Optimal Control for a Class of Complex-Valued Nonlinear Systems

  • Ruizhuo SongEmail author
  • Qinglai Wei
  • Qing Li
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 166)


In this chapter, an optimal control scheme based on ADP is developed to solve infinite-horizon optimal control problems of continuous-time complex-valued nonlinear systems. A new performance index function is established based on complex-valued state and control. Using system transformations, the complex-valued system is transformed into a real-valued one, which overcomes Cauchy–Riemann conditions effectively. Based on the transformed system and the performance index function, a new ADP method is developed to obtain the optimal control law using neural networks. A compensation controller is developed to compensate the approximation errors of neural networks. Stability properties of the nonlinear system are analyzed and convergence properties of the weights for neural networks are presented. Finally, simulation results demonstrate the performance of the developed optimal control scheme for complex-valued nonlinear systems.


  1. 1.
    Adali, T., Schreier, P., Scharf, L.: Complex-valued signal processing: the proper way to deal with impropriety. IEEE Trans. Signal Process. 59(11), 5101–5125 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fang, T., Sun, J.: Stability analysis of complex-valued impulsive system. IET Control Theory Appl. 7(8), 1152–1159 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Yang, C.: Stability and quantization of complex-valued nonlinear quantum systems. Chaos, Solitons Fractals 42, 711–723 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hu, J., Wang, J.: Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans. Neural Netw. Learn. Syst. 23(6), 853–865 (2012)CrossRefGoogle Scholar
  5. 5.
    Huang, S., Li, C., Liu, Y.: Complex-valued filtering based on the minimization of complex-error entropy. IEEE Trans. Neural Netw. Learn. Syst. 24(5), 695–708 (2013)CrossRefGoogle Scholar
  6. 6.
    Hong, X., Chen, S.: Modeling of complex-valued wiener systems using B-spline neural network. IEEE Trans. Neural Netw. 22(5), 818–825 (2011)CrossRefGoogle Scholar
  7. 7.
    Goh, S., Mandic, D.: Nonlinear adaptive prediction of complex-valued signals by complex-valued PRNN. IEEE Trans. Signal Process. 53(5), 1827–1836 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bolognani, S., Smyshlyaev, A., Krstic, M.: Adaptive output feedback control for complex-valued reaction-advection-diffusion systems, In: Proceedings of American Control Conference, Seattle, Washington, USA, pp. 961–966, (2008)Google Scholar
  9. 9.
    Hamagami, T., Shibuya, T., Shimada, S.: Complex-valued reinforcement learning. In: Proceedings of IEEE International Conference on Systems, Man, and Cybernetics, Taipei, Taiwan, pp. 4175–4179 (2006)Google Scholar
  10. 10.
    Paulraj, A., Nabar, R., Gore, D.: Introduction to Space-Time Wireless Communications. Cambridge University Press, Cambridge (2003)Google Scholar
  11. 11.
    Mandic, D.P., Goh, V.S.L.: Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models. Wiley, New York (2009)CrossRefGoogle Scholar
  12. 12.
    Vamvoudakis, K.G., Lewis, F.L.: Online actor-critic algorithm to solve the continuous-time infinite horizon optimal control problem. Automatica 46(5), 878–888 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dierks, T., Jagannathan, S.: Online optimal control of affine nonlinear discrete-time systems with unknown internal dynamics by using time-based policy update. IEEE Trans. Neural Netw. Learn. Syst. 23(7), 1118–1129 (2012)CrossRefGoogle Scholar
  14. 14.
    Khalil, H.K.: Nonlinear System. Prentice-Hall, Upper Saddle River (2002)Google Scholar
  15. 15.
    Lewis, F.L., Jagannathan, S., Yesildirek, A.: Neural Network Control of Robot Manipulators and Nonlinear Systems. Taylor & Francis, New York (1999)Google Scholar
  16. 16.
    Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two coupled dynamos system. Chaos, Solitons Fractals 33, 178–187 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.University of Science and Technology BeijingBeijingChina
  2. 2.Institute of AutomationChinese Academy of SciencesBeijingChina

Personalised recommendations