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
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, (2014), 1075. Google Scholar

[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. doi: 10.1002/nla.799. Google Scholar

[3]

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems,, SIAM Publications, (2005). doi: 10.1137/1.9780898718713. Google Scholar

[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. doi: 10.1016/S0167-6911(02)00147-0. Google Scholar

[5]

U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations,, SIAM, (1998). doi: 10.1137/1.9781611971392. Google Scholar

[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). doi: 10.1137/140980016. Google Scholar

[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. doi: 10.1007/s11075-007-9143-x. Google Scholar

[8]

P. Benner and T. Breiten, On optimality of approximate low rank solutions of large-scale matrix equations,, Systems Control Lett., 67 (2014), 55. doi: 10.1016/j.sysconle.2014.02.005. Google Scholar

[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. doi: 10.1007/s11075-012-9569-7. Google Scholar

[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. doi: 10.1080/13873954.2013.794363. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/nla.622. Google Scholar

[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, (2005). doi: 10.1007/3-540-27909-1. Google Scholar

[14]

P. Benner and T. Stykel, Numerical solution of projected algebraic Riccati equations,, SIAM J. Numer. Anal, 52 (2014), 581. doi: 10.1137/130923993. Google Scholar

[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. doi: 10.1137/S0895479892233230. Google Scholar

[16]

B. N. Datta, Numerical Methods for Linear Control Systems,, Elsevier Academic Press, (2004). Google Scholar

[17]

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics,, European Consortium for Mathematics in Industry, (1998). doi: 10.1007/978-3-663-09828-7. Google Scholar

[18]

K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞ norms,, Internat. J. Control, 39 (1984), 1115. doi: 10.1080/00207178408933239. Google Scholar

[19]

G. H. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University Press, (1983). Google Scholar

[20]

L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations,, SIAM J. Matrix Anal. Appl., 29 (2007), 870. doi: 10.1137/040618102. Google Scholar

[21]

S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/130906635. Google Scholar

[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. doi: 10.1137/070681910. Google Scholar

[23]

P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation,, in Finite Element Methods in Flow Problems (eds. J. T. Oden, (1974), 121. Google Scholar

[24]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution,, Textbooks in Mathematics, (2006). doi: 10.4171/017. Google Scholar

[25]

J. R. Li and J. White, Low rank solution of Lyapunov equations,, SIAM J. Matrix Anal. Appl., 24 (2002), 260. doi: 10.1137/S0895479801384937. Google Scholar

[26]

T. Penzl, A cyclic low rank Smith method for large sparse Lyapunov equations,, SIAM J. Sci. Comput., 21 (2000), 1401. doi: 10.1137/S1064827598347666. Google Scholar

[27]

Y. Saad, Numerical Methods for Large Eigenvalue Problems,, Manchester University Press, (1992). Google Scholar

[28]

W. H. A. Schilders, H. A. van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-78841-6. Google Scholar

[29]

V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations,, SIAM J. Sci. Comput., 29 (2007), 1268. doi: 10.1137/06066120X. Google Scholar

[30]

E. D. Sontag, Mathematical Control Theory,, 2nd edition, (1998). doi: 10.1007/978-1-4612-0577-7. Google Scholar

[31]

T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation,, Linear Algebra Appl., 415 (2006), 262. doi: 10.1016/j.laa.2004.01.015. Google Scholar

[32]

T. Stykel, Gramian-based model reduction for descriptor systems,, Math. Control Signals Systems, 16 (2004), 297. doi: 10.1007/s00498-004-0141-4. Google Scholar

[33]

M. S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system,, Internat. J. Control, 46 (1987), 1319. doi: 10.1080/00207178708933971. Google Scholar

[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. doi: 10.1137/090764566. Google Scholar

[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, (1312). Google Scholar

[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. doi: 10.1002/(SICI)1099-1239(199903)9:3<121::AID-RNC395>3.0.CO;2-1. Google Scholar

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, (2014), 1075. Google Scholar

[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. doi: 10.1002/nla.799. Google Scholar

[3]

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems,, SIAM Publications, (2005). doi: 10.1137/1.9780898718713. Google Scholar

[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. doi: 10.1016/S0167-6911(02)00147-0. Google Scholar

[5]

U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations,, SIAM, (1998). doi: 10.1137/1.9781611971392. Google Scholar

[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). doi: 10.1137/140980016. Google Scholar

[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. doi: 10.1007/s11075-007-9143-x. Google Scholar

[8]

P. Benner and T. Breiten, On optimality of approximate low rank solutions of large-scale matrix equations,, Systems Control Lett., 67 (2014), 55. doi: 10.1016/j.sysconle.2014.02.005. Google Scholar

[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. doi: 10.1007/s11075-012-9569-7. Google Scholar

[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. doi: 10.1080/13873954.2013.794363. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/nla.622. Google Scholar

[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, (2005). doi: 10.1007/3-540-27909-1. Google Scholar

[14]

P. Benner and T. Stykel, Numerical solution of projected algebraic Riccati equations,, SIAM J. Numer. Anal, 52 (2014), 581. doi: 10.1137/130923993. Google Scholar

[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. doi: 10.1137/S0895479892233230. Google Scholar

[16]

B. N. Datta, Numerical Methods for Linear Control Systems,, Elsevier Academic Press, (2004). Google Scholar

[17]

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics,, European Consortium for Mathematics in Industry, (1998). doi: 10.1007/978-3-663-09828-7. Google Scholar

[18]

K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L∞ norms,, Internat. J. Control, 39 (1984), 1115. doi: 10.1080/00207178408933239. Google Scholar

[19]

G. H. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University Press, (1983). Google Scholar

[20]

L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations,, SIAM J. Matrix Anal. Appl., 29 (2007), 870. doi: 10.1137/040618102. Google Scholar

[21]

S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/130906635. Google Scholar

[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. doi: 10.1137/070681910. Google Scholar

[23]

P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation,, in Finite Element Methods in Flow Problems (eds. J. T. Oden, (1974), 121. Google Scholar

[24]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution,, Textbooks in Mathematics, (2006). doi: 10.4171/017. Google Scholar

[25]

J. R. Li and J. White, Low rank solution of Lyapunov equations,, SIAM J. Matrix Anal. Appl., 24 (2002), 260. doi: 10.1137/S0895479801384937. Google Scholar

[26]

T. Penzl, A cyclic low rank Smith method for large sparse Lyapunov equations,, SIAM J. Sci. Comput., 21 (2000), 1401. doi: 10.1137/S1064827598347666. Google Scholar

[27]

Y. Saad, Numerical Methods for Large Eigenvalue Problems,, Manchester University Press, (1992). Google Scholar

[28]

W. H. A. Schilders, H. A. van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-78841-6. Google Scholar

[29]

V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations,, SIAM J. Sci. Comput., 29 (2007), 1268. doi: 10.1137/06066120X. Google Scholar

[30]

E. D. Sontag, Mathematical Control Theory,, 2nd edition, (1998). doi: 10.1007/978-1-4612-0577-7. Google Scholar

[31]

T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation,, Linear Algebra Appl., 415 (2006), 262. doi: 10.1016/j.laa.2004.01.015. Google Scholar

[32]

T. Stykel, Gramian-based model reduction for descriptor systems,, Math. Control Signals Systems, 16 (2004), 297. doi: 10.1007/s00498-004-0141-4. Google Scholar

[33]

M. S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system,, Internat. J. Control, 46 (1987), 1319. doi: 10.1080/00207178708933971. Google Scholar

[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. doi: 10.1137/090764566. Google Scholar

[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, (1312). Google Scholar

[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. doi: 10.1002/(SICI)1099-1239(199903)9:3<121::AID-RNC395>3.0.CO;2-1. Google Scholar

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