人群搜索和樽海鞘群的混合算法优化PID参数

doi:10.16182/j.issn1004731x.joss.20-1036

摘要/Abstract

摘要：

为解决在优化全局时人群搜索优化算法(seeker optimization algorithm，SOA)容易过早收敛的问题，提出了一种新的基于人群搜索和樽海鞘群(salp swarm algorithm，SSA)的SOA-SSA混合算法。基于双种群进化策略，种群中的部分个体由人群搜索优化算法进化，其余个体由樽海鞘群算法进化。SOA和SSA的个体都使用信息共享机制实现协同进化，增加了种群的多样性，避免了算法过早收敛。实验结果表明：该算法在高维函数和PID参数优化方面都是可行的。与其他算法相比，SOA-SSA算法的收敛速度快、精度高、鲁棒性强，有更好的优化性能。

关键词: 混合, 人群搜索算法, 樽海鞘群算法, 函数优化, PID参数优化

Abstract:

Aiming at the premature convergence of seeker optimization algorithm(SOA) during optimizing the global problems, a new SOA-SSA hybrid algorithm based on seeker optimization algorithm and salp swarm algorithm (SSA) is proposed.The SOA-SSA algorithm is based on a double population evolution strategy, in which some individuals of the population are evolved by seeker optimization algorithm and the rest are evolved from salp swarm algorithm. The individuals in SOA and SSA both employ an information sharing mechanism to realize the coevolution. These strategies increase the diversity of the population and avoid the premature convergence. The experimental results show that the proposed algorithm can be used in both the high-dimensional cases and the PID control parameter optimization. Compared to the other eleven algorithms, the SOA-SSA has the higher, convergence speed, precision and robustness, and has a better optimization performance.

Key words: hybrid, seeker optimization algorithm, salp swarm algorithm, benchmark function optimization, PID control parameter

中图分类号:

TP391

段绍米, 罗会龙, 刘海鹏. 人群搜索和樽海鞘群的混合算法优化PID参数[J]. 系统仿真学报, 2022, 34(6): 1230-1246.

Shaomi Duan, Huilong Luo, Haipeng Liu. A Hybrid Algorithm Based on Seeker Optimization Algorithm and Salp Swarm Algorithm for PID Parameters Optimization[J]. Journal of System Simulation, 2022, 34(6): 1230-1246.

图/表 10

图1

表1

基准函数

测试函数	范围	最优值
$f 1 = ∑ i = 1 n x i 2$	[-100, 100]	0
$f 2 (x) = ∑ i = 1 n x i + ∏ i = 1 n x i$	[-10, 10]	0
$f 3 (x) = ∑ i = 1 n ∑ j = 1 j x j 2$	[-100, 100]	0
$f 4 (x) = m a x i {\| x i \|, ? 1 ≤ i ≤ n}$	[-100, 100]	0
$f 5 (x) = ∑ i = 1 n - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$	[-30, 30]	0
$f 6 (x) = ∑ i = 1 n (x i + 0.5) 2$	[-100, 100]	0
$f 7 (x) = ∑ i = 1 n i x i 4 + r a n d m [0,1)$	[-1.28, 1.28]	0
$f 8 (x) = ∑ i = 1 n i x i 2$	[-10, 10]	0
$f 9 (x) = ∑ i = 1 n x i i + 1$	[-1.28, 1.28]	0
$f 10 (x) = - ∑ i = 1 n x i s i n (x i)$	[-500, 500]	-418.982 9×1 000
$f 11 (x) = ∑ i = 1 n [x i 2 - 10 c o s (2 π x i) + 10]$	[-5.12, 5.12]	0
$f 12 (x) = 20 + e - 20 e - 0.2 1 n ∑ i = 1 n x i 2 - e 1 n ∑ i = 1 n c o s (2 π x i)$	[-32, 32]	0

表1

表2

算法参数设置

算法名称	参数和设定值
PSO^[31]	惯性常数: 0.9~0.4, 第一加速度系数: 1.496 2, 第二加速度系数: 1.496 2
SA_GA^[32]	选择概率: 0.6, 交叉概率: 0.7, 变异概率: 0.05, 初始温度: 100, 温度递减系数: 0.98
DE^[33]	变异概率: 0.6, 交叉概率: 0.09
DA^[1]	s 为分离权重: 0~0.2, a 为对齐权重: 0~0.2,c 为凝聚权重: 0~0.2, f 为食物因子: 0~2,e 为敌情因子: 0~0.1, $ω$ 为惯性权重: 0.9~0.4
BSO^[34]	集群数量: 10, 集群的选择概率: 0.8, 集群中心的选择替换概率: 0.2, 使用中心的概率: 0.4
MFO^[35]	定义对数螺旋形状的常数: b=1, 随机数: t = -1~1
GSA^[36]	在第一个宇宙量子区间的引力常数: G₀=100, alfa=20，个体与个体间的引力之比为2
SCA^[5]	随机数组: r₁=0~2, r₂=0~2π, r₃=0~2, r₄=0~1
SSA^[6]	随机数组: r₁=0~2, r₂=0~1, r₃=0~1
MVO^[7]	虫洞存在概率: WEP_max =1, WEP_min=0.2, 距离移动率: TDR=0~1, 随机数组: r₁=0~1,r₂=0~1, r₃=0~1
SOA^[8]	最大隶属度值: 0.95, 最小隶属度值: 0.011 1, 最大惯性权重值: 0.8, 最小惯性权重值: 0.2
SOA-SSA	最大隶属度值: 0.95, 最小隶属度值: 0.011 1, 最大惯性权重值: 0.9, 最小惯性权重值: 0.1.随机数组: r₁=0~2, r₂=0~1, r₃=0~1

表2

表3

算法在高维基准函数中独立运行30次后的性能比较

函数	性能	PSO	SA_GA	DE	DA	BSO	MFO	GSA	SCA	MVO	SSA	SOA	SOA-SSA
f₁	平均值	9.7e+3	2.6e+6	1.3e+6	3.7e+5	1.5e+5	2.5e+6	9.8e+4	3.5e+5	3.6e+5	1.9e+5	8.4e+3	2.1e+3
	标准差	1.6e+3	5.7e+4	3.4e+4	1.8e+5	1.6e+4	5.6e+4	5.0e+3	1.4e+5	1.6e+4	9.6e+3	1.6e+3	2.9e+3
	最差适应度	1.4e+4	2.7e+6	1.4e+6	7.9e+5	1.9e+5	2.6e+6	1.1e+5	7.7e+5	4.0e+5	2.1e+5	1.3e+4	1.2e+4
	最佳适应度	7.0e+3	2.5e+6	1.2e+6	1.0e+5	1.2e+5	2.4e+6	8.7e+4	4.0e+4	3.3e+5	1.7e+5	4.8e+3	12.4
	排名	3	12	10	6	7	11	5	4	9	8	2	1
f₂	平均值	Inf	Inf	Inf	1.0e+3	N/A	Inf	1.8e+281	Inf	1.3e+273	1.1e+3	7.8e+278	78.9
	标准差	N/A	N/A	N/A	3.3e+2	N/A	N/A	Inf	N/A	Inf	20.2	Inf	6.4
	最差适应度	Inf	Inf	Inf	1.7e+3	N/A	Inf	3.8e+282	Inf	4.0e+274	1.1e+3	2.3e+280	95.9
	最佳适应度	2.9e+2	Inf	Inf	3.1e+2	N/A	Inf	4.6e+244	Inf	2.3e+209	1.0e+3	2.1e+3	67.6
	排名	2	8	8	3	12	8	7	8	6	4	5	1
f₃	平均值	3.3e+6	3.3e+7	6.2e+8	2.3e+7	1.9e+6	1.4e+7	2.0e+6	2.4e+7	6.2e+6	4.2e+6	2.4e+6	7.3e+5
	标准差	1.4e+6	1.1e+7	1.9e+7	6.0e+6	4.7e+5	2.3e+6	8.2e+5	4.7e+6	4.3e+5	1.9e+6	7.7e+5	3.5e+5
	最差适应度	8.0e+6	8.3e+7	6.5e+8	3.8e+7	3.2e+6	1.8e+7	4.8e+6	3.5e+7	7.4e+6	8.5e+6	4.0e+6	1.7e+6
	最佳适应度	1.7e+6	2.0e+7	5.7e+8	1.1e+7	1.2e+6	9.2e+6	9.6e+5	1.4e+7	5.5e+6	1.5e+6	2.0e+5	3.8e+3
	排名	6	10	12	9	4	8	3	11	7	5	2	1
f₄	平均值	18.3	99.5	95.1	74.0	43.0	99.4	28.7	99.5	97.4	43.9	37.6	30.5
	标准差	1.3	0.2	0.3	14.4	3.4	0.2	1.8	0.1	0.7	3.0	8.8	8.1
	最差适应度	21.5	99.8	95.6	88.6	50.5	99.8	33.7	99.7	98.8	53.5	44.9	39.5
	最佳适应度	16.4	99.0	94.3	19.5	36.6	99.0	25.9	99.2	95.66	39.5	12.1	2.5
	排名	3	10	8	4	6	11	5	12	9	7	2	1
f₅	平均值	2.7e+5	1.1e+10	4.0e+9	4.8e+8	8.6e+7	1.1e+10	1.7e+7	3.3e+9	6.8e+8	7.5e+7	1.0e+8	1.8e+5
	标准差	1.5e+5	3.3e+8	1.5e+8	3.9e+8	1.5e+7	3.6e+8	1.7e+6	7.0e+8	7.4e+7	8.9e+6	2.5e+7	1.9e+5
	最差适应度	7.5e+5	1.3e+10	4.2e+9	1.5e+9	1.2e+8	1.2e+10	2.2e+7	4.7e+9	8.4e+8	1.1e+8	1.5e+8	1.0e+6
	最佳适应度	1.2e+5	1.1e+10	3.5e+9	1.9e+7	5.7e+7	1.0e+10	1.5e+7	2.1e+9	5.5e+8	6.0e+7	6.3e+7	2.6e+4
	排名	2	12	10	4	5	11	3	9	8	6	7	1
f₆	平均值	1.0e+4	2.6e+6	1.3e+6	2.9e+5	1.6e+5	2.5e+6	9.9e+4	4.1e+5	3.6e+5	1.9e+5	8.0e+3	7.2e+2
	标准差	1.4e+3	4.1e+4	3.11e+4	1.2e+5	1.8e+4	4.5e+4	4.9e+3	1.3e+5	1.8e+4	1.0e+4	1.4e+3	1.2e+3
	最差适应度	1.4e+4	2.6e+6	1.4e+6	5.6e+5	2.0e+5	2.6e+6	1.1e+5	7.4e+5	4.0e+5	2.1e+5	1.2e+4	4.4e+3
	最佳适应度	6.2e+3	2.5e+6	1.2e+6	6.2e+4	1.2e+5	2.4e+6	9.1e+4	1.6e+5	3.2e+5	1.7e+5	4.8e+3	1.6e+2
	排名	3	12	10	4	6	11	5	7	9	8	2	1
f₇	平均值	1.1e+2	1.8e+5	5.2e+4	7.9e+3	6.6e+3	1.7e+5	5.3e+3	4.8e+4	8.7e+3	1.1e+3	2.0e+4	3.3e+2
	标准差	22.3	6.1e+3	2.9e+3	5.9e+3	2.1e+3	5.9e+3	5.8e+2	1.1e+4	771.6	129.1	2.7e+3	54.4
	最差适应度	1.7e+2	1.9e+5	5.9e+4	2.0e+4	1.3e+4	1.8e+5	6.9e+3	7.0e+4	1.1e+4	1.34e+3	2.6e+4	4.6e+2
	最佳适应度	81.0	1.7e+5	4.7e+4	667.2	3.2e+3	1.6e+5	4.4e+3	2.7e+4	7.3e+3	878.7	1.4e+4	2.1e+2
	排名	1	12	10	3	5	11	6	9	7	4	8	2
f₈	平均值	4.7e+4	1.3e+7	6.2e+6	1.9e+6	1.4e+6	1.2e+7	3.9e+5	1.8e+6	1.5e+6	8.6e+5	1.6e+6	6.4e+3
	标准差	6.6e+3	3.6e+5	1.9e+5	9.6e+5	1.6e+5	3.0e+5	2.2e+4	5.6e+5	6.5e+4	5.2e+4	1.1e+5	1.7e+3
	最差适应度	6.1e+4	1.3e+7	6.9e+6	3.8e+6	1.6e+6	1.26e+7	4.4e+5	2.7e+6	1.7e+6	9.9e+5	1.9e+6	1.1e+4
	最佳适应度	3.7e+4	1.1e+7	6.0e+6	6.5e+5	1.01e+6	1.1e+7	3.5e+5	4.5e+5	1.4e+6	7.9e+5	1.3e+6	3.2e+3
	排名	2	12	10	5	7	11	3	4	9	6	8	1
f₉	平均值	9.8e-9	1.6e+92	5.0e+55	4.8e+27	N/A	2.0e+90	7.7e-5	9.1e+83	1.0e+56	1.7e-6	1.4e+12	6.1e-8
	标准差	3.e-8	7.4e+92	1.0e+56	2.6e+28	N/A	1.1e+91	2. 6e-4	4.7e+84	5.5e+56	1.3e-6	5.5e+12	5.9e-8
	最差适应度	1.5e-7	4.0e+93	4.1e+56	1.4e+29	N/A	5.9e+91	1.3e-3	2.6e+85	3.0e+57	5.3e-6	2.6e+13	2.8e-7
	最佳适应度	5.5e-16	1.1e+7	1.5e+48	4.8e-4	N/A	2.3e+74	2.0e-9	9.3e+69	5.6e+38	4.1e-7	2.4e-2	7.7e-9
	排名	1	7	9	5	12	11	2	10	8	4	6	3
f₁₀	平均值	-1.7e+4	-5.8e+4	-6.0e+4	-3.2e+4	-6.6e+4	-1.1e+5	-1.5e+4	-2.3e+4	-1.3e+5	-1.2e+5	-1.3e+5	-1.6e+5
	标准差	2.6e+3	3.4e+3	1.7e+3	4.9e+3	4.3e+3	8.8e+3	2.3e+3	1.4e+3	6.0e+3	6.9e+3	2.7e+4	5.5e+4
	最差适应度	-1.0e+4	-5.3e+4	-5.7e+4	-2.4e+4	-5.5e+4	-9.4e+4	-1.1e+4	-2.0e+4	-1.2e+5	-1.18e+5	-9.1e+4	-1.1e+5
	最佳适应度	-2.2e+4	-6.7e+4	-6.5e+4	-4.2e+4	-7.4e+4	-1.3e+5	-1.8e+4	-2.7e+4	-1.5e+5	-1.4e+5	-2.0e+5	-3.7e+5
	排名	11	7	8	9	6	5	12	10	3	4	2	1
f₁₁	平均值	2.8e+3	1.6e+4	1.3e+4	94e+3	9.4e+3	1.5e+4	5.8e+3	1.8e+3	1.4e+4	6.3e+3	10.0e+3	1.3e+3
	标准差	1.7e+2	1.4e+2	84.5	1.2e+3	2.0e+2	2.4e+2	1.6e+2	8.7e+2	3.0e+2	1.6e+2	3.1e+2	3.8e+2
	最差适应度	3.3e+3	1.6e+4	1.3e+4	1.1e+4	9.8e+3	1.5e+4	6.0e+3	3.9e+3	1.4e+4	6.7e+3	1.1e+4	2.2e+3
	最佳适应度	2.5e+3	1.5e+4	1.3e+4	7.0e+3	9.1e+3	1.4e+4	5.4e+3	5.0e+2	1.3e+4	6.1e+3	9.4e+3	8.4e+2
	排名	3	12	9	6	7	11	4	1	10	5	8	2
f₁₂	平均值	4.5	20.8	19.8	15.5	14.4	20.2	10.3	18.9	20.9	14.4	10.4	4.6
	标准差	0.2	0.03	0.06	1.9	0.4	0.2	0.2	4.0	0.02	0.2	0.6	3.9
	最差适应度	5.1	20.9	19.9	18.7	15.2	20.5	10.6	20.8	21.0	14.8	11.1	11.5
	最佳适应度	4.2	20.7	19.7	11.4	13.5	20.0	9.9	8.5	20.9	14.0	9.2	0.8
	排名	2	11	9	6	7	10	5	3	12	8	4	1
平均排名		3.0	10.7	9.5	5.4	6.8	10.2	5.1	7.4	8.1	5.7	4.4	1.4
总体排名		2	12	10	6	7	11	5	8	9	4	3	1

表3

图2

图3

图4

图5

表4

图6

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函数	性能	PSO	SA_GA	DE	DA	BSO	MFO	GSA	SCA	SSA	MVO	SOA	SOA-SSA
g₁	平均值	0.22	0.31	0.26	0.14	0.43	0.07	0.45	0.09	0.06	0.25	0.19	0.05
	标准差	0.08	0.06	0.09	0.08	0.14	0.02	0.15	0.02	0.05	0.05	0.11	0.00
	最差适应度	0.42	0.43	0.47	0.30	0.88	0.11	0.86	0.11	0.33	0.26	0.42	0.06
	最佳适应度	0.04	0.10	0.06	0.05	0.26	0.04	0.27	0.04	0.04	0.05	0.05	0.04
	排名	5	10	9	6	11	3	12	4	2	7	8	1
g₂	平均值	0.35	0.50	0.50	0.29	0.68	0.10	0.64	0.10	0.26	0.39	0.28	0.10
	标准差	0.20	0.10	0.09	0.21	0.15	4.7e-4	0.12	8.77e-4	0.21	0.19	0.18	0.00
	最差适应度	0.54	0.61	0.61	0.54	1.04	0.10	0.96	0.10	0.54	0.53	0.53	0.12
	最佳适应度	0.10	0.15	0.25	0.10	0.39	0.09	0.38	0.10	0.09	0.10	0.09	0.09
	排名	8	9	10	5	12	3	11	5	1	5	4	2
g₃	平均值	58.47	62.45	60.91	58.15	62.43	52.35	60.77	24.84	30.96	59.58	42.15	11.06
	标准差	7.75	0.12	4.01	9.55	0.80	16.91	5.30	21.52	29.63	7.65	27.90	18.68
	最差适应度	62.59	62.61	62.84	62.49	63.08	62.50	64.14	62.52	62.58	62.50	62.54	56.82
	最佳适应度	36.04	62.03	42.65	14.36	58.30	7.47	42.77	0.48	0.33	32.60	0.39	0.30
	排名	8	12	9	6	11	5	10	4	2	7	3	1
g₄	平均值	1.85e+2	2.72e+2	2.71e+2	1.35e+2	2.72e+2	39.95	2.77e+2	29.04	41.75	1.09e+2	2.62e+2	12.39
	标准差	59.64	0.62	0.37	71.50	5.5362	47.83	10.30	11.98	40.96	56.67	44.81	2.72
	最差适应度	3.15e+2	2.74e+2	2.72e+2	1.98e+2	2.80e+2	1.98e+2	3.16e+2	50.00	1.50e+2	1.96e+2	2.75e+2	18.36
	最佳适应度	32.54	2.71	2.70e+2	24.91	2.44e+2	13.17	2.71e+2	14.55	9.10	20.04	26.57	8.89
	排名	9	1	11	7	10	4	12	5	3	6	8	2
g₅	平均值	1.77e+2	55.35	93.24	81.65	41.44	35.07	2.34e+2	85.19	64.19	35.72	46.10	35.20
	标准差	4.22e+2	36.00	40.91	79.81	11.38	0.40	2.18e+2	1.01e+2	41.70	1.41	26.99	0.80
	最差适应度	2.13e+3	163.65	1.96e+2	3.65e+2	90.50	36.33	1.15e+3	4.20e+2	2.10e+2	41.28	1.85e+2	37.71
	最佳适应度	34.62	34.62	40.23	34.63	35.04	34.63	58.32	34.86	34.62	34.64	34.74	34.62
	排名	2	4	11	6	10	5	12	9	3	7	8	1
平均排名		6.4	7.2	10	6	10.8	4	11.4	5.4	2.2	6.4	6.2	1.4
总体排名		7	9	10	5	11	3	12	4	2	7	6	1

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