基于NT降阶的乘积型区间二型模糊控制器结构分析

目前区间二型模糊控制器的结构分析主要基于Zadeh的取小推理和KM降阶算法。KM算法是一循环迭代过程,没有解析解,无法进行控制器的稳定性分析,且取小推理需要进行输入空间的划分,过程较为繁琐。提出了一种简化的区间二型模糊控制器分析方法,该方法首先采用乘积推理,模糊规则的激发隶属度为输入变量隶属度的乘积,统一了控制器的表达式形式,避免了输入空间的划分过程,模糊控制器的结构分析更加简单。二型模糊集合采用NT降阶算法,该算法直接利用首隶属度函数的上、下限的平均值来进行解模糊化操作,避免了迭代计算,简化了降阶过程。控制器的表达式等效于一个增量式PI（位置式PD）控制器,其比例增益、积分增益以及补偿项均为非线性可调。而且还能得到控制器的闭环表达式,易于进行区间二型模糊控制器的稳定性分析与设计等。     

一类区间二型模糊PI控制器设计算法

区间二型模糊控制器在处理不确定性方面优于传统的模糊控制器,但带来的一个问题就是区间二型模糊控制器需要降阶过程。常用的KM等迭代式降阶算法效率低下,难以用于实时性较高的场合。本文利用直接降阶算法和动态解模糊化算法,提出了一类区间二型模糊PI控制器设计算法。该算法在降阶过程中考虑偏差和偏差变化量对控制器输出的影响,避免了KM等迭代式降阶过程。通过二阶迟延对象以及一个非线性对象的仿真实验表明,本文算法能够有效降低系统超调,降低系统的稳态时间,控制器在设定值附近的输出更为平滑。     

一种等价于PI的二型模糊控制器设计算法

区间二型模糊集合将次隶属度做了简化,基于KM降阶算法的区间二型模糊控制器实现起来相对简单。虽然区间二型模糊控制器在一定程度上优于传统的一型模糊控制器或者PI控制器等,但区间二型模糊控制器并没有充分利用二型模糊集合的次隶属度信息。为解决这些问题,研究了普通二型模糊控制器的一般结构,提出了一种等价于PI的二型模糊控制器。该控制器基于普通二型模糊集合的α平面表现形式,在次隶属度函数的顶点处,将区间二型模糊集合简化为一型模糊集合。通过n阶有自平衡对象,无自平衡对象以及2个非线性对象的仿真结果表明,提出的二型模糊控制器能够得到较好的控制效果。     

一种改进的区间二型模糊控制器设计

针对二型模糊控制器设计中出现的降型计算方法损失不确定性信息的问题,提出一种改进的区间二型模糊控制器．该控制器在充分利用二型模糊推理结果的前提下,对区间模糊输出进行再次优化,其优化指标可直接与被控系统性能相关,由此可得到更有利于提高系统整体性能的准确输出量．最后,将改进的控制器用于汽车非线性悬架系统的控制,仿真结果验证了所提出方法的有效性．     

一种简化的区间二型模糊系统辨识方法

KM降阶算法是目前区间二型模糊集合常用的降阶算法,针对其效率低、难以用于实时辨识与控制的缺点,提出了一种简化的区间二型模糊系统辨识方法。该方法采用二型T-S模糊模型,前件参数为区间二型模糊集合,后件参数为普通T-S模糊模型形式。二型T-S模糊模型的解模糊化采用简化的降阶算法,提高了模型的辨识效率,可用于实时辨识与控制。仿真实例表明,所提算法在不降低辨识精度的情况下能够有效提高辨识效率。     

Type-Ⅱ型模糊控制器的结构分析与设计

全面研究了Type-Ⅱ型模糊控制器的解析结构,推导出了单输入单输出和双输入单输出Type-Ⅱ型模糊控制器的插值解析表达式,在一定程度上揭示了其本质特征.在此基础上,提出了Type-Ⅱ型模糊控制器的一般设计步骤、优化方法和一种快速精确的控制算法.     

基于包含度的区间二型模糊粗糙集

基于区间二型模糊包含度的公理化定义,给出了新的区间二型模糊包含度计算公式.进一步,通过包含度定义了区间二型模糊粗糙集,并讨论了它的一些基本性质.最后,利用区间二型模糊粗糙集研究了连续域决策信息系统的属性约简,给出了新的约简方法.实例说明了该约简方法的具体计算步骤,并且通过实验验证了该算法的有效性和可行性.     

具有加权因子的PID型模糊控制器分析及设计

为了降低模糊控制器的设计难度, 提高控制性能, 以二维模糊控制器为基础, 设计了一种具有加权因子的PID型模糊控制器, 并提出了基于普通PID控制器参数整定模糊控制器参数的方法。首先, 基于模糊控制器的解析结构推导, 从理论上证明了该类PID型模糊控制器是一个全局二维多值继电器与一个局部具有变结构PID控制器的组合; 然后, 基于平衡点处该类PID型模糊控制器与普通PID控制器之间的等效关系, 建立了模糊控制器系统化的参数设计方法; 最后, 仿真实验验证了本设计的有效性。     

Type-II型模糊控制器的结构分析与设计

全面研究了Type-II型模糊控制器的解析结构,推导出了单输入单输出和双输入单输出Type-II型模糊控制器的插值解析表达式,在一定程度上揭示了其本质特征。在此基础上,提出了Type-II型模糊控制器的一般设计步骤、优化方法和一种快速精确的控制算法。     

一种改进的区间二型模糊控制器设计

针对二型模糊控制器设计中出现的降型计算方法损失不确定性信息的问题,提出一种改进的区间二型模糊控制器．该控制器在充分利用二型模糊推理结果的前提下,对区间模糊输出进行再次优化,其优化指标可直接与被控系统性能相关,由此可得到更有利于提高系统整体性能的准确输出量．最后,将改进的控制器用于汽车非线性悬架系统的控制,仿真结果验证了所提出方法的有效性．

     

A general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and Zadeh AND operator

Deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding, insightful analysis, and more effective design of fuzzy control systems. We previously developed a technique for deriving the analytical structure of the fuzzy controllers that use Zadeh fuzzy AND operator and the symmetric, identical trapezoidal or triangular input fuzzy sets. Many fuzzy controllers use arbitrary trapezoidal/triangular input fuzzy sets that are asymmetric. At present, there exists no technique capable of deriving the analytical structure of these fuzzy controllers. Extending our original technique, we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller, Mamdani type or TS type, that employs the arbitrary trapezoidal input fuzzy sets and Zadeh fuzzy AND operator. The new technique contains our original technique as a special case. Given the importance of PID control, we focus on Mamdani fuzzy PI and PD controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy PI/PD controllers. The controllers use two arbitrary trapezoidal fuzzy sets for each input variable, four arbitrary singleton output fuzzy sets, four fuzzy rules, Zadeh fuzzy AND operator, and the centroid defuzzifier. This configuration is more general and complicated than the Mamdani fuzzy PI/PD controllers in the current literature. It actually contains them as special cases. We call this configuration the generalized fuzzy PI/PD controller.     

On the difference in control performance of interval type-2 fuzzy PI control system with different FOU shapes

Interval type-2 fuzzy logic controllers (IT2-FLCs) have been attracting a lot of attention. However, challenges in designing IT2-FLCs still remain. One of the main challenges is to choose the appropriate FOU shape for interval type-2 fuzzy sets (IT2-FSs). This paper aims to analyse the differences in control performance between three IT2 fuzzy PI controllers (IT2-F-PICs) with different FOU shapes as antecedent sets, namely the triangular top wide IT2 fuzzy set, the triangular bottom wide IT2 fuzzy set and the trapezoidal (also called parallel) IT2 fuzzy set. First, the analytical structures of these IT2-FLCs are derived and the mathematical input–output equations are obtained. Three interesting differences between the analytical structures and input–output relationship of the IT2-F-PICs are then presented. From the differences in the analytical structures of the three IT2-F-PICs and numerical simulation results, it is demonstrated that IT2-F-PICs with trapezoidal (IT2-F-PI-P) and triangular bottom wide (IT2-F-PI-BW) antecedent sets with the potential to provide faster transient response and faster settling time than the IT2-F-PICs with triangular top wide (IT2-F-PI-TW). In addition, IT2-F-PI-P is better able to handle plant uncertainties and disturbances than IT2-F-PI-BW and IT2-F-PI-TW. The contribution of this paper is to provide insights into the performance differences between different FOU shaped controllers, which in turns allowing control designers to select the appropriate FOU shape in order to meet design requirements.     

Deriving Analytical Input–Output Relationship for Fuzzy Controllers Using Arbitrary Input Fuzzy Sets and Zadeh Fuzzy AND Operator

A fuzzy controller uses either Zadeh or product fuzzy AND operator, with the former being more frequently used than the latter. We have recently published a novel technique for deriving analytical input–output relation for the fuzzy controllers that use Zadeh AND operator and arbitrary trapezoidal input fuzzy sets, including triangular ones as special cases. In this paper, we have developed a general technique based on that technique to cover arbitrary types of input fuzzy sets. Moreover, we have established some necessary and sufficient conditions to characterize general relationship between shape of input fuzzy sets and shape of input space divisions, an important and integral issue because analytical relationship differs in different regions of input space. The new technique and the shape relations are applicable to any type of fuzzy controllers (e.g., Mamdani type or Takagi–Sugeno type). The analytical structures that we have derived provide an unprecedented opportunity to insightfully and rigorously examine the advantages and shortcomings of different design choices available for various components of the fuzzy controllers. We have focused on type selection for input fuzzy sets of Mamdani fuzzy controllers. Our preliminary analysis indicates that the fuzzy controllers using trapezoidal fuzzy sets may be understood (and possibly analyzed and designed) more sensibly and easily in the context of conventional control theory than the fuzzy controllers using any other types of fuzzy sets. Our proposition is that trapezoidal fuzzy sets should be the first choice and used most of time. Possible implication for automatic learning of input fuzzy sets via neural networks or genetic algorithms is briefly discussed.     

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.     

基于连续可控T范数的模糊控制方法研究

针对传统模糊控制中MAX,MIN算子的缺陷,设计了一种由相关系数h控制的连续T范数,使得与或运算形式根据运算量的相关性来决定,而不是片面 取大或取小运算,并进一步将其推广到多元与或运算中,且应用于模糊控制,在仿真实验中利用遗传算法确定相关系数,实验结果证明了本方法的有效性。     

采用新的DNA进化算法自动设计Takagi-Sugeno模糊控制器

提出一种新颖的基于DNA的进化算法(DNA-EA)来自动设计一类Trakagi-Sugeno (TS)模糊控制器.TS模糊控制器采用带有线性规则后项的TS模糊规则,连续输 入模糊集,Zadeh模糊逻辑和常用的重心反模糊器.TS模糊控制器被证明是带有可变增 益的非线性PI控制器.DNA-EA被用于自动获取TS模糊规则,并同时优化模糊规则前 项和后项中的设计参数.DNA-EA采用由生物DNA结构启发得到的DNA编码方法来编 码模糊控制器的设计参数.在DNA-EA中,引入了受微生物进化现象启发的基因转移和细 菌变异操作.另外,也引入了基于DNA遗传操作的框构变异操作.DNA编码方法非常适 合于复杂知识的表达,基于基因水平的遗传操作也很容易引入到DNA-EA中.染色体的长 度是可变的,且可插入或删除部分碱基序列.作为示例,给出了采用DNA-EA来自动设计 TS模糊控制器用于控制一类非线性系统的方法.DNA-EA能自动地构造模糊控制器.计 算机仿真结果表明,DNA-EA是有效的,且优化得到的模糊控制器是满意的.     

一种由乘积蕴涵算子构造的模糊概念格

建立了形式背景下一种由乘积蕴涵算子构造的模糊概念格,给出了它的定义方式;讨论了它的性质和层次结构,并给出了一种计算模糊概念的算法。通过数值例子说明了此类概念格的构造方法。     

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.     

Optimal design of interval type 2 fuzzy controllers based on a simple tuning algorithm

Evolutionary algorithms are one of the most common choices reported in the literature for the tuning of fuzzy logic controllers based on either type-1 or type-2 fuzzy systems. An alternative to evolutionary algorithms is the simple tuning algorithm (STA-FLC), which is a methodology designed to improve the response of type-1 fuzzy logic controllers in a practical, intuitive and simple ways. This paper presents an extension of the simple tuning algorithm for fuzzy logic controllers based on the theory of type-2 fuzzy systems by using a parallel model implementation, it also includes a mechanism to calculate the feedback gain, new integral criteria parameters, and the effect of the AND/OR operator combinations on the fuzzy rules to improve the algorithm applicability and performance. All these improvements are demonstrated with experiments applied to different types of plants.