基于解析结构的模糊控制器的稳定性和一致逼近性分析

基于模糊控制的解析结构理论,设计了输入变量采用三角形、全交迭,等差间距分布的隶属函数,输出变量采用对称、均匀分布的单点隶属函数和线性控制规则的二维模糊控制器,推导了模糊控制器的解析表达式。利用Lyapunov第二法论证了该模糊控制系统在平衡点处的局部稳定性,并证明了模糊控制器的一致逼近性。     

模糊控制器的结构化分析及系统化设计方法

对于模糊控制器的输入变量,采用一种新型的不均匀、全交迭、三角形的隶属度函数,推导了两输入（e,△e)－输出（△u）的典型模糊控制器输出的解析表达式,并对最常用的输入变量各取5个模糊变量的情况进行分析。在此基础上提出一种模糊控制系统的系统化设计方法,可根据已有的PI／PD控制器参数设计相应的模糊控制器参数。仿真实验说明了该方法的有效性。     

π型隶属函数的典型模糊控制器的解析结构

研究了一种新型的典型模糊控制器,它的输入隶属函数采用π型样条函数,具有二阶逼近特性,而一般典型模糊控制器采用的三角形隶属函数只具有一阶逼近特性,因此研究这种新型的模糊控制器具有重要的意义.文章首先给出了该类典型模糊控制器的定义,推导了它的解析表达式,证明了该类典型模糊控制器可以等效为一个全局的二维继电器和一个局部的非线性PD控制器之和.在此基础上,给出了其极限特性和非线性特性.     

一种广义线性隶属函数的设计及应用

研究模糊控制器问题,针对广泛应用的三角形和梯形隶属函数存在的问题,为提高控制器的控制性能,设计并研究了一种将三角形和梯形隶属函数作为特例的广义线性隶属函数,并将它作为输入隶属函数,应用于两输入一输出的典型模糊控制器.基于模糊控制器的解析结构,在不增加模糊规则的情况下,增加模糊子空间的划分,使用合适的隶属函数提高模糊控制器的控制性能;同时,证明了输入为广义线性隶属函数的模糊控制系统的解析结构是全局多值继电器和局部PI控制器之和.最后,仿真结果证明了设计的有效性,并为线性隶属函数的模糊控制器起到了优化性能的作用.     

基于解析结构的模糊控制器的设计

一般常规模糊器设计中，隶属函数采用对称均匀分布曲线，此方法存在一定的不足，采用Sum-Product推理方法，等比间距不均匀分布隶属函数的模糊控制器，在控制性能上优于常规模糊控制器，可提高系统的快速性和鲁棒性，同时大大减少了隶属函数中需优化参数的个数，在解析结构理论研究的基础上对所设计的模糊控制器进行了定性和定量的分析，分析结果和仿真实验证明了该方法的优真性和有效性。     

输入采用广义梯形隶属函数的两维最简模糊控制器结构分析

总结了应用最为广泛的三角形和梯形隶属函数的共同特点, 明确定义了一种将以上两种隶属函数作为特例的广义梯形 (Generalized trapezoid-shaped, GTS) 隶属函数, 推导了输入变量采用 GTS 隶属函数的 I 类和 II 类两维最简模糊控制器的解析式. 基于此, 深入研究了模糊控制器的解析结构, 并证明了这两类模糊控制器等价于一种变结构的非线性 (或线性) PI 控制器与相应的非线性 (或定常) 控制偏置之和, 并且在其输入论域上是单调递增、连续且有界的. 最后, 将该类控制器应用于倒立摆控制系统, 通过仿真证明了其有效性, 同时揭示了此类控制器是一种更一般化的模糊控制器.     

两维最简模糊控制器结构分析

结合三角形和梯形隶属函数的共性,明确定义了一种将以上两种隶属函数作为特例的广义梯形(GTS)隶属函数．推导了输入变量采用广义梯形隶属函数的Ⅰ类和Ⅱ类两维最简Mamdani模糊控制器的解析表达式．深入研究了该模糊控制器的解析结构,并证明了这两类模糊控制器等价于一种变结构的非线性（或线性）PI控制器与相应的非线性（或定常）变控制偏置之和,并且在其输入论域上是单调递增、连续且有界的;同时揭示了此类控制器是一种更一般化的模糊控制器．     

隶属函数分布对Fuzzy-PI控制器动态性能的影响

为研究模糊控制器的隶属函数分布对Fuzzy-PI控制器动态性能的影响,达到优化Fuzzy-PI控制器动态性能的目的,以二阶系统为研究模型,用模糊控制器来修正比例系数Kp及积分系数Ki,根据PI控制器参数自整定原则,通过改变输入变量误差E及误差变化率EC的隶属函数分布,在MatlabSimulink环境下仿真Fuzzy-PI控制器的输出。通过数据分析,得出了变量误差E及误差化率EC的隶属函数分布对上升时间、最大超调量、调整时间影响的规律。     

典型T-S模糊系统是通用逼近器

研究的多输入单输出T－S模糊系统采用输入变量的线性函数作为规则后件,称为典型T－S模糊系统,在每个输入变量的模糊子集满足一致性以及隶属函数连续且分段可微的条件下,证明了典型T－S模糊系统是通和逼近器,以此为基础,提出当典型T－S模糊系统采用BP算法进行在线学习时,可以仅调节规则后件的参数,而同时仍然能够确保通用逼所性。     

工业洗衣机模糊控制器的设计

本文深入研究了工业洗衣机的模糊控制系统结构,确定了控制器的输入、输出变量,设计了输入输出变量的隶属函数,分析了输入输出变量的相关性,在正交实验的基础上建立了模糊控制规则库,给出了软件设计思想及模糊推理程序流程.     

典型模糊控制器的解析表达式及其系统化设计方法

对两输入一输出的典型模糊控制器推导了其解析表达式,并对输入变量各取５个模糊子集的情况进行分析,提出一种设计模糊控制器的系统化方法,它能保证模糊控制器的性能在工作眯附近等效于ＰＩ控制器,而在远离工作点时明显优于ＰＩ控制器。仿真实验结果验证了该方法的有效性。     

The simplest fuzzy controllers using different inference methods are different nonlinear proportional-integral controllers with variable gains

The author analytically proves that the simplest fuzzy controllers using different inference methods are different nonlinear proportional-integral (PI) controllers with proportional-gains and integral-gains changing with inputs of the controllers. The inference methods involved are Mamdani's minimum inference method, Larsen's product inference method, the drastic product inference method and the bounded product inference method. Configuration of the fuzzy controllers is minimal, which includes two input fuzzy sets, three output fuzzy sets, four control rules, Zadeh fuzzy logic AND, Lukasiewicz fuzzy logic OR and a center of gravity defuzzification algorithm. After analytically investigating properties of the nonlinear PI controllers, the author reveals that the bounded product inference method is inappropriate for the control purpose while the other three inference methods are appropriate. Dynamic and static control behaviors of the fuzzy controllers with the appropriate inference methods are analytically compared with each other, and are also compared with those of the linear PI controller. Finally, it is analytically proven that the fuzzy control systems have the same local stability at the equilibrium point as the corresponding linear PI control system does.     

Fuzzy controllers: synthesis and equivalences

It has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy. In particular, any given linear control can be achieved with a fuzzy controller for a given accuracy. The aim of this paper is to show how to automatically build this fuzzy controller. The proposed design methodology is detailed for the synthesis of a Sugeno or Mamdani type fuzzy controller precisely equivalent to a given PI controller. The main idea is to equate the output of the fuzzy controller with the output of the PI controller at some particular input values, called modal values. The rule base and the distribution of the membership functions can thus be deduced. The analytic expression of the output of the generated fuzzy controller is then established. For Sugeno-type fuzzy controllers, precise equivalence is directly obtained. For Mamdani-type fuzzy controllers, the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values. The usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed     

一类非线性系统的模糊控制与稳定性

研究了模糊控制器对一类非线性系统的可控性.利用李雅普诺夫一次近似稳定性方 法,证明了:1)若过程P可稳定控制(stabilized),则必可用T-S模糊控制器达到相同的稳定控 制性能;2)模糊控制器为泛控制器.这些结果回答了J.J.Buckley提出的有关问题.     

Combining genetic algorithms and Lyapunov-based adaptation for online design of fuzzy controllers.

This paper proposes a hybrid approach for the design of adaptive fuzzy controllers (FCs) in which two learning algorithms with different characteristics are merged together to obtain an improved method. The approach combines a genetic algorithm (GA), devised to optimize all the configuration parameters of the FC, including the number of membership functions and rules, and a Lyapunov-based adaptation law performing a local tuning of the output singletons of the controller, and guaranteeing the stability of each new controller investigated by the GA. The effectiveness of the proposed method is confirmed using both numerical simulations on a known case study and experiments on a nonlinear hardware benchmark.     

基于T- S模型的非线性系统多模型直接自适应控制

针对典型的高阶非线性系统，建立被控对象的多个论域不同的基于T—S模型的模糊控制器(TSFC)，用其加权组合控制系统的行为，并报据Lyapunov的综合方法设计一种自适应算法来调整每个TSFC的权值，形成被控对象的直接自适应模糊控制器。与采用单一TSFC的自适应模糊控制算法相比，该算法计算量小，响应速度快，能在局部上更有效地控制系统的非线性，使被控系统具有Lyapunov意义上的稳定性。仿真实验证实了算法的有效性。     

An Analytical Study on Structure, Stability and Design of General Nonlinear Takagi–Sugeno Fuzzy Control Systems

In this paper, we first study analytical structure of general nonlinear Takagi-Sugeno (TS, for short) fuzzy controllers, then establish a condition for analytically determining asymptotic stability of the fuzzy control systems at the equilibrium point, and finally use the stability condition in design of the control systems that are at least locally stable. The general TS fuzzy controllers use arbitrary input fuzzy sets, any types of fuzzy logic AND, TS fuzzy rules with linear consequent and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case. We have mathematically proved that the general TS fuzzy controllers are nonlinear controllers with variable gains continuously changing with controllers’ input variables. Using Lyapunov’s linearization method, we have established a necessary and sufficient condition for analytically determining local asymptotic stability of TS fuzzy control systems, each of which is made up of a fuzzy controller of the general class and a nonlinear plant. We show that the condition can be used in practice even when the plant model is not explicitly known. We have utilized the stability condition to design, with or without plant model, general TS fuzzy control systems that are at least locally stable. Three numerical examples are given to illustrate in detail how to use our new results. Our results offer four important practical advantages: (1) our stability condition, being a necessary and sufficient one, is the tightest possible stability condition, (2) the condition is simple and easy to use partially because it only needs the fuzzy controller structure around the equilibrium point, (3) the condition can be used for determining system local stability and designing fuzzy control systems that are stable at least around the equilibrium point even when the explicit plant models are unavailable, and (4) the condition covers a very broad range of nonlinear TS fuzzy control systems, for which a meaningful global stability condition seems impossible to establish.     

Output tracking control of MIMO fuzzy nonlinear systems using variable structure control approach

The output tracking control problem for nonlinear systems in the presence of both parameter perturbations and external disturbances is studied. Our approach is based on the Takagi-Sugeno (T-S) fuzzy modeling method and a variable structure control (VSC) technique. Therefore, the systems considered are not and need not be in the triangular and parametric strict-feedback form, which are prevalent among adaptive model following control for nonlinear systems, or in the normal form, which pervades almost all existing results in neuro-fuzzy model following control approach. We first study the problem of stabilization of T-S fuzzy systems by using a VSC technique. A method for the design of a switching surface based on linear matrix inequalities is developed and a stabilizing controller based on a reaching law concept in the presence of both parameter perturbations and external disturbances is proposed. Then, the method is extended to design controllers for output tracking of T-S fuzzy nonlinear systems in two cases, i.e. systems which possess the so-called strong passive subsystems and strong stable zero dynamics, respectively. Finally, illustrative examples are presented to demonstrate the whole design procedure from the original nonlinear systems to their fuzzification and finally to the realization of the desired controllers. Simulation results show that the goal of output tracking can be achieved by the proposed controllers.