一种基于伪Wigner-Ville分析的动态优化问题网格重构策略

doi:10.11949/j.issn.0438-1157.20180805

摘要/Abstract

摘要：

为了解决控制向量参数化方法逼近精度和计算时间之间的矛盾，提出了一种基于伪Wigner-Ville时频分析的控制向量参数化方法。该方法首先给定较少的网格进行第一次优化迭代，快速获得控制变量的大致轨迹。然后通过伪Wigner-Ville分析得出不同时间网格节点瞬时频率变化对性能指标的影响，籍此对原有网格节点进行重构，包括对时间节点的消除、细化。并且结合变时间节点控制向量参数化方法的思想，将瞬时频率为极大值时对应的时间节点作为待优化参数，与控制变量一同进行求解优化，从而找到准确的最优时间切换点。三个经典的化工反应实例用于验证所提方法，计算结果表明：与传统的控制向量参数化方法和文献结果相比，所提方法可以更有效地重构时间网格，找到准确的时间切换点，不仅计算成本低，而且计算精度更出色。

关键词: 控制, 控制向量参数化, 伪Wigner-Ville, 动态, 优化, 网格划分

Abstract:

In order to solve the contradiction between approximation accuracy and computation time of control vector parameterization (CVP) method, a control vector parameterization method based on pseudo Wigner-Ville time-frequency analysis is proposed. The method first gives a small number of meshes for the first optimization iteration, and quickly obtains the approximate trajectory of the control variables. Then, through the pseudo Wigner-Ville analysis, the influence of the instantaneous frequency change of the mesh nodes on the performance index is obtained, and the original mesh nodes are reconstructed based on above analysis, including the elimination and refinement of the time nodes. And then combined with the variable time node CVP method, the corresponding time nodes with the maximum instantaneous frequency are used as parameters, which are optimized together with the control variables to find accurate time switching points. Three classical chemical reaction examples are used to verify the proposed method. The calculation results show that compared with the traditional CVP method and literature results, the proposed method can reconstruct the time grid more effectively and find more accurate time switching points, leading to lower calculation cost and more accurate solution.

Key words: control, CVP, pseudo Wigner-Ville, dynamic, optimization, grid refinement

中图分类号:

TP 273.1

徐炜峰, 江爱朋, 王浩坤, 蒋恩辉, 丁强, 高寒寒. 一种基于伪Wigner-Ville分析的动态优化问题网格重构策略[J]. 化工学报, 2019, 70(S1): 158-167.

Weifeng XU, Aipeng JIANG, Haokun WANG, Enhui JIANG, Qiang DING, Hanhan GAO. A grid reconstruction strategy based on pseudo Wigner-Ville analysis for dynamic optimization problem[J]. CIESC Journal, 2019, 70(S1): 158-167.

图/表 28

图1 Time-Scaling技巧的示意图

Fig.1 Schematic diagram of time-scaling technique

图2 变时间节点CVP方法的示意图

Fig.2 Schematic diagram of variable time node CVP method

图3 重要时间切换点的判别示意图

Fig.3 Schematic diagram of important time switching points

图4 PWV-CVP方法的算法流程图

Fig.4 Algorithm flow chart of PWV-CVP method

图5 第一次迭代后的控制变量轨迹

Fig.5 Trajectory of control variable after the first iteration

图6 第一次迭代后控制轨迹的时频图

Fig.6 Time-frequency diagram of control trajectory after the first iteration

图7 两次迭代的时间网格对比

Fig.7 Comparison of time grid between two iterations

图8 UD-CVP与PWV-CVP的比较

Fig.8 Comparison of UD-CVP and PWV-CVP

图9 最优状态变量

Fig.9 Optimal state profiles

表1 实例一测试结果

Table 1 Test results for case 1

Method	Iteration	Number of parameters	J _max	CPU time/s
UD-CVP	1	32	0.7237498	130.4
UD-CVP	1	150	0.7238900	2499.6
PWV-CVP	1	20	0.7234708	32.3
	2	32	0.7238990	89.6

表2 实例一的不同文献方法结果对比

Reference	Method	J _max
[17]	组合模式方法CMM	0.7226
[18]	共轭梯度法CGM	0.7227
[19]	迭代动态规划IDP	0.723898
本文	PWV-CVP	0.723899

图10 第一次迭代后的控制变量轨迹

Fig.10 Trajectory of control variable after the first iteration

图11 第一次迭代后控制轨迹的时频图

Fig.11 Time-frequency diagram of control trajectory after the first iteration

图12 两次迭代的时间网格对比

Fig.12 Comparison of time grid between two iterations

图13 UD-CVP与PWV-CVP的比较

Fig.13 Comparison of UD-CVP and PWV-CVP

图14 最优状态变量

Fig.14 Optimal state profiles

表3 实例二的测试结果

Table 3 Test results for case 2

Method	Iteration	Number of parameters	J _max	CPU time/s
UD-CVP	1	15	0.4745314	8.09
UD-CVP	1	100	0.4777024	1960.2
PWV-CVP	1	20	0.4752719	18.8
	2	15	0.4777120	9.7

表4 实例二的不同文献方法结果对比

Table 4 Comparison of results from different literature methods for case 2

Reference	Method	J _max
[21]	迭代动态规划IDP	0.475272
[22]	差分进行算法DE	0.476827
[22]	三角微分进化算法TDE	0.476826
[23]	解析方法	0.477712
本文	PWV-CVP	0.477712

图15 第一次迭代后的控制变量轨迹(u 1）

Fig.15 Trajectory of control variable after the first iteration (u 1)

图16 第一次迭代后的控制变量轨迹(u 2)

Fig.16 Trajectory of control variable after the first iteration (u 2)

图17 第一次迭代后控制轨迹的时频图（u 1）

Fig.17 Time-frequency diagram of control trajectory after the first iteration (u 1)

图18 第一次迭代后控制轨迹的时频图(u 2）

Fig.18 Time-frequency diagram of control trajectory after the first iteration (u 2)

图19 两次迭代的时间网格对比

Fig.19 Comparison of time grid between two iterations

图20 UD-CVP与PWV-CVP的比较(u 1)

Fig.20 Comparison of UD-CVP and PWV-CVP (u 1)

图21 UD-CVP与PWV-CVP的比较（u 2）

Fig.21 Comparison of UD-CVP and PWV-CVP (u 2)

图22 最优状态变量

Fig.22 Optimal state profiles

表5 实例三的测试结果

Table 5 Test results for case 3

Method	Iteration	Number of parameters	J _max	CPU time/s
UD-CVP	1	86	6.151376	384.6
UD-CVP	1	200	6.151546	3746.2
PWV-CVP	1	40	6.150842	67.3
	2	86	6.151600	88.8

表6 实例三的不同文献方法结果对比

Table 6 Comparison of results from different literature methods for case 3

Reference	Method	J _max
[25]	过滤迭代动态规划FIDP	6.16
[26]	遗传算法GA	6.1504
[27]	截断牛顿法TN	6.15
[28]	MJWAT	6.15
[29]	自适应粒子群优化	6.15144
[30]	混合型智能优化方法	6.15152
本文	PWV-CVP	6.15160

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