American Institute of Mathematical Sciences

Advanced
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

Peter Benner 1, , Jens Saak 1, and  M. Monir Uddin 1,

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.
Keywords: differential algebraic system, Balanced truncation, unstable system, linearized flow model., differential index-2.
Mathematics Subject Classification: Primary: 93B40, 93B11; Secondary: 65F30, 65F5.
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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show all references

References:
[1]

in Proc. European Control Conf. ECC 2014, Strasbourg, IEEE, 2014, 1075-1079. Google Scholar

[2]

Numer. Lin. Alg. Appl., 19 (2012), 700-727. doi: 10.1002/nla.799.  Google Scholar

[3]

SIAM Publications, Philadelphia, PA, 2005. doi: 10.1137/1.9780898718713.  Google Scholar

[4]

Systems Control Lett., 46 (2002), 323-342. doi: 10.1016/S0167-6911(02)00147-0.  Google Scholar

[5]

SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611971392.  Google Scholar

[6]

SIAM J. Sci. Comput., 37 (2015), A832-A858. doi: 10.1137/140980016.  Google Scholar

[7]

Numer. Algorithms, 46 (2007), 351-368. doi: 10.1007/s11075-007-9143-x.  Google Scholar

[8]

Systems Control Lett., 67 (2014), 55-64. doi: 10.1016/j.sysconle.2014.02.005.  Google Scholar

[9]

Numer. Algorithms, 62 (2013), 225-251. doi: 10.1007/s11075-012-9569-7.  Google Scholar

[10]

Math. Comput. Model. Dyn. Syst., 19 (2013), 593-615. doi: 10.1080/13873954.2013.794363.  Google Scholar

[11]

Electron. Trans. Numer. Anal., 43 (2014), 142-162.  Google Scholar

[12]

Numer. Lin. Alg. Appl., 15 (2008), 755-777. doi: 10.1002/nla.622.  Google Scholar

[13]

Springer-Verlag, Berlin/Heidelberg, Germany, 2005. doi: 10.1007/3-540-27909-1.  Google Scholar

[14]

SIAM J. Numer. Anal, 52 (2014), 581-600. doi: 10.1137/130923993.  Google Scholar

[15]

SIAM J. Matrix Anal. Appl., 15 (1994), 1310-1318. doi: 10.1137/S0895479892233230.  Google Scholar

[16]

Elsevier Academic Press, 2004.  Google Scholar

[17]

European Consortium for Mathematics in Industry, B. G. Teubner GmbH, Stuttgart, 1998. doi: 10.1007/978-3-663-09828-7.  Google Scholar

[18]

Internat. J. Control, 39 (1984), 1115-1193. doi: 10.1080/00207178408933239.  Google Scholar

[19]

Johns Hopkins University Press, Baltimore, 1983.  Google Scholar

[20]

SIAM J. Matrix Anal. Appl., 29 (2007), 870-894. doi: 10.1137/040618102.  Google Scholar

[21]

SIAM J. Sci. Comput., 35 (2013), B1010-B1033. doi: 10.1137/130906635.  Google Scholar

[22]

SIAM J. Sci. Comput., 30 (2008), 1038-1063. doi: 10.1137/070681910.  Google Scholar

[23]

in Finite Element Methods in Flow Problems (eds. J. T. Oden, R. H. Gallagher, C. Taylor and O. C. Zienkiewicz), University of Alabama in Huntsville Press, 1974, 121-132. Google Scholar

[24]

Textbooks in Mathematics, EMS Publishing House, Zürich, Switzerland, 2006. doi: 10.4171/017.  Google Scholar

[25]

SIAM J. Matrix Anal. Appl., 24 (2002), 260-280. doi: 10.1137/S0895479801384937.  Google Scholar

[26]

SIAM J. Sci. Comput., 21 (2000), 1401-1418. doi: 10.1137/S1064827598347666.  Google Scholar

[27]

Manchester University Press, Manchester, UK, 1992.  Google Scholar

[28]

Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78841-6.  Google Scholar

[29]

SIAM J. Sci. Comput., 29 (2007), 1268-1288. doi: 10.1137/06066120X.  Google Scholar

[30]

2nd edition, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[31]

Linear Algebra Appl., 415 (2006), 262-289. doi: 10.1016/j.laa.2004.01.015.  Google Scholar

[32]

Math. Control Signals Systems, 16 (2004), 297-319. doi: 10.1007/s00498-004-0141-4.  Google Scholar

[33]

Internat. J. Control, 46 (1987), 1319-1330. doi: 10.1080/00207178708933971.  Google Scholar

[34]

SIAM J. Matrix Anal. Appl., 31 (2010), 2553-2579. doi: 10.1137/090764566.  Google Scholar

[35]

arXiv e-prints 1312.1142v1, Cornell University, 2013, http://arxiv.org/abs/1312.1142, Math. NA. Google Scholar

[36]

Internat. J. Robust and Nonlinear Cont., 9 (1999), 183-198. doi: 10.1002/(SICI)1099-1239(199903)9:3<121::AID-RNC395>3.0.CO;2-1.  Google Scholar

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