Subsystem decomposition of complex nonlinear systems

doi:10.11949/0438-1157.20201746

Abstract

Abstract:

The community detection algorithm is used to study a subsystem decomposition method of a complex nonlinear chemical system, and a distributed model predictive control design is carried out. The nodes of information graph were used to represent the state variables, input variables and output variables of nonlinear system. The weighted directed graph of nonlinear process system was then constructed. The nodes were connected by weighted edges and the weight reflects the strength of the connection between nodes. Thus, the weighted directed graph can better reflect the internal connectivity and connection strength of the system. The community structure detection algorithm was used to divide all variables into the combination of subsystems to get the subsystem decomposition of complex nonlinear system. The correlation within each group was much stronger than the interaction between different groups. For the process of continuous stirred tank reactor, the subsystem decomposition was implemented and the distributed model predictive control algorithm was designed. The results show that the proposed subsystem decomposition method can take the connection weight of subsystems into account, which is more conducive to improve the performance of chemical process system with distributed model predictive control.

Key words: process systems, process control, community detection, subsystem decomposition, model-predictive control

摘要：

利用社区发现算法研究了一种复杂非线性化工系统的子系统分解方法，并进行了分布式模型预测控制设计。引入信息图论的节点表示系统的状态、输入和输出变量，构建非线性过程系统的加权有向图，节点通过加权边连接，加权反映了节点间连接的强度，因而能够同时反映系统内部的连通性和连接强度。利用社区结构发现算法将所有变量分成子系统的群组，使得每个组内的关联比不同组间的相互作用强，从而得到复杂化工过程系统的子系统分解。针对连续搅拌反应釜过程，实施子系统分解，并设计分布式模型预测控制算法，结果表明，所提出的子系统分解方法更能考虑子系统之间的连接权重，得到更有利于分布式模型预测控制的子系统划分，提升系统控制的性能。

关键词: 过程系统, 过程控制, 社区发现, 子系统分解, 模型预测控制

CLC Number:

XIE Miaomiao, ZHANG Langwen, XIE Wei. Subsystem decomposition of complex nonlinear systems[J]. CIESC Journal, 2021, 72(3): 1557-1566.

谢苗苗, 张浪文, 谢巍. 复杂非线性系统的子系统分解方法[J]. 化工学报, 2021, 72(3): 1557-1566.

Figures/Tables 8

Fig.1 Flowchart of the proposed subsystem decomposition method of nonlinear systems.

Fig.2 Two CSTRs and a flash tank with recycle stream

Table 1 Parameter values in Eq.(29) and Eq.(30)

参数值	单位
T₁₀=300，T₂₀=300	K
F₁₀=5，F₂₀=5，F_r=1.9	m³/h
C_A10=4，C_A20=3	kmol/m³
V₁=1.0，V₂=0.5，V₃=1.0	m³
E₁=5×10⁴，E₂=5.5×10⁴	kJ/kmol
k₁=3×10⁶，k₂=3×10⁶	h^-1
ΔH₁=-5×10⁴，ΔH₂=-5.3×10⁴	kJ/kmol
H_vap=5	kJ/kmol
C_P=0.231	kJ/(kg·K)
R=8.314	kJ/(kmol·K)
$ρ = 1000$	kg/m³
$α A = 2, α B = 1, α C = 1.5, α D = 3$
MW_A=50，MW_B=50，MW_C=50	kg/kmol

Table 1 Parameter values in Eq.(29) and Eq.(30)

参数值	单位
T₁₀=300，T₂₀=300	K
F₁₀=5，F₂₀=5，F_r=1.9	m³/h
C_A10=4，C_A20=3	kmol/m³
V₁=1.0，V₂=0.5，V₃=1.0	m³
E₁=5×10⁴，E₂=5.5×10⁴	kJ/kmol
k₁=3×10⁶，k₂=3×10⁶	h^-1
ΔH₁=-5×10⁴，ΔH₂=-5.3×10⁴	kJ/kmol
H_vap=5	kJ/kmol
C_P=0.231	kJ/(kg·K)
R=8.314	kJ/(kmol·K)
$ρ = 1000$	kg/m³
$α A = 2, α B = 1, α C = 1.5, α D = 3$
MW_A=50，MW_B=50，MW_C=50	kg/kmol

Table 2 Three decompositions for system (29)

分解	子系统	状态	输入	输出
分解1	1	T₁，C_A1，C_A2，C_A3	Q₁	T₁
	2	T₂，C_B1，C_B2，C_B3	Q₂	T₂
	3	T₃，C_C1，C_C2，C_C3	Q₃	T₃
分解2	1	T₁，C_A1，C_B1，C_C1	Q₁	T₁
	2	T₂，C_A2，C_B2，C_C2	Q₂	T₂
	3	T₃，C_A3，C_B3，C_C3	Q₃	T₃
分解3	1	T₁，C_A1，C_A3	Q₁	T₁
	2	T₂，C_A2	Q₂	T₂
	3	C_B1，C_C1，C_B2，C_C2，T₃，C_B3，C_C3	Q₃	T₃

Fig.3 Temperature and input trajectories of the nominal closed-loop system under Decomposition 1 (blue dash dotted line), Decomposition 2 (red solid line), and Decomposition 3 (black dotted line)

Table 3 Performance index (J) and CPU time of iterative distributed MPC with different decomposition models and centralized MPC

分解	J	CPU time/s
分解1	298.6897	26.622806
分解2	297.3096	25.406688
分解3	295.0141	25.270445
集中式	276.6084	24.151277

Fig.4 The temperature and input trajectories of distributed LMPC when iteration times is 1(red solid line), 2 (black dotted line) and 3 (blue dash dotted line)

Table 5 Performance index and CPU time of distributed LMPC with different iteration times

迭代次数	J	CPU time/s
1次	338.5985	12.626477
2次	297.3096	25.406688
3次	295.3047	41.282411

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