Evolutionary Control of CML Systems

  • Ivan Zelinka
Part of the Studies in Computational Intelligence book series (SCI, volume 267)


This chapter is a continuation of an investigation on deterministic spatiotemporal chaos real-time control by means of selected evolutionary techniques. Real-time like behavior is specially defined and simulated with spatiotemporal chaos model based on mutually nonlineary joined n equations, so called Coupled Map Lattices. In total five evolutionary algorithms has been used for chaos control: differential evolution, self-organizingmigrating algorithm, genetic algorithm, simulated annealing and evolutionary strategies in a total of 15 versions. For modeling of spatiotemporal chaos behavior, the so called coupled map lattices were used based on logistic equation to generate chaos. The main aim of this investigation was to show that evolutionary algorithms, under certain conditions, are capable of controlling of CML deterministic chaos, when the cost function is properly defined alongside the parameters of selected evolutionary algorithms. Investigation consists of four different case studies with increasing simulation complexity. For all algorithms each simulation was evaluated 100 times in order to show and check robustness of used methods. All data were processed and used in order to get summarized results and graphs.


Cost Function Simulated Annealing Evolutionary Algorithm Differential Evolution Chaotic System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Beyer, H.: Theory of Evolution Strategies. Springer, New York (2001)Google Scholar
  2. 2.
    Cerny, V.: Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. J. Opt. Theory Appl. 45(1), 41–51 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, G.: Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton (2000)zbMATHGoogle Scholar
  4. 4.
    Chen, G., Dong, X.: From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore (1998)zbMATHCrossRefGoogle Scholar
  5. 5.
    Clerc, M.: Particle Swarm Optimization. ISTE Publishing Company (2006)Google Scholar
  6. 6.
    Das, S., Konar, A.: A swarm intelligence approach to the synthesis of two-dimensional IIR filters. Eng. Appl. Artif. Intell. 20(8), 1086–1096 (2007)CrossRefGoogle Scholar
  7. 7.
    Dashora, Y.: Improved and generalized learning strategies for dynamically fast and statistically robust evolutionary algorithms. Eng. Appl. Artif. Intel. (2007), doi:10.1016/j.engappai.2007.06.005Google Scholar
  8. 8.
    Davis, L.: Handbook of Genetic Algorithms. Van Nostrand Reinhold, Berlin (1996)Google Scholar
  9. 9.
    Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Chichester (2001)zbMATHGoogle Scholar
  10. 10.
    Deilami, M., Rahmani, C., Motlagh, M.: Control of spatio-temporal on-off intermittency in random driving diffusively coupled map lattices, Chaos, Solitons & Fractals, December 21 (2007)Google Scholar
  11. 11.
    Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, Japan, pp. 39–43 (1995)Google Scholar
  12. 12.
    Gilmore, R., Lefranc, M.: The Topology of Chaos: Alice in Stretch and Squeezeland. Wiley-Interscience, New York (2002)zbMATHGoogle Scholar
  13. 13.
    Grebogi, C., Lai, Y.C.: Controlling chaos. In: Schuster, H. (ed.) Handbook of Chaos Control. Wiley-VCH, New York (1999)Google Scholar
  14. 14.
    He, Q., Wang, L.: An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng. Appl. Artif. Intell. 20(1), 89–99 (2007)CrossRefGoogle Scholar
  15. 15.
    Hilborn, R.: Chaos and Nonlinear Dynamics. Oxford University Press, Oxford (1994)zbMATHGoogle Scholar
  16. 16.
    Holland, J.: Adaptation in Natural and Artificial Systems. Univ. Michigan Press, Ann Arbor (1975)Google Scholar
  17. 17.
    Hu, G., Xie, F., Xiao, J., Yang, J., Qu, Z.: Control of patterns and spatiotemporal chaos and its application. In: Schuster, H. (ed.) Handbook of Chaos Control. Wiley-VCH, New York (1999)Google Scholar
  18. 18.
    Hwang, G.-H.-H., Dong-Wan, K., Jae-Hyun, L., Young-Joo, A.: Design of fuzzy power system stabilizer using adaptive evolutionary algorithm. Eng. Appl. Artif. Intell. 21(1), 86–96 (2007)CrossRefGoogle Scholar
  19. 19.
    Just, W.: Principles of time delayed feedback control. In: Schuster, H. (ed.) Handbook of Chaos Control. Wiley-VCH, New York (1999)Google Scholar
  20. 20.
    Just, W., Benner, H., Reibold, E.: Theoretical and experimental aspects of chaos control by time-delayed feedback. Chaos 13, 259–266 (2003)CrossRefGoogle Scholar
  21. 21.
    Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Liu, L., Liu, W., Cartes, D.: Particle swarm optimization-based parameter identification applied to permanent magnet synchronous motors. Eng. Appl. Artif. Intell. (2007), doi:10.1016/j.engappai.2007.10.002Google Scholar
  23. 23.
    Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)CrossRefMathSciNetGoogle Scholar
  24. 24.
    May, R.: Simple mathematical model with very complicated dynamics. Nature 261, 45–67 (1976)CrossRefGoogle Scholar
  25. 25.
    Nolle, L., Goodyear, A., Hopgood, A.A., Picton, P.D., Braithwaite, N.StJ.: On Step Width Adaptation in Simulated Annealing for Continuous Parameter Optimisation. In: Reusch, B. (ed.) Fuzzy Days 2001. LNCS, vol. 2206, pp. 589–598. Springer, Heidelberg (2001)Google Scholar
  26. 26.
    Nolle, L., Zelinka, I., Hopgood, A., Goodyear, A.: Comparison of an self organizing migration algorithm with simulated annealing and differential evolution for automated waveform tuning. Adv. Eng. Software 36(10), 645–653 (2005)CrossRefGoogle Scholar
  27. 27.
    Ott, E., Grebogi, C., Yorke, J.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Price, K.: An Introduction to Differential Evolution. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization, pp. 79–108. McGraw-Hill, London (1999)Google Scholar
  29. 29.
    Richter, H.: An evolutionary algorithm for controlling chaos: The use of multiobjective fitness functions. In: Guervós, J.J.M., Adamidis, P.A., Beyer, H.-G., Fernández-Villacañas, J.-L., Schwefel, H.-P. (eds.) PPSN 2002. LNCS, vol. 2439, pp. 308–317. Springer, Heidelberg (2002)Google Scholar
  30. 30.
    Richter, H., Reinschke, K.: Optimization of local control of chaos by an evolutionary algorithm. Physica D 144, 309–334 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Schuster, H.: Handbook of Chaos Control. Wiley-VCH, New York (1999)zbMATHCrossRefGoogle Scholar
  32. 32.
    Stewart, I.: The Lorenz attractor exists. Nature 406, 948–949 (2000)CrossRefGoogle Scholar
  33. 33.
    Wang, X., Chen, G.: Chaotification via arbitrarily small feedback controls: Theory, method, and applications. Int. J. of Bifur. Chaos 10, 549–570 (2000)zbMATHGoogle Scholar
  34. 34.
    Zahra, R., Motlagh, M.: Control of spatiotemporal chaos in coupled map lattice by discrete-time variable structure control. Phys. Lett. A 370(3–4), 302–305 (2007)Google Scholar
  35. 35.
    Zelinka, I.: SOMA — Self Organizing Migrating Algorithm. In: Babu, B., Onwubolu, G. (eds.) New Optimization Techniques in Engineering, pp. 167–218. Springer, New York (2004)Google Scholar
  36. 36.
    Zelinka, I.: Investigation on Evolutionary Deterministic Chaos Control. In: IFAC, Prague 2005 (2005a)Google Scholar
  37. 37.
    Zelinka, I.: Investigation on Evolutionary Deterministic Chaos Control — Extended Study. In: 19th International Conference on Simulation and Modeling, ECMS 2005, Riga, Latvia, June 1–4 (2005b)Google Scholar
  38. 38.
    Zelinka, I.: Investigation on realtime deterministic chaos control by means of evolutionary algorithms. In: Proc. First IFAC Conference on Analysis and Control of Chaotic Systems, Reims, France, pp. 211–217 (2006)Google Scholar
  39. 39.
    Zelinka, I.: Real-time deterministic chaos control by means of selected evolutionary algorithms. Eng. Appl. Artif. Intell (2008), doi:10.1016/j.engappai.2008.07.008Google Scholar
  40. 40.
    Zelinka, I., Nolle, L.: Plasma reactor optimizing using differential evolution. In: Price, K., Lampinen, J., Storn, R. (eds.) Differential Evolution: A Practical Approach to Global Optimization, pp. 499–512. Springer, New York (2006)Google Scholar
  41. 41.
    Zelinka, I., Senkerik, R., Navratil, E.: Investigation on Evolutionary Optimitazion of Chaos Control, Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.07.045Google Scholar
  42. 42.
    Zou, Y., Luo, X., Chen, G.: Pole placement method of controlling chaos in DC-DC buck converters. Chinese Phys. 15, 1719–1724 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ivan Zelinka
    • 1
    • 2
  1. 1.Faculty of Applied InformaticsTomas Bata University in ZlinZlinCzech Republic
  2. 2.Faculty of Electrical Engineering and Computer ScienceVSB-TUOOstrava-PorubaCzech Republic

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