Abstract
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.
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References
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)
Fang, T., Sun, J.: Stability analysis of complex-valued impulsive system. IET Control Theory Appl. 7(8), 1152–1159 (2013)
Yang, C.: Stability and quantization of complex-valued nonlinear quantum systems. Chaos, Solitons Fractals 42, 711–723 (2009)
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)
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)
Hong, X., Chen, S.: Modeling of complex-valued wiener systems using B-spline neural network. IEEE Trans. Neural Netw. 22(5), 818–825 (2011)
Goh, S., Mandic, D.: Nonlinear adaptive prediction of complex-valued signals by complex-valued PRNN. IEEE Trans. Signal Process. 53(5), 1827–1836 (2005)
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)
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)
Paulraj, A., Nabar, R., Gore, D.: Introduction to Space-Time Wireless Communications. Cambridge University Press, Cambridge (2003)
Mandic, D.P., Goh, V.S.L.: Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models. Wiley, New York (2009)
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)
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)
Khalil, H.K.: Nonlinear System. Prentice-Hall, Upper Saddle River (2002)
Lewis, F.L., Jagannathan, S., Yesildirek, A.: Neural Network Control of Robot Manipulators and Nonlinear Systems. Taylor & Francis, New York (1999)
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)
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Song, R., Wei, Q., Li, Q. (2019). Optimal Control for a Class of Complex-Valued Nonlinear Systems. In: Adaptive Dynamic Programming: Single and Multiple Controllers. Studies in Systems, Decision and Control, vol 166. Springer, Singapore. https://doi.org/10.1007/978-981-13-1712-5_6
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DOI: https://doi.org/10.1007/978-981-13-1712-5_6
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