**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.

**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:

- Representation of dynamical systems using polynomial matrices.
- 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.
- - 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.
- 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.

**Schedule**

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, G(1) =Grade after first exam, G(2) =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

G(1) =(HW+Rep+Pres+2*WrEx)/5 and G(2) =(HW+Rep+Pres+2*WrExR)/5 otherwise G(1) =min(HW,WrEx) and G(2) =min(HW,WrExR).

**Lecture Notes/Literature**

Prof. Felix Schwenninger (UT, f.l.schwenninger@utwente.nl)

Prof. Stephan Trenn (RUG, s.trenn@rug.nl)

- Docent: Felix Schwenninger
- Docent: Stephan Trenn