are introduced along with their relations. Important examples of such representation are input-output representations that reveal that some variables may be unrestricted by the equations, and state space representation.
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Controllability and observability are fundamental system theoretic concepts and will be introduced and characterized for general behavior in kernel representation and for state-space models. The theory of controllability and observability forms one of the highlights of the course.
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Stability can be an important and desirable property of a system. Stabilization by static or dynamic feedback is one of the key features of Systems and Control. In the pole placement theorem linear algebraic methods and the notion of controllability are used in their full strength. It forms one of the most elegant results of the course and indeed of the field of Systems and Control.
Course structure
The course consists of two main parts: 1) Guided self-study of the material (in the first 8 weeks) and 2) working on a project as a group of two or three students during an intensive week (during Week 9 of the course), 3) final presentation of the project results (Week 11)
Part 1: Guided self-study
The self-study is supported by summaries of the relevant content in four bi-weekly lectures in Utrecht (mandatory attendance). Furthermore, corresponding homework assignments will be given which are discussed during the lectures. Additionally, short videos about specific topics will be provided by the lecturers. In addition to the homework grades, the assessment will be done via an exam (end November/beginning December).
Part 2: Project
The content obtained via the guided self-study will be deepened and applied by working on a project during an intensive week at the University of Twente. On the first day of the intensive week the last homework assignment will be discussed and the projects will be assigned and groups will be formed. The remaining days of the intensive week the students will work on the project and the lecturers will be present and available for questions. Attendance at the intensive week is mandatory and the accommodation costs will be covered by mastermath. The assessment will be done via a presentation (in Groningen, one week after the intensive week) and a written report.
Summary of course structure
Lecture 1 (week 1 of course): Introduction, Summary of Chapters 1+2
Lecture 2 (week 3 of course): Discussion and of HW1, Summary of Chapters 3+4
Lecture 3 (week 5 of course): Discussion of HW2, Summary of Chapters 5+6
Lecture 4 (week 7 of course): Discussion of HW3, Summary of Chapters 7,9+10
Intensive week at University of Twente (week 9 of the course, attendance mandatory):
Monday afternoon: Discussion of HW4, Project selection
Tuesday – Thursday: Working on project
Friday morning: Feedback on presentation draft
Final presentations of projects in Groningen (on the Monday of week 11 of the course) and project report hands-in one week later.
Final exam (end of November/beginning December)
Rules about Homework/Exam
The final grade is determined by the results of four homework assignments, the oral presentation, the report and the written exam. The grades of the homework assignments and presentations are based on guided peer review. The report and the exam are graded by the lecturers.
The four homework assignments concern all chapters of the book with the exception of Chapter 8. The report and presentation will be based on additional material that will be assigned during the intensive week. Presentation and report will be prepared in small groups. All six items will be graded separately. The final grade is determined by the six sub-grades and the written exam. With HW=HomeWork, G1=Grade after first exam, G2=Grade after resit (if applicable), Rep=Report, WrEx=Written Exam, WrExR=Retake of the Written Exam.
For a pass it is required that WrEx>=5 or, if applicable, WrExR>=5, and HW>=5, where HW= (H1+H2+H3+H4)/4. If these conditions are satisfied then G1=(HW+Rep+Pres+2*WrEx)/5 and G2=(HW+Rep+Pres+2*WrExR)/5 otherwise G1=min(HW,WrEx) and G2=min(HW,WrExR).