On the region of attraction of dynamical systems: Application to Lorenz equations

• Autorzy: Hammami M. A. , Rettab N. H.

Identyfikatory

DOI

10.24425/acs.2020.134671

• Języki publikacji: EN

Abstrakty

EN

Many nonlinear dynamical systems can present a challenge for the stability analysis in particular the estimation of the region of attraction of an equilibrium point. The usual methodis based on Lyapunov techniques. For the validity of the analysis it should be supposed that the initial conditions lie in the domain of attraction. In this paper, we investigate such problem for a class of dynamical systems where the origin is not necessarily an equilibrium point. In this case, a small compact neighborhood of the origin can be estimated as an attractor for the system. We give a method to estimate the basin of attraction based on the construction of a suitable Lyapunov function. Furthermore, an application to Lorenz system is given to verify the effectiveness of the proposed method.

Słowa kluczowe

EN

nonlinear dynamical systems; Lyapunov function; basin of attraction; Lorenz equations

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2020
• Tom: Vol. 30, No. 3
• Strony: 389--409
• Opis fizyczny: Bibliogr. 21 poz., rys., wykr., wzory

Twórcy

autor

Hammami M. A.

• University of Sfax Tunisia, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia

autor

Rettab N. H.

• University of Sfax Tunisia, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia

Bibliografia

• [1] A. Ben Abdallah, I. Ellouze, and M. A. Hammami: Practical stability ofnonlinear time-varying cascade systems, J. Dyn. Control Sys.15(1) (2009), 45–62.
• [2] A. Ben Abdallah, M. Dlalaand M. A. Hammami: A new Lyapunov function for stability of time-varying nonlinear perturbed systems, Systems Control Lett., 56(3) (2007), 179–187.
• [3] A. Ben Makhloufand, M.A. Hammami: A nonlinear inequality and application to global asymptotic stability of perturbed systems, Math. Methods Appl. Sci., 38(12) (2015), 2496–2505.
• [4] C. F. Chuang, Y. J. Sun, and W. J. Wang: A novel synchronization scheme with a simple linear control and guaranteed convergence time for generalized Lorenz chaotic systems, Chaos, 22(4) (2012), 043108, 7 pp.
• [5] M. Corlessand, G. Leitmann: Controller design for uncertain systems via Lyapunov functions, Proceedings of the 1988 American Control Conference, Atlanta, Georgia, 1988.
• [6] M. Corless: Guaranteed rates of exponential convergence for uncertain systems, Journal of Optimization Theory and Applications, 64(3) (1990), 481-494.
• [7] H. Damak, M. A. Hammami and Y. J. Sun: The existence of limit cycle for perturbed bilinear systems, Kybernetika (Prague), 48(2) (2012), 177-189.
• [8] D. Krokavec and A. Filasova: A new D-stability area for linear discrete-time systems, Arch. Control Sci., 29(1) (2019), 5-23.
• [9] R. Gao: A novel track control for Lorenz system with single state feedback, Chaos Solitons Fractals,122(2019), 236-244.
• [10] F. Garofalo and G. Leitmann: Guaranteeing ultimate boundedness and exponential rate of convergence for a class of nominally linear uncertain systems, Journal of Dynamic Systems, Measurement, and Control,111(1989), 584-588.
• [11] B. Ghanmi, N. HadjTaieb and M. A. Hammami: Growth conditions forexponential stability of time-varying perturbed systems, Internat. J. Control, 86(6) (2013), 1086-1097.
• [12] W. Hahn:Stability of motion, Springer Verlag, New York, 1967.
• [13] M. A. Hammami: On the stability of nonlinear control systems with uncertainty, Journal of Dynamical and Control Systems, 7(2) (2001), 171-179.
• [14] M. Hammi and M. A. Hammami: Non-linear integral inequalities and applications to asymptotic stability, IMA J. Math. Control Inform, 32(4) (2015), 717-735.
• [15] H. Khalil: Nonlinear systems, Second Edition, Prentice Hall, 2002.
• [16] Y. Liu, Z. Wei, C. Li, A. Liu and L. Li: Attractor and bifurcation of forced Lorenz-84 system, Int. J. Geom. Methods Mod. Phys.,16(1) (2019),1950002, 20 pp.
• [17] A. M. Lyapunov: The general problem of the stability of motion, International Journal of Control, 55(3) (1992), 521-790.
• [18] R. Martinez-Guerra, J. C. Cruz-Victoria, R. Gonzalez-Galan, and R. Aguilar-Lopez: A new reduced-order observer design for the synchronization of Lorenz systems, Chaos Solitons Fractals, 28(2) (2006), 511-517.
• [19] Y.J. Sun: A simple observer design of the generalized Lorenz chaotic systems, Physics Letters A, 374(7) (2010), 933-937.
• [20] Y. J. Sun: Robust Stabilization for a Class of Uncertain Nonlinear Systems via a Novel Hybrid Control Applicable to Mechanical Systems, Advances in Mechanical Engineering, (2014), doi:10.1155/2014/952342.
• [21] Vaidyanathan Sundarapandian, Sambas Aceng, Mamat Mustafa, and W. S. Mada Sanjaya: A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot, Arch. Control Sci., 27(4) (2017), 4, 541-554.