Abstract
Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.
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References
Mullis, C., Roberts, R.A.: Synthesis of minimum roundoff noise fixed point digital filters. IEEE Trans. Circuits Syst. 23(9), 551–562 (1976)
Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Auto. Control 26(1), 17–32 (1981)
Liu, W.Q., Sreeram, V., Teo, K.L.: Model reduction for state-space symmetric systems. Systems Control Lett. 34, 209–215 (1998)
Opmeer, M.R., Reis, T.: A lower bound for the balanced truncation error for MIMO systems. IEEE Trans. Auto. Control 60(8), 2207–2212 (2015)
Antoulas, A.C.: Approximation of large-scale dynamical systems. SIAM, Philadelphia (2005)
Pernebo, L., Silverman, L.: Model reduction via balanced state space representations. IEEE Trans. Auto. Control 27(2), 382–387 (1982)
Enns, D.F.: Model reduction with balanced realizations: an error bound and a frequency weighted generalization. In: 23rd IEEE Conference on decision and control, Las Vegas, NV, pp. 127–132 (1984)
Wilson, D., Kumar, A.: Symmetry properties of balanced systems. IEEE Trans. Auto. Control 28(9), 927–929 (1983)
Ober, R.J.: Balanced realizations: canonical form, parametrization, model reduction. Int. J. Control 46(2), 643–670 (1987)
Ober, R., McFarlane, D.: Balanced canonical forms for minimal systems: a normalized coprime factor approach. Linear Algebra Appl. 122, 23–64 (1989)
Maciejowski, J.M., Ober, R.J.: Balanced parametrizations and canonical forms for system identification. IFAC Proceedings 21(9), 701–708 (1988)
Lu, T.-T., Shiou, S.-H.: Inverses of 2\(\times \) 2 block matrices. Comput. Math. with Appl. 43(1–2), 119–129 (2002)
Liu, Y., Anderson, B.D.O.: Singular perturbation approximation of balanced systems. Int. J. Control 50(4), 1379–1405 (1989)
Fernando, K., Nicholson, H.: On the cross-Gramian for symmetric MIMO systems. IEEE Trans. Circuits Syst. 32(5), 487–489 (1985)
Min, H., Paganini, F., Mallada, E.: Accurate reduced order models for coherent synchronous generators. In: 57th Annual allerton Conference on Communication, Control, and Computing, Monticello, IL, pp. 316–317 (2019)
Min, H., Paganini, F., Mallada, E.: Accurate reduced-order models for heterogeneous coherent generators. IEEE Control Systems Letters 5(5), 1741–1746 (2020)
Paganini, F., Mallada, E.: Global analysis of synchronization performance for power systems: bridging the theory-practice gap. IEEE Trans. Auto. Control 65(7), 3007–3022 (2020)
Salkuyeh, D.K., Beik, F.P.A.: An explicit formula for the inverse of arrowhead and doubly arrow matrices. Int. J. Appl. Comput. Math. 4(3), 1–8 (2018)
Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. Johns Hopkins University Press, Baltimore (2012)
Zhou, K., Doyle, J.C., Glover, K.: Robust and optimal control. Prentice Hall, Upper Saddle River, NJ (1996)
Acknowledgements
The authors thank Christopher Beattie and Vassilis Kekatos for helpful discussions, Christian Himpe for bringing reference [4] to our attention, and the referees for numerous comments that improved our presentation.
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This work was supported by the US National Science Foundation grant AMPS-1923221.
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Communicated by: Tobias Breiten
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Reiter, S., Damm, T., Embree, M. et al. On the balanced truncation error bound and sign parameters from arrowhead realizations. Adv Comput Math 50, 10 (2024). https://doi.org/10.1007/s10444-024-10105-y
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DOI: https://doi.org/10.1007/s10444-024-10105-y
Keywords
- Model reduction
- Balanced truncation
- Error bound
- Arrowhead matrix
- Sign parameters
- Sign symmetry
- Power systems