Volume 16, Issue 2 (Journal of Control, V.16, N.2 Summer 2022)                   JoC 2022, 16(2): 1-10 | Back to browse issues page


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Delavar A, Haeri M. Determination of Gain and Phase Margins in Lur’e Nonlinear Systems using Extended Circle Criterion. JoC 2022; 16 (2) :1-10
URL: http://joc.kntu.ac.ir/article-1-843-en.html
1- Sharif u. of T.
Abstract:   (4643 Views)
Nonlinearity is one of the main behaviors of systems in the real world. Therefore, it seems necessary to introduce a method to determine the stability margin of these systems. Although the gain and phase margins are established criteria for the analysis of linear systems, finding a specific way to determine the true value of these margins in nonlinear systems in general is an ongoing research in the literature. The main goal of this paper is to introduce a new method to determine these margins more precisely for particular type of nonlinearities. Based on Lyapunov theorem, stability conditions are obtained and the stability margins of Lur'e systems are determined by applying the extended circle criterion. By providing numerical examples, the correctness of the obtained results is checked and by comparing them with other methods, the accuracy of the obtained results is evaluated.
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Type of Article: Research paper | Subject: Special
Received: 2021/02/23 | Accepted: 2021/12/23 | ePublished ahead of print: 2022/01/19

References
1. [1] R. Dorf, R. Bishop, Modern Control System, 2011. [DOI:10.1016/B978-1-85617-695-8.00053-1]
2. [2] Z.Y. Nie, M. Wu, Q.G. Wang, Y. He, "A novel computational method for loop gain and phase margins of TITO systems", Journal of Franklin Institute, vol. 350, pp. 503-520, 2013. [DOI:10.1016/j.jfranklin.2012.12.011]
3. [3] M. G. Safonov, "Stability margins of diagonally perturbed multivariable feedback systems," 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, pp. 1472-1478, 1981. [DOI:10.1109/CDC.1981.269503]
4. [4] J.C. Doyle, "Analysis of feedback systems with structured uncertainties", IEEE Proceedings Part D, vol. 129, pp. 242-250, 1982. [DOI:10.1049/ip-d.1982.0053]
5. [5] L. Arnold, V. Wihstutz, Lyapunov exponents: A survey, Lecture Notes in Math, Springer-Verlag, New York, pp. 1186-1985, 2006. [DOI:10.1007/BFb0076829]
6. [6] X. Yang, J.J. Zhu, "Singular perturbation margin assessment of linear time-invariant systems via the Bauer-Fike theorems", 51st IEEE Conference on Decision and Control (CDC), pp. 6521-6528, 2013. [DOI:10.1109/CDC.2012.6426813]
7. [7] X. Yang, J.J. Zhu, "Singular perturbation margin and generalized gain margin for nonlinear time-invariant systems", International Journal of Control, vol. 89, pp. 451-468, 2016. [DOI:10.1080/00207179.2015.1079738]
8. [8] A. Rosales, L. Ibarra, P. Ponce, A. Molina, "Fuzzy sliding mode control design based on stability margins", Journal of the Franklin Institute, vol. 356, pp. 5260-5273, 2019. [DOI:10.1016/j.jfranklin.2019.04.035]
9. [9] X. Yang, J.J. Zhu, "Generalized gain margin for nonlinear systems", American Control Conference, pp. 3316-3321, 2012.
10. [10] A. Rosales, Y. Shtessel, L. Fridman, "Analysis and design of systems driven by finite-time convergent controllers", International Journal of Control, vol. 91, pp. 2563-2572, 2017. [DOI:10.1080/00207179.2016.1255354]
11. [11] C.B. Panathula, Y. Shtessel, "Practical stability margins in continuous higher order sliding mode control systems", Journal of the Franklin Institute, vol. 357, pp. 106-120, 2020. [DOI:10.1016/j.jfranklin.2019.09.034]
12. [12] J. Zhou, "Interpreting Popov criteria in Lur'e systems with complex scaling stability analysis", Communication Nonlinear Science Numerical Simulation, vol. 59, pp. 306-318, 2018. [DOI:10.1016/j.cnsns.2017.11.029]
13. [13] S.S. Das, Y. Shtessel, F. Plestan, "Phase and gain stability margins for a class of nonlinear systems", IFAC, vol. 51, pp. 263-268, 2018. [DOI:10.1016/j.ifacol.2018.11.116]
14. [14] S.S. Das, Y. Shtessel, F. Plestan, "Gain margins in a class of nonlinear systems: Lyapunov approach", IEEE Conference on Control Technology and Applications (CCTA), pp. 839-844, 2020. [DOI:10.1109/CCTA41146.2020.9206380]
15. [15] C. Guiver, H. Logemann, "A circle criterion for strong integral input-to-state stability", Automatica, vol. 111, 10.1016/j.automatica.2019.108641, 2020. [DOI:10.1016/j.automatica.2019.108641]
16. [16] B. Jayawardhana, H. Logemann, E.P. Ryan, "The circle criterion and input-to-state stability", IEEE Control Systems Magazine, vol. 31, pp. 32-67, 2011. [DOI:10.1109/MCS.2011.941143]
17. [17] H. Khalil, Nonlinear Systems, 3rd edition, 2002.
18. [18] D. Materassi, G. Innocenti, R. Genesio, M. Basso, "A composite circle criterion", 46th IEEE Conference on Decision and Control (CDC), USA, pp. 4459-4464, 2007. [DOI:10.1109/CDC.2007.4434762]
19. [19] T.D. Quoc, S. Gumussoy, W. Michiels, M. Diehl, "Combining convex-concave decompositions and linearization approaches for solving BMIs, with application to static output feedback", IEEE Transactions on Automatic Control, vol. 57, pp. 1377-1390, 2011. [DOI:10.1109/TAC.2011.2176154]

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