JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2011, Vol. 46 ›› Issue (10): 45-56.

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Stage-by-Stage optimization approach to optimal control for general timedelay systems

ZHANG Huan-shui1, SONG  Xin-min1, XIE Li-hua2   

  1. 1. School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong,  China;
    2. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798.
  • Received:2011-06-22 Online:2011-10-20 Published:2011-10-18
  • About author:ZHANG Huanshui(1963- ),Male, Professor, His interests include optimal estimation, robust filtering and control,time delay systems, singular systems, wireless communication, and signal processing. Email:
  • Supported by:

    Supported by the Taishan Scholar Construction Engineering by Shandong Government;the National Natural Science Foundation for Distinguished Young Scholars of China (60825304);the Major State Basic Research Development Program of China (973 Program) (2009cb320600)


As it is well known that the optimal control of delay-free systems has been well solved with dynamic programming since 60s of last century, however the classical dynamic programming has not been successfully applied to time delay systems so as to derive the controller step by step as for delayfree systems. In this paper, we will propose a stage-by-stage optimization approach to the optimal control problem for general delayed systems by introducing an associated dual backward stochastic system and applying inner-product theory. The proposed approach allows us to complete the square of linear quadratic (LQ) form step by step for the general time delay systems and then present the analytical solution to the controller. It is interesting to show that the controller design for time delay systems is equivalent to the estimator design of different signals from the dual backward stochastic time delay systems.
The proposed general approach is powerful as it allows us to reduce the optimal control for time delay systems from one of determining an entire control sequence at once to one of determining the elements of the sequences singly and recursively. It is trustful that the proposed stage-by-stage approach can be  applied to solve other related complex control problems such as the necessary and sufficient solution to H∞ control and so on.

Key words: Time delay, optimal control, completing the square, dynamic programming, multiplicative noise

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