Abstract
In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov equations, which are observed in the study of controllability and observability of the state vector in deterministic bilinear systems. The same equations are used in the stability analysis and stabilization of stochastic linear control systems. To calculate these spectral decompositions, an iterative algorithm is proposed that uses the residues of the resolvent of the dynamics matrix. This algorithm converges for any initial guess, for a non-singular and stable dynamical system. The practical significance of the obtained results is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations to the asymptotic dynamics of the perturbation energy in deterministic bilinear and stochastic linear systems. In particular, the norm of the obtained eigen-components increases when frequencies of the corresponding oscillating modes approximate each other. Thus, the proposed decompositions provide a new fundamental approach for quantifying resonant modal interactions in a large and important class of weakly nonlinear systems.
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REFERENCES
B. T. Polyak, M. V. Khlebnikov, and L. B. Rapoport, Mathematical Theory of Automatic Control (LENAND, Moscow, 2019).
S. N. Vassilyev and A. A. Kosov, Autom. Remote Control 72 (6), 1163–1183 (2011).
U. Baur, P. Benner, and L. Feng, Arch. Comput. Methods Eng. 21 (4), 331–358 (2014).
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems: Advances in Design and Control (SIAM, Philadelphia, 2005).
I. B. Yadykin, Autom. Remote Control 71 (6), 1011–1021 (2010).
I. B. Yadykin and A. B. Iskakov, Dokl. Math. 95 (1), 103–107 (2017).
N. E. Zubov, E. Yu. Zybin, E. A. Mikrin, M. Sh. Misrikhanov, and V. N. Ryabchenko, Izv. Ross. Akad. Nauk, Teor. Sist. Upr. 56 (1), 1–18 (2017).
R. R. Mohler and W. J. Kolodziej, IEEE Trans. Syst. Man Cybern. SMC 10 (10), 683–688 (1980).
K. A. Pupkov, V. I. Kapalin, and A. S. Yushchenko, Functional Series in the Theory of Nonlinear Systems (Nauka, Moscow, 1976) [in Russian].
W. S. Gray and J. Mesko, “Energy functions and algebraic Gramians for bilinear systems,” in Preprint of the 4th IFAC Nonlinear Control Systems Design Symposium, Enschede (1998).
S. Al-Baiyat, A. S. Farag, and M. Bettayeb, Electr. Power Syst. Res. 26 (1), 11–19 (1993).
L. Zhang and J. Lam, Automatica 38, 205–216 (2002).
T. Damm, Numer. Linear Algebra Appl. 15, 853–871 (2008).
P. Benner and T. Damm, SIAM J. Control Optim. 49 (2), 686–711 (2011).
H. Schneider, Numer. Math. 7, 11–17 (1965).
ACKNOWLEDGMENTS
This work was supported by the Russian Foundation for Basic Research, project no. 17-08-01107-a.
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Yadykin, I.B., Iskakov, A.B. Spectral Decompositions for the Solutions of Lyapunov Equations for Bilinear Dynamical Systems. Dokl. Math. 100, 501–504 (2019). https://doi.org/10.1134/S1064562419050259
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DOI: https://doi.org/10.1134/S1064562419050259