[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

Estimation of Solution of Discrete Linear Time-Varying System 325
3 Conclusion
In this paper we have obtained upper and lower bounds for norm of solution of
linear discrete time-varying system in terms of coefficients of matrices A(n). Itis
interesting that to calculate our bounds we do not need to know eigenvalues or spectral
norm of matrices because we can simply calculate it by matrix coefficients. It
is also worth to notice that our bounds are also valid in case of singular matrices.
Moreover, in the paper, as the corollary of our results, we have obtained an upper
estimate for stationary system.
Presented solutions can be widely used in the algorithms of data transmission
[19], [20] and control developed by the research team from Silesian University of
Technology. The proposed method can be particularly useful for high-compression
algorithms where the key parameter is the time of computation. A time series analysis
can bring a significant increase of efficiency of the communication system between
the UAV and mobile control station. The presented methods could also be
used for communication channel modeling applications. In particular important is
the ability to apply to applications related to cryptography and signal variability
analysis and validation of quality of encryption algorithms.
Acknowledgements. The research presented here was done by first author as a part of
the project funded by the National Science Centre granted according to decision DEC-
2012/05/B/ST7/ 00065. The research of the second author was co-financed by the European
Union within European Social Fund for SWIFT project POKL. 08.02.01-24-005/10.
References
1. Adrianova, L.Y.: Introduction to Linear Systems of Differential Equations, vol. 146.
AMS Bookstore (1995)
2. Arnold, L., Crauel, H., Eckmann, J.P. (eds.): Lyapunov Exponents. Proceedings of a Conference,
held in Oberwolfach, May 28-June 2. Lecture Notes in Mathematics, vol. 1486.
Springer, Berlin (1991)
3. Awad, L.R., El-Kholy, E.M., El-Bendary, S.: On the Estimation of Solutions for Some
Linear Systems of Differential Equations. Acta Mathematica Sinica, New Series 14(1),
41–46 (1998)
4. Barreira, L., Pesin, Y.: Lyapunov exponents and Smooth Ergodic Theory. Univ. Lecture
Ser., vol, 23. Amer. Math Soc., Providence (2002)
5. Li, C., Chen, G.: Estimating the Lyapunov exponents of discrete systems. Chaos 14(2),
343–346 (2004)
6. Li, C., Xia, X.: On the bound of the Lyapunov exponents for continuous systems.
Chaos 14(3), 557–561 (2004)
7. Czornik, A., Jurgas, P.: Set of possible values ofmaximal Lyapunov exponents of discrete
time-varying linear system. Automatica 44(2), 580–583 (2008)
8. Czornik, A., Nawrat, A.: On new estimates for Lyapunov exponents of discrete timevarying
linear systems. Automatica 46(4), 775–778 (2010)