ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2020): 1.045
A quadratic bilinear equation arising from the quadratic dynamical system; pp. 248–259
PDF | 10.3176/proc.2021.3.04

Authors
Bo Yu, Ning Dong, Qiong Tang
Abstract

A quadratic dynamical system with practical applications is taken into consideration. This system is transformed into a new bilinear system with Hadamard products by means of the implicit matrix structure. The corresponding quadratic bilinear equation is subsequently established via the Volterra series. Under proper conditions, the existence of the solution to the equation is proved by using a fixed-point iteration. Numerical experiments verify the proposed theory of the solution.

References

1. Antoulas, A. C. Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia, PA, 2005.
https://doi.org/10.1137/1.9780898718713

2. Astrid, P., Weiland, S., Willcox, K. and Backx, T. Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Automat. Control, 2008, 53(10), 2237–2251.
https://doi.org/10.1109/TAC.2008.2006102

3. Barrault, M., Maday, Y., Nguyen, N. C. and Patera, A. T. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math., 2004, 339(9), 667–672.
https://doi.org/10.1016/j.crma.2004.08.006

4.Benner, P. and Goyal, P. Balanced truncation model order reduction for quadratic-bilinear control systems. 2017, arXiv:1705.00160v1 [math.OC].

5. Bhatia, R. Matrix analysis, Graduate Texts in Mathematics. Springer, Berlin, 1997.
https://doi.org/10.1007/978-1-4612-0653-8

6. Brennan, C., Condon, M. and Ivanov, R. Model order reduction of nonlinear dynamical systems. In Progress in Industrial 

Mathematics at ECMI 2004 (Di Bucchianico, A., Mattheij, R. and Peletier, M., eds), vol. 8. Springer, Berlin, Heidelberg, 2006, 114–118.

7. Bubnicki, Z. Modern Control Theory. Springer, Berlin, Heidelberg, 2005.

8. Damm, T. Rational Matrix Equations in Stochastic Control. Springer, Berlin, Heidelberg, 2004.

9. Damm, T. and Hinrichsen, D. Newton’s method for a rational matrix equation occuring in stochastic control. Linear Algebra Appl., 2001, 332–334(3), 81–109.
https://doi.org/10.1016/S0024-3795(00)00144-0

10. Fan, H.-Y., Weng P. C.-Y. and Chu, E. K.-W. Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control. Numer. Algorithms, 2016, 71, 245–272.
https://doi.org/10.1007/s11075-015-9991-8

11. Horn, R. A. and Johnson, C. R. The Hadamard product. In Topics in Matrix Analysis. Cambridge University Press, 1991, 298–381.
https://doi.org/10.1017/CBO9780511840371.006

12. Gu, C. QLMOR: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlin- ear systems. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 2011, 30(9), 1307–1320.
https://doi.org/10.1109/TCAD.2011.2142184

13. Gosea, V., Petreczky, M., Antoulas, A. C. and Fiter, C. Balanced truncation for linear switched systems. Adv. Comput. Math., 2018, 44(11), 1845–1886.
https://doi.org/10.1007/s10444-018-9610-z

14. Guo, C.-H. Iterative solution of a matrix Riccati equation arising in stochastic control. Oper. Theory Adv. Appl., 2001, 130, 209–221.
https://doi.org/10.1007/978-3-0348-8181-4_16

15. Lancaster, P. and Rodman, L. Algebraic Riccati Equations. Clarendon Press, Oxford, 1995.

16. Ledermann, W. Issai Schur and his school in Berlin. Bull. London Math. Soc., 1983, 15(2), 97–106.
https://doi.org/10.1112/blms/15.2.97

17. Li, T.-X., Weng, P. C.-Y., Chu, E. K.-W. and Lin, W.-W. Large-scale Stein and Lyapunov equations, Smith method, and applications. Numer. Algorithms, 2013, 63(4), 727–752.
https://doi.org/10.1007/s11075-012-9650-2

18. Ogata, K. Modern Control Engineering. Fifth edition. Pearson Education, Harlow, 2010.

19. Kolda, T. G. and Bader, B. W. Tensor decompositions and applications. SIAM Rev., 2009, 51(3), 455–500.
https://doi.org/10.1137/07070111X

20. Ivanov, I. G. Iterations for solving a rational Riccati equation arising in stochastic control. Comput. Math. Appl., 2007, 53(6), 977–988.
https://doi.org/10.1016/j.camwa.2006.12.009

21. Krasnoselskii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Ya. B. and Stetsenko, V. Ya. Approximate Solution of Operator Equations. Springer, Dordrecht, 1972.
https://doi.org/10.1007/978-94-010-2715-1

22. Rewienski M. and White J. A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 2003, 22(2), 155–170.
https://doi.org/10.1109/TCAD.2002.806601

23. Schetzen, M. The Volterra and Wiener Theories of Nonlinear Systems. Wiley-Interscience, New York, NY, 1980.

24. Schilders, W. H. A., van der Vorst, H. A. and Rommes, J. Model Order Reduction: Theory, Research Aspects and Applications. Springer, Berlin, Heidelberg, 2008. 
https://doi.org/10.1007/978-3-540-78841-6

Back to Issue