## Advanced Systems Theory, 2017

### Description:

This course deals with linear input-output systems in state space form. The course starts with a review of the most important concepts on linear systems that were already taught in the second year course Integrating Project System Theory. We review the notions of internal and external stability, controllability and observability with special emphasis on their geometric characterizations. We also review the notions of stabilizability and detectability and discuss their relevance in the design of stabilizing controllers and observer design.
We then shift to systems with disturbance inputs, and formulate the design problems of disturbance decoupling by state feedback, both with and without the requirement of internal stability. We give necessary and sufficient conditions for solvability of these problems in terms of controlled invariant subspaces and stabilizability subspaces.
Next, we consider the more general formulations of these problems where only part of the state is available for feedback. In this case we are forced to apply dynamic output feedback, and the solution of the decoupling problems in addition uses the concepts of conditioned invariant and detectability subspaces.
We also discuss the classical problem of tracking and regulation that requires the design of dynamic controllers such that the output of the controlled system asymptotically follows a given time signal, despite the fact that the systems is affected by external disturbances. This problem is studied by means of solvability of linear matrix equations.
As a final subject, we consider linear networked multi-agent systems, which deals with controlling (possibly large numbers of) interconnected linear input-output systems. The interconnection structure of such networks are described by graphs, representing the communication between the agents. We discuss the notions of consensus and synchronization for such systems, and study under what conditions controllers exist that achieve consensus.
### Literature:

H.L. Trentelman, A.A. Stoorvogel and M.L.J. Hautus, "Control Theory for Linear Systems", Springer Verlag, 2001. Unfortunately, the first edition is sold out, and a second edition is not available. However, the complete pdf file of the book can be downloaded from the website www.math.rug.nl/~trentelman
under 'Publications'. Additional lecture notes will be handed out during the course.
### Organization:

Lectures: Tuesday 09:00 -11:00, 5115-0020, Wednesday 09:00-11:00 in 5171-0603, Exercise Class on Thursday 11:00-13:00, 5115.0020.
### Examination

During the the course, there will be two `takehome-exams', each consisting of a number of problems on the theory treated in the course. Solutions to the problems of these take-home exams should be handed in before a given date (no exceptions!). After the course there will be an written examination. The final grade is determined on the basis of the two take-home exams and the written examination.