پیوندهای مفید
اخبار و اعلانات
آمار
| تعداد نشریات | 31 |
| تعداد شمارهها | 834 |
| تعداد مقالات | 8,015 |
| تعداد مشاهده مقاله | 14,853,756 |
| تعداد دریافت فایل اصل مقاله | 9,587,194 |
A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems | ||
| International Journal of Industrial Electronics Control and Optimization | ||
| مقاله 7، دوره 4، شماره 3، آبان 2021، صفحه 333-340 اصل مقاله (774.92 K) | ||
| نوع مقاله: Research Articles | ||
| شناسه دیجیتال (DOI): 10.22111/ieco.2021.35409.1300 | ||
| نویسندگان | ||
| Majid Akbarian1؛ Naser Pariz* 2 | ||
| 1Department of Electrical Engineering, Ferdowsi University of Mashhad, 91775-1111, Iran | ||
| 2Ferdowsi University of Mashhad | ||
| چکیده | ||
| Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability. | ||
| کلیدواژهها | ||
| Stability؛ Smooth Lyapunov functions؛ Stable equilibrium points؛ Limit cycles | ||
| مراجع | ||
|
[1] H. K. Khalil and J. Grizzle, "Nonlinear systems, vol. 3," Prentice hall Upper Saddle River, 2002. [2] S. Jafari Fesharaki, F. Sheikholeslam, M. Kamali, and A. Talebi, "Tractable robust model predictive control with adaptive sliding mode for uncertain nonlinear systems," International Journal of Systems Science, vol. 51, no. 12, pp. 2204-2216, 2020. [3] T. S. Gardner, C. R. Cantor, and J. J. Collins, "Construction of a genetic toggle switch in Escherichia coli," Nature, vol. 403, no. 6767, pp. 339-342, 2000. [4] L. Li, H. Zhang, and Y. Wang, "Stabilization and optimal control of discrete-time systems with multiplicative noise and multiple input delays," Systems & Control Letters, vol. 147, p. 104833, 2021. [5] Z. Shen, C. Li, H. Li, and Z. Cao, "Estimation of the domain of attraction for discrete-time linear impulsive control systems with input saturation," Applied Mathematics and Computation, vol. 362, p. 124502, 2019. [6] X.-F. Wang, A. R. Teel, K.-Z. Liu, and X.-M. Sun, "Stability analysis of distributed convex optimization under persistent attacks: A hybrid systems approach," Automatica, vol. 111, p. 108607, 2020. [7] J.J. E. Slotine and W. Li, Applied nonlinear control (no. 1). Prentice hall Englewood Cliffs, NJ, 1991. [8] A. I. Doban and M. Lazar, "Computation of Lyapunov functions for nonlinear differential equations via a Yoshizawa—type construction," IFAC-PapersOnLine, vol. 49, no. 18, pp. 29-34, 2016. [9] A. I. Doban and M. Lazar, "Computation of Lyapunov functions for nonlinear differential equations via a Massera-type construction," IEEE Transactions on Automatic Control, vol. 63, no. 5, pp. 1259-1272, 2017. [10] P. Giesl and S. Hafstein, "Review on computational methods for Lyapunov functions," Discrete and Continuous Dynamical Systems-Series B, vol. 20, no. 8, pp. 2291-2331, 2015. [11] G. Bidari, N. Pariz, and A. Karimpour, "Sufficient conditions for stabilization of interval uncertain LTI switched systems with unstable subsystems," International Journal of Industrial Electronics, Control and Optimization, vol. 2, no. 1, pp. 1-6, 2019. [12] M. Akbarian, N. Eghbal, and N. Pariz, "A Novel Method for Optimal Control of Piecewise Affine Systems Using Semi-Definite Programming," International Journal of Industrial Electronics, Control and Optimization, vol. 3, no. 1, pp. 59-68, 2020. [13] S. S. S. Farahani and S. Fakhimi Derakhshan, "LMI-based congestion control algorithms for a delayed network," vol. 2, no. 2, pp. 91-98, 2019. [14] K. Persidskii, "On a theorem of Liapunov," in CR (Dokl.) Acad. Sci. URSS, 1937, vol. 14, pp. 541-543. [15] C. M. Kellett, "Classical converse theorems in Lyapunov's second method," arXiv preprint arXiv:1502.04809, 2015. [16] A. Bacciotti, L. Rosier, and Z. Lin, "Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences 267," Appl. Mech. Rev., vol. 55, no. 5, pp. B88-B89, 2002. [17] J. Auslander and P. Seibert, "Prolongations and stability in dynamical systems," in Annales de l'institut Fourier, 1964, vol. 14, no. 2, pp. 237-267. [18] A. A. Ahmadi, M. Krstic, and P. A. Parrilo, "A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function," in 2011 50th IEEE Conference on Decision and Control and European Control Conference, 2011, pp. 7579-7580: IEEE. [19] A. A. Ahmadi and B. El Khadir, "A globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial Lyapunov function," Systems & Control Letters, vol. 121, pp. 50-53, 2018. [20] A. A. Ahmadi and B. El Khadir, "On algebraic proofs of stability for homogeneous vector fields," IEEE Transactions on Automatic Control, 2019. [21] W. Hahn, Stability of motion. Springer, 1967. [22] F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory. Springer Science & Business Media, 2008. [23] A. Bacciotti and L. Rosier, Liapunov functions and stability in control theory. Springer Science & Business Media, 2006. [24] A. Bacciotti and L. Rosier, "Regularity of Liapunov functions for stable systems," Systems & control letters, vol. 41, no. 4, pp. 265-270, 2000. [25] J. LaSalle, "Some extensions of Liapunov's second method," IRE Transactions on circuit theory, vol. 7, no. 4, pp. 520-527, 1960. | ||
|
آمار تعداد مشاهده مقاله: 656 تعداد دریافت فایل اصل مقاله: 441 |
||