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摘要: 针对间歇过程的实时优化问题, 提出了一种基于自优化控制的批内优化方法. 以测量变量的线性组合为被控变量, 在单批次内跟踪控制被控变量实现间歇过程的实时优化. 根据是否在间歇过程的不同阶段切换被控变量, 给出了两种自优化控制策略, 对每种策略又分别提出两种设定轨线选取方案. 为求解这些情形下的最优被控变量(组合矩阵), 以最小化平均经济损失为目标, 推导了组合矩阵和经济损失之间的函数关系, 分别将其描述为相应的非线性规划问题. 在此基础上, 进一步引入了扩张组合矩阵, 将这些非线性规划问题归纳为求解扩张组合矩阵的一致形式(扩张组合矩阵具有不同的结构约束), 并推导得到了其中一种方案的解析解计算方法. 以一个间歇反应器为研究对象, 验证了方法的有效性.Abstract: For real-time optimization of uncertain batch processes, a within-batch self-optimizing control (SOC) approach is proposed. In this approach, measurement combinations are selected as controlled variables, which are tracked at time-varying setpoints along batch operations. Regarding whether the controlled variables are switched at different batch phases, two self-optimizing control strategies are given, both of which further contain two schemes in terms of the computation of setpoint trajectory. To solve the optimal controlled variables (combination matrices), the average economic loss is considered as the to-be-minimized cost function, then the nonlinear programming problems are formulated by establishing the relationships between the loss and combination matrices. Based on these results, the extended combination matrix is introduced, then different nonlinear programming problems are uniformed as one problem for solving the optimal extended combination matrix (with different structural constraints). Among these cases, the analytical solution of one option is further derived. A batch reactor is studied to show the effectiveness of the proposed approach.
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图 1 间歇过程的离散化变量及自优化控制策略
Fig. 1 Discretization of batch processes and self-optimizing control strategy
图 2 最优扩张组合矩阵
$\bar H$ 的求解步骤Fig. 2 Procedure for solving the optimal extended combination matrix
$\bar H$ 
图 5 批内自优化控制效果
$( k_1 $ : +20%,$ k_2 $ : −20%)Fig. 5 Within-batch self-optimizing performance
$ (k_1 $ : +20%,$ k_2 $ : −20%)
图 6 批内自优化控制效果
$( k_1 $ : +40%,$ k_2 $ : −40%)Fig. 6 Within-batch self-optimizing performance
$( k_1 $ : +40%,$ k_2 $ : −40%)表 1 间歇反应器参数及标称值
Table 1 Parameters for the reactor model and nominal values
符号 物理含义 标称值 $ k_1 $ 主反应的反应常数 0.053 L·mol/min $ k_2 $ 副反应的反应常数 0.128 L·mol/min $ u_L $ $ u $下限 0 L/min $ u_U $ $ u $上限 0.001 L/min $ c_{Bin} $ B 进料浓度 5 mol/L $ c_{Ao} $ A 初始浓度 0.72 mol/L $ c_{Bo} $ B 初始浓度 0.0614 mol/L $ V_o $ V 初始值 1 L $ t_f $ 批次运行时间 250 min 
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表 2 损失函数
$ L_{\rm{av}} $ Table 2 Loss function
$ L_{\rm{av}} $ 策略及方案 $ N = 2 $ $ N = 20 $ 策略 2 (方案 1) 0.0371 0.0083 策略 2 (方案 2) 0.03423 0.0024 策略 3 (方案 3) 0.0368 0.0069 策略 3 (方案 4) 0.03420 0.0022
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表 3 100组随机扰动下的非线性损失统计量
Table 3 Statistics of nonlinear losses for 100 groups of random disturbances
方案 平均损失 最大损失 标准差 标称操作 0.0036 0.0227 0.0068 控制$ c_B $ 0.0042 0.0165 0.0054 策略 2 (方案 1) 0.0026 0.0167 0.0050 策略 2 (方案 2) 0.0007 0.0053 0.0016
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