# Numerical Methods for Control Systems

This research topic mainly deals with numerical algorithms for robust control and stabilization of descriptor systems. One focus is the construction of (sub-)optimal H-infinity controllers or the computation of system norms by using spectral information of certain structured matrix pencils. Another important point is the development of efficient and robust software to solve these problems.

Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well.

## Internal Projects

### Mixed Integer Nonlinear Programming for Damping Optimization

Vibrations are a typical and a mostly unwanted phenomenon in mechanical systems. Resonance and sustained oscillations have undesired effects such as energy waste, noise creation and even structural damage. The way to reduce dangerous vibrations is through damping. Damping is also responsible for many important system properties such as stability and dissipativity. Our investigation is devoted to damping optimization.
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### Optimal Damping of Vibrating Systems

In real physical systems which possess elasticity and mass, dangerous vibrations are a typical phenomenon which have been widely studied in the past. Up to this day it is an intensively investigated phenomenon.
For the majority of engineering applications, resonance and sustained oscillations can cause structural damage. The way to reduce dangerous vibrations is through damping.
Damping can also produce undesirable effects such as energy waste or heat production. On the other hand, damping is also responsible for many important system properties such as stability and dissipativity. Our investigation is devoted to optimization of damping of vibrating systems which is a very demanding problem.
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### Numerical algorithms for generalized eigenvalue problems of even structure with application in robust control of descriptor systems

The aim of the project is the development of efficient structure-preserving numerical methods for the solution of generalized eigenvalue problems of even structure, as well as their use in the construction of efficient numerical procedures for the robust control of continuous-time and discrete-time general descriptor systems (under inclusion of time retardations). more

### Periodic Control Systems: Analysis, Efficient Model Reduction and Development of Numerical Algorithms

Periodic systems arise in various scientific fields such as aerospace realm, wind turbines, pumps, fans, control of industrial processes and communication systems, modeling of periodic time-varying filters and networks, wave motion in periodic media such as ultra-cold atoms in an optical lattice, phenomenon of parametric resonance in forced oscillators and nonlinear systems linearized about a periodic trajectory. more

### Pendulum control

Multiple pendulum systems are illustrative but at the same time challenging academic examples for applied control theory. Using the Hasomed triple pendulum, which is installed in the Foyer of the MPI, different strategies for the stabilization and trajectory tracking of single, double, and triple pendulums are implemented and investigated.
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### Development of the Systems and Control Library SLICOT

in cooperation with the maintainer SynOptio GmbH, Matthias Voigt is contributing code enhancements and additions since February 2011.

## Concluded Projects

### Numerical Algorithms for Matrix Equations and Structured Eigenvalue Problems

Promotion of scientific exchange between scientists in Germany and scientists in Middle and Eastern Europe.
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