A grid reconstruction strategy based on pseudo Wigner-Ville analysis for dynamic optimization problem

doi:10.11949/j.issn.0438-1157.20180805

Abstract

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

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

CLC Number:

TP 273.1

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.

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.

Figures/Tables 28

Fig.1 Schematic diagram of time-scaling technique

Fig.2 Schematic diagram of variable time node CVP method

Fig.3 Schematic diagram of important time switching points

Fig.4 Algorithm flow chart of PWV-CVP method

Fig.5 Trajectory of control variable after the first iteration

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

Fig.7 Comparison of time grid between two iterations

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

Fig.9 Optimal state profiles

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

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

Fig.10 Trajectory of control variable after the first iteration

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

Fig.12 Comparison of time grid between two iterations

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

Fig.14 Optimal state profiles

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

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

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

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

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

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

Fig.19 Comparison of time grid between two iterations

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

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

Fig.22 Optimal state profiles

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

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

References 30

1	彭鑫, 祁荣宾, 杜文莉, 等 . 一种改进的知识进化算法及其在化工动态优化中的应用[J].化工学报, 2012, 63(3): 841-850.
	Peng X , Qi R B , Du W L , et al . An improved knowledge evolution algorithm and its application to chemical process dynamic optimization[J]. CIESC Journal, 2012, 63(3): 841-850.
2	罗雄麟, 夏车奎, 孙琳 . 有旁路换热网络全周期节能的动态优化控制实现方法[J].化工学报, 2013, 64(4): 1340-1350.
	Luo X L , Xia C K , Sun L . A dynamic optimization control approach of life cycle energy saving for heat exchanger network with bypasses[J]. CIESC Journal, 2013, 64(4): 1340-1350
3	Zhang B , Chen D , Wu X . Graded optimization strategy and its application to chemical dynamic optimization with fixed boundary[J]. Journal of Chemical Industry & Engineering, 2005, 56(7): 1276-1280.
4	Rein L . Optimal control by dynamic programming using systematic reduction in grid size[J]. International Journal of Control, 1990, 51(5): 995-1013.
5	Kirk D E . Optimal control theory: an introduction[J]. Positively Aware the Monthly Journal of the Test Positive Aware Network, 2004, 23(2): 13-5.
6	Srinivasan B , Palanki S , Bonvin D . Erratum to “Dynamic optimization of batch processes I. Characterization of the nominal solution”: [Computers and Chemical Engineering 27 (2003) 1-26][J]. Computers & Chemical Engineering, 2003, 27(5): 759-760.
7	Szymkat M , Korytowski A . Method of monotone structural evolution for control and state constrained optimal control problems[C]// European Control Conference. IEEE, 2015: 294-299.
8	Liu P , Li G , Liu X , et al . Novel non-uniform adaptive grid refinement control parameterization approach for biochemical processes optimization[J]. Biochemical Engineering Journal, 2016, 111: 63-74.
9	Binder T , Cruse A , Villar C A C , et al . Dynamic optimization using a wavelet based adaptive control vector parameterization strategy[J]. Computers & Chemical Engineering, 2000, 24(2): 1201-1207.
10	Schlegel M , Stockmann K , Binder T , et al . Dynamic optimization using adaptive control vector parameterization[J]. Computers & Chemical Engineering, 2005, 29(8): 1731-1751.
11	Liu P , Liu X . Empirical mode decomposition-based time grid refinement optimization approach for optimal control problems[J]. Optimization Letters, 2017, 11(7): 1243-1256.
12	Liu P , Li G , Liu X , et al . A novel non-uniform control vector parameterization approach with time grid refinement for flight level tracking optimal control problems[J]. ISA Transactions, 2018, 73: 66-78.
13	Li G , Liu P , Liu X . A control parameterization approach with variable time nodes for optimal control problems[J]. Asian Journal of Control, 2016, 18(3): 976-984.
14	Teo K L , Jennings L S , Lee H W J , et al . The control parameterization enhancing transform for constrained optimal control problems[J]. Journal of the Australian Mathematical Society, 1999, 40(3): 314-335.
15	Loxton R C , Teo K L , Rehbock V , et al . Optimal control problems with a continuous inequality constraint on the state and the control ☆[J]. Automatica, 2009, 45(10): 2250-2257.
16	Loxton R C , Teo K L , Rehbock V . Optimal control problems with multiple characteristic time points in the objective and constraints ☆[J]. Automatica, 2008, 44(11): 2923-2929.
17	Daniel Y C K , Stevens W F . Studies of singular solutions in dynamic optimization (Ⅱ): Optimal singular design of a plug-flow tubular reactor[J]. AIChE Journal, 1971, 17(1): 160-166.
18	Reddy K V , Husain A . Computation of optimal control policy with singular subarc[J]. Canadian Journal of Chemical Engineering, 1981, 59(4): 557-559.
19	Luus R . Iterative Dynamic Programming[M]. New York: Chapman & Hall/CRC, 2000: 767-767.
20	Gunn D J , Thomas W J . Mass transport and chemical reaction in multifunctional catalyst systems[J]. Chemical Engineering Science, 1965, 20(2): 89-100.
21	Dadebo S A , Mcauley K B . Dynamic optimization of constrained chemical engineering problems using dynamic programming [J]. Computers & Chemical Engineering, 1995, 19(5): 513-525.
22	Angira R , Santosh A . Optimization of dynamic systems: a trigonometric differential evolution approach[J]. Computers & Chemical Engineering, 2007, 31(9): 1055-1063.
23	Li G , Liu X . Comments on “optimal use of mixed catalysts for two successive chemical reactions”[J]. Journal of Optimization Theory & Applications, 2015, 165(2): 678-692.
24	Lee J , Ramirez W F . Optimal fed‐batch control of induced foreign protein production by recombinant bacteria[J]. Chemical Engineering Science, 1994, 40(5): 899-907.
25	Tholudur A , Fredramirez W . Obtaining smoother singular arc policies using a modified iterative dynamic programming algorithm[J]. International Journal of Control, 1997, 68(5): 1115-1128.
26	Sarkar D , Modak J M . Optimization of fed-batch bioreactors using genetic algorithms: two control variables[J]. Chemical Engineering Science, 2003, 14(11): 1127-1132.
27	Balsa C E , Banga J R , Alonso A A , et al . Dynamic optimization of chemical and biochemical processes using restricted second-order information[J]. Computers & Chemical Engineering, 2001, 25(4/5/6): 539-546.
28	Shelokar P S , Jayaraman V K . Multicanonical jump walk annealing assisted by tabu for dynamic optimization of chemical engineering processes[J]. European Journal of Operational Research, 2008, 185(3): 1213-1229.
29	Zhou Y , Liu X . Control parameterization-based adaptive particle swarm approach for solving chemical dynamic optimization problems[J]. Chemical Engineering & Technology, 2014, 37(4): 692-702.
30	Zhang P , Liu X , Ma L . Optimal control vector parameterization approach with a hybrid intelligent algorithm for nonlinear chemical dynamic optimization problems[J]. Chemical Engineering & Technology, 2015, 38(11): 2067-2078.

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