Abstract
In this chapter, an online adaptive dynamic programming (ADP) based optimal control scheme is developed for continuous-time chaotic systems. The idea is to use ADP algorithm to obtain the optimal control input which makes the performance index function reach the optimum. The expression of the performance index function for the chaotic system is first presented. The online ADP algorithm is presented to get the optimal control law. In the ADP structure, the neural networks are used to construct the critic network and action network, which can obtain the approximate performance index function and the control input, respectively. It is proven that the critic parameter error dynamics and the closed-loop chaotic systems are uniformly ultimately bounded exponentially. Simulation results are given to illustrate the performance of the established optimal control method.
References
Chen, S., Lü, J.: Synchronization of an uncertain unified chaotic system via adaptive control. Chaos Solitons Fractals 14, 643 (2002)
Feng, J., Zhang, Q., Wang, W., Hao, S.: Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems. Chin. Phys. B 22(9), 090503 (2013)
Fu, J., He, H., Zhou, X.: Adaptive learning and control for MIMO system based on adaptive dynamic programming. IEEE Trans. Neural Netw. 22(7), 1133–1148 (2011)
Giuseppe, G.: Continuous-time chaotic systems: Arbitrary full-state hybrid projective synchronization via a scalar signal. Chin. Phys. B 22(8), 080505 (2013)
Khalil, H.: Nonlinear System. Prentice-Hall, NJ (2002)
Lewis, F., Vamvoudakis, K.: Reinforcement learning for partially observable dynamic processes: adaptive dynamic programming using measured output data. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 41(1), 14–25 (2011)
Liu, S., Chen, L.: Chaos synchronization of a chain network based on a sliding mode control. Chin. Phys. B 22, 100506 (2013)
Luo, Y., Du, M.: A self-adapting image encryption algorithm based on spatiotemporal chaos and ergodic matrix. Chin. Phys. B 22(8), 080503 (2013)
Ma, T., Fu, J.: Global exponential synchronization between Lü system and Chen system with unknown parameters and channel time-delay. Chin. Phys. B 20, 050511 (2011)
Ma, T., Fu, J., Sun, Y.: An improved impulsive control approach to robust lag synchronization between two different chaotic systems. Chin. Phys. B 19, 090502 (2010)
Ma, T., Zhang, H., Fu, J.: Exponential synchronization of stochastic impulsive perturbed chaotic Lur’e systems with time-varying delay and parametric uncertainty. Chin. Phys. B 17(12), 4407 (2008)
Murray, J., Cox, C., Lendaris, G., Saeks, R.: Adaptive dynamic programming. IEEE Trans. Syst. Man. Cybern. Part C: Appl. Rev. 32, 140–153 (2002)
Song, R., Xiao, W., Sun, C.: Optimal tracking control for a class of unknown discrete-time systems with actuator saturation via data-based ADP algorithm. Acta Automatica Sinica 39, 1413–1420 (2013)
Song, R., Xiao, W., Sun, C., Wei, Q.: Approximation-error-ADP-based optimal tracking control for chaotic systems with convergence proof. Chin. Phys. B 22(9), 090502 (2013)
Song, R., Xiao, W., Wei, Q.: A new approach of optimal control for a class of continuous-time chaotic systems by an online ADP algorithm. Chin. Phys. B 23(5), 050504 (2014)
Song, R., Xiao, W., Zhang, H.: Multi-objective optimal control for a class of unknown nonlinear systems based on finite-approximation-error ADP algorithm. Neurocomputing 119, 212–221 (2013)
Song, R., Zhang, H., Luo, Y., Wei, Q.: Optimal control laws for time-delay systems with saturating actuators based on heuristic dynamic programming. Neurocomputing 73, 3020–3027 (2010)
Vamvoudakis, K., Lewis, F.: Online actor-critic algorithm to solve the continuous-time infinite horizon optimal control problem. Automatica 46, 878–888 (2010)
Zhang, H., Huang, W., Wang, Z., Wang, Z., Chai, T.: Adaptive synchronization between two different chaotic systems with unknown parameters. Phys. Lett. A 350, 363–366 (2006)
Zhang, H., Lewis, F.: Multi-agent differential graphical games: Online adaptive learning solution for synchronization with optimality. Automatica 48(8), 1598–1611 (2012)
Zhang, H., Ma, T., Fu, J., Tong, S.: Robust lag synchronization between two different chaotic systems via dual-stage impulsive control. Chin. Phys. B 18, 3751 (2009)
Zhang, H., Wang, Z., Liu, D.: Chaotifying fuzzy hypertolic model using adaotive inverse optimal control approach. Int. J. Bifur. Chaos 14, 3505 (2004). doi:10.1142/S0218127404011442
Zhang, H., Wei, Q., Liu, D.: An iterative adaptive dynamic programming method for solving a class of nonlinear zero-sum differential games. Automatica 47(1), 207–214 (2011)
Zhang, H., Wei, Q., Luo, Y.: A novel infinite-time optimal tracking control scheme for a class of discrete-time nonlinear systems via the greedy HDP iteration algorithm. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 38(4), 937–942 (2008)
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Wei, Q., Song, R., Li, B., Lin, X. (2018). A New Approach for a Class of Continuous-Time Chaotic Systems Optimal Control by Online ADP Algorithm. In: Self-Learning Optimal Control of Nonlinear Systems. Studies in Systems, Decision and Control, vol 103. Springer, Singapore. https://doi.org/10.1007/978-981-10-4080-1_8
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DOI: https://doi.org/10.1007/978-981-10-4080-1_8
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