American Institute of Mathematical Sciences

Advanced
2016, 6(4): 413-435. doi: 10.3934/naco.2016018

A POD projection method for large-scale algebraic Riccati equations

Boris Kramer 1, and  John R. Singler 2,

1. 

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,Cambridge, MA 02139, United States
2. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, United States

Received  February 2016 Revised  September 2016 Published  December 2016

The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also briefly discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.
Keywords: Algebraic Riccati equations, large-scale, proper orthogonal decomposition, control theory., reduced-order modeling.
Mathematics Subject Classification: Primary: 49; Secondary: 6.
Citation: Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
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[2]

Numerical Linear Algebra with Applications, 19 (2012), 700-727. doi: 10.1002/nla.799.  Google Scholar

[3]

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[4]

Proceedings of the IEEE, 72 (1984), 1746-1754. Google Scholar

[5]

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[6]

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[7]

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[8]

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[9]

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[10]

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[11]

in Proceedings of the 45th IEEE Conference on Decision and Control, 2006. Google Scholar

[12]

in 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, (2014), 257-262. Google Scholar

[13]

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

[14]

Preprint SPP1253-090, DFG Priority Programme 1253 Optimization with Partial Differential Equations, 2010, http://www.am.uni-erlangen.de/home/spp1253/wiki/images/2/28/Preprint-SPP1253-090.pdf. Google Scholar

[15]

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[16]

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[17]

Numer. Algebra Control Optim., 6 (2016), 1-20. doi: 10.3934/naco.2016.6.1.  Google Scholar

[18]

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[22]

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[23]

CRC Press, Boca Raton, FL, 2014.  Google Scholar

[24]

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[25]

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SIAM Journal on Matrix Analysis and Applications, 19 (1998), 755-771. doi: 10.1137/S0895479895292400.  Google Scholar

[31]

Math. Comput. Modelling, 34 (2001), 91-107. doi: 10.1016/S0895-7177(01)00051-6.  Google Scholar

[32]

SIAM J. Control Optim., 29 (1991), 1-37. doi: 10.1137/0329001.  Google Scholar

[33]

Numer. Math., 101 (2005), 87-100. doi: 10.1007/s00211-005-0615-4.  Google Scholar

[34]

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

[35]

J. Fluid Mech., 629 (2009), 41-72. doi: 10.1017/S0022112009006363.  Google Scholar

[36]

Electron. Trans. Numer. Anal., 33 (2009), 53-62.  Google Scholar

[37]

2nd edition, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9780511919701.  Google Scholar

[38]

SIAM J. Numer. Anal., 31 (1994), 227-251. doi: 10.1137/0731012.  Google Scholar

[39]

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Applied Mathematics Letters, 19 (2006), 437-444. doi: 10.1016/j.aml.2005.07.001.  Google Scholar

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in Proceedings of the 1972 IEEE Conference on Decision and Control and 11th Symposium on Adaptive Processes., vol. 11, (1972), 219-223. Google Scholar

[43]

IEEE Trans. Automat. Control, 58 (2013), 2522-2535. doi: 10.1109/TAC.2013.2266870.  Google Scholar

[44]

Mathematics of Control, Signals and Systems, 3 (1990), 211-224. doi: 10.1007/BF02551369.  Google Scholar

[45]

in Proceedings of the 19th IFAC World Congress, (2014), 7767-7772. Google Scholar

[46]

SIAM Journal on Numerical analysis, 40 (2002), 492-515. doi: 10.1137/S0036142900382612.  Google Scholar

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[51]

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[53]

International Journal for Numerical Methods in Engineering, 2016. Google Scholar

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[62]

Numer. Math., 121 (2012), 127-164. doi: 10.1007/s00211-011-0424-x.  Google Scholar

[63]

Comp. Opt. and Appl., 53 (2012), 227-248. doi: 10.1007/s10589-011-9451-x.  Google Scholar

[64]

IMA Journal of Numerical Analysis, 31 (2011), 1468-1496. doi: 10.1093/imanum/drq028.  Google Scholar

[65]

Quart. Appl. Math., 45 (1987), 561-571.  Google Scholar

[66]

Numerische Mathematik, 76 (1997), 249-263. doi: 10.1007/s002110050262.  Google Scholar

[67]

Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[68]

in Proc. 24th Midwest Symp. Circ. Syst., Albuquerque, NM, USA, (1981), 365-369. Google Scholar

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Lecture Notes, University of Konstanz, 2013, http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/scripts.php. Google Scholar

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Lecture Notes, University of Konstanz, 2013, http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/scripts.php. Google Scholar

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PhD thesis, Virginia Polytechnic Institute and State University, 2004, https://vtechworks.lib.vt.edu/handle/10919/27404.  Google Scholar

[72]

Numer. Algebra Control Optim., 3 (2013), 491-518. doi: 10.3934/naco.2013.3.491.  Google Scholar

[73]

J. Comput. Appl. Math., 260 (2014), 364-374. doi: 10.1016/j.cam.2013.09.074.  Google Scholar

[74]

AIAA Journal, 40 (2015), 2323-2330. Google Scholar

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