Problemy redukcji złożonych modeli obiektów sterowania

• Autorzy: Rydel M. , Stanisławski W.

Warianty tytułu

EN

Problems of a complex plant models reduction

• Języki publikacji: PL

Abstrakty

PL

W artykule zaprezentowano zagadnienia związane z redukcją złożonych modeli obiektów sterowania. Przedstawiono podstawowe grupy metod redukcji wraz z omówieniem ich wad i zalet dla celów redukcji złożonych modeli obiektów sterowania. Zaproponowano wprowadzenie nowej miary błędu aproksymacji dla oceny dokładności aproksymacji modelu zredukowanego. Dobór parametrów metod redukcji pod kątem minimalizacji tej miary błędu umożliwia uzyskanego znacznie lepszych rezultatów redukcji niż minimalizacja normy Hankela błędu aproksymacji.

EN

Model order reduction problems of complex control model are presented in this paper. The article presents basic method group of complex control models reduction. Out of the many methods known from the literature, the methods based on balancing a system and then reducing it (SVD method) have become of great significance [1, 2, 5, 6, 10, 11, 13, 17, 23]. Also Krylov-based approximation methods are widely used, which are based on moment matching of the impulse response of the reduced system [1-3, 7, 9, 20]. Most of the MIMO models are characterized by large differences in gain characteristic and gain variability in frequency function (Fig. 1). The Hankel norm of approximation error (4) can't be used to estimate quality of approximation gain characteristic (Fig. 2). Authors propose in the paper new approximation error measure for reduced models of control objects (5). Minimizing this approximation error for selection of method reduction parameters gives better results than minimizing Hankel norm of approximation error (Tab. 1, Fig. 2). The reduction results of a structural model of component 1r (Russian service module) of the International Space Station [4] are also described in the paper [4].

Słowa kluczowe

PL

redukcja modelu; modele złożone

EN

model reduction; complex models

• Wydawca: Wydawnictwo PAK
• Czasopismo: Pomiary Automatyka Kontrola
• Rocznik: 2010
• Tom: R. 56, nr 2
• Strony: 197--200
• Opis fizyczny: Bibliogr. 24 poz., rys., tab., wzory

Twórcy

autor

Rydel M.

autor

Stanisławski W.

• Wydział Elektrotechniki, Automatyki i Informatyki, Politechnika Opolska,

Bibliografia

• [1] Antoulas A., Sorensen D.: Approximation of Large-Scale Dynamical System: An overview. Int. J. Appl. Comput. Sci., vol. 11, no 5, 2001, pp. 1093-1121.
• [2] Antoulas A.: Approximation of Large-Scale Dynamical System, Society for Industrial and Applied Mathematics, Philadelphia 2005.
• [3] Boley D.: Krylov space methods on state-space control models, Circuits Syst. Signal Process, vol. 13, no 6, 1994, pp. 733-758.
• [4] Chahlaoui Y., Van Dooren P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems, SLICOT Working Note 2002-2, 2002.
• [5] Chiu T. Y.: Model Reduction by the Low-Frequency Ap-proximation Balancing Method for Unstable Systems, IEEE Trans. Automat. Contr., vol. 41, no 7, 1996, pp 995-997.
• [6] Enns D.: Model reduction with balanced realizations: An error bound and frequency weighted generalization, Proc 23rd IEEE Conf. Decision and Control, 1984, pp. 127-132.
• [7] Feldmann P., Freund R. W.: Efficient linear circuit analysis by Pade approximation via the Lanczos process, IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, vol. 14 no 5, 1995, pp. 639-649.
• [8] Fortuna L., Nunnari G., Gallo A.: Model order reduction techniques with Applications in electrical engineering, Springer-Verlag, London 1992.
• [9] Freund R. W.: Model reduction methods based on Krylov sub-spaces, Acta Numerica vol. 12, 2003, pp. 267-319.
• [10] Gawroński W., Juang J.: Model reduction in limited time and frequency intervals, Int. J. System Sci., vol. 21, no 2, 1990, pp. 349-376.
• [11] Glover K.: All optimal Hankel-norm approximations of linear multivariable systems and their Linf error bounds, Int. J. Contr., vol. 39, no 6, 1984, pp. 1115-1193.
• [12] Gugercin S., Antoulas A.: A Comparative Study of 7 Algorithms for Model Reduction. Proc. of the 39th IEEE Conf. Decision and Control, vol. 3, 2000, pp. 2367-2372.
• [13] Gugercin S., Antoulas A.: A survey of model reduction by balanced truncation and some new results, Int. J. of Control, vol. 77, no 8, 2004, pp. 748-766.
• [14] Gugercin S., Willcox K.: Krylov Projection Framework for Fourier Model Reduction. Automatica, vol. 44, no 1, 2008, pp. 209-215.
• [15] Li J. R., White J.: Low-Rank Solution of Lyapunov Equa-tions, SIAM Review, vol. 46, no 4, 2004, pp. 693-713.
• [16] Liu Y., Anderson B.: Singular perturbation approximation of balanced system, Proc. of 28th IEEE Conf. Decision and Control, vol. 2, 1989, pp. 1355-1360.
• [17] Moore B.: Principal component analysis in linear systems: Controllability, observability and model reduction. IEEE Trans. Automat. Contr., vol. AC-26, no 1, 1981, pp. 17-32.
• [18] Obinata G., Anderson B.: Model Reduction for Control System Design, Springer-Verlag London 2001.
• [19] Penzl T.: Algorithms for Model Reduction of Large Dynamical Systems, Linear Algebra and its Applications, vol. 415, 2006, pp. 322-343.
• [20] Pillage L. T., Rohrer R. A.: Asymptotic Waveform Evaluation for Timing Analysis, IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, vol. 9 no 4, 1990, pp. 352-366.
• [21] Rydel M: Zredukowane hierarchiczne modele złożonych obiektów sterowania na przykładzie kotła energetycznego, Rozprawa doktorska, Opole 2009.
• [22] Singler J.: Approximate low rank solutions of Lapunov equations via Proper Orthogonal Decomposition, American Control Conference, 2008, pp. 267-272.
• [23] Wang G., Sreeram V., Liu W. Q.: A New Frequency-Weighted Balanced Truncation Method and an Error Bound, IEEE Trans. Automat. Contr., vol. 44, no 9, 1999, pp. 1734-1737.
• [24] Willcox K., Magretski A.: Fourier Series for Accurate, Stable, Reduced-Order Models in Large-Scale Applications, SIAM Journal for Scientific Computing, vol. 26, no 3, 2005, pp. 944-962.