75 On Gramian-Based Techniques for Minimal Realization of Large-Scale Mechanical Systems 805 Table 75.3 Efficiency analysis of different approaches to treat moderate-size problems Time effort (s) Relative error (%) M6 21.46 0.003 M5b 28.33 0.006 M5a 26.79 0.001 M3 82.26 0.001 M1 133.02 0.001 75.7 Conclusion In this paper, a review of balancing related and Gramian-based minimal realization algorithms were presented. Also the ill-condition and inefficiency problem that typically arises in balancing of large-scale realizations were addressed. A hybrid modal-balanced algorithm to treat non-minimal realization of very large second-order systems with dense clusters of close eigenvalues was proposed. The method benefits the effectiveness of balancing techniques in treatment of nonminimal realizations in combination with the computational efficiency of modal techniques to treat large-scale problems. The advantages of the method were discussed through an illustrative example and also a moderate-size FE model problem. References 1. Gilbert EG (1963) Controllability and observability in multi-variable control systems. SIAM J Control 1(2):128–151 2. Kalman RE (1963) Mathematical description of linear dymanical systems. SIAM J Control 1(2):152–192 3. Van Dooren PM (1981) The generalized eigenstructure problem in linear system theory. IEEE Trans Autom Control AC-26(1):111–129 4. Moore BC (1981) Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Trans Autom Control AC-26(1):17–32 5. Laub AJ, Heath MT, Paige CC, Ward RC (1987) Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans Autom Control AC-32(2):115–122 6. Tombs MS, Postlethwaite I (1987) Truncated balanced realization of a stable non-minimal state-space system. Int J Control 46(4):1319–1330 7. Safonov MG, Chiang RY (1989) A Schur method for balanced-truncation. IEEE Trans Autom Control 34(7):729–733 8. Fernando KV, Nicholson H (1982) Minimality of SISO linear systems. Proc IEEE 70(10):1241–1242 9. Fernando KV, Nicholson H (1985) On the cross-Grammian for symmetric MIMO systems. IEEE Trans Circuit Syst 32(5):487–489 10. Sorensen DC, Antoulas AC, (2002) A Sylvester equation and approximate balanced reduction. Linear Algebra Appl 351/352:671–700 11. Varga A (1991) Efficient minimal realization procedure based on balancing. In: Proceedings of IMACS/IFAC symposium on modelling and control of technological systems, Lille, pp 42–47 12. Khorsand Vakilzadeh M, Rahrovani S, Abrahamsson T (2012) An improved modal approach for model reduction based on input-output relation. In: Proceedings of International conference on noise and vibration engineering, Leuven, Belgium pp 3451–3460 13. Rahrovani S, Khorsand Vakilzadeh M, Abrahamsson T (2013) A metric for modal truncation in model reduction problems. In: Proceedings of conference and exposition on structural dynamics, IMAC XXXI, Garden Grove, CA, USA 14. Skelton RE (1980) Cost decomposition of linear systems with application to model reduction. Int J Control 32:1031–1055 15. Glover K (1984) All optimal Hankel-norm approximations of linear multivariable systems and their L 1error bounds. Int J Control 39: 1115–1193 16. Enns D (1984) Model reduction with balanced realizations: an error bound and a frequency weighted generalization. In: Proceedings of of the 23rd IEEE conference on decision and control, Las Vegas 17. Kabamba PT (1985) Balanced gains and their significance for L2 model reduction. IEEE Trans Autom Control 30(7):690–693 18. Hammarling SJ (1982) Numerical solution of the stable non-negative definite Lyapunov equation. J Numer Anal 2:303–323 19. Antoulas AC, Sorensen DC, Gugercin S (2003) A modified low-rank Smith method for large-scale Lyapunov equations. Numer Algorithms 32:27–55
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