Common Solutions to Stein Inequalities

Year 2023, Volume: 27 Issue: 5, 1097 - 1103, 18.10.2023

Şerife YILMAZ Birgül AKSOY

https://doi.org/10.16984/saufenbilder.1260438

Abstract

In this paper for linear discrete time switched systems, the problem of existence of a common solution to Stein inequalities is considered. A sufficient condition for robust Schur stability of a matrix polyope by using Schur complement lemma and a necessary and sufficient condition for the existence of a common solution of Stein equation are given. As in the case of continuous time systems, the problem of existence of a common solution is reduced to a convex optimization one. An efficient solution algorithm which requires solving a linear minimax problem at each step is suggested. The algorithm is supported with a number of examples from the literature and observed that the method desired results fastly.

Keywords

Discrete-time system, Schur stability, Stein equation, common quadratic Lyapunov function, Schur complement

References

• M. Akar, K. S. Narendra, “On the existence of common quadratic Lyapunov functions for second-order linear time-invariant discrete-time systems,” International Journal of Adaptive Control and Signal Processing, vol. 16, pp. 729-751, 2002.
• J. C. Geromel, M. C. de Oliveira, L. Hsu, “LMI characterization of structural and robust stability,” Linear Algebra and its Applications, vol. 285, pp. 69-80, 1998.
• O. Taussky, “Matrices C with C^n→0,” Journal of Algebra, vol. 1, pp. 5-10, 1964.
• K. S. Narendra, J. A. Balakrishnan, “Common Lyapunov function for stable LTI systems with commuting A-matrices,” IEEE Transactions on Automatic Control, vol. 39(12), pp. 2469-2471, 1994.
• S. P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, “Some standard problems involving LMIs” in Linear Matrix Inequalities in System and Control Theory, Philadelphia, PA, USA: SIAM, 1994, ch. 2, pp. 7-35.
• D. Liberzon, J. P. Hespanha, A. S. Morse, “Stability of switched systems: a Lie-algebraic condition,” Systems & Control Letters, vol. 37, pp. 117–122, 1999.
• R. N. Shorten, K. S. Narendra, “Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for M stable second order linear time-invariant systems,” in Proceedings of the American Control Conference, Chicago, IL, USA, 2000, pp. 359–363.
• V. Dzhafarov, T. Büyükköroğlu, “On one inner point algorithm for common Lyapunov functions,” Systems & Control Letters, vol. 167, pp. 1-4, 2022.
• E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978.
• Ş. Yılmaz, “Hurwitz Stability of Matrix Segment and The Common Solution Set of 2 and 3-Dimensional Lyapunov Equations,” Sakarya University Journal of Science, vol. 24, no. 2, pp. 357-364, 2020.
• Ş. Yılmaz, “Common qudratic Lyapunov functions for two stable matrices,” Eskişehir Technical University Journal of Science and Technology B - Theoretical Sciences, vol. 10, no.1, pp. 18-26, 2022.