×

Accuracy-enhancing methods for balancing-related frequency-weighted model and controller reduction. (English) Zbl 1045.93010

For a given \(n\)th-order original continuous-time-space model \[ {\mathbf G}= (A,B,C,D), \] let \({\mathbf G}_r= (A_r, B_r, C_r, D_r)\) be an \(r\)th-order approximation of the original model \((r< n)\). Thus, the methods for frequency-weighted model reduction try to minimize a weighted approximation error in the \(L\)-infinity norm and involving weights \(W_0\) and \(W_i\) which are suitably chosen to respectively weight the output and input transfer-function matrices.
On the other hand, in a controller reduction problem, for a given stabilizing controller of order \(n_c\), say \({\mathbf K}= (A_c, B_c, C_c, D_c)\) one wants to find \({\mathbf K}_r\), an \(r_c\)th-order approximation with the same number of unstable poles as \({\mathbf K}\), such that a similar weighted error is needed to be minimized.
Here, the square-root and balancing-free accuracy-enhancing techniques are extended to this kind of frequency-weighted problems. For this purpose, solving reduced-order Lyapunov equations is required by directly computing the grammians in terms of their Cholesky factors.

MSC:

93B11 System structure simplification
93B40 Computational methods in systems theory (MSC2010)
93B20 Minimal systems representations

Software:

SLICOT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, B. D.O.; Liu, Y., Controller reductionConcepts and approaches, IEEE Transactions of Automatic Control, 34, 802-812 (1989) · Zbl 0698.93034
[2] Enns, D. (1984). Model reduction for control systems design; Enns, D. (1984). Model reduction for control systems design
[3] Gill, P. E.; Golub, G. H.; Murray, W.; Saunders, M. A., Methods for modifying matrix factorizations, Mathematics of Computation, 28, 505-535 (1974) · Zbl 0289.65021
[4] Hammarling, S. J., Numerical solution of the stable, non-negative definite Lyapunov equation, IMA Journal of Numerical Analysis, 2, 303-323 (1982) · Zbl 0492.65017
[5] Lin, C.-A.; Chiu, T.-Y., Model reduction via frequency weighted balanced realization, CONTROL—Theory and Advanced Technology, 8, 341-351 (1992)
[6] Liu, Y.; Anderson, B. D.O., Singular perturbation approximation of balanced systems, International Journal of Control, 50, 1379-1405 (1989) · Zbl 0688.93008
[7] Liu, Y.; Anderson, B. D.O.; Ly, U. L., Coprime factorization controller reduction with Bezout identity induced frequency weighting, Automatica, 26, 233-249 (1990) · Zbl 0708.93029
[8] Moore, B. C., Principal component analysis in linear systemControllability, observability and model reduction, IEEE Transactions on Automatic Control, 26, 17-32 (1981) · Zbl 0464.93022
[9] Pernebo, L.; Silverman, L. M., Model reduction via balanced state space representations, IEEE Transactions of Automatic Control, 27, 382-387 (1982) · Zbl 0482.93024
[10] Safonov, M. G.; Chiang, R. Y., A Schur method for balanced-truncation model reduction, IEEE Transactions of Automatic Control, 34, 729-733 (1989) · Zbl 0687.93027
[11] Schelfhout, G.; De Moor, B., A note on closed-loop balanced truncation, IEEE Transactions of Automatic Control, 41, 1498-1500 (1996) · Zbl 0871.93008
[12] Tombs, M. S.; Postlethwaite, I., Truncated balanced realization of a stable non-minimal state-space system, International Journal of Control, 46, 1319-1330 (1987) · Zbl 0642.93015
[13] Varga, A. (1991a). Balancing-free square-root algorithm for computing singular perturbation approximations (pp. 1062-1065). Proceedings of the 30th IEEE CDC; Varga, A. (1991a). Balancing-free square-root algorithm for computing singular perturbation approximations (pp. 1062-1065). Proceedings of the 30th IEEE CDC
[14] Varga, A. (1991b). Efficient minimal realization procedure based on balancing. In A. El Moudni, P. Borne, & S. G. Tzafestas (Eds.), Preprints of IMACS Symposium on Modelling and Control of Technological Systems; Varga, A. (1991b). Efficient minimal realization procedure based on balancing. In A. El Moudni, P. Borne, & S. G. Tzafestas (Eds.), Preprints of IMACS Symposium on Modelling and Control of Technological Systems
[15] Varga, A. (2001). Model reduction software in the SLICOT library. In B. N. Datta, (Ed.), Applied and computational controlsignals and circuits; Varga, A. (2001). Model reduction software in the SLICOT library. In B. N. Datta, (Ed.), Applied and computational controlsignals and circuits
[16] Varga, A. (2002). Numerical software in SLICOT for low order controller design. In Proceedings of the CACSD’2002 Symposium. Glasgow, UK.; Varga, A. (2002). Numerical software in SLICOT for low order controller design. In Proceedings of the CACSD’2002 Symposium. Glasgow, UK.
[17] Varga, A., & Anderson, B. D. O. (2001). Square-root balancing-free methods for the frequency-weighted balancing related model reduction. Proceedings of the CDC’2001; Varga, A., & Anderson, B. D. O. (2001). Square-root balancing-free methods for the frequency-weighted balancing related model reduction. Proceedings of the CDC’2001
[18] Varga, A., & Anderson, B. D. O. (2002). Frequency-weighted balancing related controller reduction. Proceedings of the IFAC’2002 congress; Varga, A., & Anderson, B. D. O. (2002). Frequency-weighted balancing related controller reduction. Proceedings of the IFAC’2002 congress
[19] Wang, G.; Sreeram, V.; Liu, W. Q., A new frequency-weighted balanced truncation method and error bound, IEEE Transactions of Automatic Control, 44, 1734-1737 (1999) · Zbl 0958.93020
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.