# Teaching Activities

Current lectures and general information

##### Important Links

- Ilias
- Campus

Prof. Scherer - Campus

Department of Mathematics - Engineering Cybernetics (German)

Specialization "Mathematical Methods in Cybernetics" - Specialisiations in Mathenatics - Optimiation and Control Theory (in German only)

### Winter Term 2020

**Linear Control Theory **

Concerning Teaching during the WinterTerm please check with the Campus System and the Teaching Information from the University of Stuttgart

## Regularly Offered Courses

**Description**

Many processes that evolve over time (such as the movement of a robot) can be described by means of dynamical systems via ordinary differential equations. This lecture provides the fundamental knowledge for investigating the qualitative solution behavior of a dynamic system.

**Contents of the lecture**

The lecture covers the following topics:

- Linearisation and theory of linear differential equations
- Periodic differential equations, planar systems
- Explicit methods
- Existence und uniqueness of solutions
- Dependency of solutions on parameters and initial condition
- Stability of solutions
- Lyapunov functions and theorems of Lyapunov and Lasalle
- Invariant manifolds, bifurcation theory, normal forms of nonlinear systems, control systems

**Literature **

- H. Amann, Gewöhnliche Differentialgleichungen, de Gruyter, 1995
- L. Grüne and O. Junge, Gewöhnliche Differentialgleichungen, Springer Spektrum, 2016
- H. Logemann and E. P. Ryan, Ordinary Differential Equations, Springer, 2014

**Target audience**

Mathematics-, SimTech-, cybernetics- and engineering students

**Expected knowledge**

Linear algebra I+II and calculus I+II or higher mathematics I-III

**Further lectures**

Special lectures

**Description**

The evolution of many physical, biological or social processes is described by difference or differential equation. This introduction to control theory yields the mathematical foundation for one of the most fascinating field of applications of mathematics. Control theory is about the systematic and purposeful manipulation of the underlying systems. So it is not the goal to find or characterize solutions of differential equations, but to change the free components of the system via a controller such a way that the solutions admit desired properties.

The presented techniques form the foundation for a wide range of applications in various fields such as automotive and space industry, optimal administration of drugs in medical science or the coordination of electric supply networks.

**Contents of the lecture**

The lecture covers the following topics:

- State-space description of multivariable linear systems, blockdiagrams
- Linearisation, equilibria, Lyapunov funktions and the Lyapunov equation
- Responses of linear systems, modes, matrix exponential und convolution integral
- Transfer functions, realization theory and normal forms
- Controllability, stabilizability, uncontrollable modes and pole placement
- Linear-quadratic optimization, algebraic Riccati equation, robustness
- Observability, detectability, unobservable modes, state estimation
- Feedback controllers, separation principle
- Reference- and disturbance models as well as the "Internal Model Principle"
- Balanced realisations and model reduction
- H
_{2}-optimal control via output feedback

**Literature**

- T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, 1980
- H. W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie, Springer-Verlag Berlin 1985
- E. D. Sontag, Mathematical Control Theory, Springer, New York 1998
- B. Friedland, Control System Design: An Introduction to State-space Methods, Dover Publications, 2005
- K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, Princeton and Oxford, 2009

**Target audience**

Mathematics-, SimTech-, cybernetics- and engineering students

**Expected knowledge**

Linear algebra I+II and calculus I+II or higher mathematics I-III

**Further lectures**

- Robust control
- Linear matrix inequalities in control
- Spezial lectures

**Description**

In practice, a mathematical model often deviates from the underlying real system (e.g., caused by unknown system parameters or by unmodeled dynamics). Therefore, in control engineering it is required to design robust controllers to cope with these discrepancies. This lecture provides basics for modeling uncertainties, methods for analyzing robust controllers, as well as a general framework for optimization-based controller design.

**Contents of the lecture**

The lecture covers the following topics:

- Selected mathematical background for robust control
- Introduction to uncertainty descriptions (unstructured uncertainties, structured uncertainties and uncertainties, ...)
- The generalized plant framework
- Robust stability and performance analysis of uncertain dynamical systems
- Structured singular value theory
- Theory of optimal H
_{∞}controller design - Application of modern controller design methods (H
_{∞}control and μ-synthesis) to concrete examples - Algebraic approach to control
- Youla parameterization
- Structured controller synthesis

**Literature**

- C. W. Scherer, Theory of Robust Control, Lecture Notes
- G. E. Dullerud and F. Paganini, A Course in Robust Control, Springer-Verlag, 1999
- S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis & Design, Wiley, 2005

**Target audience**

Mathematics-, SimTech-, cybernetics- and engineering students (second year and above)

**Expected knowledge**

Linear algebra I+II and calculus I+II or higher mathematics I-III.

With regards to content, it is recommended to attend the lecture control theory beforehand. However, this is not necessary.

**Further lectures**

- Linear matrix inequalities in control
- Special lectures

**Description**

Optimization plays an important role in mathematical systems theory and industrial practice, e.g., to control dynamical systems in an energy or time-efficient fashion. Particularly in recent years, a large number of the underlying questions have been recast as specific convex optimization problems, namely as so-called semi-definite programs, by modelling the constraints via linear matrix inequalities. The structure of these programs allows their highly efficient numerical solution even in complex situations. This advanced lecture provides mathematical methods to translate problems of feedback design and robust control into semi-definite programs.

**Contents of the lecture**

The lecture covers the following topics:

- Introduction to (convex) optimization and semi-definite programming
- Theory of dissipative dynamical systems
- Analysis of systems under various performance criteria
- From analysis to synthesis: A general framework
- Multi-objective control
- Robustness analysis against time-varying parametric uncertainties
- Linear Fractional Representations and Integral Quadratic Constraints
- Robust controlller synthesis: State feedback, state estimation and output feedback
- Controller synthesis for linear parameter-varying systems and gain-scheduling

**Literature**

- S. P. Boyd et al., Linear matrix inequalities in system and control theory, SIAM 1994
- L. El Ghaoui and S.I. Niculescu, Eds., Advances in Linear Matrix Inequality Methods in Control, SIAM 2000
- A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization. Philadelphia, SIAM, 2001
- S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004
- C. W. Scherer and S. Weiland, DISC Lecture Notes LMI's in Control

**Target audience**

Mathematics-, SimTech-, cybernetics- and engineering students (second year and above)

**Expected knowledge**

Linear algebra I+II and calculus I+II or higher mathematics I-III.

With regards to content, it is recommended to attend the lecture control theory and robust control beforehand. However, this is not necessary.

**Further lectures**

Special lectures

Special lectures and seminars take place - just have a look on our current semester website

## Thesis

**Topics**

The following concrete topics are presently available:

- Distributed optimal control of complex interconnected systems
- Adaptive controller design based on dynamic multipliers
- Parameterization of Zames-Falb multipliers for repeated nonlinearities
- Simulation based design of a robust feedback controller for a quadrotor

Please contact us for further information or concerning other topics of your interest.

**Expected knowledge**

- Bachelor thesis: Control theory
- Master thesis: Robust control and linear matrix inequalities in control

## SimTech Projects

see pdf information - (in German only)

## Previous semesters

## Miscellaneous

###### Model Reduction for Large-Scale Systems: Theory, Numerics and Applications

Instructor: Thanos Antoulas, Rice University, Houston

April 2-5, 2012, Universität Stuttgart, Pfaffenwaldring 5A (new SimTechBuilding), R. 0.015,

(PWR 57, R. 7.143 for the exercises in the afternoon)

**Topics**

- Overview: model reduction for large-scale systems
- Linear System Theory (Overview)
- Eigenvalue Theory and Algorithms
- Moment Matching Algorithms
- Balanced Truncation Model Reduction
- Numerical Solution of Lyapunov Equations
- Nonlinear Model Reduction

**Time schedule**

Monday-Wednesday: 9 am – 3 pm (with breaks)

Thursday: 9 am – 10.15 am

Exercises: Monday-Wednesday: 3.45 pm – 5.15 pm

**Colloquium Presentation**

Right after the course Prof. Antoulas will as well give a colloquium presentation entitled

"Recent advances in model reduction of large-scale systems"

Thursday April 5, 2 p.m. Pfaffenwaldring 57, room 8.122

### Contact Prof. Scherer

**Professor Dr.**

### Carsten W. Scherer

#### Head of Institute and Chairholder mst / Erasmus-Coordinator Dep. of Mathematics

- Profile page
- +49 711 685-60025
- Write e-mail
- Carsten W. Scherer

[Photo: Uni Stuttgart]