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A POD projection method for large-scale algebraic Riccati equations
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 |
References:
[1] |
I. Akhtar, J. Borggaard, M. Stoyanov and L. Zietsman, On commutation of reduction and control: linear feedback control of a von Kármán street, in 5th AIAA Flow Control Conference, (2010), 1-14. |
[2] |
L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation, Numerical Linear Algebra with Applications, 19 (2012), 700-727.
doi: 10.1002/nla.799. |
[3] |
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005.
doi: 10.1137/1.9780898718713. |
[4] |
W. F. Arnold and A. J. Laub, Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proceedings of the IEEE, 72 (1984), 1746-1754. |
[5] |
J. A. Atwell, J. Borggaard and B. King, Reduced order controllers for Burgers' equation with a nonlinear observer, International Journal of Applied Mathematics and Computer Science, 11 (2001), 1311-1330. |
[6] |
J. A. Atwell and B. B. King, Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations, Math. Comput. Modelling, 33 (2001), 1-19.
doi: 10.1016/S0895-7177(00)00225-9. |
[7] |
J. Baker, M. Embree and J. Sabino, Fast singular value decay for Lyapunov solutions with nonnormal coefficients, SIAM J. Matrix Anal. Appl., 36 (2015), 656-668.
doi: 10.1137/140993867. |
[8] |
H. T. Banks, S. C. Beeler, G. M. Kepler and H. T. Tran, Reduced order modeling and control of thin film growth in an HPCVD reactor, SIAM J. Appl. Math., 62 (2002), 1251-1280.
doi: 10.1137/S0036139901383280. |
[9] |
H. T. Banks, R. C. H. del Rosario and R. C. Smith, Reduced-order model feedback control design: numerical implementation in a thin shell model, IEEE Trans. Automat. Control, 45 (2000), 1312-1324.
doi: 10.1109/9.867024. |
[10] |
H. T. Banks and K. Ito, A numerical algorithm for optimal feedback gains in high-dimensional linear quadratic regulator problems, SIAM J. Control Optim., 29 (1991), 499-515.
doi: 10.1137/0329029. |
[11] |
C. A. Beattie, J. Borggaard, S. Gugercin and T. Iliescu, A domain decomposition approach to POD, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006. |
[12] |
P. Benner, Balancing-related model reduction for parabolic control systems, in 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, (2014), 257-262. |
[13] |
P. Benner, J.-R. Li and T. Penzl, Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Algebra Appl., 15 (2008), 755-777.
doi: 10.1002/nla.622. |
[14] |
P. Benner and J. Saak, A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations, 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. |
[15] |
P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52.
doi: 10.1002/gamm.201310003. |
[16] |
P. Benner and Z. Bujanović, On the solution of large-scale algebraic Riccati equations by using low-dimensional invariant subspaces, Linear Algebra and its Applications, 488 (2016), 430-459.
doi: 10.1016/j.laa.2015.09.027. |
[17] |
P. Benner, J. Saak and M. M. Uddin, Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control, Numer. Algebra Control Optim., 6 (2016), 1-20.
doi: 10.3934/naco.2016.6.1. |
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J. Borggaard and M. Stoyanov, An efficient long-time integrator for Chandrasekhar equations, in Proceedings of the 47th IEEE Conference on Decision and Control, (2008), 3983-3988. |
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J. A. Burns and K. P. Hulsing, Numerical methods for approximating functional gains in LQR boundary control problems, Math. Comput. Modelling, 33 (2001), 89-100. Computation and Control, VI (Bozeman, MT, 1998).
doi: 10.1016/S0895-7177(00)00231-4. |
[21] |
J. A. Burns, E. W. Sachs and L. Zietsman, Mesh independence of Kleinman-Newton iterations for Riccati equations in Hilbert space, SIAM J. Control Optim., 47 (2008), 2663-2692.
doi: 10.1137/060653962. |
[22] |
J. Burns, K. Ito and R. Powers, Chandrasekhar equations and computational algorithms for distributed parameter systems, in Proceedings of the 23rd IEEE Conference on Decision and Control., 23 (1984), 262-267. |
[23] |
J. A. Burns, Introduction to theCalculus of Variations and Control-With Modern Applications, CRC Press, Boca Raton, FL, 2014. |
[24] |
D. H. Chambers, R. J. Adrian, P. Moin, D. S. Stewart and H. J. Sung, Karhunen-Love expansion of Burgers model of turbulence, Phys. Fluids, 31 (1988), 2573-2582. |
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S. Chaturantabut and D. C. Sorensen, Discrete empirical interpolation for nonlinear model reduction, in Proceedings of the 48th IEEE Conference on Decision and Control, IEEE, (2009), 4316-4321. |
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K. K. Chen and C. W. Rowley, Fluid flow control applications of H2 optimal actuator and sensor placement, in Proceedings of the American Control Conference, (2014), 4044-4049. |
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K. Chen and C. W. Rowley, H2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system, J. Fluid Mech., 681 (2011), 241-260.
doi: 10.1017/jfm.2011.195. |
[28] |
N. Darivandi, K. Morris and A. Khajepour, An algorithm for LQ optimal actuator location, Smart Materials and Structures, 22 (2013), 035001. |
[29] |
B. T. Dickinson and J. R. Singler, Nonlinear model reduction using group proper orthogonal decomposition, Int. J. Numer. Anal. Model., 7 (2010), 356-372. |
[30] |
V. Druskin and L. Knizhnerman, Extended Krylov subspaces: approximation of the matrix square root and related functions, SIAM Journal on Matrix Analysis and Applications, 19 (1998), 755-771.
doi: 10.1137/S0895479895292400. |
[31] |
M. Fahl, Computation of POD basis functions for fluid flows with Lanczos methods, Math. Comput. Modelling, 34 (2001), 91-107.
doi: 10.1016/S0895-7177(01)00051-6. |
[32] |
J. S. Gibson and A. Adamian, Approximation theory for linear-quadratic-Gaussian optimal control of flexible structures, SIAM J. Control Optim., 29 (1991), 1-37.
doi: 10.1137/0329001. |
[33] |
L. Giraud, J. Langou, M. Rozložník and J. van den Eshof, Rounding error analysis of the classical Gram-Schmidt orthogonalization process, Numer. Math., 101 (2005), 87-100.
doi: 10.1007/s00211-005-0615-4. |
[34] |
G. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University, ().
|
[35] |
A. Hay, J. Borggaard and D. Pelletier, Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech., 629 (2009), 41-72.
doi: 10.1017/S0022112009006363. |
[36] |
M. Heyouni and K. Jbilou, An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation, Electron. Trans. Numer. Anal., 33 (2009), 53-62. |
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P. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edition, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 2012.
doi: 10.1017/CBO9780511919701. |
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K. Jbilou, Block Krylov subspace methods for large algebraic Riccati equations, Numerical Algorithms, 34 (2003), 339-353.
doi: 10.1023/B:NUMA.0000005349.18793.28. |
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K. Jbilou, An Arnoldi based algorithm for large algebraic Riccati equations, Applied Mathematics Letters, 19 (2006), 437-444.
doi: 10.1016/j.aml.2005.07.001. |
[41] |
K. Jbilou and A. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra and its Applications, Special Issue on Order Reduction of Large-Scale Systems, 415 (2006), 344-358.
doi: 10.1016/j.laa.2004.11.004. |
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T. Kailath, Some Chandrasekhar-type algorithms for quadratic regulators, in Proceedings of the 1972 IEEE Conference on Decision and Control and 11th Symposium on Adaptive Processes., vol. 11, (1972), 219-223. |
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D. Kasinathan and K. Morris, H∞-optimal actuator location, IEEE Trans. Automat. Control, 58 (2013), 2522-2535.
doi: 10.1109/TAC.2013.2266870. |
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show all references
References:
[1] |
I. Akhtar, J. Borggaard, M. Stoyanov and L. Zietsman, On commutation of reduction and control: linear feedback control of a von Kármán street, in 5th AIAA Flow Control Conference, (2010), 1-14. |
[2] |
L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation, Numerical Linear Algebra with Applications, 19 (2012), 700-727.
doi: 10.1002/nla.799. |
[3] |
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005.
doi: 10.1137/1.9780898718713. |
[4] |
W. F. Arnold and A. J. Laub, Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proceedings of the IEEE, 72 (1984), 1746-1754. |
[5] |
J. A. Atwell, J. Borggaard and B. King, Reduced order controllers for Burgers' equation with a nonlinear observer, International Journal of Applied Mathematics and Computer Science, 11 (2001), 1311-1330. |
[6] |
J. A. Atwell and B. B. King, Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations, Math. Comput. Modelling, 33 (2001), 1-19.
doi: 10.1016/S0895-7177(00)00225-9. |
[7] |
J. Baker, M. Embree and J. Sabino, Fast singular value decay for Lyapunov solutions with nonnormal coefficients, SIAM J. Matrix Anal. Appl., 36 (2015), 656-668.
doi: 10.1137/140993867. |
[8] |
H. T. Banks, S. C. Beeler, G. M. Kepler and H. T. Tran, Reduced order modeling and control of thin film growth in an HPCVD reactor, SIAM J. Appl. Math., 62 (2002), 1251-1280.
doi: 10.1137/S0036139901383280. |
[9] |
H. T. Banks, R. C. H. del Rosario and R. C. Smith, Reduced-order model feedback control design: numerical implementation in a thin shell model, IEEE Trans. Automat. Control, 45 (2000), 1312-1324.
doi: 10.1109/9.867024. |
[10] |
H. T. Banks and K. Ito, A numerical algorithm for optimal feedback gains in high-dimensional linear quadratic regulator problems, SIAM J. Control Optim., 29 (1991), 499-515.
doi: 10.1137/0329029. |
[11] |
C. A. Beattie, J. Borggaard, S. Gugercin and T. Iliescu, A domain decomposition approach to POD, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006. |
[12] |
P. Benner, Balancing-related model reduction for parabolic control systems, in 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, (2014), 257-262. |
[13] |
P. Benner, J.-R. Li and T. Penzl, Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Algebra Appl., 15 (2008), 755-777.
doi: 10.1002/nla.622. |
[14] |
P. Benner and J. Saak, A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations, 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. |
[15] |
P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52.
doi: 10.1002/gamm.201310003. |
[16] |
P. Benner and Z. Bujanović, On the solution of large-scale algebraic Riccati equations by using low-dimensional invariant subspaces, Linear Algebra and its Applications, 488 (2016), 430-459.
doi: 10.1016/j.laa.2015.09.027. |
[17] |
P. Benner, J. Saak and M. M. Uddin, Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control, Numer. Algebra Control Optim., 6 (2016), 1-20.
doi: 10.3934/naco.2016.6.1. |
[18] |
D. Bini, B. Iannazzo and B. Meini, Numerical Solution of Algebraic Riccati Equations, Fundamentals of algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, 2012. |
[19] |
J. Borggaard and M. Stoyanov, An efficient long-time integrator for Chandrasekhar equations, in Proceedings of the 47th IEEE Conference on Decision and Control, (2008), 3983-3988. |
[20] |
J. A. Burns and K. P. Hulsing, Numerical methods for approximating functional gains in LQR boundary control problems, Math. Comput. Modelling, 33 (2001), 89-100. Computation and Control, VI (Bozeman, MT, 1998).
doi: 10.1016/S0895-7177(00)00231-4. |
[21] |
J. A. Burns, E. W. Sachs and L. Zietsman, Mesh independence of Kleinman-Newton iterations for Riccati equations in Hilbert space, SIAM J. Control Optim., 47 (2008), 2663-2692.
doi: 10.1137/060653962. |
[22] |
J. Burns, K. Ito and R. Powers, Chandrasekhar equations and computational algorithms for distributed parameter systems, in Proceedings of the 23rd IEEE Conference on Decision and Control., 23 (1984), 262-267. |
[23] |
J. A. Burns, Introduction to theCalculus of Variations and Control-With Modern Applications, CRC Press, Boca Raton, FL, 2014. |
[24] |
D. H. Chambers, R. J. Adrian, P. Moin, D. S. Stewart and H. J. Sung, Karhunen-Love expansion of Burgers model of turbulence, Phys. Fluids, 31 (1988), 2573-2582. |
[25] |
S. Chaturantabut and D. C. Sorensen, Discrete empirical interpolation for nonlinear model reduction, in Proceedings of the 48th IEEE Conference on Decision and Control, IEEE, (2009), 4316-4321. |
[26] |
K. K. Chen and C. W. Rowley, Fluid flow control applications of H2 optimal actuator and sensor placement, in Proceedings of the American Control Conference, (2014), 4044-4049. |
[27] |
K. Chen and C. W. Rowley, H2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system, J. Fluid Mech., 681 (2011), 241-260.
doi: 10.1017/jfm.2011.195. |
[28] |
N. Darivandi, K. Morris and A. Khajepour, An algorithm for LQ optimal actuator location, Smart Materials and Structures, 22 (2013), 035001. |
[29] |
B. T. Dickinson and J. R. Singler, Nonlinear model reduction using group proper orthogonal decomposition, Int. J. Numer. Anal. Model., 7 (2010), 356-372. |
[30] |
V. Druskin and L. Knizhnerman, Extended Krylov subspaces: approximation of the matrix square root and related functions, SIAM Journal on Matrix Analysis and Applications, 19 (1998), 755-771.
doi: 10.1137/S0895479895292400. |
[31] |
M. Fahl, Computation of POD basis functions for fluid flows with Lanczos methods, Math. Comput. Modelling, 34 (2001), 91-107.
doi: 10.1016/S0895-7177(01)00051-6. |
[32] |
J. S. Gibson and A. Adamian, Approximation theory for linear-quadratic-Gaussian optimal control of flexible structures, SIAM J. Control Optim., 29 (1991), 1-37.
doi: 10.1137/0329001. |
[33] |
L. Giraud, J. Langou, M. Rozložník and J. van den Eshof, Rounding error analysis of the classical Gram-Schmidt orthogonalization process, Numer. Math., 101 (2005), 87-100.
doi: 10.1007/s00211-005-0615-4. |
[34] |
G. Golub and C. F. Van Loan, Matrix Computations,, Johns Hopkins University, ().
|
[35] |
A. Hay, J. Borggaard and D. Pelletier, Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech., 629 (2009), 41-72.
doi: 10.1017/S0022112009006363. |
[36] |
M. Heyouni and K. Jbilou, An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation, Electron. Trans. Numer. Anal., 33 (2009), 53-62. |
[37] |
P. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edition, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 2012.
doi: 10.1017/CBO9780511919701. |
[38] |
I. Jaimoukha and E. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31 (1994), 227-251.
doi: 10.1137/0731012. |
[39] |
K. Jbilou, Block Krylov subspace methods for large algebraic Riccati equations, Numerical Algorithms, 34 (2003), 339-353.
doi: 10.1023/B:NUMA.0000005349.18793.28. |
[40] |
K. Jbilou, An Arnoldi based algorithm for large algebraic Riccati equations, Applied Mathematics Letters, 19 (2006), 437-444.
doi: 10.1016/j.aml.2005.07.001. |
[41] |
K. Jbilou and A. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra and its Applications, Special Issue on Order Reduction of Large-Scale Systems, 415 (2006), 344-358.
doi: 10.1016/j.laa.2004.11.004. |
[42] |
T. Kailath, Some Chandrasekhar-type algorithms for quadratic regulators, in Proceedings of the 1972 IEEE Conference on Decision and Control and 11th Symposium on Adaptive Processes., vol. 11, (1972), 219-223. |
[43] |
D. Kasinathan and K. Morris, H∞-optimal actuator location, IEEE Trans. Automat. Control, 58 (2013), 2522-2535.
doi: 10.1109/TAC.2013.2266870. |
[44] |
C. Kenney, A. Laub and M. Wette, Error bounds for Newton refinement of solutions to algebraic Riccati equations, Mathematics of Control, Signals and Systems, 3 (1990), 211-224.
doi: 10.1007/BF02551369. |
[45] |
B. Kramer, Solving algebraic Riccati equations via proper orthogonal decomposition, in Proceedings of the 19th IFAC World Congress, (2014), 7767-7772. |
[46] |
K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM Journal on Numerical analysis, 40 (2002), 492-515.
doi: 10.1137/S0036142900382612. |
[47] |
H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972. |
[48] |
C. H. Lee and H. T. Tran, Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation, J. Comput. Appl. Math., 173 (2005), 1-19.
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