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

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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

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

References:

[1]	M. I. Ahmad and P. Benner, Interpolatory model reduction techniques for linear second-order descriptor systems, in Proc. European Control Conf. ECC 2014, Strasbourg, IEEE, 2014, 1075-1079.
[2]	L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation, Numer. Lin. Alg. Appl., 19 (2012), 700-727. doi: 10.1002/nla.799.
[3]	A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM Publications, Philadelphia, PA, 2005. doi: 10.1137/1.9780898718713.
[4]	A. C. Antoulas, D. C. Sorensen and Y. Zhou, On the decay rate of Hankel singular values and related issues, Systems Control Lett., 46 (2002), 323-342. doi: 10.1016/S0167-6911(02)00147-0.
[5]	U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611971392.
[6]	E. Bänsch, P. Benner, J. Saak and H. K. Weichelt, Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flows, SIAM J. Sci. Comput., 37 (2015), A832-A858. doi: 10.1137/140980016.
[7]	S. Barrachina, P. Benner and E. S. Quintana-Ortí, Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function, Numer. Algorithms, 46 (2007), 351-368. doi: 10.1007/s11075-007-9143-x.
[8]	P. Benner and T. Breiten, On optimality of approximate low rank solutions of large-scale matrix equations, Systems Control Lett., 67 (2014), 55-64. doi: 10.1016/j.sysconle.2014.02.005.
[9]	P. Benner, P. Kürschner and J. Saak, Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method, Numer. Algorithms, 62 (2013), 225-251. doi: 10.1007/s11075-012-9569-7.
[10]	P. Benner, P. Kürschner and J. Saak, An improved numerical method for balanced truncation for symmetric second order systems, Math. Comput. Model. Dyn. Syst., 19 (2013), 593-615. doi: 10.1080/13873954.2013.794363.
[11]	P. Benner, P. Kürschner and J. Saak, Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations, Electron. Trans. Numer. Anal., 43 (2014), 142-162.
[12]	P. Benner, J. R. Li and T. Penzl, Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems, Numer. Lin. Alg. Appl., 15 (2008), 755-777. doi: 10.1002/nla.622.
[13]	P. Benner, V. Mehrmann and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng., Springer-Verlag, Berlin/Heidelberg, Germany, 2005. doi: 10.1007/3-540-27909-1.
[14]	P. Benner and T. Stykel, Numerical solution of projected algebraic Riccati equations, SIAM J. Numer. Anal, 52 (2014), 581-600. doi: 10.1137/130923993.
[15]	K. A. Cliffe, T. J. Garratt and A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics, SIAM J. Matrix Anal. Appl., 15 (1994), 1310-1318. doi: 10.1137/S0895479892233230.
[16]	B. N. Datta, Numerical Methods for Linear Control Systems, Elsevier Academic Press, 2004.
[17]	E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics, European Consortium for Mathematics in Industry, B. G. Teubner GmbH, Stuttgart, 1998. doi: 10.1007/978-3-663-09828-7.
[18]	K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L^∞ norms, Internat. J. Control, 39 (1984), 1115-1193. doi: 10.1080/00207178408933239.
[19]	G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1983.
[20]	L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations, SIAM J. Matrix Anal. Appl., 29 (2007), 870-894. doi: 10.1137/040618102.
[21]	S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput., 35 (2013), B1010-B1033. doi: 10.1137/130906635.
[22]	M. Heinkenschloss, D. C. Sorensen and K. Sun, Balanced truncation model reduction for a class of descriptor systems with applications to the Oseen equations, SIAM J. Sci. Comput., 30 (2008), 1038-1063. doi: 10.1137/070681910.
[23]	P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation, 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.
[24]	P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, Textbooks in Mathematics, EMS Publishing House, Zürich, Switzerland, 2006. doi: 10.4171/017.
[25]	J. R. Li and J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24 (2002), 260-280. doi: 10.1137/S0895479801384937.
[26]	T. Penzl, A cyclic low rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21 (2000), 1401-1418. doi: 10.1137/S1064827598347666.
[27]	Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, UK, 1992.
[28]	W. H. A. Schilders, H. A. van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78841-6.
[29]	V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput., 29 (2007), 1268-1288. doi: 10.1137/06066120X.
[30]	E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.
[31]	T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation, Linear Algebra Appl., 415 (2006), 262-289. doi: 10.1016/j.laa.2004.01.015.
[32]	T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16 (2004), 297-319. doi: 10.1007/s00498-004-0141-4.
[33]	M. S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system, Internat. J. Control, 46 (1987), 1319-1330. doi: 10.1080/00207178708933971.
[34]	B. Vandereycken and S. Vandewalle, A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations, SIAM J. Matrix Anal. Appl., 31 (2010), 2553-2579. doi: 10.1137/090764566.
[35]	T. Wolf, H. K. F. Panzer and B. Lohmann, ADI iteration for Lyapunov equations: a tangential approach and adaptive shift selection, arXiv e-prints 1312.1142v1, Cornell University, 2013, http://arxiv.org/abs/1312.1142, Math. NA.
[36]	K. Zhou, G. Salomon and E. Wu, Balanced realization and model reduction for unstable systems, 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.

show all references

References:

[1]	M. I. Ahmad and P. Benner, Interpolatory model reduction techniques for linear second-order descriptor systems, in Proc. European Control Conf. ECC 2014, Strasbourg, IEEE, 2014, 1075-1079.
[2]	L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation, Numer. Lin. Alg. Appl., 19 (2012), 700-727. doi: 10.1002/nla.799.
[3]	A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM Publications, Philadelphia, PA, 2005. doi: 10.1137/1.9780898718713.
[4]	A. C. Antoulas, D. C. Sorensen and Y. Zhou, On the decay rate of Hankel singular values and related issues, Systems Control Lett., 46 (2002), 323-342. doi: 10.1016/S0167-6911(02)00147-0.
[5]	U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611971392.
[6]	E. Bänsch, P. Benner, J. Saak and H. K. Weichelt, Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flows, SIAM J. Sci. Comput., 37 (2015), A832-A858. doi: 10.1137/140980016.
[7]	S. Barrachina, P. Benner and E. S. Quintana-Ortí, Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function, Numer. Algorithms, 46 (2007), 351-368. doi: 10.1007/s11075-007-9143-x.
[8]	P. Benner and T. Breiten, On optimality of approximate low rank solutions of large-scale matrix equations, Systems Control Lett., 67 (2014), 55-64. doi: 10.1016/j.sysconle.2014.02.005.
[9]	P. Benner, P. Kürschner and J. Saak, Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method, Numer. Algorithms, 62 (2013), 225-251. doi: 10.1007/s11075-012-9569-7.
[10]	P. Benner, P. Kürschner and J. Saak, An improved numerical method for balanced truncation for symmetric second order systems, Math. Comput. Model. Dyn. Syst., 19 (2013), 593-615. doi: 10.1080/13873954.2013.794363.
[11]	P. Benner, P. Kürschner and J. Saak, Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations, Electron. Trans. Numer. Anal., 43 (2014), 142-162.
[12]	P. Benner, J. R. Li and T. Penzl, Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems, Numer. Lin. Alg. Appl., 15 (2008), 755-777. doi: 10.1002/nla.622.
[13]	P. Benner, V. Mehrmann and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng., Springer-Verlag, Berlin/Heidelberg, Germany, 2005. doi: 10.1007/3-540-27909-1.
[14]	P. Benner and T. Stykel, Numerical solution of projected algebraic Riccati equations, SIAM J. Numer. Anal, 52 (2014), 581-600. doi: 10.1137/130923993.
[15]	K. A. Cliffe, T. J. Garratt and A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics, SIAM J. Matrix Anal. Appl., 15 (1994), 1310-1318. doi: 10.1137/S0895479892233230.
[16]	B. N. Datta, Numerical Methods for Linear Control Systems, Elsevier Academic Press, 2004.
[17]	E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics, European Consortium for Mathematics in Industry, B. G. Teubner GmbH, Stuttgart, 1998. doi: 10.1007/978-3-663-09828-7.
[18]	K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L^∞ norms, Internat. J. Control, 39 (1984), 1115-1193. doi: 10.1080/00207178408933239.
[19]	G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1983.
[20]	L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations, SIAM J. Matrix Anal. Appl., 29 (2007), 870-894. doi: 10.1137/040618102.
[21]	S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput., 35 (2013), B1010-B1033. doi: 10.1137/130906635.
[22]	M. Heinkenschloss, D. C. Sorensen and K. Sun, Balanced truncation model reduction for a class of descriptor systems with applications to the Oseen equations, SIAM J. Sci. Comput., 30 (2008), 1038-1063. doi: 10.1137/070681910.
[23]	P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation, 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.
[24]	P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, Textbooks in Mathematics, EMS Publishing House, Zürich, Switzerland, 2006. doi: 10.4171/017.
[25]	J. R. Li and J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24 (2002), 260-280. doi: 10.1137/S0895479801384937.
[26]	T. Penzl, A cyclic low rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21 (2000), 1401-1418. doi: 10.1137/S1064827598347666.
[27]	Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, UK, 1992.
[28]	W. H. A. Schilders, H. A. van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78841-6.
[29]	V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput., 29 (2007), 1268-1288. doi: 10.1137/06066120X.
[30]	E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.
[31]	T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation, Linear Algebra Appl., 415 (2006), 262-289. doi: 10.1016/j.laa.2004.01.015.
[32]	T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16 (2004), 297-319. doi: 10.1007/s00498-004-0141-4.
[33]	M. S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system, Internat. J. Control, 46 (1987), 1319-1330. doi: 10.1080/00207178708933971.
[34]	B. Vandereycken and S. Vandewalle, A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations, SIAM J. Matrix Anal. Appl., 31 (2010), 2553-2579. doi: 10.1137/090764566.
[35]	T. Wolf, H. K. F. Panzer and B. Lohmann, ADI iteration for Lyapunov equations: a tangential approach and adaptive shift selection, arXiv e-prints 1312.1142v1, Cornell University, 2013, http://arxiv.org/abs/1312.1142, Math. NA.
[36]	K. Zhou, G. Salomon and E. Wu, Balanced realization and model reduction for unstable systems, 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.

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