DOI:https://doi.org/10.15407/kvt186.04.030
KVT, 2016, Issue 186, pp.30-46
UDC 519.71
PROBLEM OF MODEL ORDER REDUCTION FOR LINEAR LARGE-SCALE TIME-INVARIANT SYSTEM
Space Research Institute NAS Ukraine and SSA Ukraine, Kiev, Ukraine
Introduction. Very significant for application model reduction problem of large-scale time-invariant system to more simple small order is considered and developed in the paper. Real and approximate models fitting is determined by norms which establish the difference between impulse response of these two models.
The purpose of the article is to propose a new approach of setting the model reduction problem and to develop methods based on variational principle of its solving.
Methods. It is proposed to set model reduction problem as optimization. For this initial state space model was transformed to equivalent description in form of input-output relation using analytical expression for impulse response. Such form allows to apply conception of fit between real system and its low-order approximation widely used in identification. Parameters of approximate model and its dimention are determined from optimization problem with different measure of fit writing as norm. Algorithms of numerical solving the optimization problems and needed for this data are considered in the paper. Besides the modified subspace method that permits to construct the observability matrix directly from output data using SVD factorization is proposed and described. Singular values of SVD-decomposition indicate as the best way to truncate full model.
Results. Some results dealing with mutual disposition of eigenvalues of real model and reduced one are demonstrated.
Conclusion. Developed methods may be used both for systems with scalar input and output and for multi-input and multi-output system as well. Results obtained by modelling show efficiency of all worked out methods.
Keywords: model reduction, approximation, optimization, model fit, state-space model.
Reference
1 Mischenko E.F., Rozov N.H. Differential equations with small parameter and relaxation oscillations. M.: Nauka. 1975, 248 p. (in Russian).
2 Antoulas A.C., Sorenson D.C., Gugercin S.A. A survey of model reduction methods for large-scale systems. Contemporary Mathematics. 2001, Vol. 280, pp. 193–219. https://doi.org/10.1090/conm/280/04630
3 Reis T., Stykel T. A survey on model reduction of coupled systems. Model order Reduction. Theory, Research Aspects and Applications. 2008.
4 Gubarev V.F. Method of iterative identification of multivariable systems over inexact data. Part 1. Theoretical aspects. Journal of Automation and Information Sciences, 2006, No5,pp. 16–31 (in Russian).
5 Gubarev V.F. and Melnichuk S.V. Identification of multivariable systems using steady-state parameters. Journal of Automation and Information Sciences, 2012, No 5, pp. 26–42 (in Russian). https://doi.org/10.1615/JAutomatInfScien.v44.i9.30
6 Rainer S., Price K. Differential evolution – a simple and efficient adaptive scheme for global optimization over continuous spaces. Journal of Global Optimization, 1997, No 11, pp. 341–359.
7 Golub G.H. and Van Loan Ch.F. Matrix Computations. Baltimore and London: John Hopkins University Press, 1999, 550 p.
8 Melnichuk S.V. Method of structural parametric multivariable systems identification using frequency characteristics. Kibernetika i vycislitel’naa tehnika, 2015, No 181, p. 66–79 (in Russian).
Received 15.09.16