Welcome to the PhD course on Linear Dynamical Systems, Spring 2026!

There will be one session per week (approximately 2.5 hours), devoted to tutorial lectures, problem solving and discussions on the text. Exercise questions are mandatory to solve prior the sessions as preparation. On sessions, active participation is required: contributing to the discussions and presenting solutions (of selected problems) on white board.

Why to study FSSY055?
• Many mathematical and engineering problem makes use of linear theory
• We know a lot about linear systems.
• We do know a lot about linear system theory, and we can even analytically calculate related
control and estimation solutions.

What to study in FSSY055?
1. Introduction. State equations to LTV. Linearization. State transition matrix. [Chapters 1-4]
2. Time invariant and periodic systems. Floquet decompostion. Discrete time state equations. [5, 20, 21]
3. Uniform internal stability, Lyapunov transformation. Lyapunov criteria. [6-8]
4. Controllability and observability of LTV systems. Canonical state space models. [9, 13, 25]
5. Realizability and minimal realizations. [10, 11, 26]
6. Linear state feedback. [14, 28]
7. State observers. [15, 29]
8. Geometric theory [18,19]

How to study in FSSY055?
There will be one session per week (approximately 2.5 hours), devoted to tutorial lectures, problem solving and discussions on the text. Exercise questions are mandatory to solve prior the sessions as preparation. On sessions, active participation is required: contributing to the discussions and presenting solutions (of selected problems) on white board.

Instructor, place, time and occurrence
Prof. Balazs Kulcsar, Automatic Control Group, Systems and Control Division, Department of Electrical Engineering, Chalmers, kulcsar@chalmers.se

First meeting: January 23, 2026, 10:00 am (sharp!).

How can I prepare to the first lecture?

Read Chapter 1 from the Rugh book (see below).
Solve the warm up questions: 1.4, 1.15, 1.17.


Literature
Wilson J Rugh: Linear System Theory, 2nd edition, Prentice-Hall, 1996.

Prerequisites, examination, credit
Basic course in automatic control, including state space models. A course in matrix analysis is recommended but not mandatory. The course gives 7.5 hp based on passed written examination, a satisfactory number of successfully solved assignments during the course with active participation in course sessions.

Assignments, exercises, scheduling

Here