Hurwitz Stability of Matrix Segment and The Common Solution Set of 2 and 3-Dimensional Lyapunov Equations

Year 2020, Volume: 24 Issue: 2, 357 - 364, 01.04.2020

Şerife Yılmaz

https://doi.org/10.16984/saufenbilder.573551

Cited By: 1

Abstract

In this study, a necessary and sufficient condition is given for the stability of the convex combinations of n-dimensional two Hurwitz stable matrices. There is a close relationship between Hurwitz stability of the matrix segment and common solution to the Lyapunov equations corresponding to those matrices. Therefore, the results obtained in this area are important. In the case of existence, an algorithm that determines common solutions set is also given. A number of illustrative examples using this algorithm are given.

Keywords

Hurwitz stability, matrix segment, common quadratic Lyapunov function

References

• R. Bellman, Introduction to Matrix Analysis, SIAM, Philadelphia, 1997.
• H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, (2002).
• D. Liberzon, Switching in System and Control, Birkhauser, Boston, (2003).
• H. Lin, P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control, Vol.54, N.2, pp. 308-322, 2009.
• R. N. Shorten, K. S. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems, International Journal of Adaptive Control and Signal Processing, Vol.16, pp. 709-728, 2002.
• V. Dzhafarov, T. Büyükköroğlu, Ş. Yılmaz, On one application of convex optimization to stability of linear systems, Trudy Inst. Mat. i Mekh. UrO RAN, Vol.21, N.2, pp. 320-328, 2015.
• D. Liberzon, R. Tempo, Common Lyapunov functions and gradient algorithms, IEEE Transactions on Automatic Control, Vol.49, N.6, pp. 990-994, 2004.
• M. Góra, Some methods of construction of a common Lyapunov solution to a finite set of complex systems,Linear Algebra and its Applications, Vol.530, pp. 77-93, 2017.
• C. King, M. Nathanson, On the existence of a common quadratic Lyapunov function for a rank one difference, Linear Algebra and its Applications, Vol.419, pp. 400-416, 2006.
• T.J. Laffey, H. Šmigoc, Common solution to the Lyapunov equation for 2×2 complex matrices, Linear Algebra and its Applications, Vol.420, pp. 609-624, 2007.
• A. T. Fuller, Conditions for a matrix to have only characteristics roots with negative real parts, Journal of Mathematical Analysis and Applications, Vol.23, pp. 71-98, 1968.
• W. Govarets, B. Sijnave, Matrix manifolds and Jordan Structure of the bialternate matrix product, Linear Algebra and its Applications, Vol.292, pp. 245-266, 1999.
• L. Elsner, V. Monov, The bialternate matrix product revisited, Linear Algebra and its Applications, Vol.434, pp. 1058-1066, 2011.
• R. K. Yedevalli, Robust Control of Uncertain Dynamic Systems: A linear state space approach, Springer, New York, 2014.
• B. R. Barmish, New Tools for Robustness of Linear Systems, Macmillan, New York, 1994.
• L. H. Keel and S. P. Bhattacharyya, Robust stability via sign-definite decomposition, IEEE Transactions on Automatic Control, Vol.56, N.1, pp. 140-145, 2011.