Analytical structures and analysis of the simplest fuzzy PI controllers

This paper deals with simplest fuzzy PI controllers which employ two fuzzy numbers on the universe of discourse (UOD) of each input variable, and three fuzzy numbers on the UOD of output variable. Analytical structures of such controllers are derived using triangular membership functions for fuzzification, different combinations of T-norms and T-conorms, different inference methods, and center of area (COA) method for defuzzification. Properties of these controllers are investigated. A comparative study is made on (i) the fuzzy PI controllers derived, and (ii) on the fuzzy PI controllers and their counterpart—conventional PI controller. Moreover, sufficient conditions for bounded-input bounded-output (BIBO) stability of fuzzy PI control systems are established using the well-known small gain theorem.     

The simplest fuzzy PID controllers: mathematical models and stability analysis

This paper reveals mathematical models for the simplest fuzzy PID controllers which employ two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. Mathematical models are derived via left and right trapezoidal membership functions for each input, singleton or triangular membership functions for output, algebraic product triangular norm, different combinations of triangular co-norms and inference methods, and center of sums (COS) defuzzification method. Properties of these structures are studied to examine their suitability for control application. For the structure which is suitable for control, bounded-input bounded-output (BIBO) stability proof is presented. An approach to design fuzzy PID controllers is given. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controllers.     

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.     

Multiple Fuzzy Relation and Its Application to Coupled Fuzzy Control

Using semi‐tensor product (STP) of matrix, this paper investigates the fuzzy relation of multiple fuzzy and uses this to design coupled fuzzy control is designed. First of all, under the assumption that the universe of discourse is finite, a fuzzy logical variable can be expressed as a vector, which unifies the expression of elements, subsets, and fuzzy subsets of a universe of discourse. Then, the matrix expression of set mappings is naturally extended to fuzzy sets. Second, based on STP, logic‐based matrix addition and product are proposed. These are particulary useful for the calculation of compounded fuzzy relations. Third, a dual fuzzy structure is introduced, which assures the finiteness of the universe of discourse, and is used for fuzzification and defuzzification. Finally, using the results obtained, a new technique is developed to design a coupled fuzzy controller for multi‐input multi‐output (MIMO) systems with coupled multiple fuzzy relations.     

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.     

Quick Design of Fuzzy Controllers With Good Interpretability in Mobile Robotics

This paper presents a methodology for the design of fuzzy controllers with good interpretability in mobile robotics. It is composed of a technique to automatically generate a training data set plus an efficient algorithm to learn fuzzy controllers. The proposed approach obtains a highly interpretable knowledge base in a very reduced time, and the designer only has to define the number of membership functions and the universe of discourse of each variable, together with a scoring function. In addition, the learned fuzzy controllers are general because the training set is composed of a number of automatically generated examples that cover the universe of discourse of each variable uniformly and with a predefined precision. The methodology has been applied to the design of a wall-following and moving object following behavior. Several tests in simulated environments using the Nomad 200 robot software and a comparison with another learning method show the performance and advantages of the proposed approach.     

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。     

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.     

Structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear PID control with variable gains

The popular linear PID controller is mostly effective for linear or nearly linear control problems. Nonlinear PID controllers, however, are needed in order to satisfactorily control (highly) nonlinear plants, time-varying plants, or plants with significant time delay. This paper extends our previous papers in which we show rigorously that some fuzzy controllers are actually nonlinear PI, PD, and PID controllers with variable gains that can outperform their linear counterparts. In the present paper, we study the analytical structure of an important class of two- and three-dimensional fuzzy controllers. We link the entire class, as opposed to one controller at a time, to nonlinear PI, PD, and PID controllers with variable gains by establishing the conditions for the former to structurally become the latter. Unlike the results in the literature, which are exclusively for the fuzzy controllers using linear fuzzy sets for the input variables, this class of fuzzy controllers employs nonlinear input fuzzy sets of arbitrary types. Our structural results are thus more general and contain the existing ones as special cases. Two concrete examples are provided to illustrate the usefulness of the new results.     

Constructing nonlinear variable gain controllers via theTakagi-Sugeno fuzzy control

We investigated the analytical structure of the Takagi-Sugeno (TS) type of fuzzy controllers, which was unavailable in the literature. The TS fuzzy controllers we studied employ a new and simplified TS control rule scheme in which all the rule consequent use a common function and are proportional to one another, greatly reducing the number of parameters needed in the rules. Other components of the fuzzy controllers are general: arbitrary input fuzzy sets, any type of fuzzy logic, and the generalized defuzzifier, which contains the popular centroid defuzzifier as a special case. We proved that all these TS fuzzy controllers were nonlinear variable gain controllers and characteristics of the gain variation were parametrized and governed by the rule proportionality. We conducted an in-depth analysis on a class of nonlinear variable gain proportional-derivative (PD) controllers. We present the results to show: (1) how to analyze the characteristics of the variable gains in the context of control; (2) why the nonlinear variable gain PD controllers can outperform their linear counterpart; and (3) how to generate various gain variation characteristics through the manipulation of the rule proportionality     

Analytical structure and stability analysis of a fuzzy PID controller

Analytical structure for a fuzzy PID controller is introduced by employing two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. This structure is derived via left and right trapezoidal membership functions for inputs, trapezoidal membership functions for output, algebraic product triangular norm, bounded sum triangular co-norm, Mamdani minimum inference method, and center of sums (COS) defuzzification method. Conditions for bounded-input bounded-output (BIBO) stability are derived using the Small Gain Theorem. Finally, two numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controller.     

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。     

一类乘积型区间二型模糊控制器的解析结构

区间二型模糊控制器的降型算法需要使用迭代计算,是导致其解析结构推导困难的主要原因.针对乘积型区间二型模糊控制器,本文提出了一种新的解析结构推导方法.区间二型模糊控制器的配置为:三角形输入模糊集,一型输出模糊单值,集合中心法降型器,平均法解模糊器和基于乘积型"与"操作的规则前件.通过对比传统PID控制器的解析结构,证明了区间二型模糊控制器等效于两个PI(或PD)控制器之和.利用KM算法的迭代终止条件,提出了6步骤IC划分法,保证了激活子空间的正确划分.叠加各个子空间,即可得出全局IC划分图.为了避免重复求解符号数学方程,提出了IC边界线的直接定义法,改进了6步骤IC划分法的便利性.本文方法避开了降型算法的迭代计算,可以保证推导出区间二型模糊控制器的闭环解析表达式.     

New time-varying fuzzy sets based on a PSO midpoint of the universe of discourse

The paper presents a robust parallel distributed compensation(PDC) fuzzy controller for a nonlinear and certain system in continuous time described by the Takagi-Sugeno(T-S) fuzzy model. This controller is based on a new type of time-varying fuzzy sets(TVFS). These fuzzy sets are characterized by displacement of the kernels to the right or left of the universe of discourse, and they are directed by a well-defined criterion. In this work, we only focused on the movement of midpoint of the universe. The movements of this midpoint are optimized by particle swarm optimization(PSO) approach.     

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.     

Rough fuzzy approximations on two universes of discourse

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various fuzzy generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a covering and three relations are induced to a single universe of discourse. Based on the induced notions, four pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized, and some interesting properties are investigated. Finally, the connections of these approximation operators are made, and conditions under which some of these approximation operators are equivalent are obtained.     

Genetic algorithm for the design of a class of fuzzy controllers:an alternative approach

A simple, easy to implement alternative method for designing fuzzy logic controllers (FLCs) with symmetrically distributed fuzzy sets in a universe of discourse is introduced. The design parameters include the parameters of the membership functions of the inputs and outputs and the rule base. The method is based on a network implementation of the FLC with real and binary weights with constraints. Due to the presence of the binary weights the backpropagation technique cannot be used. The learning problem is cast as a mixed integer constrained dynamic optimization problem and solved using the genetic algorithm (GA). The crossover and mutation are slightly disrupted in order to cope with the constraints on the binary weights. Training of the controller is carried out in a closed-loop simulation with the controller in the loop     

Design of a fuzzy controller in mobile robotics using genetic algorithms

The design of fuzzy controllers for the implementation of behaviors in mobile robotics is a complex and highly time-consuming task. The use of machine learning techniques, such as evolutionary algorithms or artificial neural networks for the learning of these controllers allows to automate the design process. In this paper, the automated design of a fuzzy controller using genetic algorithms for the implementation of the wall-following behavior in a mobile robot is described. The algorithm is based on the Iterative Rule Learning (IRL) approach, and a parameter (δ) is defined with the aim of selecting the relation between the number of rules and the quality and accuracy of the controller. The designer has to define the universe of discourse and the precision of each variable, and also the scoring function. No restrictions are placed neither in the number of linguistic labels nor in the values that define the membership functions.     

Design of Mamdani fuzzy logic controllers with rule base minimisation using genetic algorithm

This paper presents a design procedure for Mamdani fuzzy logic controller including rule base minimisation. The rules are modelled with binary weights on which constraints are imposed in order to ensure consistency. A genetic algorithm is used for finding stabilising controllers that minimise the number of rules. The cost function includes a stability/performance coefficient which insures that stable, performance satisfying controllers are given the highest possible fitness. The number of fuzzy sets for the input and the control variables are set by the user and the design procedure is concerned only with the rule base and the distribution of the fuzzy sets in the universes of discourses. Two examples were studied: the control of the pole and cart system and the control of the concentration in CSTR. In both cases, the fuzzy sets were isosceles triangles evenly distributed, in the universe of discourses.     

Vague controller: A generalization of fuzzy logic controller

In this research, a vague controller (VC) is synthesized by using the notion of vague sets, which are a generalization of fuzzy sets and characterized by a truth-membership function and a falsity-membership function. The vague sets follow the basic set operations and logic operations defined for fuzzy sets, and are superior to fuzzy sets in that they could deal with the uncertainty encountered in real-world applications in a more natural way. Depending on the vague sets, the VC is developed as a generalization of fuzzy logic controller (FLC). The design procedures of the VC, which allow an arbitrary number of input variables, and each variable could have a distinct number of linguistic values, are outlined in this paper. In order to compensate the effort in constructing two series of membership functions for vague sets and to ease the difficulties in designing VCs, a new means of designating membership functions for VCs is also presented in this article. This method constructs a set of membership functions systematically by using only two parameters: number of linguistic values of a linguistic variable and shrinking factor. The membership functions generated by this method, shrinking-span membership functions (SSMFs), have different spans over the universe of discourse and, therefore, are more rational and more practical from the human expert's point of view.