不确定性环境下维纳模型的随机变分贝叶斯学习

刘切; 李俊豪; 王浩; 曾建学; 柴毅

doi:10.16383/j.aas.c210925

不确定性环境下维纳模型的随机变分贝叶斯学习

doi: 10.16383/j.aas.c210925

刘切^1,,
李俊豪^1,,
王浩^1,,
曾建学^1,,
柴毅^1,

1.
重庆大学自动化学院重庆 400044

基金项目: 国家重点研发计划(2021YFB1715000), 国家自然科学基金(61903051, U2034209)资助

详细信息

作者简介:
刘切：重庆大学自动化学院副教授. 2016年获得北京化工大学控制科学与工程专业博士学位. 主要研究方向为人工智能及其在复杂过程的控制和优化中的应用. 本文通信作者. E-mail: qieliu@cqu.edu.cn

李俊豪：重庆大学自动化学院硕士研究生. 2020年获得西安理工大学自动化与信息工程学院学士学位. 主要研究方向为系统辨识与人工智能. E-mail: 202013021042@cqu.edu.cn

王浩：重庆大学自动化学院硕士研究生. 2021年获得安徽师范大学物理与电子信息学院学士学位. 主要研究方向为模型预测与人工智能. E-mail: 202113021007@cqu.edu.cn

曾建学：重庆大学自动化学院硕士研究生. 2018年获得华北科技学院电子信息工程学院学士学位. 主要研究方向为容器技术与人工智能. E-mail: lc9zjx@126.com

柴毅：重庆大学自动化学院教授. 2001年获得重庆大学博士学位. 主要研究方向为信息融合, 故障诊断, 智能控制系统. E-mail: chaiyi@cqu.edu.cn

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出版历程
- 收稿日期: 2021-09-27
- 录用日期: 2022-03-01
- 网络出版日期: 2022-05-08
- 刊出日期: 2024-06-20

Stochastic Variational Bayesian Learning of Wiener Model in the Presence of Uncertainty

LIU Qie^1
,,
LI Jun-Hao^1
,,
WANG Hao^1
,,
ZENG Jian-Xue^1
,,
CHAI Yi^1
,

1.
School of Automation, Chongqing University, Chongqing 400044

Funds: Supported by National Key Research and Development Program of China (2021YFB1715000) and National Natural Science Foundation of China (61903051, U2034209)

More Information

Author Bio:
LIU Qie　Associate professor at the School of Automation, Chongqing University. He received his Ph.D. degree from Beijing University of Chemical Technology in 2016. His research interest covers artificial intelligence and its applications on the control and optimization of complex processes. Corresponding author of this paper

LI Jun-Hao　Master student at the School of Automation, Chongqing University. He received his bachelor degree from the School of Automation and Information Engineering, Xi＇an University of Technology in 2020. His research interest covers system identification and artificial intelligence

WANG Hao　Master student at the School of Automation, Chongqing University. He received his bachelor degree from the School of Physics and Electronic Information, Anhui Normal University in 2021. His research interest covers model prediction and artificial intelligence

ZENG Jian-Xue　Master student at the School of Automation, Chongqing University. He received his bachelor degree from the School of Electronic Information Engineering, North China Institute of Science and Technology in 2018. His research interest covers container and artificial intelligence

CHAI Yi　Professor at the School of Automation, Chongqing University. He received his Ph.D. degree from Chongqing University in 2001. His research interest covers information fusion, fault diagnosis, and intelligent control system

摘要

摘要: 多重不确定性环境下的非线性系统辨识是一个开放问题. 贝叶斯学习在描述、处理不确定性方面具有显著优势, 已在线性系统辨识方面得到广泛应用, 但在非线性系统辨识的应用较少, 且面临概率估计复杂、计算量大等难题. 针对上述问题, 以典型维纳(Wiener)非线性过程为对象, 提出基于随机变分贝叶斯的非线性系统辨识方法. 首先对过程噪声、测量噪声以及参数不确定性进行概率描述; 然后利用随机变分贝叶斯方法对模型参数进行后验估计. 在估计过程中, 利用随机优化思想, 仅利用部分中间变量概率信息估计模型参数分布的自然梯度期望, 与利用所有中间变量概率信息估计模型参数比较, 显著降低了计算复杂性. 该方法是首次在系统辨识领域中的应用. 最后, 利用一个仿真实例和一个维纳模型的Benchmark问题, 证明了该方法在对大规模数据下非线性系统辨识的有效性.
- 非线性系统辨识 /
- 随机优化 /
- 变分贝叶斯 /
- 维纳模型
Abstract: Nonlinear system identification in multiple uncertain environment is an open problem. Bayesian learning has significant advantages in describing and dealing with uncertainties and has been widely used in linear system identification. However, the use of Bayesian learning for nonlinear system identification has not been well studied, confronted with the complexity of the estimation of the probability and the high computational cost. Motivated by these problems, this paper proposes a nonlinear system identification method based on stochastic variational Bayesian for Wiener model, a typical nonlinear model. First, the process noise, measurement noise and parameter uncertainty are described in terms of probability distribution. Then, the posterior estimation of model parameters is carried out by using the stochastic variational Bayesian approach. In this framework, only a few intermediate variables are used to estimate the natural gradient of the lower bound function of the likelihood function based on the stochastic optimization idea. Compared with classical variational Bayesian approach, where the estimation of model parameters depends on the information of all the intermediate variables, the computational complexity is significantly reduced for the proposed method since it only depends on the information of a few intermediate variables. To the best of our knowledge, it is the first time to use the stochastic variational Bayesian to system identification. A numerical example and a Benchmark problem of Wiener model are used to show the effectiveness of this method in the nonlinear system identification in the presence of large-scale data.
- Nonlinear system identification /
- stochastic optimization /
- variational Bayesian /
- Wiener model

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图 1 维纳模型结构示意图

Fig. 1 The structure of Wiener model

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图 2 SVBI中目标函数的更新示意图

Fig. 2 The update process of the objective function in SVBI

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图 3 辨识参数的收敛状况

Fig. 3 Convergence of identified parameters

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图 4 下界函数的收敛过程

Fig. 4 Convergence process of the lower bound function

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图 5 预测输出与实际输出比较

Fig. 5 Comparison of predicted output with actual output

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图 6 系统预测输出与实际输出

Fig. 6 Predicted output and actual output of the system

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表 1 不同子采样数据点对应的参数辨识情况

Table 1 Identification of parameters corresponding to different sub-sampling data points

	$ \langle \theta _0 \rangle $	$ \langle \theta _1 \rangle $	$ \langle \theta _2 \rangle $	$ \langle \theta _3 \rangle $	$ \langle \theta _4 \rangle $	$ \langle \lambda _0 \rangle $	$ \langle \lambda _1 \rangle $	$ \langle \lambda _2 \rangle $	时间(s)
真实值	1	−0.5000	0.2500	−0.1250	0.0625	0	1	1	—
采样1个点	1±0	−0.5463±0.3604	0.2507±0.2471	−0.2446±0.2655	0.0358±0.2882	0.5434±0.4180	0.6625±0.2907	0.3803±0.2185	0.6005
采样5%	1±0	−0.5060±0.0330	0.2693±0.0497	−0.1252±0.0323	0.0633±0.0323	0.0908±0.2707	0.9871±0.1480	0.9103±0.1246	3.1829
采样10%	1±0	−0.5055±0.0248	0.2571±0.0257	−0.1341±0.0255	0.0594±0.0256	0.0631±0.0504	0.9684±0.0498	0.9499±0.0459	7.7402
采样20%	1±0	−0.5077±0.0204	0.2544±0.0202	−0.1287±0.0289	0.0659±0.0291	0.0575±0.0540	0.9813±0.0518	0.9574±0.0451	11.4620
采样全部	1±0	−0.5078±0.0278	0.2541±0.0283	−0.1299±0.0271	0.0685±0.0246	0.0777±0.0726	0.9439±0.1183	0.9252±0.1326	9.0772

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表 2 不同异常值存在时的参数辨识情况

Table 2 Parameter identification when different outliers exist

	$ \langle \theta _0 \rangle $	$ \langle \theta _1 \rangle $	$ \langle \theta _2 \rangle $	$ \langle \theta _3 \rangle $	$ \langle \theta _4 \rangle $	$ \langle \theta _5 \rangle $	时间 (s)
真实值	1	−0.5000	0.2500	−0.1250	0.0625	−0.03125	—
无异常值	1±0	−0.4989±0.0292	0.2495±0.0293	−0.1254±0.0223	0.0611±0.0257	−0.0338±0.0262	2.9369
2% 异常值	1±0	−0.5097±0.0389	0.2672±0.0497	−0.1305±0.0426	0.0652±0.0452	−0.0291±0.0494	2.9480
5% 异常值	1±0	−0.5060±0.0330	0.2693±0.0497	−0.1252±0.0323	0.0633±0.0323	−0.0314±0.0523	3.1829
10% 异常值	1±0	−0.5349±0.0325	0.2627±0.0323	−0.1314±0.0330	0.0685±0.0389	−0.0377±0.0355	2.9057

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表 3 不同辨识方法的性能比较

Table 3 Performance comparison of different recognition methods

		$ b_0 $	$ a_1 $	$ \langle \lambda _0 \rangle(\lambda_0) $	$ \langle \lambda _1 \rangle(\lambda_1) $	$ \langle \lambda _2 \rangle(\lambda_2) $	均方误差	时间(s)
真实值		1	0.5	0	1	1	—	—
无异常值	SVBI	—	—	0.0648±0.0620	0.9633±0.0509	0.9766±0.0626	0.9136	2.936 9
	VBEM	—	—	0.0503±0.0346	0.9411±0.0393	0.9655±0.0459	0.8978	9.7046
	MLE	1±0	0.5102±0.0136	0.1054±0.0405	1.0154±0.0464	0.9490±0.0411	0.9130	9.0350
	PEM	1±0	0.4948±0.0172	0.0828±0.0524	0.9905±0.0373	1.0072±0.0449	0.9132	0.6474
5% 异常值	SVBI	—	—	0.0575±0.0540	0.9813±0.0520	0.9573±0.0450	5.4540	2.9352
	VBEM	—	—	0.0503±0.0411	0.9770±0.0532	0.9748±0.0518	3.8695	9.7709
	MLE	1±0	0.4150±0.0711	−0.9407±0.1253	1.0019±0.1839	1.3715±0.1895	3.9574	9.6693
	PEM	1±0	0.4999±0.0549	0.1072±0.1871	0.9646±0.1926	0.9878±0.1558	3.8374	0.6580
10% 异常值	SVBI	—	—	0.1439±0.1065	0.9163±0.0924	0.8416±0.0924	7.5364	2.9057
	VBEM	—	—	0.0556±0.0468	0.9711±0.0538	0.9568±0.0553	5.5110	9.9245
	MLE	—	—	—	—	—	—	—
	PEM	1±0	0.4723±0.2004	0.1458±0.5211	0.9746±0.3091	1.0030±0.3253	5.4992	0.6620

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表 4 式(52)部分参数辨识结果

Table 4 The identification results of the part parameters of the process (52)

参数	$\theta_0$	$\theta_1$	$\theta_2$	$\theta_3$	$\theta_4$	$\theta_5$	$\theta_6$	$\theta_7$	$\theta_8$	$\theta_9$	$c_0$	$c_1$	$c_2$	$Q$	$R$
结果值	−0.0390	0.0648	−0.0547	0.0856	−0.0462	0.2613	0.0501	0.2041	0.3396	0.4154	−0.0188	0.1035	−0.0030	0.0034	0.0014

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表 5 不同方法的性能比较

Table 5 Performance comparison of different methods

采样点数	方法	均方误差(V)	参数个数	时间(s)
2 000	SVBI	0.056 95	25	256.12
	VBEM	0.062 83	25	1 211.27
	SVBI	0.034 07	40	264.27
	VBEM	0.034 25	40	1 214.55
10 000	SVBI	0.061 79	25	1 299.99
	VBEM	0.093 34	25	6 347.28
	SVBI	0.033 85	40	1 332.31
	VBEM	0.034 04	40	6 442.98

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