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A trust-region method for optimal \(H_2\) model reduction of discrete-time dynamical systems. (English) Zbl 1405.93055

Summary: In this paper, we present an optimal \(H_2\) model reduction method on the basis of the trust-region technique for linear time-invariant discrete-time dynamical systems. First, based on the poles and residues, the \(H_2\) error norm for single-input and single-output discrete-time systems is investigated, which leads to the \(H_2\) error gradient and Hessian. Next, for multiple-input and multiple-out discrete-time systems, the gradient and Hessian of the \(H_2\) error norm are accordingly derived. Then, the \(H_2\) error gradient and Hessian are employed to establish the trust-region method for optimal \(H_2\) model reduction. Moreover, it is shown that the proposed method can produce a decreasing sequence. The construction of the state space realization of the reduced order system is studied concerning the divisions of the resulting residues. Model reduction is investigated for nonlinear discrete-time system. Finally, an illustrative example is used to demonstrate the performance.

MSC:

93B11 System structure simplification
93B60 Eigenvalue problems
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
93C10 Nonlinear systems in control theory
93C35 Multivariable systems, multidimensional control systems
93-04 Software, source code, etc. for problems pertaining to systems and control theory
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References:

[1] Trust-region proper orthogonal decomposition for flow control, Technical Report 2000-25, ICASE, 2000
[2] Bai, Z. J.; Skoogh, D., A projection method for model reduction of bilinear dynamical system, Linear Algebra Appl., 415, 406-425, (2006) · Zbl 1107.93012
[3] A trust region method for optimal \(##?##\) model reduction, Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, 2009, pp. 5370–5375
[4] Benner, P.; Breiten, T., interpolation-based \(##?##\) model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl., 33, 859-885, (2012) · Zbl 1256.93027
[5] Benner, P.; Breiten, T.; Damm, T., Krylov subspace methods for model order reduction of bilinear discrete-time control systems, Proc. Appl. Math. Mech., 10, 601-602, (2010)
[6] Bunse-Gerstner, A.; Kubalinska, D.; Vossen, G.; Wilczek, D., \(##?##\)-norm optimal model reduction for large-scale discrete dynamical MIMO systems, J. Comput. Appl. Math., 233, 1202-1216, (2010) · Zbl 1178.93032
[7] Chen, X.; Akella, S.; Navon, I. M., A dual-weighted trust-region adaptive POD 4-D var applied to a finite-volume shallow water equations model on the sphere, Int. J. Numer. Meth. Fluids., 68, 377-402, (2012) · Zbl 1426.76347
[8] Chen, L. J.; Narendra, K. S., identification and control of a nonlinear discrete-time system based on its linearization: a unified framework, IEEE Trans. Neural Networks, 15, 663-673, (2004)
[9] Chen, X.; Navon, I. M.; Fang, F., A dual-weighted trust-region adaptive POD 4D-VAR applied to a finite-element shallow-water equations model, Int. J. Numer. Meth. Fluids., 65, 520-541, (2011) · Zbl 1428.76145
[10] Chu, C. C.; Lai, M. H.; Feng, W. S., model-order reductions for MIMO systems using global Krylov subspace methods, Math. Comput. Simul., 79, 1153-1164, (2008) · Zbl 1165.65011
[11] Conway, J. B., Functions of One Complex Variable, (1978), Springer-Verlag, New York
[12] Du, J.; Navon, I. M.; Zhu, J.; Fang, F. X.; Alekseev, A. K., reduced order modeling based on POD of a parabolized Navier–Stokes equations model II: trust region POD 4D VAR data assimilation, Comput. Math. Appl., 65, 380-394, (2013) · Zbl 1319.76030
[13] Ebihara, Y.; Hagiwara, T., on \(##?##\) model reduction using LMIs, IEEE Trans. Automat. Contr., 49, 1187-1191, (2004) · Zbl 1365.93132
[14] Gugercin, S.; Antoulas, A. C.; Beattie, C., \(##?##\) model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl., 30, 609-638, (2008) · Zbl 1159.93318
[15] Hyland, D. C.; Bernstein, D. S., the optimal projection equations for model reduction and relationship among the methods of Wilson, skelton and Moore, IEEE Trans. Automat. Contr., 30, 609-638, (1985) · Zbl 0583.93004
[16] Jiang, Y. L., Model Order Reduction Methods, (2010), Science Press, Beijing
[17] Jiang, Y. L.; Chen, H. B., time domain model order reduction of general orthogonal polynomials for linear input–output systems, IEEE Trans. Automat. Contr., 57, 330-343, (2012) · Zbl 1369.93117
[18] Knockaert, L.; De Zutter, D., Laguerre-SVD reduced-order modeling, IEEE Trans. Microw. Theory Tech., 48, 1469-1475, (2000)
[19] Moore, B. C., principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automat. Contr., 26, 17-31, (1981) · Zbl 0464.93022
[20] Nocedal, J.; Wright, S. J., Numerical Optimization, (2006), Springer-Verlag, New York · Zbl 1104.65059
[21] Pernebo, L.; Silverman, L. M., model reduction via balanced state space representations, IEEE Trans. Automat. Contr., 27, 382-387, (1982) · Zbl 0482.93024
[22] Steihaug, T., the conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20, 626-637, (1983) · Zbl 0518.65042
[23] Wang, X. L.; Jiang, Y. L., model reduction of bilinear systems based on Laguerre series expansion, J. Franklin Inst., 349, 1231-1246, (2012) · Zbl 1273.93034
[24] Wang, X. L.; Jiang, Y. L., model reduction of discrete-time bilinear systems by a Laguerre expansion technique, Appl. Math. Model., 40, 6650-6662, (2016)
[25] Wilson, D., optimum solution of model-reduction problem, Proc. Inst. Elec. Eng., 117, 1161-1165, (1970)
[26] Xiao, Z. H.; Jiang, Y. L., model order reduction of MIMO bilinear systems by multi-order Arnoldi method, Syst. Control Lett., 94, 1-10, (2016) · Zbl 1344.93025
[27] Xu, K. L.; Jiang, Y. L., reduced \(##?##\) optimal models via cross Gramian for continuous linear time-invariant systems, IET Circuits Devices Syst., 12, 25-32, (2018)
[28] Yan, W. Y.; Lam, J., an approximate approach to \(##?##\) optimal model reduction, IEEE Trans. Automat. Contr., 44, 1341-1358, (1999) · Zbl 0958.60040
[29] Zhang, L. Q.; Lam, J.; Huang, B.; Yang, G. H., on gramians and balanced truncation of discrete-time bilinear systems, Int. J. Control., 76, 414-427, (2003) · Zbl 1048.93015
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