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A new switching time optimization technique for multi-switching systems

  • *Corresponding author: Changjun Yu

    *Corresponding author: Changjun Yu 

This work is supported by National Natural Science Foundation of China(NSFC), Grant No.11871039, Science and Technology Commission of Shanghai Municipality(STCSM), Grant No. 20JC1413900 and Australian Research Council Discovery Grant

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  • For a multi-switching system, it consists of multiple parallel switching systems. The optimal control problem controlled by a multi-switching system is to determine the optimal time sequence for each of the parallel switching systems. The time-scaling transformation is a well-known switching time optimization approach which has been widely used in various problem settings. However, the time-scaling transformation requires that these parallel switching systems have the same number of subsystems and are designed to switch simultaneously. Thus, it is not applicable for solving optimal control problems of general multi-switching systems. This paper presents a new technique for optimizing the switching times of multi-switching systems.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  The structure of a switching system

    Figure 2.  The structure of a multi-switching system

    Figure 3.  The time-scaling function $ \mu(s\mid\boldsymbol\theta) $

    Figure 4.  The variable switching times to be optimized

    Figure 5.  The first switching time transformation process

    Figure 6.  The second switching time transformation process

    Figure 7.  Optimal state trajectories for Example 1 obtained by using the two techniques

    Figure 8.  Optimal state trajectories for Case 1 of Example 2 obtained by using the two techniques

    Figure 9.  Optimal state trajectories for Case 2 and Case 3 of Example 2 obtained by using the traditional time scaling transformation and our proposed methods

    Figure 10.  Optimal state trajectories for Example 3

    Table 1.  Optimal costs for Example 1 obtained by using the two techniques

    Time-scaling method Proposed method
    Optimal cost $ g_0^\star $ 17.0752 14.2028
    Optimal switching times $ \boldsymbol\tau_1^\star, \boldsymbol\tau_2^\star=[0.61, 0.86] $ $ \boldsymbol\tau_1^\star=[0.46, 0.94] $
    $ \boldsymbol\tau_2^\star=[0.20, 1.17] $
     | Show Table
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    Table 2.  Optimal costs obtained by using the two techniques for Example 2

    Optimal cost $ g_0^* $
    Using method $ Case\; 1 $ $ Case \; 2 $ $ Case\; 3 $
    Time-scaling method 1.6632$ \times 10^4 $ 4.0932$ \times 10^4 $ -
    Proposed method 1.6632$ \times10^4 $ 4.0932$ \times 10^4 $ 1.6650$ \times10^4 $
     | Show Table
    DownLoad: CSV
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