Simplification of a fuzzy mechanism with rule combination

 In this paper, a systematic approach to reduce the complexity of a fuzzy controller with the rule combination of a fuzzy rule base is presented. The complexity of a fuzzy controller is defined to be the computation load in this work. The proposed rule combination approach can be applied to the fuzzy mechanisms with product–sum and min–max inferences. With the input membership functions indexed in sequence for each input variable, the n-dimensional fuzzy rule table is represented as vectors so that the combination of the fuzzy rule base is realizable. Then the adjacent fuzzy rules with the same output consequent are combined to have smaller size of fuzzy rule base. The fuzzy mechanism with the combined rule table is shown to have the same output with the original fuzzy mechanism (without rule combination). Thus, in many applications, the rule combination approach presented in this paper can be used to reduce the complexity of the fuzzy mechanism without degrading the performances. Moreover, the Don't Care fuzzy rules are defined and it is indicated that the number of the necessary fuzzy rules might be decreased when the Don't Care fuzzy rules are taken into consideration. Further, the properties of the simplification approach for the fuzzy rule base of the fuzzy mechanism are discussed.     

Perfect tracking control of discrete-time quadratic TS fuzzy systems via feedback linearisation

Perfect tracking control is an important and frequently encountered requirement in various industries (e.g. robotic control). We developed a novel systematic framework for designing a fuzzy controller via feedback linearisation to control a class of discrete-time Takagi–Sugeno (TS) fuzzy systems with quadratic rule consequents to achieve such tracking. We established a necessary condition for its local stability and a necessary and sufficient condition for the boundedness of the controller. The feedback linearisation is known to fail to work in certain systems due to the unboundedness of the tracking controller output. To address this issue, we developed a method to check whether any given quadratic TS fuzzy system will cause such a failure. We developed a scheme to ensure that the output of the controller designed for any failure-causing system will be bounded and the resulting controller will attain nearly perfect tracking performance. Applying feedback linearisation to the quadratic fuzzy systems is innovative relative to the literature exclusively dealing with the TS fuzzy systems with linear rule consequents (including our previous results), which are now generalised by the new findings. Two numerical examples are provided to illustrate the effectiveness and utility of our new theoretical results.     

Reduction of fuzzy rule base via singular value decomposition

Introduces a singular value-based method for reducing a given fuzzy rule set. The method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membership functions to form new ones for the reduced set. The present work characterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO). Algorithms to preserve the SN and NN conditions in the new membership functions are presented. Preservation of the NO condition relates to a high-dimensional convex hull problem and is not always feasible in which case a closed-to-NO solution may be sought. The proposed method is applicable regardless of the adopted inference paradigms. With product-sum-gravity inference and singleton support fuzzy rule base, output errors between the full and reduced fuzzy set are bounded by the sum of the discarded singular values. The work discusses three specific applications of fuzzy reduction: fuzzy rule base with singleton support, fuzzy rule base with nonsingleton support (which includes the case of missing rules), and the Takagi-Sugeno-Kang (TSK) model. Numerical examples are presented to illustrate the reduction process     

Exact analytical inverse mapping of decomposable TS fuzzy systems with singleton and linear consequents

In this study, a new method is proposed for the exact analytical inverse mapping of Takagi–Sugeno fuzzy systems with singleton and linear consequents where the input variables are described by using strong triangular partitions. These fuzzy systems can be decomposed into several fuzzy subsystems. The output of the fuzzy subsystem results in multi-linear form in singleton consequent case or multi-variate second order polynomial form in linear consequent case. Since there exist explicit analytical formulas for the solutions of first and second order equations, the exact analytical inverse solutions can be obtained for decomposable Takagi–Sugeno fuzzy systems with singleton and linear consequents. In the proposed method, the output of the fuzzy subsystem is represented by using the matrix multiplication form. The parametric inverse definition of the fuzzy subsystem is obtained by using appropriate matrix partitioning with respect to the inversion variable. The inverse mapping of each fuzzy subsystem can then easily be calculated by substituting appropriate parameters of the fuzzy subsystem into this parametric inverse definition. So, it becomes very easy to find the analytical inverse mapping of the overall Takagi–Sugeno fuzzy system by composing inverse mappings of all fuzzy subsystems. The exactness and the effectiveness of the proposed inversion method are demonstrated on trajectory tracking problems by simulations.     

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．     

Quasi sliding mode‐based single input fuzzy self‐tuning decoupled fuzzy PI control for robot manipulators with uncertainty

This paper presents a robust adaptive control strategy for robot manipulators, based on the coupling of the fuzzy logic control with the so‐called sliding mode control (SMC) approach. The motivation for using SMC in robotics mainly relies on its appreciable features. However, the drawbacks of the conventional SMC, such as chattering effect and required a priori knowledge of the bounds of uncertainties can be destructive. In this paper, these problems are suitably circumvented by adopting a reduced rule base single input fuzzy self tuning decoupled fuzzy proportional integral sliding mode control approach. In this new approach a decoupled fuzzy proportional integral control is used and a reduced rule base single input fuzzy self‐tuning controller as a supervisory fuzzy system is added to adaptively tune the output control gain of the decoupled fuzzy proportional integral control. Moreover, it is proved that the fuzzy control surface of the single‐input fuzzy rule base is very close to the input/output relation of a straight line. Therefore, a varying output gain decoupled fuzzy proportional integral sliding mode control approach using an approximate line equation is then proposed. The stability of the system is guaranteed in the sense of the Lyapunov theorem. Simulations using the dynamic model of a 3DOF planar manipulator with uncertainties show the effectiveness of the approach in high speed trajectory tracking problems. The simulation results that are compared with the results of conventional SMC indicate that the control performance of the robot system is satisfactory and the proposed approach can achieve favorable tracking performance, and it is robust with regard to uncertainties and disturbances. Copyright © 2011 John Wiley & Sons, Ltd.     

Predictive control based on local linear fuzzy models

This paper deals with predictive control based on fuzzy models. A novel algorithm (LOLIMOT) is proposed for the construction of Takagi-Sugeno fuzzy models. The rule consequents are optimized by a local orthogonal least-squares method that selects the significant regressors. The rule premises are optimized by a tree construction algorithm which partitions the input space in hyper-rectangles. A generalized predictive controller (GPC) and a dynamic matrix controller (DMC) are designed. Both controllers require the extraction of a linear model from the Takagi-Sugeno fuzzy model. For the GPC a new technique called local dynamic linearization is proposed that exploits the special structure of the local linear models. The DMC is based on the evaluation of a step response. The effectiveness of both the identification algorithm and the predictive controllers is shown by application to temperature control of an industrial-scale cross-flow heat exchanger.     

Online global learning in direct fuzzy controllers

A novel approach to achieve real-time global learning in fuzzy controllers is proposed. Both the rule consequents and the membership functions defined in the premises of the fuzzy rules are tuned using a one-step algorithm, which is capable of controlling nonlinear plants with no prior offline training. Direct control is achieved by means of two auxiliary systems: The first one is responsible for adapting the consequents of the main controller's rules to minimize the error arising at the plant output, while the second auxiliary system compiles real input-output data obtained from the plant. The system then learns in real time from these data taking into account, not the current state of the plant but rather the global identification performed. Simulation results show that this approach leads to an enhanced control policy thanks to the global learning performed, avoiding overfitting.     

Comments on “Reduction of fuzzy rule base via singular valuedecomposition”

Several comments are presented for the reduction of the fuzzy rule base in the paper of Yam et al. (1999). In their paper, the approach to determine the number of singular values necessary for the reduction process to obtain the effective and most efficient fuzzy rule base is not provided. Although the output error of the fuzzy controller is bounded, the performance of the system output may not be satisfied. Moreover, the computation load is increased for each input of the fuzzy mechanism since the input membership functions are modified     

An approach to fuzzy default reasoning for function approximation

This paper discusses fuzzy reasoning for approximately realizing nonlinear functions by a small number of fuzzy if-then rules with different specificity levels. Our fuzzy rule base is a mixture of general and specific rules, which overlap with each other in the input space. General rules work as default rules in our fuzzy rule base. First, we briefly describe existing approaches to the handling of default rules in the framework of possibility theory. Next, we show that standard interpolation-based fuzzy reasoning leads to counterintuitive results when general rules include specific rules with different consequents. Then, we demonstrate that intuitively acceptable results are obtained from a non-standard inclusion-based fuzzy reasoning method. Our approach is based on the preference for more specific rules, which is a commonly used idea in the field of default reasoning. When a general rule includes a specific rule and they are both compatible with an input vector, the weight of the general rule is discounted in fuzzy reasoning. We also discuss the case where general rules do not perfectly but partially include specific rules. Then we propose a genetics-based machine learning (GBML) algorithm for extracting a small number of fuzzy if-then rules with different specificity levels from numerical data using our inclusion-based fuzzy reasoning method. Finally, we describe how our approach can be applied to the approximate realization of fuzzy number-valued nonlinear functions     

比例型T-S模糊控制系统稳定性分析与设计

讨论了比例型T-S模糊控制系统(TSS)的稳定性与设计问题. 利用具有乘积可换性状态矩阵的T-S系统的公共P阵构造方法及鲁棒稳定域条件, 给出了TSS系统的满足全局Lyapunov稳定的正定矩阵P的递推求解方法, 并提出一种TSS模糊控制系统设计和稳定性分析的规范化方法.     

SIGNIFICANT FUZZY RULES EXTRACTION BY AN SVD–QR-BASED APPROACH

ABSTRACT

In this article, an SVD–QR-based approach is proposed to extract the important fuzzy rules from a rule base with several fuzzy rule tables to design an appropriate fuzzy system directly from some input-output data of the identified system. A fuzzy system with fuzzy rule tables is defined to approach the input-output pairs of an identified system. In the rule base of the defined fuzzy system, each fuzzy rule table corresponds to a partition of an input space. In order to extract the important fuzzy rules from the rule base of the defined fuzzy system, a firing strength matrix determined by the membership functions of the premise fuzzy sets is constructed. According to the firing strength matrix, the number of important fuzzy rules is determined by the Singular Value Decomposition SVD, and the important fuzzy rules are selected by the SVD–QR-based method. Consequently, a reconstructed fuzzy rule base composed of significant fuzzy rules is determined by the firing strength matrix. Furthermore, the recursive least-squares method is applied to determine the consequent part of the reconstructed fuzzy system according to the gathered input-output data so that a fine fuzzy system is determined by the proposed method. Finally, three nonlinear systems illustrate the efficiency of the proposed method.     

Genetic algorithm-based optimal fuzzy controller design in the linguistic space

In this paper, a genetic algorithm (GA) based optimal fuzzy controller design is proposed. The design procedure is accomplished by establishing an index function as the consequent part of the fuzzy control rule. The inputs of the controller, after scaling, are utilized by the index function for computing the output linguistic value. This linguistic value can then be used to map the suitable fuzzy control actions. This proposed novel fuzzy control rule has crisp input and fuzzified output characteristics. The index function plays a role in mapping the desired fuzzy sets for defuzzification resulting in a controlled hypersurface in the linguistic space formed by the input fuzzy variables. Two types of index functions, both linear and nonlinear, are introduced for controlling systems with different degrees of nonlinearity. The parameters of the index function are obtained by applying a simple GA with a suitable fitness function. Various controlled systems result in various parameter sets depending on their dynamics. Under the acquired optimal parameter set the optimal index function can be used to generate the desired control actions. Several simulation examples are given to verify the performance of the proposed GA-based fuzzy controller.     

Fuzzy controllers with conditionally firing rules

Mamdani (1975) controller was successfully used in many applications. One of its interpretations is that it uses a fuzzy relation as an approximation of the desirable input-output correspondence. We analyze mathematical properties of Mamdani controller and notice that it has lower computational complexity when compared to the residuum-based controller. However, we show that in standard situations, both these fuzzy controllers do not represent the rule base properly in the sense of finding a solution to the related system of fuzzy relational equations. First, we consider the premises and consequents as typical inputs and outputs, and we want their correspondence to be kept. Next, we require that each normal input produces an output that bears nontrivial information. These two conditions appear to be almost contradictory to the previous controllers. We suggest a generalization of Mamdani controller which allows us to satisfy these requirements. The theory and experiments suggest that it performs better without any change of rule base and without a substantial increase of complexity     

基于矩阵测度的非线性广义时滞系统鲁棒模糊控制

研究T-S模糊广义时滞系统的鲁棒控制问题.不同于传统的寻求公共正定矩阵的方法,基于矩阵测度给出保证系统鲁棒稳定的充分条件,并将此条件进一步转化为线性矩阵不等式.通过求解线性矩阵不等式,得到状态反馈控制器和静态输出反馈控制器.最后通过算例仿真验证了方法的有效性.     

Adaptive hierarchical tuning of fuzzy controllers

Fuzzy controller design includes both linear and non-linear dynamic analysis. The knowledge base parameters associated within the fuzzy rule base influence the non-linear control dynamics while the linear parameters associated within the fuzzy output signal influence the overall control dynamics. For distinct identification of tuning levels, an equivalent linear controller output and a normalized non-linear controller output are defined. A linear proportional-integral-derivative (PID) controller analogy is used for determining the linear tuning parameters. Non-linear tuning is derived from the locally defined control properties in the non-linear fuzzy output. The non-linearity in the fuzzy output is then represented in a graphical form for achieving the necessary non-linear tuning. Three different tuning strategies are evaluated. The first strategy uses a genetic algorithm to simultaneously tune both linear and non-linear parameters. In the second strategy the non-linear parameters are initially selected on the basis of some desired non-linear control characteristics and the linear tuning is then performed using a trial and error approach. In the third method the linear tuning is initially performed off-line using an existing linear PID law and an adaptive non-linear tuning is then performed online in a hierarchical fashion. The control performance of each design is compared against its corresponding linear PID system. The controllers based on the first two design methods show superior performance when they are implemented on the estimated process system. However, in the presence of process uncertainties and external disturbances these controllers fail to perform any better than linear controllers. In the hierarchical control architecture, the non-linear fuzzy control method adapts to process uncertainties and disturbances to produce superior performance.     

Evolution of a Negative-Rule Fuzzy Obstacle Avoidance Controller for an Autonomous Vehicle

A fuzzy obstacle avoidance controller is designed for an autonomous vehicle. The controller is given the capability for obstacle avoidance by using negative fuzzy rules in conjunction with traditional positive ones. Negative fuzzy rules prescribe actions to be avoided rather than performed. A rule base of positive rules is specified by an expert for directing the vehicle to the target in the absence of obstacles, while a rule base of negative rules is experimentally determined from expert operation of the vehicle in the presence of obstacles. The consequents of the negative-rule system are codified into a chromosome, and this chromosome is evolved using an evolutionary algorithm. The resulting fuzzy system has far fewer rules than would be necessary for an obstacle avoidance controller using purely positive rules, while in addition retaining greater interpretability.     

A systematic method for design of multivariable fuzzy logic controlsystems

This paper proposes a systematic method to design a multivariable fuzzy logic controller for large-scale nonlinear systems. In designing a fuzzy logic controller, the major task is to determine fuzzy rule bases, membership functions of input/output variables, and input/output scaling factors. In this work, the fuzzy rule base is generated by a rule-generated function, which is based on the negative gradient of a system performance index; the membership functions of isosceles triangle of input/output variables are fixed in the same cardinality and only the input/output scaling factors are generated from a genetic algorithm based on a fitness function. As a result, the searching space of parameters is narrowed down to a small space, the multivariable fuzzy logic controller can quickly constructed, and the fuzzy rules and the scaling factors can easily be determined. The performance of the proposed method is examined by computer simulations on a Puma 560 system and a two-inverted pendulum system     

Delay-dependent guaranteed cost control for T-S fuzzy systems with time delays

This study introduces a guaranteed cost control method for nonlinear systems with time-delays which can be represented by Takagi-Sugeno (T-S) fuzzy models with time-delays. The state feedback and generalized dynamic output feedback approaches are considered. The generalized dynamic output feedback controller is presented by a new fuzzy controller architecture which is of dual indexed rule base. It considers both the dynamic part and the output part of T-S fuzzy model which guarantees that the controller without any delay information can stabilize time-delay T-S fuzzy systems. Based on delay-dependent Lyapunov functional approach, some sufficient conditions for the existence of state feedback controller are provided via parallel distributed compensation (PDC) first. Second, the corresponding conditions are extended into the generalized dynamic output feedback closed-loop system via so-called generalized PDC technique. The upper bound of time-delay can be obtained using convex optimization such that the system can be stabilized for all time-delays whose sizes are not larger than the bound. The minimizing method is also proposed to search the suboptimal upper bound of guaranteed cost function. The effectiveness of the proposed method can be shown by the simulation examples.     

Fuzzy observer design using linear matrix inequalities for fuzzy closed-loop control systems

Linear observers play an important role in modern control theory and practice. A systematic design method of fuzzy observers would be important for fuzzy control as well. The fuzzy observer is designed by solving linear matrix inequalities (LMI) that represent control performance such as disturbance rejection and robust stability. Our approach in designing the fuzzy observer is based on the LMI formulation of the stability conditions for closed-loop Takagi-Sugeno (T-S) fuzzy systems, when states are not available for measurement of feedback. We present a new approach, which is to design an observer based on fuzzy implications, with fuzzy sets in the antecedents, and an asymptotic observer in the consequents. Each fuzzy rule is responsible for observing the states of a locally linear subsystem. An example illustrates the design procedure.