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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2026, Vol. 60 Issue (10): 2259-2277    DOI: 10.3785/j.issn.1008-973X.2026.10.018
    
Review of fuzzy modeling and control for multivariable nonlinear complex systems
Zhuang KANG1,2(),Limin JIA1,2,*()
1. State Key Laboratory of Advanced Rail Autonomous Operation, Beijing Jiaotong University, Beijing 100044, China
2. School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China
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Abstract  

In response to the challenges of high difficulty in modeling and the low control efficiency in multivariable nonlinear complex systems, fuzzy control has been identified as a promising research approach. Significant progress has been made over the years. Advancements in fuzzy control theory for these systems have been reviewed. Firstly, relevant knowledge of fuzzy theory and the basic components of fuzzy controllers were presented. Secondly, various fuzzy models and control methods for multivariable nonlinear complex systems were discussed, including fuzzy control systems based on semantic rules, model-based fuzzy control systems, and others. Special emphasis was placed on model-based fuzzy control systems, such as the T-S fuzzy model, fuzzy cell-to-cell mapping model, fuzzy neural network model, and fuzzy models based on the variation relationship between process input and output data. Detailed summaries and analyses of both model structure identification and parameter identification were provided. Finally, challenges related to the universality, interpretability, stability, and applicability of these systems were explored, along with potential directions for future research and development.

Key words： multivariable nonlinear complex system      fuzzy control      fuzzy modelling      structure identification      parameter identification     
Received: 22 May 2025      Published: 29 July 2026
CLC:  TP 273  
Fund:  中央高校基本科研业务费专项资金资助项目（科技领军人才团队项目）（2022JBXT009）.
Corresponding Authors: Limin JIA     E-mail: 21114026@bjtu.edu.cn;lmjia@bjtu.edu.cn
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Cite this article:

Zhuang KANG,Limin JIA. Review of fuzzy modeling and control for multivariable nonlinear complex systems. Journal of ZheJiang University (Engineering Science), 2026, 60(10): 2259-2277.

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https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2026.10.018     OR     https://www.zjujournals.com/eng/Y2026/V60/I10/2259


多变量非线性复杂系统的模糊建模与控制综述

针对多变量非线性复杂系统建模难度高和控制效率低的问题，模糊控制提供了新的研究途径，经过多年发展，已取得大量研究成果. 本研究主要综述模糊控制理论在多变量非线性复杂系统中的研究进展. 介绍模糊理论相关知识和模糊控制器的基本组成部分；分析多变量非线性复杂系统的各类模糊模型与控制方法，包括基于语义规则的模糊控制系统、基于模型的模糊控制系统和其他模糊控制系统. 着重讨论基于模型的控制系统，包括T-S模糊模型、模糊穴-穴映射模型、模糊神经网络模型和基于过程输入输出数据变化关系的模糊模型，并从模型的结构辨识和参数辨识2方面进行总结与研究. 探讨多变量非线性复杂系统在普适性、可解释性、稳定性、应用性等方面存在的问题及未来可能的发展方向.


关键词： 多变量非线性复杂系统,  模糊控制,  模糊建模,  结构辨识,  参数辨识 
Fig.1 Basic structure of fuzzy controller
Fig.2 Two-stage fuzzy curve algorithm flowchart
内容方法优点缺点
输入变量
选择		模糊搜索树有效地通过RC（规则性准则）优化输入变量选择，能减少输入输出关系的复杂性.计算量较大，尤其在处理复杂或多维数据时，需要较多的计算资源.
灰色关联法适用于处理多输入变量系统，能够评估输入变量与输出之间的关联度，直观且简单易用.对数据噪声敏感，特别是数据中存在干扰因素时，结果误差较大.
灵敏度分析法通过量化输入对系统输出的影响，能有效识别关键输入变量，适用于多变量系统.对高维度系统的计算要求高，耗时大，且忽视了变量之间的相互影响.
双阶段模糊
曲线法		对非线性系统尤其有效，能够处理不确定性较高的输入输出关系，适合MIMO系统.需要良好的数据结构，并且对初始参数选择较为敏感，系统复杂时可能需要调参.
模糊空间划分模糊网格法简单直观，适应性强，适合规则且均匀分布的输入空间划分，计算高效.划分精度有限，无法适应复杂数据；高维情况下计算量大，处理复杂数据困难.
模糊聚类法能处理复杂的非线性关系，灵活性高，鲁棒性强，对噪声数据容忍度高.计算要求高，对初始参数敏感，尤其是初始中心的选择，且容易陷入局部最优解.
模糊树法通过树形结构划分输入空间，结构灵活，适合高维数据的处理，适用于多层次复杂问题.模型复杂，容易过拟合，需要较多计算资源，训练过程较为耗时.
Tab.1 Comparison of advantages and disadvantages of various methods for structural identification of fuzzy models
Fig.3 Hierarchical fuzzy system structure with p inputs
Fig.4 FCM-I system structure
Fig.5 FLCON structure
Fig.6 ANFIS system structure
模型处理高维MIMO系统优点缺点
模糊规则数复杂度应对策略
T-S模糊模型规则数为kn（n为输入维度，k为模糊集数）高规则数随输入维度呈指数增长. 通过引入规则约简方法（如SVD[57]）来减少冗余规则能与状态空间控制理论紧密结合，后件线性结构适合稳定性分析与控制律设计，适用于非线性系统近似建模高维系统面临维数爆炸，且须解决多Lyapunov方程组求解难题，前件/后件辨识分离导致实现复杂
模糊穴-穴映射模型规则数为pn（n为输入维度，p为模糊穴数）较高规则数随输入和模糊穴数增加而增长，计算复杂度较高. 可通过$ \alpha $-cut滤波[47]来优化规则库将MIMO模糊系统建模简化为二维映射问题，建模逻辑直观，推理效率高，规则冗余问题得到缓解辨识精度过于依赖样本质量，易陷入局部最优；稳定性分析理论尚不完善，控制系统的设计受限
神经网络模糊模型规则数较少，主要由神经网络决定较低（依赖
网络训练）		神经网络通过学习有效的规则权重和连接减少规则数量[58]，避免传统模糊系统规则爆炸问题具备自学习能力，适合未知系统建模；模型泛化强、精度高，适用数据驱动的大规模非线性复杂系统结构复杂、参数众多；模型稳定性与可解释性差，训练样本需求高，须防范过拟合与训练效率瓶颈
输入输出变化关系模型规则数依赖于系统输入和输出变化模式中等通过输入输出的变化模式生成规则，可以灵活调整规则数量并避免规则爆炸建模结构简单，参数少，适用于数据丰富、变化规律清晰的工业系统，有利于在线更新与实时控制模型构建依赖高质量数据，泛化能力受限；在强耦合、高非线性系统中表现不如其他方法稳定
Tab.2 Complexity analysis of handling high-dimensional MIMO systems by various fuzzy models and comparison of advantages and disadvantages of each fuzzy model
神经网络模糊推理系统
难以使用先验知识可以使用先验知识
从零开始学习无法学习（语言知识）
黑匣子可解释（IF-THEN规则）
复杂的学习算法可解释性的简单算法
难以提取知识可以提取经验知识
Tab.3 Comparison of characteristics between neural networks and fuzzy inference systems
模糊控制器优点缺点
模糊PID控制利用模糊推理优化PID参数，提升非线性复杂系统的自适应能力，结构清晰简单，易于工程实现高维MIMO系统中规则库膨胀显著，参数调试工作繁重，难以应对剧烈工况变化
模糊解耦控制能有效分解多变量系统中的耦合关系，适用于复杂MIMO系统；适配多种解耦机制模糊规则增长快，对系统结构依赖性强，实时计算代价高，工程落地难度大
模糊滑模控制结合鲁棒性控制优势，适用于不确定性与干扰显著的系统；收敛性能强，稳定性可分析控制器理论设计要求高，尤其是控制律设计复杂，须平衡收敛速度与抖振消除，计算负荷大
模糊神经
网络控制		拥有强大的自学习和非线性逼近能力，兼具自学习与可解释性，架构多样，适应性强模型训练需大量数据，且依赖高质量数据，计算资源消耗高，系统稳定性分析复杂
自适应模
糊控制		能够在线调整参数应对系统变化，鲁棒性强；可融合多种控制机制提高性能实现难度大，算法设计与稳定性分析要求高，对硬件资源有较高依赖
Tab.4 Comparison of advantages and disadvantages of fuzzy controllers
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