# MOR by Rewriting to (Quadratic-)Bilinear System Order Reduction

### Researcher:

- Tobias Breiten

Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg,

Computational Methods in Systems and Control Theory,

Tel: +49 (0)391-6110-806

E-mail: breiten@mpi-magdeburg.mpg.de

### Project Description

An important tool in the analysis of complex physical phenomena is the simulation of the underlying mathematical models which are often given by systems of ordinary and/or partial differential equations. As one is interested in models as accurate as possible, linear models are often insufficient such that one is faced with large-scale nonlinear systems. Frequently, these cannot be handled efficiently, necessitating model order reduction, i.e., the construction of a smaller system approximating the original one.While most existing nonlinear reduction methods like proper orthogonal decomposition (POD) and trajectory piecewise linear (TPWL) require specific training inputs and thus are input dependent, here we study Krylov-based reduction techniques for so-called quadratic-bilinear systems which overcome this drawback.

The advantage of quadratic-bilinear systems is that they allow the exact representation of a large class of nonlinear systems. Moreover, the quadratic-bilinearization is often achieved with an acceptable increase of the state dimension. The initial idea using quadratic-bilinear systems for nonlinear model reduction emerged quite recently and can be found in [1].

### Duration and Funding

- since September 2010: MPI Magdeburg

### Related Publications

158 | Krylov-Subspace Based Model Reduction of Nonlinear Circuit Models Using Bilinear and Quadratic-Linear Approximations; Benner, Peter; Breiten, Tobias; in M. Günther, A. Bartel, M. Brunk, S. Schöps, M. Striebel: Progress in Industrial Mathematics at ECMI 2010 : Mathematics in Industry, Vol. 17, pp. 153-159;Springer-Verlag, Berlin; 2012. ISBN/ISSN: 978-3-642-25099-6 |

### References

# Dynamic, Analysis and Control of Anaerobic Digestions Processes for Biogas Production

### Researcher:

- Astrid Bornhöft

Otto-von-Guericke University Magdeburg,

Chair Process Systems Engineering,

Tel: +49 (0)391 67-54637

E-mail: abornhoeft@mpi-magdeburg.mpg.de

- Tobias Breiten

Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg,

Computational Methods in Systems and Control Theory,

Tel: +49 (0)391-6110-806

E-mail: breiten@mpi-magdeburg.mpg.de

- Richard Hanke-Rauschenbach

Otto-von-Guericke University Magdeburg,

Chair Process Systems Engineering,

Tel: +49 (0)391 67-54630

E-mail: hanke-rauschenbach@mpi-magdeburg.mpg.de

### Project Description

In cooperation with the group of Prof. Dr. Kai Sundmacher, we investigate the above mentioned nonlinear model reduction technique by means of a real-life application arising in digstion processes in the context of biogas production. Before applying model order reduction, as a first step the nonlinear dynamics have to be transformed into a qudratic-bilinear control systems with additional algebraic constraints that make the reduction itself more complicated. Subsequently, we want to apply the model reduction approach and give an interpretation of the reduced system.### Duration and Funding

- since September 2010: MPI Magdeburg

### Related Publications

Steady-state analysis of the anaerobic digestion model No. 1 (ADM1); A. Bornhöft, R. Hanke-Rauschenbach, and K. Sundmacher; submitted to Journal of Biotechnology; 2011. |