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Time-delay optimal control of a fed-batch production involving multiple feeds

  • * Corresponding author: Chongyang Liu

    * Corresponding author: Chongyang Liu 
Abstract Full Text(HTML) Figure(3) / Table(3) Related Papers Cited by
  • In this paper, we consider time-delay optimal control of 1, 3-propan-ediol (1, 3-PD) fed-batch production involving multiple feeds. First, we propose a nonlinear time-delay system involving feeds of glycerol and alkali to formulate the production process. Then, taking the feeding rates of glycerol and alkali as well as the terminal time of process as the controls, we present a time-delay optimal control model subject to control and state constraints to maximize 1, 3-PD productivity. By a time-scaling transformation, we convert the optimal control problem into an equivalent problem with fixed terminal time. Furthermore, by applying control parameterization and constraint transcription techniques, we approximate the equivalent problem by a sequence of finite-dimensional optimization problems. An improved particle swarm optimization algorithm is developed to solve the resulting optimization problems. Finally, numerical results show that 1, 3-PD productivity increases considerably using the obtained optimal control strategy.

    Mathematics Subject Classification: Primary: 49J21, 49M37; Secondary: 34K34.

    Citation:

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  • Figure 1.  Optimal feeding rates of glycerol and alkali in Phs. Ⅱ-Ⅵ

    Figure 2.  Concentration changes of biomass, glycerol and 1, 3-PD with respect to fermentation time

    Figure 3.  1, 3-PD productivity changes with respect to fermentation time. Stars represent the 1, 3-PD productivity in experiment [21], and solid line denotes the 1, 3-PD productivity in this work

    Table 1.  Phase characteristics in fed-batch process [18]

    Phase Start time (h) End time (h) Number of processes Process duration (s)
    Feeding Batch Feeding Batch
    0 5.3300 0 1 0 19188
    5.3300 6.1078 28 28 5 95
    6.1078 7.1356 37 37 7 93
    7.1356 8.8300 61 61 8 92
    8.8300 12.1356 119 119 7 93
    12.1356 15.8300 133 133 6 94
    15.8300 18.0800 81 81 4 96
    18.0800 19.8300 63 63 3 97
    19.8300 23.8300 144 144 2 98
    23.8300 24.1633 12 12 1 99
     | Show Table
    DownLoad: CSV

    Table 2.  The kinetic parameters and critical concentrations in system (1) [14]

    $ \Delta_1 $ $ k_1 $ $ m_2 $ $ Y_2 $ $ \Delta_2 $ $ k_2 $ $ m_3 $
    0.8 0.28 1.927 0.0063 6.8489 17.7296 -3.2819
    $ Y_3 $ $ \Delta_3 $ $ k_3 $ $ m_4 $ $ Y_4 $ $ \Delta_4 $ $ k_4 $
    80.6096 10.3687 15.50 -0.97 33.07 5.74 85.71
    $ c_1 $ $ c_2 $ $ c_3 $ $ c_4 $ $ x_{*1} $ $ x_{*2} $ $ x_{*3} $
    0.025 0.06 2.81 65.5226 0.01 0 0
    $ x_{*4} $ $ x_{*5} $ $ x^{*}_1 $ $ x^{*}_2 $ $ x^{*}_3 $ $ x^{*}_4 $ $ x^{*}_5 $
    0 0 9 2039 1036 1026 360.9
     | Show Table
    DownLoad: CSV

    Table 3.  The bounds of feeding rates in Phs.Ⅱ-Ⅹ [21]

    Phases Upper bounds ($ u_1 $, $ u_2 $) Lower bounds ($ u_1 $, $ u_2 $)
    Ⅱ-Ⅲ 0.2524 0.1682
    0.2390 0.1594
    Ⅴ-Ⅵ 0.2524 0.1682
    0.2657 0.1771
    0.2924 0.1949
    Ⅸ-Ⅹ 0.3058 0.2038
     | Show Table
    DownLoad: CSV
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