×

Finite-time balanced truncation for linear systems via shifted Legendre polynomials. (English) Zbl 1425.93067

Summary: In this paper, we present a finite-time model order reduction method for linear systems via shifted Legendre polynomials. The main idea of the approach is to use finite-time empirical Gramians, which are constructed from impulse responses by solving block tridiagonal linear systems, to generate approximate balanced system for the large-scale system. The balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted Legendre polynomials, without individual reduction of the Gramians and a separate eigenvector solve. Then, the reduced-order model is constructed by truncating the states corresponding to the small approximate Hankel singular values (HSVs). The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, we modify the reduction procedure to alleviate the shortcomings of the above method, which may unexpectedly lead to unstable systems even though the original one is stable. Furthermore, the properties of the resulting reduced models are considered. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods.

MSC:

93B11 System structure simplification
93A15 Large-scale systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C05 Linear systems in control theory
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Software:

CSparse; emgr
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antoulas, A. C., Approximation of Large-Scale Dynamical Systems (2005), SIAM · Zbl 1112.93002
[2] Ionescu, T. C.; Astolfi, A.; Colaneri, P., Families of moment matching based, low order approximations for linear systems, Systems Control Lett., 64, 47-56 (2014) · Zbl 1283.93060
[3] Benner, P.; Gugercin, S.; Willcox, K., A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57, 4, 483-531 (2015) · Zbl 1339.37089
[4] Gugercin, S.; Antoulas, A. C., A survey of model reduction by balanced truncation and some new results, Internat. J. Control, 77, 8, 748-766 (2004) · Zbl 1061.93022
[5] Pernebo, L.; Silverman, L., Model reduction via balanced state space representations, IEEE Trans. Automat. Control, 27, 2, 382-387 (1982) · Zbl 0482.93024
[6] Glover, K., All optimal Hankel-norm approximations of linear multivariable systems and their \(L_\infty \)-error bounds, Internat. J. Control, 39, 6, 1115-1193 (1984) · Zbl 0543.93036
[7] Moore, B., Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26, 1, 17-32 (1981) · Zbl 0464.93022
[8] Lall, S.; Marsden, J. E.; Glavaški, S., A subspace approach to balanced truncation for model reduction of nonlinear control systems, Internat. J. Robust Nonlinear Control, 12, 6, 519-535 (2002) · Zbl 1006.93010
[9] Lall, S.; Marsden, J. E.; Glavaški, S., Empirical model reduction of controlled nonlinear systems, IFAC Proc., 32, 2, 2598-2603 (1999)
[10] Hahn, J.; Edgar, T. F., An improved method for nonlinear model reduction using balancing of empirical Gramians, Comput. Chem. Eng., 26, 10, 1379-1397 (2002)
[11] Baur, U.; Benner, P.; Feng, L., Model order reduction for linear and nonlinear systems: A system-theoretic perspective, Arch. Comput. Methods Eng., 21, 4, 331-358 (2014) · Zbl 1348.93075
[12] Himpe, C., Emgr-The empirical Gramian framework, Algorithms, 11, 7, 91 (2018) · Zbl 1461.93059
[13] Rathinam, M.; Petzold, L. R., A new look at proper orthogonal decomposition, SIAM J. Numer. Anal., 41, 5, 1893-1925 (2003) · Zbl 1053.65106
[14] Willcox, K.; Peraire, J., Balanced model reduction via the proper orthogonal decomposition, AIAA J., 40, 11, 2323-2330 (2002)
[15] Rowley, C. W., Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcation Chaos, 15, 03, 997-1013 (2005) · Zbl 1140.76443
[16] Opmeer, M. R., Model order reduction by balanced proper orthogonal decomposition and by rational interpolation, IEEE Trans. Automat. Control, 57, 2, 472-477 (2012) · Zbl 1369.93119
[17] Montier, L.; Henneron, T.; Goursaud, B.; Clnet, S., Balanced proper orthogonal decomposition applied to magnetoquasi-static problems through a stabilization methodology, IEEE Trans. Magn., 53, 7, 1-10 (2017)
[18] Jiang, Y.-L.; Chen, H.-B., Time domain model order reduction of general orthogonal polynomials for linear input-output systems, IEEE Trans. Automat. Control, 57, 2, 330-343 (2012) · Zbl 1369.93117
[19] Jiang, Y.-L.; Chen, H.-B., Application of general orthogonal polynomials to fast simulation of nonlinear descriptor systems through piecewise-linear approximation, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 31, 5, 804-808 (2012)
[20] Xiao, Z.-H.; Jiang, Y.-L., Model order reduction of MIMO bilinear systems by multi-order arnoldi method, Systems Control Lett., 94, 1-10 (2016) · Zbl 1344.93025
[21] Gawronski, W.; Juang, J.-N., Model reduction in limited time and frequency intervals, Internat. J. Systems Sci., 21, 2, 349-376 (1990) · Zbl 0692.93007
[22] Gugercin, S.; Antoulas, A., A time-limited balanced reduction method, (Decision and Control, 2003. Decision and Control, 2003, Proceedings. 42nd IEEE Conference on, vol. 5 (2003), IEEE), 5250-5253
[23] Datta, K. B.; Mohan, B. M., Orthogonal Functions in Systems and Control, vol. 9 (1995), World Scientific · Zbl 0866.93003
[24] Shih, D.-H.; Kung, F.-C., Optimal control of deterministic systems via shifted Legendre polynomials, IEEE Trans. Automat. Control, 31, 5, 451-454 (1986) · Zbl 0586.93037
[25] Benner, P.; Kürschner, P.; Saak, J., Frequency-limited balanced truncation with low-rank approximations, SIAM J. Sci. Comput., 38, 1, A471-A499 (2016) · Zbl 1391.65123
[26] Ghafoor, A.; Sreeram, V., Model reduction via limited frequency interval Gramians, IEEE Trans. Circuits Syst. I. Regul. Pap., 55, 9, 2806-2812 (2008)
[27] Li, X.; Yin, S.; Gao, H., Passivity-preserving model reduction with finite frequency \(H_\infty\) approximation performance, Automatica, 50, 9, 2294-2303 (2014) · Zbl 1297.93043
[28] Li, X.; Yu, C.; Gao, H., Frequency-limited \(H_\infty\) model reduction for positive systems, IEEE Trans. Automat. Control, 60, 4, 1093-1098 (2015) · Zbl 1360.93223
[29] Petersson, D.; Löfberg, J., Model reduction using a frequency-limited \(H_2\)-cost, Systems Control Lett., 67, 32-39 (2014) · Zbl 1288.93031
[30] Shen, J.; Lam, J., Improved results on \(H_\infty\) model reduction for continuous-time linear systems over finite frequency ranges, Automatica, 53, 79-84 (2015) · Zbl 1371.93170
[31] Haider, K. S.; Ghafoor, A.; Imran, M.; Malik, F. M., Model reduction of large scale descriptor systems using time limited Gramians, Asian J. Control, 19, 3, 1217-1227 (2017) · Zbl 1366.93090
[32] Tahavori, M.; Shaker, H. R., Model reduction via time-interval balanced stochastic truncation for linear time invariant systems, Internat. J. Systems Sci., 44, 3, 493-501 (2013) · Zbl 1307.93092
[33] Gugercin, S.; Sorensen, D. C.; Antoulas, A. C., A modified low-rank smith method for large-scale Lyapunov equations, Numer. Algorithms, 32, 1, 27-55 (2003) · Zbl 1034.93020
[34] Hammarling, S. J., Numerical solution of the stable non-negative definite Lyapunov equation, IMA J. Numer. Anal., 2, 3, 303-323 (1982) · Zbl 0492.65017
[35] Kürschner, P., Balanced truncation model order reduction in limited time intervals for large systems, Adv. Comput. Math. (2018), [Online]. Available: https://doi.org/10.1007/s10444-018-9608-6 · Zbl 1453.65093
[36] Redmann, M.; Kürschner, P., An output error bound for time-limited balanced truncation, Systems Control Lett., 121, 1-6 (2018) · Zbl 1408.93036
[37] Davis, T. A., Direct Methods for Sparse Linear Systems (2006), SIAM · Zbl 1119.65021
[38] van der Vorst, H. A., Large tridiagonal and block tridiagonal linear systems on vector and parallel computers, Parallel Comput., 5, 1-2, 45-54 (1987) · Zbl 0632.65034
[39] Rossi, T.; Toivanen, J., A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension, SIAM J. Sci. Comput., 20, 5, 1778-1793 (1999) · Zbl 0931.65020
[40] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042
[41] Van der Vorst, H. A., Iterative Krylov Methods for Large Linear Systems, vol. 13 (2003), Cambridge University Press · Zbl 1023.65027
[42] Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018
[43] Van der Vorst, H. A., Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13, 2, 631-644 (1992) · Zbl 0761.65023
[44] Heller, D., Some aspects of the cyclic reduction algorithm for block tridiagonal linear systems, SIAM J. Numer. Anal., 13, 4, 484-496 (1976) · Zbl 0347.65019
[45] Amsallem, D.; Farhat, C., Stabilization of projection-based reduced-order models, Internat. J. Numer. Methods Engrg., 91, 4, 358-377 (2012) · Zbl 1253.90184
[46] Penzl, T., Algorithms for model reduction of large dynamical systems, Linear Algebra Appl., 415, 2-3, 322-343 (2006) · Zbl 1092.65053
[47] Li, J.-R.; White, J., Reduction of large circuit models via low rank approximate Gramians, Int. J. Appl. Math. Comput. Sci., 11, 1151-1171 (2001) · Zbl 0995.93027
[48] Szegö, G., Orthogonal Polynomials (1939), American Mathematical Society: American Mathematical Society New York · JFM 65.0286.02
[49] Baur, U.; Benner, P., Gramian-based model reduction for data-sparse systems, SIAM J. Sci. Comput., 31, 1, 776-798 (2008) · Zbl 1183.93045
[50] Benner, P.; Saak, J., A semi-discretized heat transfer model for optimal cooling of steel profiles, (Benner, P.; Sorensen, D. C.; Mehrmann, V., Dimension Reduction of Large-Scale Systems (2005), Springer: Springer Berlin, Heidelberg), 353-356 · Zbl 1170.80341
[51] Varga, A., Balancing free square-root algorithm for computing singular perturbation approximations, Proceedings of the 30th IEEE Conference on Decision and Control, 1062-1065 (1991)
[52] Korvink, J. G.; Rudnyi, E. B., Oberwolfach benchmark collection, (Dimension Reduction of Large-Scale Systems (2005), Springer: Springer Berlin, Heidelberg), 311-315
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.