# American Institute of Mathematical Sciences

2016, 6(1): 1-20. doi: 10.3934/naco.2016.6.1

## Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control

 1 Max Planck Institute for Dynamics of Complex Technical Systems, Computational Methods in Systems and Control Theory, Magdeburg, Germany, Germany, Germany

Received  November 2014 Revised  January 2016 Published  January 2016

Stabilizing a flow around an unstable equilibrium is a typical problem in flow control. Model-based designed of modern controllers like LQR/LQG or $H_\infty$ compensators is often limited by the large-scale of the discretized flow models. Therefore, model reduction is usually needed before designing such a controller. Here we suggest an approach based on applying balanced truncation for unstable systems to the linearized flow equations usually used for compensator design. For this purpose, we modify the ADI iteration for Lyapunov equations to deal with the index-2 structure of the underlying descriptor system efficiently in an implicit way. The resulting algorithm is tested for model reduction and control design of a linearized Navier-Stokes system describing von Kármán vortex shedding.
Citation: Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
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