Lyapunov-based Methods for Maximizing the Domain of Attraction


  • Houssem Mahmoud JERBI University of Hail
  • Faiçal HAMIDI ISIMG - Tunisia
  • Sondess BEN AOUN College of Computer Science and Engineering - Department of Computer Engineering, University of Hail
  • Severus Constantin OLTEANU
  • Dumitru POPESCU


Lyapunov function, nonlinear model, asymptotic stability, equilibrium point, genetic algorithm, threshold accepted algorithm, LMI


This paper investigates Lyapunov approaches to expand the domain of attraction (DA) of nonlinear autonomous models. These techniques had been examined for creating generic numerical procedures centred on the search of rational and quadratic Lyapunov functions. The outcomes are derived from all investigated methods: the method of estimation via Threshold Accepted Algorithm (TAA), the method of estimation via a Zubov technique and the method of estimation via a linear matrix inequality (LMI) optimization and genetic algorithms (GA). These methods are effective for a large group of nonlinear models, they have a significant ability of improvement of the attraction domain area and they are distinguished by an apparent propriety of direct application for compact and nonlinear models of high degree. The validity and the effectiveness of the examined techniques are established based on a simulation case analysis. The effectiveness of the presented methods is evaluated and discussed through the study of the renowned Van der Pol model.

Author Biographies

Houssem Mahmoud JERBI, University of Hail

Dr, Eng.

Faiçal HAMIDI, ISIMG - Tunisia


Sondess BEN AOUN, College of Computer Science and Engineering - Department of Computer Engineering, University of Hail


Severus Constantin OLTEANU





Alshammari, O., Mahyuddin, M. N., and Jerbi, H. (2020). A Neural Network-Based Adaptive Backstepping Control Law With Covariance Resetting for Asymptotic Output Tracking of a CSTR Plant. IEEE Access, 8, 29755-29766, 2020.

Alshammari, O., Mahyuddin, M. N., and Jerbi, H. (2019). An Advanced PID Based Control Technique With Adaptive Parameter Scheduling for A Nonlinear CSTR Plant. IEEE Access, 7, 158085-158094, 2019.

Alshammari, O., Mahyuddin, M. N., and Jerbi, H. (2018). A survey on control techniques of a benchmarked continuous stirred tank reactor. Journal of Engineering Science and Technology, 13(10), 3277-3296, 2018.

Bacha, A., Jerbi, H., and Braiek, N. B. (2006, October). An approach of asymptotic stability domain estimation of discrete polynomial systems. In The Proceedings of the Multiconference on†Computational Engineering in Systems Applications†(Vol. 1, pp. 288-292). IEEE, 2006.

Borne, P., Popescu, D., Filip, F. G., and Stefanoiu, D. (2013). Optimisation en sciences de l’ingnieur: mthodes exactes. Herms sciences publications, 2013.

Chesi, G., and Colaneri, P. (2017). Homogeneous rational Lyapunov functions for performance analysis of switched systems with arbitrary switching and dwell time constraints. IEEE Transactions on Automatic Control, 62(10), 5124-5137, 2017.

Chesi, G. (2013). Rational Lyapunov functions for estimating and controlling the robust domain of attraction. Automatica, 49(4), 1051-1057, 2013.

Chesi, G. (2009). Estimating the domain of attraction for non-polynomial systems via LMI optimizations. Automatica, 45(6), 1536-1541, 2009.

Chesi, G. (2008). Optimal representation matrices for solving polynomial systems via LMI. International Journal of Pure and Applied Mathematics, 45(3), 397, 2008.

Chesi, G. (2007). On the gap between positive polynomials and SOS of polynomials. IEEE Transactions on Automatic Control, 52(6), 1066-1072, 2007.

Chesi, G., Garulli, A., Tesi, A., and Vicino, A. (2005). Polynomially parameter-dependent

Lyapunov functions for robust stability of polytopic systems: an LMI approach. IEEE transactions on Automatic Control, 50(3), 365-370, 2005.

Chesi, G. (2004). Computing output feedback controllers to enlarge the domain of attraction in polynomial systems. IEEE Transactions on Automatic Control, 49(10), 1846-1853, 2004.

Chesi, G., Garulli, A., Tesi, A., and Vicino, A. (2003). Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach. International Journal of Robust and Nonlinear Control: IFAC Affiliated Journal, 13(13), 1239-1257, 2003.

Chesi, G., Garulli, A., Tesi, A., and Vicino, A. (2003). Solving quadratic distance problems: an LMI-based approach. IEEE Transactions on Automatic Control, 48(2), 200-212, 2003.

Chesi, G., Garulli, A., Tesi, A., and Vicino, A. (2003). Homogeneous Lyapunov functions for systems with structured uncertainties. Automatica, 39(6), 1027-1035, 2003.

Camilli, F., Grne, L., and Wirth, F. (2009, August). Domains of attraction of interconnected systems: A Zubov method approach. In 2009 European Control Conference (ECC) (pp. 91-96). IEEE, 2009.

Chermnykh, S. V. (2016). Carleman linearization and normal forms for differential systems with quasi-periodic coefficients. SpringerPlus, 5(1), 1347, 2016.

Dauphin-Tanguy, G., Foulloy, L., and Popescu, D. (2004). Modlisation, identification et commande des systmes. Ed. Academiei romne, 2004.

Deutscher, J. (2003, September). Asymptotically exact input-output linearization using Carleman linearization. In 2003 European Control Conference (ECC) (pp. 1726-1731). IEEE, 2003.

Fujisaki, Y., and Sakuwa, R. (2006). Estimation of asymptotic stability regions via homogeneous polynomial Lyapunov functions. International Journal of Control, 79(06), 617-623, 2006.

Genesio, R., Tartaglia, M., and Vicino, A. (1985). On the estimation of asymptotic stability regions: State of the art and new proposals. IEEE Transactions on automatic control, 30(8), 747-755, 1985.

Hamidi, F., Jerbi, H., Olteanu, S. C., and Popescu, D. (2019). An Enhanced Stabilizing Strategy for Switched Nonlinear Systems. Studies in Informatics and Control, 28(4), 391-400, 2019.

Hamidi, F., Jerbi, H., Aggoune, W., Djemai, M., and Abdelkrim, M. N. (2013). Enlarging the domain of attraction in nonlinear polynomial systems. International Journal of Computers Communications and Control, 8(4), 538-547, 2013.

Hamidi, F., Jerbi, H., Aggoune, W., Djemai, M., and Abdkrim, M. N. (2011, March). Enlarging region of attraction via LMI-based approach and Genetic Algorithm. In 2011 International Conference on Communications, Computing and Control Applications (CCCA) (pp. 1-6). IEEE, 2011.

Hamidi, F., and Jerbi, H. (2009, March). On the estimation of a maximal Lyapunov function and domain of attraction determination via a genetic algorithm. In 2009 6th International Multi-Conference on Systems, Signals and Devices (pp. 1-6). IEEE, 2009.

Henrion, D., and Korda, M. (2013). Convex computation of the region of attraction of polynomial control systems. IEEE Transactions on Automatic Control, 59(2), 297-312, 2013.

Hachicho, O. (2007). A novel LMI-based optimization algorithm for the guaranteed estimation of the domain of attraction using rational Lyapunov functions. Journal of the Franklin Institute, 344(5), 535-552, 2007.

Hernandez, C. N.,and Banks, S. P. (2004). A generalization of lyapunov’s equation to nonlinear systems. IFAC Proceedings Volumes, 37(13), 745-750, 2004.

Hachicho, O., and Tibken, B. (2002, December). Estimating domains of attraction of a class of nonlinear dynamical systems with LMI methods based on the theory of moments. In Proceedings of the 41st IEEE Conference on Decision and Control, 2002. (Vol. 3, pp.3150-3155). IEEE, 2002.

Jerbi, H. (2017). Estimations of the domains of attraction for classes of nonlinear continuous polynomial systems. Arabian Journal for Science and Engineering, 42(7), 2829-2837, 2017.

Khalil, H. K. (2009). Lyapunov stability. Control Systems, Robotics and Automation, Volume XII: Nonlinear, Distributed, and Time Delay Systems-I, 115, 2009.

Khalil, H. K., and Grizzle, J. W. (2002). Nonlinear systems (Vol. 3). Upper Saddle River, NJ: Prentice hall, 2002.

Loccufier, M., and Noldus, E. (2000). A new trajectory reversing method for estimating stability regions of autonomous nonlinear systems. Nonlinear dynamics, 21(3), 265-288, 2000.

Loccufier, M., and Noldus, E. (1995). On the estimation of asymptotic stability regions for autonomous nonlinear systems. IMA Journal of Mathematical Control and Information, 12(2), 91-109, 1995.

Matallana, L. G., Blanco, A. M., and Bandoni, J. A. (2010). Estimation of domains of attraction: A global optimization approach. Mathematical and Computer Modelling, 52(3-4), 574-585, 2010.

Najafi, E., Babuka, R.,and Lopes, G. A. (2016). A fast sampling method for estimating the domain of attraction. Nonlinear Dynamics, 86(2), 823-834, 2016.

Panikhom, S., Sarasiri, N., and Sujitjorn, S. (2010). Hybrid bacterial foraging and tabu search optimization (BTSO) algorithms for Lyapunovs stability analysis of nonlinear systems. International Journal of Mathematics and Computers in Simulation, 3(4), 81-89, 2010.

Popescu, D., and Dion, J. M. (2000). Commande Optimale Optimisation des Systmes, 2000.

Rozgonyi, S., Hangos, K., and Szederknyi, G. (2010). Determining the domain of attraction of hybrid non-linear systems using maximal Lyapunov functions. Kybernetika, 46(1), 19-37, 2010.

Topcu, U., Packard, A., and Seiler, P. (2008). Local stability analysis using simulations and sum-of-squares programming. Automatica, 44(10), 2669-2675, 2008.

Vannelli, A., and Vidyasagar, M. (1985). Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica, 21(1), 69-80, 1985.

Additional Files



Most read articles by the same author(s)

Obs.: This plugin requires at least one statistics/report plugin to be enabled. If your statistics plugins provide more than one metric then please also select a main metric on the admin's site settings page and/or on the journal manager's settings pages.