基于期望最大化方法的非线性SSM黑箱鲁棒辨识

doi:10.16182/j.issn1004731x.joss.25-0736

系统仿真学报 ›› 2026, Vol. 38 ›› Issue (5): 1146-1158.doi: 10.16182/j.issn1004731x.joss.25-0736

• 论文 • 上一篇

基于期望最大化方法的非线性SSM黑箱鲁棒辨识

李校男¹^,², 晁涛¹^,³, 马萍¹^,³, 杨明¹^,³, 王玉轩²^,⁴

^1.哈尔滨工业大学控制与仿真中心，黑龙江哈尔滨 150001
^2.空基信息感知与融合全国重点实验室，河南洛阳 471009
^3.复杂系统建模与仿真全国重点实验室，黑龙江哈尔滨 150001
^4.中国空空导弹研究院，河南洛阳 471009

收稿日期:2025-07-31 修回日期:2025-09-02 出版日期:2026-05-21 发布日期:2026-05-29
通讯作者: 晁涛
第一作者简介:李校男(1996-)，男，博士生，研究方向为系统建模及状态估计。
基金资助:
国家自然科学基金(62273119);航空科学基金联合资助(20240001077001)

Robust Identification of Black-box Nonlinear SSM Using Expectation-maximization

Li Xiaonan¹^,², Chao Tao¹^,³, Ma Ping¹^,³, Yang Ming¹^,³, Wang Yuxuan²^,⁴

^1.Control and Simulation Center, Harbin Institute of Technology, Harbin 150001, China
^2.National Key Laboratory of Air-based Information Perception and Fusion, Luoyang 471009, China
^3.National Key Laboratory of Modeling and Simulation for Complex Systems, Harbin 150001, China
^4.China Airborne Missile Academy, Luoyang 471009, China

Received:2025-07-31 Revised:2025-09-02 Online:2026-05-21 Published:2026-05-29
Contact: Chao Tao

摘要/Abstract

摘要：

针对观测存在异常、缺失且状态方程未知的非线性SSM鲁棒辨识问题，提出一种基于“本征函数展开‒高斯过程状态空间模型(Gaussian-process state-space models, GP-SSM)-Student-t分布”的建模方法。采用本征函数对状态转移函数进行建模，将基函数系数通过GP-SSM预先编码以提升模型参数化灵活性；将观测建模为含未知参数的Student-t分布，增强对异常值的鲁棒性；提出扩展粒子Gibbs祖先采样(extended particle gibbs with ancestor sampling, EPGAS)算法，以适配观测值缺失场景下的状态估计；基于期望最大化(expectation maximization, EM)方法推导模型未知参数。仿真算例与基准模型测试结果表明：所提方法相较现有方法性能更优，可明显提升观测存在随机异常与缺失时的模型辨识精度。

关键词: 鲁棒辨识, 系统动态未知, Student-t分布, 缺失和异常观测, 期望最大化

Abstract:

To address the robust identification problem of nonlinear state space models (SSM) with outliers, missing observations, and unknown state equations, this paper proposes a modeling method based on eigenfunction expansion, Gaussian-process state-space models (GP-SSM), and Student-t distribution. The proposed approach consists of: modeling the state transition function using eigenfunctions and pre-encoding the priors of basis function coefficients via GP-SSM to enhance flexibility; modeling observations as a Student-t distribution with unknown parameters to enhance robustness against outliers; proposing the enhanced particle Gibbs with ancestor sampling (EPGAS) algorithm to adapt to state estimation in scenarios with missing observations; and deriving unknown model parameters based on the expectation maximization (EM) method. The simulation examples and benchmark model test results show that the proposed method has better performance compared to existing literature methods, and can significantly improve the model identification accuracy when there are outliers and missing observations.

Key words: robust identification, unknown system dynamics, Student-t distribution, missing and abnormal observations, expectation-maximization

中图分类号:

TP391.9

李校男,晁涛,马萍等 . 基于期望最大化方法的非线性SSM黑箱鲁棒辨识[J]. 系统仿真学报, 2026, 38(5): 1146-1158.

Li Xiaonan,Chao Tao,Ma Ping,et al . Robust Identification of Black-box Nonlinear SSM Using Expectation-maximization[J]. Journal of System Simulation, 2026, 38(5): 1146-1158.

图/表 10

图1

表1

算法复杂度

算法名称	复杂度
EPGAS方法	$O (T N (d x + d y 3))$
鲁棒辨识方法	$O (M K 3 + M T N (d x + d y 3))$

表1

表2

不同迭代次数对应的误差

指标	20	50	80	100	200	300
$M A E$	2.12	1.77	1.60	1.51	1.50	1.50
$R M S E$	2.73	2.31	2.08	1.97	1.95	1.95

表2

图2

表3

不同异常比例下的误差对比

指标	方法	$p a b n$
指标	方法	0.05	0.10	0.20
$M A E$	Svensson-RML	1.81	2.49	3.56
	Svensson-Bayesian	1.67	2.16	2.78
	Adhau	1.62	2.11	2.69
	本文	1.33	1.50	1.72
$R M S E$	Svensson-RML	2.58	3.71	4.70
	Svensson-Bayesian	2.25	3.01	3.78
	Adhau	2.18	2.98	3.66
	本文	1.74	1.95	2.28

表3

表4

不同缺失、异常比例下的误差对比

指标	方法	$p a b n = 0.1 p m i s = 0.1$	$p a b n = 0.2 p m i s = 0.2$
$M A E$	Svensson-RML	2.78	4.52
	Svensson-Bayesian	2.55	3.44
	Adhau	2.48	3.29
	本文	1.61	1.98
$R M S E$	Svensson-RML	3.88	6.07
	Svensson-Bayesian	3.64	4.60
	Adhau	3.53	4.41
	本文	2.12	2.61

表4

图3

表5

不同方法的运行时间

方法	运行时间/s
Svensson-RML	$54$
Svensson-Bayesian	$55$
Adhau	$82$
本文	$60$

表5

表6

不同方法的辨识误差

误差	方法	含20%异常值
$M A E$	Svensson-RML	0.27
	Svensson-Bayesian	0.23
	Adhau	0.22
	本文	0.14
$R M S E$	Svensson-RML	0.34
	Svensson-Bayesian	0.30
	Adhau	0.29
	本文	0.18

表6

表7

含20%异常及10%缺失下不同算法的误差

方法	$R M S E$
Svensson-RML	0.36
Svensson-Bayesian	0.32
Adhau	0.31
本文	0.21

表7

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基于期望最大化方法的非线性SSM黑箱鲁棒辨识

Robust Identification of Black-box Nonlinear SSM Using Expectation-maximization

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