H₂ optimal model order reduction on the Stiefel manifold for the MIMO discrete system by the cross Gramian

References

• Y.L. Jiang, Model Order Reduction Methods, Science Press, Beijing, 2010. ISBN 9787030274373.
   Google Scholar
• Z.Z. Qi, Y.L. Jiang, and Z.H. Xiao, Structure-preserving model order reduction based on laguerre-SVD for coupled systems, Math. Comput.Model, Dyn. Syst. 21 (6) (2015), pp. 573–590. doi:10.1080/13873954.2015.1065279.
   Web of Science ®Google Scholar
• T. Ishizaki and K. Kashima, Model reduction and clusterization of large-scale bidirectional network, IEEE Trans. Autom. Contr. 59 (1) (2014), pp. 48–63. doi:10.1109/TAC.2013.2275891.
   Web of Science ®Google Scholar
• X.L. Wang and Y.L. Jiang, Two-sided projection methods for model reduction of MIMO bilinear systems, Math. Comput. Model, Dyn. Syst. 19 (6) (2013), pp. 575–592. doi:10.1080/13873954.2013.805145.
   Web of Science ®Google Scholar
• K. Milner, T.W. Becker, and L. Boschi, New software framework to share research tools, EOS. Trans. AGU. 90 (12) (2013), pp. 104. doi:10.1029/2009EO120005.
   Google Scholar
• B.F. Rarrell and P.J. Ioannou, Sate estimation using a reduced order kalman filter, J. Atmos. Sci. 58 (23) (2001), pp. 3666–3680. doi:10.1175/1520-0469(2001)058<3666:SEUARO>2.0.CO;2.
   Web of Science ®Google Scholar
• Z.H. Xiao and Y.L. Jiang, Dimension reduction for second-order systems by general orthogonal polynomials, Math. Comput. Model, Dyn. Syst. 20 (4) (2014), pp. 414–432. doi:10.1080/13873954.2013.867274.
   Web of Science ®Google Scholar
• Y.L. Jiang and H.B. Chen, Time domain model order reduction of general orthogonal polynomials for linear input-output systems, IEEE Trans. Autom. Contr. 55 (99) (2012), pp. 330–343. doi:10.1109/TAC.2011.2161839.
   Google Scholar
• J.W. Lee and G.E. Dullerud, Optimal disturbance attenuation for discrete-time switched and Markovian jump linear systems, SIAM J. Control Optim. 45 (4) (2006), pp. 1329–1358. doi:10.1137/050627538.
   Web of Science ®Google Scholar
• J.W. Lee and P.P. Khargonekar, Optimal output regulation for discrete-time switched and Markovian jump linear systems, SIAM J. Control Optim. 47 (1) (2008), pp. 40–72. doi:10.1137/060662290.
   Web of Science ®Google Scholar
• A.C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005.
   Google Scholar
• L.Q. Zhang, L. James, B. Huang, and G.H. Yang, On gramians and balanced truncation of discrete time bilinear systems, Int. J. Contr 76 (6) (2003), pp. 414–427. doi:10.1080/0020717031000082540.
   Google Scholar
• R.W. Freund, Krylov-subspace methods for reduced-order modeling in circuit simulation, J. Comput. Appl. Math. 123 (1) (2000), pp. 395–421. doi:10.1016/S0377-0427(00)00396-4.
   Web of Science ®Google Scholar
• K. Gallivan and P.V. Dooren, A rational Lanczos algorithm for model reduction, Numer. Algorithms 12 (1) (1996), pp. 33–63. doi:10.1007/BF02141740.
   Web of Science ®Google Scholar
• X.L. Wang and Y.L. Jiang, Model reduction of discrete-time bilinear systems by a Laguerre expansion technique, Appl. Math. Model. 40 (14) (2016), pp. 6650–6662. doi:10.1016/j.apm.2016.02.015.
   Web of Science ®Google Scholar
• S. Rahrovani, M.K. Vakilzadeh, and T. Abrahamsson, On Gramian-Based Techniques for Minimal Realization of Large-Scale Mechanical Systems, Proc. 31st IMAC, Conf. Structural Dyn. 7 (2013), pp. 797–805.
   Google Scholar
• U. Al-Saggaf and G. Franklin, An error bound for a discrete reduced order model of a linear multivariable system, IEEE Trans. Autom. Contr. 32 (9) (2003), pp. 815–819. doi:10.1109/TAC.1987.1104712.
   Google Scholar
• P. Kerfriden, O. Goury, T. Rabczuk, et. al, A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics, Comput. Methods Appl. Mech. Eng. 256 (5) (2013), pp. 169–188. doi:10.1016/j.cma.2012.12.004.
   PubMedGoogle Scholar
• D.A. Wilson, Optimum solution of model reduction problem, Proc. IEEE 117 (1970), pp. 1161–1165.
   Web of Science ®Google Scholar
• S. Gugercin, A.C. Antoulas, and C. Beattie, H2 model reduction for large-scale linear dynamical systems, SIAM Matrix Anal. Appl. 30 (2) (2008), pp. 609–638. doi:10.1137/060666123.
   Web of Science ®Google Scholar
• P.V. Dooren, K.A. Gallivanb, and P.A. Absil, H2 optimal model reduction of MIMO systems, Appl. Math. Lett. 21 (12) (2008), pp. 1267–1273. doi:10.1016/j.aml.2007.09.015.
   Web of Science ®Google Scholar
• A. Bunse-Gerstner, D. Kubalinska, G. Vossen, and D. Wilczek, H2-norm optimal model reduction for large scale discrete dynamical MIMO systems, J. Comput. Appl. Math. 233 (5) (2010), pp. 1202–1216. doi:10.1016/j.cam.2008.12.029.
   Web of Science ®Google Scholar
• W.Y. Yan and J. Lam, An approximate approach to H2 optimal model reduction, IEEE Trans. Autom. Contr. 40 (1999), pp. 1515–1521.
   Google Scholar
• H. Sato and K. Sato, Riemannian trust-region methods for H2 optimal model reduction, IEEE 54th Annual CDC. (2015), pp. 4648–4655.
   Google Scholar
• Y.L. Jiang and K.,.L. Xu, H2 optimal reduced models of general MIMO LTI systems via the cross Gramian on the Stiefel manifold, J. Frankl. Inst. 354 (8) (2017), pp. 3210–3224. doi:10.1016/j.jfranklin.2017.02.019.
   Web of Science ®Google Scholar
• C. Himpe and M. Ohlberger, A note on the cross Gramian for non-symmetric systems, Systems Sci. Control Engineer. Ens. 4 (1) (2015), pp. 199–208. doi:10.1080/21642583.2016.1215273.
   Web of Science ®Google Scholar
• K.V. Fernando and H. Nicholson, Minimality of SISO linear systems, Proc. IEEE 70 (10) (1982), pp. 1241–1242. doi:10.1109/PROC.1982.12460.
   Web of Science ®Google Scholar
• R.W. Aldhaheri and A.H. Bamani, Balanced realization of non-minimal MIMO discrete-time systems, Int. J. Syst. Sci. 25 (1) (1994), pp. 173–182. doi:10.1080/00207729408928951.
   Web of Science ®Google Scholar
• Y. Chahlaoui, A posteriori error bounds for discrete balanced truncation, Linear A Lgebra Appl. 436 (8) (2012), pp. 2744–2763. doi:10.1016/j.laa.2011.07.025.
   Web of Science ®Google Scholar
• A. Edelman, T.A. Arias, and S.T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20 (2) (1999), pp. 303–353. doi:10.1137/S0895479895290954.
   Web of Science ®Google Scholar
• W. Huang and P.A. Absil, A Riemannian symmetric rank-one trust-region method, Math. Program. 150 (2) (2015), pp. 179–216. doi:10.1007/s10107-014-0765-1.
   Web of Science ®Google Scholar
• P.A. Absil and C.G. Baker, Trust-region methods on Riemannian manifolds, Found. Comput. Math 7 (3) (2007), pp. 303–330. doi:10.1007/s10208-005-0179-9.
   Web of Science ®Google Scholar
• P.A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Prinection, 2008.
   Google Scholar
• X. Zhu, A Riemannian conjugate gradient method for optimization on the Stiefel manifold, Comput. Optim. Appl. 67 (1) (2017), pp. 73–110. doi:10.1007/s10589-016-9883-4.
   Web of Science ®Google Scholar
• H. Sato, A Dai-Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions, Comput. Optim. Appl. 64 (1) (2016), pp. 101–118. doi:10.1007/s10589-015-9801-1.
   Web of Science ®Google Scholar
• Y. Chahlaoui and P.V. Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, SLICOT Working Note 2002-2, 2002 .
   Google Scholar