# Extracting second-order structures from single-input state-space models: Application to model order reduction

Jérôme Guillet; Benjamin Mourllion; Abderazik Birouche; Michel Basset

International Journal of Applied Mathematics and Computer Science (2011)

- Volume: 21, Issue: 3, page 509-519
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topJérôme Guillet, et al. "Extracting second-order structures from single-input state-space models: Application to model order reduction." International Journal of Applied Mathematics and Computer Science 21.3 (2011): 509-519. <http://eudml.org/doc/208065>.

@article{JérômeGuillet2011,

abstract = {This paper focuses on the model order reduction problem of second-order form models. The aim is to provide a reduction procedure which guarantees the preservation of the physical structural conditions of second-order form models. To solve this problem, a new approach has been developed to transform a second-order form model from a state-space realization which ensures the preservation of the structural conditions. This new approach is designed for controllable single-input state-space realizations with real matrices and has been applied to reduce a single-input second-order form model by balanced truncation and modal truncation.},

author = {Jérôme Guillet, Benjamin Mourllion, Abderazik Birouche, Michel Basset},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {second-order form model; preservation of the structural conditions; balanced truncation; modal truncation},

language = {eng},

number = {3},

pages = {509-519},

title = {Extracting second-order structures from single-input state-space models: Application to model order reduction},

url = {http://eudml.org/doc/208065},

volume = {21},

year = {2011},

}

TY - JOUR

AU - Jérôme Guillet

AU - Benjamin Mourllion

AU - Abderazik Birouche

AU - Michel Basset

TI - Extracting second-order structures from single-input state-space models: Application to model order reduction

JO - International Journal of Applied Mathematics and Computer Science

PY - 2011

VL - 21

IS - 3

SP - 509

EP - 519

AB - This paper focuses on the model order reduction problem of second-order form models. The aim is to provide a reduction procedure which guarantees the preservation of the physical structural conditions of second-order form models. To solve this problem, a new approach has been developed to transform a second-order form model from a state-space realization which ensures the preservation of the structural conditions. This new approach is designed for controllable single-input state-space realizations with real matrices and has been applied to reduce a single-input second-order form model by balanced truncation and modal truncation.

LA - eng

KW - second-order form model; preservation of the structural conditions; balanced truncation; modal truncation

UR - http://eudml.org/doc/208065

ER -

## References

top- Antoulas, A.C. (2005). Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, SIAM, Philadelphia, PA. Zbl1112.93002
- Bai, Z., Li, R.-C. and Su, Y. (2008). A unified Krylov projection framework for structure-preserving model reduction, in W.H. Schilders, H.A. van der Vorst and J. Rommres (Eds.) Model Order Reduction: Theory, Research Aspects and Applications, Springer, Berlin/Heidelberg, pp. 75-94. Zbl1154.93010
- Chahlaoui, Y., Lemonnier, D., Meerbergen, K., Vandendorpe, A. and Dooren, P.V. (2002). Model reduction of second order systems, Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems of Notre Dame, South Bend, IN, USA.
- Chahlaoui, Y., Lemonnier, D., Vandendorpe, A. and Dooren, P. V. (2006). Second-order balanced truncation, Linear Algebra and Its Applications 415(2-3): 373-384. Zbl1102.93008
- Dorf, R.C. and Bishop, R.H. (2008). Modern Control Systems, 11th Edn., Prentice Hall, Upper Saddle River, NJ. Zbl0907.93001
- Ersal, T., Fathy, T.H., Louca, L., Rideout, D. and Stein, J. (2007). A review of proper modeling techniques, Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Seattle, WA, USA.
- Fortuna, L., Nunnari, G. and Gallo, A. (1992). Model Order Reduction Techniques with Applications in Electrical Engineering, Springer-Verlag, Berlin/Heidelberg.
- Freund, R.W. (2005). Padé-type model reduction of secondorder and higher-order linear dynamical systems, in V.M. P. Benner and D. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, Vol. 45, Springer-Verlag, Berlin/Heidelberg, pp. 191-223. Zbl1079.65532
- Friswell, M.I. (1999). Extracting second-order system from state-space representations, American Institute of Aeronautics and Astronautics Journal 37(1): 132-135.
- Friswell, M.I., Garvey, S.D. and Penny, J.E.T. (1995). Model reduction using dynamic and iterated IRS techniques, Journal of Sound and Vibration 186(2): 311-323. Zbl1049.74725
- Glover, K. (1984). All optimal Hankel-norm approximation of linear multivariable systems and their ${L}_{\infty}$-error bounds, International Journal of Control 39(6): 1115-1193. Zbl0543.93036
- Gohberg, I., Lancaster, P. and Rodman, L. (1982). Matrix Polynomials, Academic Press, New York, NY. Zbl0482.15001
- Gugercin, S. (2004). A survey off-road model reduction by balanced truncation and some new results, International Journal of Control 77(8): 748-766. Zbl1061.93022
- Guyan, R. (1964). Reduction of stiffness and mass matrices, American Institute of Aeronautics and Astronautics Journal 3(2): 380.
- Houlston, P.R. (2006). Extracting second order system matrices from state space system, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 220(8): 1147-1149.
- Hughes, P. and Skelton, R. (1980). Controllability and observability of linear matrix-second-order systems, Journal of Applied Mechanics 47(2): 415-420. Zbl0441.73117
- Koutsovasilis, P. and Beitelschmidt, M. (2008). Comparison of model reduction techniques for large mechanical systems, Multibody System Dynamics 20(2): 111-128. Zbl1332.70024
- Li, J.-R. and White, J. (2001). Reduction of large circuit models via low rank approximate gramians, International Journal of Applied Mathematics and Computer Science 11(5): 1151-1171. Zbl0995.93027
- Li, R.-C. and Bai, Z. (2006). Structure-preserving model reduction, in J. Dongarra, K. Madsen and J. Waśniewski (Eds.) PARA 2004, Lecture Notes in Computer Science, Vol. 3732, Springer-Verlag, Berlin/Heidelberg, pp. 323-332.
- Meyer, D.G. and Sirnivasan, S. (1996). Balancing and model reduction for second-order form linear systems, IEEE Transactions on Automatic Control 41(11): 1632-644.
- Moore, B. (1981). Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Transactions on Automatic Control ac-26(1): 17-32. Zbl0464.93022
- Prells, U. and Lancaster, P. (2005). Isospectral vibrating systems. Part 2: Structure preserving transformation, Operator Theory 163: 275-298. Zbl1095.15017
- Reis, T. and Stykel, T. (2007). Balanced truncation model reduction of second-order systems, Technical report, DFG Research Center Matheon, Berlin. Zbl1151.93010
- Salimbahrami, S.B. (2005). Structure Preserving Order Reduction of Large Scale Second Order Models, Ph.D. thesis, Technical University of Munchen, Munchen.
- Schilders, W.H.A. (2008). Introduction to model order reduction, in W.H. Schilders, H.A. van der Vorst and J. Ronnres (Eds.) Model Order Reduction: Theory, Research Aspects and Applications, Springer, Berlin/Heidelberg, pp. 3-32. Zbl1154.93322
- Sorensen, D. and Antoulas, A. (2004). Gramians of structured systems and an error bound for structure-preserving model reduction, in V.M.P. Benner and D. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, Vol. 45, Springer-Verlag, Heidelberg/Berlin, pp. 117-130.
- Stykel, T. (2006). Balanced truncation model reduction of second-order systems, Proceedings of 5th MATHMOD, Vienna, Austria. Zbl1102.65075
- Tisseur, F. and Meerbergen, K. (2001). The quadratic eigenvalue problem, Society for Industrial and Applied Mathematics Review 43(2): 235-286. Zbl0985.65028
- Yan, B., Tan, S.-D. and Gaughy, B.M. (2008). Second-order balanced truncation for passive order reduction of RLCK circuits, IEEE Transactions on Circuits and Systems II 55(9): 942-946.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.