Design Of a New Optimal Controller for a Particular Class of Chaotic Systems Using the Artificial Bee Colony Algorithm (ABC)

Document Type : Original Article


Department of Electrical Engineering, Langarud Branch, Islamic Azad University, Langarud, Iran


The primary aim of this paper is to devise an optimal regulator for stabilizing a distinct class of chaotic systems through a systematic two-step approach. Initially, the chaotic system undergoes transformation into state-dependent equations. Subsequently, the State Dependent Riccati Equation (SDRE) is tackled via the power series method, facilitating the determination of the optimal control law. Ensuring a suitable regulatory response involves the utilization of an intuitive optimization algorithm of a naturalistic nature, with a focus on optimizing the weight matrices within the SDRE equation. Employing the Artificial Bee Colony (ABC) algorithm, we derive the weighted matrices, leveraging the honey bee algorithm to fine-tune the gain coefficients by minimizing the chosen fitness function. The fitness function, represented as the sum of squares of system state errors, proves instrumental in achieving effective stabilization of the chaotic system, minimizing error, enhancing response speed, and reducing control costs. Through simulation, we scrutinize the effectiveness of regulators designed to stabilize and control chaotic systems, particularly comparing the regulatory performance of this algorithm against the SDRE method.


  • Kellert, S. H. (1994). In the wake of chaos: Unpredictable order in dynamical systems. University of Chicago press.
  • Werndl, C. (2013). What are the new implications of chaos for unpredictability? Retrieved from
  • Boeing, G. (2016). Visual analysis of nonlinear dynamical systems: Chaos, fractals, self-similarity and the limits of prediction. Systems, 4(4), 37.
  • Akhavan, A., Samsudin, A., & Akhshani, A. (2011). A symmetric image encryption scheme based on combination of nonlinear chaotic maps. Journal of the Franklin Institute, 348(8), 1797–1813. doi:10.1016/j.jfranklin.2011.05.001
  • Behnia, S., Akhshani, A., Mahmodi, H., &Akhavan, A. (2008). A novel algorithm for image encryption based on mixture of chaotic maps. Chaos, Solitons& Fractals, 35(2), 408-419.
  • Nehmzow, U., & Walker, K. (2005). Quantitative description of robot–environment interaction using chaos theory. Robotics and Autonomous Systems, 53(3–4), 177–193. doi:10.1016/j.robot.2005.09.009
  • Goswami, A., Thuilot, B., & Espiau, B. (1998). A study of the passive gait of a compass-like biped robot: Symmetry and chaos. The International Journal of Robotics Research, 17(12), 1282–1301.
  • Sivakumar, B. (2000). Chaos theory in hydrology: important issues and interpretations. Journal of Hydrology, 227(1–4), 1–20. doi:10.1016/s0022-1694(99)00186-9
  • Steven, S. (2003). SYNC, The emerging science of spontaneous order. New York: Theia.
  • Chlouverakis, K. E., & Sprott, J. C. (2006). Chaotic hyperjerk systems. Chaos, Solitons & Fractals, 28(3), 739-746.
  • Coelho, S., & Luis De Andrade, D. (2009). PID control design for chaotic synchronization using a tribesoptimization approach”. Chaos, Solitons and Fractals, 42, 634–640.
  • Li, X., Xu, W., & Li, R. (2009). Chaos synchronization of the energy resource system”. Chaos, Solitons and Fractals, 40, 642–652.
  • Yan, H., Fahroo, F., & Ross, I. M. (2001). Optimal feedback control laws by Legendre pseudospectral approximations. Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148). Presented at the Proceedings of American Control Conference, Arlington, VA, USA. doi:10.1109/acc.2001.946110
  • Çimen, T. (2010). Systematic and effective design of nonlinear feedback controllers via the optimal control problems. J. Guid. Control Dyn, 31(4), 927–936.
  • Beeler, S. C., & Cox, D. E. (2004). State-dependent Riccati equation regulation of systems with state and control nonlinearities(No. NASA/CR-2004-213245).
  • Katebi, M. R., Grimble, M. J., & Zhang, Y. (1997). H∞ robust control design for dynamic ship positioning. IEE Proceedings - Control Theory and Applications, 144(2), 110–120. doi:10.1049/ip-cta:19971030
  • Lin, F. (n.d.). Robust Control Design : An Optimal Control Approach. 137–140.
  • Karaboga, D. (2005). An Idea Based on Honey Bee Swarm for Numerical Optimization. 1–10.
  • Wan, L. X., & Mei, A. (2013). An efficient and robust artificial bee colony algorithm for numerical optimization. Computers & Operations Research, 1256–1265. doi:10.1016/j.cor.2012.12.006
  • Gao, W., Liu, S., & Huang, L. (2012). A global best artificial bee colony algorithm for global optimization. Journal of Computational and Applied Mathematics, 236(11), 2741-2753. org/10.1016/
  • Kiran, M. S., Hakli, H., Gunduz, M., & Uguz, H. (2015). Artificial bee colony algorithm with variable search strategy for continuous optimization. Information Sciences, 300, 140-157. org/10.1016/j.ins.2014.12.043
  • Karaboga, D., Gorkemli, B., Ozturk, C., & Karaboga, N. (2014). A comprehensive survey: artificial bee colony (ABC) algorithm and applications. Artificial intelligence review, 42, 21-57. org/10.1007/s10462-012-9328-0
  • Karaboga, D., & Kaya, E. (2016). An adaptive and hybrid artificial bee colony algorithm (aABC) for ANFIS training, Appl. Appl. Soft Comput, 49, 423–436. doi:10.1016/j.asoc.2016.07.039
  • Kalayci, C. B., Ertenlice, O., Akyer, H., & Aygoren, H. (2017). An artificial bee colony algorithm with feasibility enforcement and infeasibilitytoleration procedures for cardinality constrained portfolio optimization. Expert Syst. Appl, 85, 61–75.
  • Karaboga, Dervis, & Akay, B. (2009). A comparative study of Artificial Bee Colony algorithm. Applied Mathematics and Computation, 214(1), 108–132. doi:10.1016/j.amc.2009.03.090
  • Liao, T., Aydın, D., & Stützle, T. (2013). Artificial bee colonies for continuous optimization: Experimental analysis and improvements. Swarm Intelligence, 7(4), 327–356. doi:10.1007/s11721-013-0088-5
  • BOLAJI, A. L. A., Khader, A. T., Al-Betar, M. A., & Awadallah, M. A. (2013). Artificial bee colony algorithm, its variants and applications: A survey. Journal of Theoretical & Applied Information Technology, 47(2).
  • Ozkis, A., & Babalik, A. (2013). Accelerated ABC (A-ABC) algorithm for continuous optimization problems. Lecture Notes on Software Engineering, 1(3), 262.
  • Akgul, A., Hussain, S., & Pehlivan, I. (2016). A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik, 127(18), 7062-7071. org/10.1016/j.ijleo.2016.05.010