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.     

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.     

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.     

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。     

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.     

Parallel distributed compensator design of tank level control based on fuzzy Takagi–Sugeno model

In this paper, a fuzzy controller is designed based on parallel distributed compensation (PDC) method and it is implemented in an experimental tank level control system. Firstly, a mathematical model of the system is obtained experimentally. An important feature of the plant is its nonlinearity. To control the level of water in the tank over the whole range, the nonlinear model of the system is linearized around three different operating points. Then, three PI controllers are designed for the operating points, using Skogestad's method. By using the PDC method, an overall fuzzy controller is designed by the fuzzy blending of the three PI-controllers. To evaluate the practical performance of the PDC-based fuzzy controller, the control system is implemented in the experimental system. The evaluation criteria considered are step response and disturbance rejection. The comparison results showed the superiority of the PDC-controller over the classical PI-controller.     

Design of variable structure control for fuzzy nonlinear systems

In this paper, the variable structure control problem is presented for Takagi–Sugeno fuzzy systems with uncertainties and external disturbances. The sliding surfaces for the T–S fuzzy system are proposed by using a Lyapunov function and a fuzzy Lyapunov function, respectively. And we design the variable structure controllers such that the global T–S fuzzy system confined on the sliding surfaces is asymptotically stable. Two examples are given to illustrate the effectiveness of our proposed methods.     

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.     

Active Control of a Piezoelectric Actuated Four-Bar Mechanism Deployed in Robotics Applications

This work presents a new micro-positioning system that is implemented in an inchworm robot to move into desired locations. The system consists of four-bar mechanism; one link is fixed, and each one of the remaining links carries a piezoelectric actuator (PZT). PZTs are specifically chosen since they provide fast response and small displacements; up to ±30 µm for ±100 Volts. The system’s mathematical model is derived and is numerically simulated by MATLAB. Three fuzzy PI controllers, which are tuned automatically by genetic algorithm, are designed to control the system. Results indicate an error of less than 1% although disturbances present.     

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.     

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.     

Output feedback controller design of Takagi–Sugeno fuzzy systems

This paper presents new systematic design methods of two types of output feedback controllers for Takagi–Sugeno (T–S) fuzzy systems, one of which is constructed with a fuzzy regulator and a fuzzy observer, while the other is an output direct feedback controller. In order to use the structural information in the rule base to decrease the conservatism of the stability analysis, the standard fuzzy partition (SFP) is employed to the premise variables of fuzzy systems. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. The concept of parallel distributed compensation (PDC) is employed to design fuzzy regulators and fuzzy observers from the T–S fuzzy models. New stability analysis and design methods of output direct feedback controllers are also presented. The output feedback controllers design and simulation results for a nonlinear mass-spring-damper mechanical system show that these methods are effective.     

Design of fuzzy PID controllers using modified triangular membership functions

A design method for fuzzy proportional-integral-derivative (PID) controllers is investigated in this study. Based on conventional triangular membership functions used in fuzzy inference systems, the modified triangular membership functions are proposed to improve a system’s performance according to knowledge-based reasonings. The parameters of the considered controllers are tuned by means of genetic algorithms (GAs) using a fitness function associated with the system’s performance indices. The merits of the proposed controllers are illustrated by considering a model of the induction motor control system and a higher-order numerical model.     

基于模糊和PI控制的MIG焊接电源设计

针对模拟控制和单片机控制的脉冲MIG(Metal Inert Gas)弧焊电源控制系统灵活性差、控制精度低和可靠性差等缺点,设计了基于模糊和PI控制的MIG焊接电源控制系统。为了提高焊接电流的控制精度,控制焊接电流的PI参数在一个周期的不同阶段应该是不同,所以该系统的焊接电流控制采用变参数PI控制方法。在不同焊接条件下的PI参数由专家系统确定。为了提高电弧弧长的稳定性,电弧电压控制采用模糊控制方法。模糊控制和变参数PI控制算法分别由数字信号控制器(DSC)和现场可编程门阵列(FPGA)实现。最后,介绍了系统的硬件电路设计和软件流程。利用焊接铝板对该系统进行了测试,测试结果表明,基于模糊和PI控制的MIG焊接电源控制系统动态响应快、可靠性高、弧长控制稳定。     

一种解析的模糊PI自整定方法

针对模糊PID控制器缺乏系统的整定方法的问题,提出了一种解析的基于增益裕度和相位裕度的模糊PI控制器的参数自整定方法。首先推导出模糊PI控制器的解析模型,该解析模型包括线性控制器和非线性补偿控制器2个部分。参数整定时,将非线性补偿控制器看作过程的扰动,由线性控制器和被控对象的一阶纯时滞模型,基于系统的增益裕度和相位裕度,导出模糊PI控制器的参数。仿真结果表明,对于时变高阶系统,和传统的PI控制器相比,模糊PI控制器具有鲁棒性强,超调小,调整时间短等优点。     

Fuzzy arithmetic-based interpolative reasoning for nonlinear dynamic fuzzy systems

FAIR (fuzzy arithmetic-based interpolative reasoning)—a fuzzy reasoning scheme based on fuzzy arithmetic, is presented here. Linguistic rules of the Mamdani type, with fuzzy numbers as consequents, are used in an inference mechanism similar to that of a Takagi–Sugeno model. The inference result is a weighted sum of fuzzy numbers, calculated by means of the extension principle. Both fuzzy and crisp inputs and outputs can be used, and the chaining of rule bases is supported without increasing the spread of the output fuzzy sets in each step. This provides a setting for modeling dynamic fuzzy systems using fuzzy recursion. The matching in the rule antecedents is done by means of a compatibility measure that can be selected to suit the application at hand. Different compatibility measures can be used for different antecedent variables, and reasoning with sparse rule bases is supported. The application of FAIR to the modeling of a nonlinear dynamic system based on a combination of knowledge-driven and data-driven approaches is presented as an example.     

Emergence of self‐learning fuzzy systems by a new virus DNA–based evolutionary algorithm

In this article, we propose a new approach to the virus DNA–based evolutionary algorithm (VDNA‐EA) to implement self‐learning of a class of Takagi‐Sugeno (T‐S) fuzzy controllers. The fuzzy controllers use T‐S fuzzy rules with linear consequent, the generalized input fuzzy sets, Zadeh fuzzy logic and operators, and the generalized defuzzifier. The fuzzy controllers are proved to be nonlinear proportional‐integral (PI) controllers with variable gains. The fuzzy rules are discovered automatically and the design parameters in the input fuzzy sets and the linear rule consequent are optimized simultaneously by the VDNA‐EA. The VDNA‐EA uses the VDNA encoding method that stemmed from the structure of the VDNA to encode the design parameters of the fuzzy controllers. We use the frameshift decoding method of the VDNA to decode the DNA chromosome into the design parameters of the fuzzy controllers. In addition, the gene transfer operation and bacterial mutation operation inspired by a microbial evolution phenomenon are introduced into the VDNA‐EA. Moreover, frameshift mutation operations based on the DNA genetic operations are used in the VDNA‐EA to add and delete adaptively fuzzy rules. Our encoding method can significantly shorten the code length of the DNA chromosomes and improve the encoding efficiency. The length of the chromosome is variable and it is easy to insert and delete parts of the chromosome. It is suitable for complex knowledge representation and is easy for the genetic operations at gene level to be introduced into the VDNA‐EA. We show how to implement the new method to self‐learn a T‐S fuzzy controller in the control of a nonlinear system. The fuzzy controller can be constructed automatically by the VDNA‐EA. Computer simulation results indicate that the new method is effective and the designed fuzzy controller is satisfactory. © 2003 Wiley Periodicals, Inc.     

A novel hybrid DEPS optimized fuzzy PI/PID controller for load frequency control of multi-area interconnected power systems

In this paper, a novel hybrid Differential Evolution (DE) and Pattern Search (PS) optimized fuzzy PI/PID controller is proposed for Load Frequency Control (LFC) of multi-area power system. Initially a two-area non-reheat thermal system is considered and the optimum gains of the fuzzy PI/PID controller are optimized employing a hybrid DE and PS (hDEPS) optimization technique. The superiority of the proposed controller is demonstrated by comparing the results with some recently published modern heuristic optimization techniques such as DE, Bacteria Foraging Optimization Algorithm (BFOA), Genetic Algorithm (GA) and conventional Ziegler Nichols (ZN) based PI controllers for the same interconnected power system. Furthermore, robustness analysis is performed by varying the system parameters and operating load conditions from their nominal values. It is observed that the optimum gains of the proposed controller need not be reset even if the system is subjected to wide variation in loading condition and system parameters. Additionally, the proposed approach is further extended to multi-area multi-source power system with/without HVDC link and the gains of fuzzy PID controllers are optimized using hDEPS algorithm. The superiority of the proposed approach is shown by comparing the results with recently published DE optimized PID controller and conventional optimal output feedback controller for the same power systems. Finally, Reheat turbine, Generation Rate Constraint (GRC) and time delay are included in the system model to demonstrate the ability of the proposed approach to handle nonlinearity and physical constraints in the system model.