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
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- 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.
Abstract
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.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | October 5, 2023 |
| Publication Date | October 18, 2023 |
| Submission Date | March 5, 2023 |
| Acceptance Date | July 25, 2023 |
| Published in Issue | Year 2023 Volume: 27 Issue: 5 |
Cite
Sakarya University Journal of Science (SAUJS)