Relaxations of Conditions of Lyapunov Functions

Abstract

In this study, stability conditions are given for nonlinear time varying systems using the classical Lyapunov 2nd Method and its arguments. A novel approach is utilized and so that uniform stability can also be proved by using an unclassical Lyapunov Function. In contrast with the studies in the literature, Lyapunov Function is allowed to be negative definite and increasing through the system. To construct a classical Lyapunov Function, we use a reverse time approach methodology for the intervals where the unclassical one is increasing. So we prove the stability using a new Lyapunov Function construction methodology. The main result shows that the existence of such a function guarantees the stability of the origin. Some numerical examples are also given to demonstrate the efficiency of the method we use.

Keywords

• Nonlinear Systems
• Stability Analysis
• Lyapunov 2nd Method
• Lyapunov Function
• Time-varying Systems
• Uniform Stability

Lyapunov Fonksiyonun Koşullarının Gevşetilmesi

Abstract

Bu çalışmada, klasik Lyapunov 2. Metodu ve bu metoda dair argümanlar kullanılarak, zamanla değişen yapıdaki Doğrusal Olmayan Sistemler için kararlı olma koşulları verilmektedir. Özgün bir yaklaşım kullanılmış ve böylece düzgün kararlılık, klasik olmayan bir Lyapunov Fonksiyonu kullanılarak da ayrıca ispatlanabilmiştir. Literatürdeki çalışmaların aksine, kullandığımız klasik olmayan Lyapunov Fonksiyonunun bazı aralıklar için sistem boyunca artan ve negatif tanımlı olmasına izin verilmiştir. Klasik Lyapunov Fonksiyonu’nu inşaa etmek için, klasik olmayan Lyapunov Fonksiyonu’nun artan olduğu aralıklarda ters zaman yaklaşımını kullanıyoruz. Böylece yeni bir Lyapunov Fonksiyonu inşa etme yaklaşımı kullanarak kararlığı ispatlamış oluyoruz. Ana sonuç böyle bir fonksiyonun varlığının, orjinin kararlılığını garantilediğini gösterir. Yaklaşımın efektif olduğunu göstermek için ayrıca bir takım nümerik örnekler de verilmiştir.

Keywords

• Nonlineer Sistemler
• Kararlılık Analizi
• Lyapunov 2. Metodu
• Lyapunov Fonksiyonu
• Zamanla değişen Sistemler
• Düzgün Kararlılık

References

1. [1] Khalil H. 2002. Nonlinear systems. Macmillan Publishing Company, New Jersey, 750p.
2. [2] Vidyasagar, M. 1993. Nonlinear Systems Analysis. SIAM series, Prentice Hall, New Jersey, 498p.
3. [3] Bayrak, A . 2017. Sliding Mode Based Self-Tuning PID Controller for Second Order Systems. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi , 21 (3) , 866-872 . K. ed. 1999.
4. [4] Butz A.R. 1969. Higher order derivatives of Liapunov functions. IEEE Trans.Automatic Control, AC-14, 111-112.
5. [5] Ahmadi A.A. 2008. Non-monotonic Lyapunov Functions for Stability of Nonlinear and Switched Systems: Theory and Computation. Ms Thesis, MIT, USA.
6. [6] Meigoli V. and Nikravesh S.K.Y. 2009. A new theorem on higher order derivatives of Lyapunov functions. ISA Transactions, 48, 173-179.
7. [7] Meigoli V. and Nikravesh S.K.Y. 2012. Stability analysis of nonlinear systems using higher order derivatives of Lyapunov function candidates. Systems and Control Letters, 61, 973-979.
8. [8] Lee, D. H., Park, J. B., and Joo, Y. H. 2011. Fundamental connections among the stability conditions using higher-order time derivatives of Lyapunov functions for the case of linear time-invariant systems. Systems and Control Letters, 60(9), 778-785.

9. [9] Lacerda M.J. and Seiler P. 2017. Stability of uncertain systems using Lyapunov func. with non-monotonic terms. Automatica, 82, 187-193.
10. [10] Tanaka K., Hori T. and Wang H.O. 2001. A Fuzzy Lyapunov Approach to Fuzzy Control System Design. Proc. of the American Control Conference, Arlington VA, USA, 4790-4795.
11. [11] Tanaka K., Hori T. and Wang H.O. 2003. A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems. IEEE Trans. On Fuzzy Systems, 11(4), 582-589.
12. [12] Rhee, BJ and Won, S. 2006. A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design. Fuzzy Sets and Systems, 157(9), 1211-1228.
13. [13] Gao, H.; Liu, X.; Lam, J. 2009. Stability Analysis and Stabilization for DiscreteTime Fuzzy Systems with Time-Varying Delay. IEEE Trans. on Systems Man and Cybernetics Part B-Cybernetics, 39(2), 306-317.
14. [14] Zhang H. and Xie X. 2011. Relaxed Stability Conditions for Continuous-Time T-S Fuzzy-Control Systems Via Augmented Multi-Indexed Matrix Approach. IEEE Transactions on Fuzzy Systems, 19(3) 478-492.
15. [15] Ying-J.C., Motoyasu T., Kohei I., Hiroshi O., Kazuo T., Thierry M., Alexandre K., Hua O.W. 2014. A nonmonotonically decreasing relaxation approach of Lyapunov functions to guaranteed cost control for discrete fuzzy systems. IET Control Theory and Applications, 8(16), 1716-1722.
16. [16] Ahmadi A.A. and Parrilo P.A. 2008. Non-monotonic Lyapunov Functions for Stability of Discrete Time Nonlinear and Switched Systems. Proc. of the 47th IEEE Conference on Decision and Control, 2008, Cancun, Mexico, 614-621.
17. [17] Jungers R., Ahmadi A.A., Parrilo A.P. and Roozbehani M. 2017. A Characterization of Lyapunov Inequalities for Stability of Switched Systems. IEEE Trans.on Aut.Control, 62(6) 3062-3067.
18. [18] Liberzon D., Ying C., Zharnitsky V. 2014. On almost Lyapunov functions. 53rd IEEE Conference on Decision and Control. Los Angeles, California, USA, 3083-3088.
19. [19] Liu S., Liberzon D. Zharnitsky V. 2016. On almost Lyapunov functions for Nonvanishing Vector Fields. IEEE 55th Conference on Decision and Control, Las Vegas, USA, 5557-5562.
20. [20] Defoort M., Djemai M. and Trenn S. 2014. Nondecreasing Lyapunov Functions. 21st Int. Symp. On Math. Theory of Networks and Systems, Groningen, Netherlands, 1038-1043.
21. [21] Chen G., Yang Y. 2016. New stability conditions for a class of linear time-varying systems. Automatica (71), 342-347.
22. [22] Michel A.N., Hou L., Liu D. 2015. Stability of Dynamical Systems: On the Role of Monotonic and Non-Monotonic Lyapunov Functions. Birkhauser Basel.
23. [23] Wang Z. and Hu G. 2017. Economic MPC of nonlinear systems with nonmonotonic Lyapunov functions and its application to HVAC control. Int. J. of Robust and Nonlinear Control 28, 2513-2527.
24. [24] Papachristodoulou A., Prajna S. 2002. On the Construction of Lyapunov Functions using the Sum of Squares Decomposition. IEEE 41st Conference on Decision and Control, ThP10-1, Las Vegas, USA.
25. [25] Zhang H., Zhongkui L., Zhihua Q., Lewis F. 2015. On constructing Lyapunov functions for multi-agent systems. Automatica (58), 39-42.