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.     

Analytical structures and analysis of the simplest fuzzy PDcontrollers

This paper deals with simplest fuzzy PD controllers which employ only two fuzzy sets on the universe of discourse of each input variable, and three fuzzy sets on the universe of discourse of output variable. First, analytical structures of the simplest fuzzy PD controllers are derived via triangular membership functions for fuzzification, intersection T-norm, Lukasiewicz OR and Zadeh (1965) OR T-conorms, Mamdani's minimum, Larsen's product and drastic product inference methods, and center of area method for defuzzification. Properties of such fuzzy PD controllers are investigated. Based on these properties a comparative study is made on fuzzy controllers derived, and also on the fuzzy controllers and their counterpart-conventional linear PD controller. Finally, sufficient conditions for bounded-input bounded-output stability of fuzzy PD control systems are established using the well known small gain theorem.     

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.     

一类模糊P I D 控制器的鲁棒优化设计

研究一类模糊 PID控制器的鲁棒设计。以小增益定理分析得到该模糊 PID控制系统稳定性条件。针对参数摄动系统的“最坏点”,用该稳定性条件作为约束 ,采用遗传算法对标称系统的性能进行优化 ,求得优化鲁棒控制器。以倒立摆为例进行鲁棒模糊 PID控制器的设计 ,实验结果表明了该方法的有效性     

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.     

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.     

模糊PID控制器的稳定性分析

构造出一种PID型模糊控制器，并证明了这种模糊控制器近似于一种变参数的PID控制器，以PID模型为基础，基于无源性定量对模型PID控制器的稳定性进行分析，导出了使模糊PID控制器稳定的充分条件，为设计稳定的模糊PID控制器提供了理论指导。     

Design and optimization of fuzzy controllers based on the operator's knowledge

Industrial processes are normally operated by skilled humans who have cumulative and logical information about the systems. Fuzzy control has been investigated for many applications. Intelligent control approaches based on fuzzy logic might be able to include human thinking. This article describes a modeling approach based upon an operator's knowledge without a mathematical model of the system and optimization of the controller. The test system applied was constructed to send a ball into the goal position using wind from two DC motors in a predefined path. A vision camera to mimic human eyes detects the ball's position. The system used in this experiment was difficult to model by mathematical methods, and could not easily be controlled by conventional methods. The controller is designed based on the input-output data as well as experimental knowledge obtained by trials, and optimized under predefined performance criteria. This work was presented, in part, at the Sixth International Symposium on Artifical Life and Robotics, Tokyo, Japan, January 15–17, 2001     

PI-FUZZY模糊控制器的设计及应用

本文介绍了一种以PLC为核心的铝板初轧机自动位置控制系统中控制器的设计方法，对传统的PI调节器和Fuzzy控制器分别进行了理论分析和仿真研究，给出了系统的软、硬件结构，PLC配置及现场运行结果，本系统的特点是成本低、工作可靠，操作简单，定位精度可达0．02mm。     

模糊逻辑补偿的网络PI控制系统及稳定性分析

针对现代工业过程中常用的网络PI控制系统, 分析了网络所诱导的时滞对PI控制系统性能的影响, 设计了一个模糊逻辑补偿器来实现对网络所诱导的时滞的补偿. 利用Hermite-Biehler引理在准多项式稳定性问题上的推广, 研究了具有模糊逻辑补偿的网络PI控制系统的稳定性问题, 给出了使闭环系统稳定的模糊逻辑调节参数的取值范围. 仿真结果验证了理论分析结果的有效性.     

两款通用模糊控制器的设计与实现

介绍了两款基于高性能单片机的通用模糊控制器的设计与实现,对其系统结构和各功能模块作了详细的描述。这两款通用模糊控制器成本低、可靠性高,为锅炉燃烧控制等实际应用提供了合适的硬件平台。     

Hierarchical aggregation of multiple fuzzy controllers for global complex control problems

In this paper, we illustrate a proposed method for control that combines the outputs of several individual controllers to improve global control of complex nonlinear plants. In the first part of this paper, we illustrate the proposed method that consists of two levels, where in the top level a fuzzy system represents a superior control that is designed for adjusting the behavior of the individual fuzzy controllers at the lower level. To test the approach, we consider the problem of flight control because it requires several individual controllers. Also a comparison is performed, where the hierarchical control strategy is compared with a simple control approach using the t student test. In this paper, we show that the proposed method outperforms the conventional fuzzy control approach. In the optimal design of the proposed control architecture a genetic algorithm was also applied to tune the parameters of the fuzzy systems in an optimal fashion.     

Genetic algorithm design of neural network and fuzzy logic controllers

This paper discusses the design of neural network and fuzzy logic controllers using genetic algorithms, for real-time control of flows in sewerage networks. The soft controllers operate in a critical control range, with a simple set-point strategy governing “easy” cases. The genetic algorithm designs controllers and set-points by repeated application of a simulator. A comparison between neural network, fuzzy logic and benchmark controller performance is presented. Global and local control strategies are compared. Methods to reduce execution time of the genetic algorithm, including the use of a Tabu algorithm for training data selection, are also discussed. The results indicate that local control is superior to global control, and that the genetic algorithm design of soft controllers is feasible even for complex flow systems of a realistic scale. Neural network and fuzzy logic controllers have comparable performance, although neural networks can be successfully optimised more consistently.     

典型模糊控制器的模糊规则函数与极限结构分析

讨论一般模糊控制器的解析形式。在以数值示例分析原有模糊规则解析描述的基础上，提出模糊规则函数(FRF)的概念，并证明了模糊规则函数的连续性和无穷性是控制器的极限结构定理的充要条件，统一了模糊控制器的极限结构及其函数逼近能力。这使得设计模糊控制器时，不必担心由于提高逼近精度而影响到已有的控制规则。     

Bidirectional active control of structures with type-2 fuzzy PD and PID

Proportional-derivative and proportional-integral-derivative (PD/PID) controllers are popular algorithms in structure vibration control. In order to maintain minimum regulation error, the PD/PID control require big proportional and derivative gains. The control performances are not satisfied because of the big uncertainties in the buildings. In this paper, type-2 fuzzy system is applied to compensate the unknown uncertainties, and is combined with the PD/PID control. We prove the stability of these fuzzy PD and PID controllers. The sufficient conditions can be used for choosing the gains of PD/PID. The theory results are verified by a two-storey building prototype. The experimental results validate our analysis.     

Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition

In this paper, we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max-min composition in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with Ordinary Inequality constraints (denoted by x_OI in this paper). We have presented an algorithm to generate such optimal solutions. Also two examples are given to illustrate the algorithm presented and some real application of this work.     

PI-Fuzzy controllers for integral plants to ensure robust stability

New PI-fuzzy controllers comprising Takagi-Sugeno and Mamdani fuzzy systems to control a class of integral plants specific to servo systems are proposed in this paper. Linear PI controllers are designed in the first phase. They are tuned using the Extended Symmetrical Optimum method to ensure the desired overshoot and settling time with respect to step setpoint modifications and three types of load disturbance inputs. This type of PI controller design guarantees robust stability of the closed-loop system in response to parametric variations in a controlled plant. Following this, based on the results from the linear case and the modal equivalence principle, an original development method for PI-fuzzy controllers is proposed. Some experimental results are included to illustrate the effectiveness of the design process.     

Stabilizing sets of PI/PID controllers for unstable second order delay system

In this paper, the problem of stabilizing an unstable second order delay system using classical proportional-integralderivative(PID) controller is considered. An extension of the Hermite-Biehler theorem, which is applicable to quasi-polynomials, is used to seek the set of complete stabilizing proportional-integral/proportional-integral-derivative(PI/PID) parameters. The range of admissible proportional gains is determined in closed form. For each proportional gain, the stabilizing set in the space of the integral and derivative gains is shown to be a triangle.     

T-S模糊系统输出反馈控制器的稳定性分析与设计

输出反馈控制是T-S模糊控制系统设计的一种重要方法．本文提出了一类由模糊状态观测器和模糊调节器构成的输出反馈控制器稳定性分析和解析设计的新方法．为了减小稳定性分析的保守性和难度,本文充分利用了模糊规则前件变量模糊隶属度函数的结构信息,对前件变量采用标准模糊分划的T-S模糊系统输出反馈控制器进行了研究,获得了一些新的稳定性条件．然后采用平行分布补偿法(PDC)和线性矩阵不等式方法(LMI),研究了该类输出反馈控制器的解析设计方法．通过一个非线性质量块-弹簧-阻尼器系统输出反馈控制器的设计和计算机仿真,验证了本文方法的有效性．