D ERIVATIVE ACTION IN PID-FUZZY CONTROLLERS

Fuzzy control has made possible the establishment of intelligent control However fuzzy logic controllers FLCs are used only in simply configura tions and analytic knowledge about them is poor In this paper a quantita tive study of fuzzy controllers is done for the most complete case of a fuzzy PID Although the derivative action increases the complexity it makes it possible to complete a study of FLCs in a similar way to the study of conventional controllers The derivative term can improve the stability and gives more flexibility The analytic performance of a fuzzy PID is summa rized in terms of its three input variables These equations yield initial     

A systematic study of fuzzy PID controllers-function-basedevaluation approach

A function-based evaluation approach is proposed for a systematic study of fuzzy proportional-integral-derivative (PID)-like controllers. This approach is applied for deriving process-independent design guidelines from addressing two issues: simplicity and nonlinearity. To examine the simplicity of fuzzy PID controllers, we conclude that direct-action controllers exhibit simpler design properties than gain-scheduling controllers. Then, we evaluate the inference structures of direct-action controllers in five criteria: control-action composition, input coupling, gain dependency, gain-role change, and rule/parameter growth. Three types of fuzzy PID controllers, using one-, two- and three-input inference structures, are analyzed. The results, according to the criteria, demonstrate some shortcomings in Mamdani's two-input controllers. For keeping the simplicity feature like a linear PID controller, a one-input fuzzy PID controller with "one-to-three" mapping inference engine is recommended. We discuss three evaluation approaches in a nonlinear approximation study: function-estimation-based, generalization-capability-based and nonlinearity-variation-based approximations. The study focuses on the last approach. A nonlinearity evaluation is then performed for several one-input fuzzy PID controllers based on two measures: nonlinearity variation index and linearity approximation index. Using these quantitative indices, one can make a reasonable selection of fuzzy reasoning mechanisms and membership functions without requiring any process information. From the study we observed that the Zadeh-Mamdani's "max-min-gravity" scheme produces the highest score in terms of nonlinearity variations, which is superior to other schemes, such as Mizumoto's "product-sum-gravity" and "Takagi-Sugeno-Kang" schemes     

Robust fuzzy control for a plant with fuzzy linear model

A robust complexity reduced proportional-integral-derivative (PID)-like fuzzy controllers is designed for a plant with fuzzy linear model. The plant model is described with the expert's linguistic information involved. The linguistic information for the plant model is represented as fuzzy sets. In order to design a robust fuzzy controller for a plant model with fuzzy sets, an approach is developed to implement the best crisp approximation of fuzzy sets into intervals. Then, Kharitonov's Theorem is applied to construct a robust fuzzy controller for the fuzzy uncertain plant with interval model. With the linear combination of input variables as a new input variable, the complexity of the fuzzy mechanism of PID-like fuzzy controller is significantly reduced. The parameters in the robust fuzzy controller are determined to satisfy the stability conditions. The robustness of the designed fuzzy controller is discussed. Also, with the provided definition of relative robustness, the robustness of the complexity reduced fuzzy controller is compared to the classical PID controller for a second-order plant with fuzzy linear model. The simulation results are included to show the effectiveness of the designed PID-like robust fuzzy controller with the complexity reduced fuzzy mechanism.     

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.     

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.     

Flexible complexity reduced PID-like fuzzy controllers

In this paper, a flexible complexity reduced design approach for PID-like fuzzy controllers is proposed. With the linear combination of input variables as a new input variable, the complexity of the fuzzy mechanism of PID-like fuzzy controllers is significantly reduced. However, the performance of the complexity reduced fuzzy PID controller may be degraded since the degree of freedom is decreased by the combination of input variables. To alleviate the drawback and improve the performance of the complexity reduced PID-like fuzzy controller, a flexible complexity reduced design approach is introduced in which the functional scaling factors are heuristically generated. Since the functional scaling factors are heuristically created, they can be easily adjusted for the flexible complexity reduced PID-like fuzzy controller without a priori knowledge of the exact mathematical model of the plant. Moreover, heuristic scaling factors are implemented as functionals. Therefore, the complexity of the flexible PID-like fuzzy controller will not be increased. Further, the stability of the fuzzy control system with a flexible complexity reduced PID-like fuzzy controller is discussed. Finally, the simulation results are also included to show the effectiveness of the PID-like fuzzy controller designed with the flexible complexity reduced approach.     

A NEW DESIGN APPROACH FOR FUZZY‐LEARNING FUZZY CONTROLLERS

In this paper, a new approach to designing fuzzy‐learning fuzzy controllers for a system plant without an exact mathematical model is presented. The cost function is defined as the square of the sliding function to alleviate the difficulty of overshoot when on‐line learning is conducted. The learning mechanism of a fuzzy controller is constructed so as to minimize the cost function with a set of linguistic rules. Moreover, to reduce the complexity of the fuzzy‐learning fuzzy controller, the fuzzy mechanism used for learning and the fuzzy mechanism contained in the fuzzy controller are designed so as to have the identical structures. Finally, simulations are included to show the effectiveness of the fuzzy‐learning fuzzy controllers.     

过热汽温二级模糊鲁棒自调整PID控制器设计

许多过程控制对象的模型结构及参数常常具有随设备运行状态改变而产生较大幅度 变化的特性，或者具有某些不确定性，文章探讨了采用增益配置型模糊PID控制器的参数自 调整的鲁棒设计问题，提出利用二级模糊推理结合系统特征在线辨识实现PID参数动态调整 的控制器设计方法，实例仿真表明该控制器能较好地适应对象动态模型的大幅度变化，保持 较优的鲁棒调节性能．     

Fundamental limitation on achievable decentralized performance

A commonly accepted fact is that the diagonal structure of the decentralized controller poses fundamental limitations on the achievable performance, but few quantitative results are available for measuring these limitations. This paper provides a lower bound on the achievable quality of disturbance rejection using a decentralized controller for stable discrete time linear systems with time delays, which do not contain any finite zeros on or outside the unit circle. The proposed result is useful for assessing when full multivariable controllers can provide significantly improved performance, as compared to decentralized controllers. The results are also extended to the case, where the individual subcontrollers are restricted to be PID controllers.     

New methodology for analytical and optimal design of fuzzy PIDcontrollers

Describes a methodology for the systematic design of fuzzy PID controllers based on theoretical fuzzy analysis and, genetic-based optimization. An important feature of the proposed controller is its simple structure. It uses a one-input fuzzy inference with three rules and at most six tuning parameters. A closed-form solution for the control action is defined in terms of the nonlinear tuning parameters. The nonlinear proportional gain is explicitly derived in the error domain. A conservative design strategy is proposed for realizing a guaranteed-PID-performance (GPP) fuzzy controller. This strategy suggests that a fuzzy PID controller should be able to produce a linear function from its nonlinearity tuning of the system. The proposed PID system is able to produce a close approximation of a linear function for approximating the GPP system. This GPP system, incorporated with a genetic solver for the optimization, will provide the performance no worse than the corresponding linear controller with respect to the specific performance criteria. Two indexes, linearity approximation index (LAI) and nonlinearity variation index (NVI), are suggested for evaluating the nonlinear design of fuzzy controllers. The proposed control system has been applied to several first-order, second-order, and fifth-order processes. Simulation results show that the proposed fuzzy PID controller produces superior control performance to the conventional PID controllers, particularly in handling nonlinearities due to time delay and saturation     

Fuzzy Constrained Min-Max Model Predictive Control Based on Piecewise Lyapunov Functions

This paper proposes two novel stable fuzzy model predictive controllers based on piecewise Lyapunov functions and the min-max optimization of a quasi-worst case infinite horizon objective function. The main idea is to design state feedback control laws that minimize the worst case objective function based on fuzzy model prediction, and thus to obtain the optimal transient control performance, which is of great importance in industrial process control. Moreover, in both of these predictive controllers, piecewise Lyapunov functions have been used in order to reduce the conservatism of those existent predictive controllers based on common Lyapunov functions. It is shown that the asymptotic stability of the resulting closed-loop discrete-time fuzzy predictive control systems can be established by solving a set of linear matrix inequalities. Moreover, the controller designs of the closed-loop control systems with desired decay rate and input constraints are also considered. Simulations on a numerical example and a highly nonlinear benchmark system are presented to demonstrate the performance of the proposed fuzzy predictive controllers.     

Real-time control of a water-gas shift reactor by a model-based fuzzy gain scheduling technique

A model-based fuzzy gain scheduling technique is proposed. Fuzzy gain scheduling is a form of variable gain scheduling which involves implementing several linear controllers over a partitioned process space. A higher-level rule-based controller determines which local controller is executed. Unlike conventional gain scheduling, a controller with fuzzy gain scheduling uses fuzzy logic to dynamically interpolate controller parameters near region boundaries based on known local controller parameters. Model-based fuzzy gain scheduling (MFGS) was applied to PID controllers to control a laboratory-scale water-gas shift reactor. The experimental results were compared with those obtained by PID with standard fuzzy gain scheduling, PID with conventional gain scheduling, simple PID and a nonlinear model predictive control (NMPC) strategy. The MFGS technique performed comparably to the NMPC method. It exhibited excellent control behaviour over the desired operating space, which spanned a wide temperature range. The other three PID-based techniques were adequate only within a limited range of the same operating space. Due to the simple algorithm involved, the MFGS technique provides a low cost alternative to other computationally intensive control algorithms such as NMPC.     

一种新型的二维PID模糊控制器

通过对常规PID控制器的结构分析,设计出一种新型的二维PID模糊控制器,其结构形式简称为fuzzy PD+ fuzzy ID型.根据模糊规则的图解分析,提出fuzzy ID控制嚣的输入变量(偏差和偏差变化加速率)与输出变量之间的控制结构,并确定两控制器的模糊控制规则的相似性.通过对该PID模糊控制器的结构分析,给出与常...     

Analysis of direct action fuzzy PID controller structures

The majority of the research work on fuzzy PID controllers focuses on the conventional two-input PI or PD type controller proposed by Mamdani (1974). However, fuzzy PID controller design is still a complex task due to the involvement of a large number of parameters in defining the fuzzy rule base. This paper investigates different fuzzy PID controller structures, including the Mamdani-type controller. By expressing the fuzzy rules in different forms, each PLD structure is distinctly identified. For purpose of analysis, a linear-like fuzzy controller is defined. A simple analytical procedure is developed to deduce the closed form solution for a three-input fuzzy inference. This solution is used to identify the fuzzy PID action of each structure type in the dissociated form. The solution for single-input-single-output nonlinear fuzzy inferences illustrates the effect of nonlinearity tuning. The design of a fuzzy PID controller is then treated as a two-level tuning problem. The first level tunes the nonlinear PID gains and the second level tunes the linear gains, including scale factors of fuzzy variables. By assigning a minimum number of rules to each type, the linear and nonlinear gains are deduced and explicitly presented. The tuning characteristics of different fuzzy PID structures are evaluated with respect to their functional behaviors. The rule decoupled and one-input rule structures proposed in this paper provide greater flexibility and better functional properties than the conventional fuzzy PHD structures.     

Design and analysis of region-wise linear fuzzy controllers

In this paper, a methodology to reduce the complexity of a robust controller based on fuzzy if-then rules is proposed. The motivation and the design of this complexity-reduced fuzzy controller are presented. This fuzzy controller with the triangular membership functions and fuzzy partition methods used here leads to a region-wise linear fuzzy controller (RLFC). The properties of the region-wise linear fuzzy controllers are discussed and the reasons why they in general perform better than the PD controllers are also provided. And the simulation results based on a second order plant are included to show that the region-wise linear fuzzy controller outperforms the PD controller. We also show that the region-wise linear fuzzy controller and original fuzzy controller have similar performances.     

Decomposed fuzzy proportional–integral–derivative controllers

In this paper, several types of decomposed proportional–integral–derivative fuzzy logic controllers (PID FLCs) are tested and compared. An important feature of decomposed PID FLCs are their simple structures. In its simplest version, the decomposed PID FLC uses three one-input one-output inferences with three separate rule bases. Behaviours of proportional, integral and derivative PID FLC parts are defined with simple rules in proportional rule base, integral rule base and derivative rule base. The proposed decomposed PID FLC has been compared with several PID FLCs structures. All PID FLCs have been realised by the same hardware and software tools and have been applied as a real-time controller to a simple magnetic suspension system.     

Fuzzy PID Control of a Five DOF Robot Arm

This paper studies the application of fuzzy logic control on a five degrees of freedom (DOF) robot arm, the Maker 100 of U.S. Robots. The elaboration of the fuzzy control laws is based on two structures of coupled rules fuzzy PID controllers. The fuzzy PID controllers are numerically simulated and the simulation results confirm the success of the fuzzy PID control in trajectory tracking problems. Seeking a performance optimization, a systematic study of the choice of tuning parameters of the controllers is done. The success of the proposed fuzzy control law is again affirmed by a comparative evaluation with respect to the computed torque control method and the direct adaptive control method, the last two controls being also numerically implemented using the same dynamic model of the robot arm.     

模糊PID控制技术研究发展回顾及其面临的若干重要问题

简要回顾了模糊PID控制器的研究发展状况,系统地归纳了模糊PID控制器的 基本形式.讨论了阻碍模糊控制技术发展的几个重要理论问题,包括系统化设计方法,控制 性能优于传统PID控制器的模糊控制器设计,模糊控制器非线性逼近能力及其量化指标,模 糊系统的规则爆炸.文中试图对有关问题的内涵进行重新定义,以及探求解决难题的关键. 如将模糊系统中的"规则爆炸"问题更正为"参数效率"问题.最后,提出了对模糊控制 理论与技术发展的思考,其中特别强调应把"简单性"作为智能控制器设计中的基本策略.     

Controlling Chua's circuits using computational verb controllers

In this paper, computational verb controllers were used to control Chua's circuits that were chaotic. The computational verb rule‐wise linear models of Chua's circuits were used to approximately segment the dynamics of Chua's circuits into four qualitatively different clusters; namely, the dynamics in the inner region, in the outer region and at boundaries of both regions. Then the stable verb controllers were designed by using linear matrix inequalities. Simulation results are presented to show the soundness of the design method. Copyright © 2008 John Wiley & Sons, Ltd.     

Combining fuzzy, PID and regulation control for an autonomous mini-helicopter

This paper reports on the synthesis of different flight controllers for an X-Cell mini-helicopter. They are developed on the basis of the most realistic mathematical model currently available. Two hybrid intelligent control systems, combining computational intelligence methodologies with other control techniques, are investigated. For both systems, Mamdani-type fuzzy controllers determine the set points for altitude/attitude control. These fuzzy controllers are designed using a simple rule base. The first scheme consists of conventional SISO PID controllers for z-position and roll, pitch and yaw angles. In the second scheme, two of the previous PID controllers are used for roll and pitch, and a linear regulator is added to control altitude and yaw angle. These control schemes mimic the action of an expert pilot. The designed controllers are tested via simulations. It is shown that the designed controllers exhibit good performance for hover flight and control positioning at slow speed.