An Error Bound for Low Order Approximation of Dynamical Systems Subjected to Initial Conditions

Gabriel Pedro Ramos Maciel, Roberto Spinola Barbosa

Abstract


In recent years, a great effort has been taken focused on the development of reduced order modeling techniques of dynamical systems. This necessity is pushed by the requirement for efficient numerical techniques for simulations of dynamical systems arising from structural dynamics, controller design, circuit simulation, fluid dynamics and micro electromechanical systems.

We introduce a method to calculate the minimum upper $\mathcal{L}_2$ error bound of a linear time invaritant reduced order model considering any possible unitary initial conditions (IC). As a consequence, the proposed method calculates the unitary IC vector which leads to the maximum $\mathcal{L}_2$ norm of the error. Based on this error bound, it is discussed the capacity of a reduced order system to approximate the free transient response in the worst case scenario.


Keywords


model reduction; dynamical systems; error bound

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References


A. C. Antoulas, Approximation of large-scale dynamical systems, vol. 6. Siam, 2005.

A. C. Antoulas, D. C. Sorensen, and S. Gugercin, A survey of model reduction methods for large-scale systems, Contemporary mathematics, vol. 280,pp. 193220, 2001.

Z. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Applied numerical mathematics, vol. 43, no. 1-2, pp. 944, 2002.

P. Benner, V. Mehrmann, and D. C. Sorensen, Dimension reduction of large- scale systems, vol. 45. Springer, 2005.

R. W. Freund, Model reduction methods based on krylov subspaces, Acta Numerica, vol. 12, pp. 267319, 2003.

G. Obinata and B. D. Anderson, Model reduction for control system design. Springer Science & Business Media, 2012.

P. Benner, Numerical linear algebra for model reduction in control and simulation, vol. 29. Wiley Online Library, 2006.

D. Wilson, Optimum solution of model-reduction problem, in Proceedings of the Institution of Electrical Engineers, vol. 117, pp. 11611165, IET, 1970.

A. Varga, On modal techniques for model reduction, tech. rep., Technical Report TR R136-93, Institute of Robotics and System Dynamics, DLR Oberpfaenhofen, PO Box 1116, D-82230 Wessling, Germany, 1993.

M. Green and D. J. Limebeer, Linear robust control. Courier Corporation, 2012.

W. K. Gawronski, Dynamics and control of structures: A modal approach. Springer Science & Business Media, 2004.

S. Rahrovani, M. K. Vakilzadeh, and T. Abrahamsson, Modal dominancy analysis based on modal contribution to frequency response function H2-norm, Mechanical Systems and Signal Processing, vol. 48, no. 1, pp. 218231, 2014.

K. Glover, All optimal hankel-norm approximations of linear multivariable systems and their L1-error bounds, International journal of control, vol. 39, no. 6, pp. 11151193, 1984.

A. Megretski, H-innity model reduction with guaranteed suboptimality bound, in American Control Conference, 2006, pp. 6pp, IEEE, 2006.

M. Imran, A. Ghafoor, and V. Sreeram, A frequency weighted model order reduction technique and error bounds, Automatica, vol. 50, no. 12, pp. 3304 3309, 2014.

N. van de Wouw, W. Michiels, and B. Besselink, Model reduction for delay dierential equations with guaranteed stability and error bound, Automatica, vol. 55, pp. 132139, 2015.

S. M. Cox and A. Roberts, Initial conditions for models of dynamical systems, Physica D: Nonlinear Phenomena, vol. 85, no. 1, pp. 126141, 1995.

B. P. . F. L. Baur, U., Model order reduction for linear and nonlinear systems: A system-theoretic perspective, Arch Computat Methods Eng, vol. 21, no. 4, pp. 331358, 2014.

G. P. R. Maciel, Métodos de redução de graus de liberdade em sistemas dinâmicos lineares, Master's thesis, Escola politécnica da Universidade de São Paulo, 2015.

P. Lancaster and M. Tismenetsky, The theory of matrices: with applications. Elsevier, 1985

E. Davison, A method for simplifying linear dynamic systems, IEEE Transactions on automatic control, vol. 11, no. 1, pp. 93101, 1966.

C. Gregory, Reduction of large exible spacecraft models using internal balancing theory, Journal of Guidance, Control, and Dynamics, vol. 7, no. 6, pp. 725732, 1984.

L. Pernebo and L. Silverman, Model reduction via balanced state space representations, IEEE Transactions on Automatic Control, vol. 27, no. 2, pp. 382 387, 1982.

L. Meirovitch, Computational methods in structural dynamics, vol. 5. Sijtho & Noordho International Pub, 1980.

U. Baur and P. Benner, Factorized solution of lyapunov equations based on hierarchical matrix arithmetic, Computing, vol. 78, no. 3, pp. 211234, 2006.

B. N. Datta, Numerical methods for linear control systems: design and analysis, vol. 1. Academic Press, 2004.




DOI: https://doi.org/10.5540/tema.2018.019.02.197

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TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
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