Nearly Finite-Horizon Optimal Control for Nonaffine Time-Delay 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, a novel ADP algorithm is developed to solve the nearly optimal finite-horizon control problem for a class of deterministic nonaffine nonlinear time-delay systems. The idea is to use ADP technique to obtain the nearly optimal control which makes the optimal performance index function close to the greatest lower bound of all performance index functions within finite time. The proposed algorithm contains two cases with respective different initial iterations. In the first case, there exists control policy which makes arbitrary state of the system reach to zero in one time step. In the second case, there exists a control sequence which makes the system reach to zero in multiple time steps. The state updating is used to determine the optimal state. Convergence analysis of the performance index function is given. Furthermore, the relationship between the iteration steps and the length of the control sequence is presented. Two neural networks are used to approximate the performance index function and compute the optimal control policy for facilitating the implementation of ADP iteration algorithm. At last, two examples are used to demonstrate the effectiveness of the proposed ADP iteration algorithm.


  1. 1.
    Niculescu, S.: Delay Effects on Stability: A Robust Control Approach. Springer, Berlin (2001)zbMATHGoogle Scholar
  2. 2.
    Gu, K., Kharitonov, V., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  3. 3.
    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(16–18), 3020–3027 (2010)CrossRefGoogle Scholar
  4. 4.
    Huang, J., Lewis, F.: Neural-network predictive control for nonlinear dynamic systems with time-delay. IEEE Trans. Neural Netw. 14(2), 377–389 (2003)CrossRefGoogle Scholar
  5. 5.
    Chyung, D.: On the controllability of linear systems with delay in control. IEEE Trans. Autom. Control 15(2), 255–257 (1970)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Phat, V.: Controllability of discrete-time systems with multiple delays on controls and states. Int. J. Control 49(5), 1645–1654 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chyung, D.: Discrete optimal systems with time delay. IEEE Trans. Autom. Control 13(1), 117 (1968)CrossRefGoogle Scholar
  8. 8.
    Chyung, D., Lee, E.: Linear optimal systems with time delays. SIAM J. Control 4(3), 548–575 (1966)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, F., Jin, N., Liu, D., Wei, Q.: Adaptive dynamic programming for finite-horizon optimal control of discrete-time nonlinear systems with \(\epsilon \)-error bound. IEEE Trans. Neural Netw. 22, 24–36 (2011)CrossRefGoogle Scholar
  10. 10.
    Manu, M., Mohammad, J.: Time-Delay Systems Analysis, Optimization and Applications. North-Holland, New York (1987)Google 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