A reduction approach for fuzzy rule bases of fuzzy controllers

In this paper, a new approach to reducing the number of rules in a given fuzzy rule base of a fuzzy controller is presented. The fuzzy mechanism of the fuzzy controller under consideration consists of the product-sum inference, singleton output consequents and centroid defuzzification. The output consequents in the cells of the rule table are collected and represented as an output consequent matrix. The feature of the output consequent matrix is extracted by the singular values of the matrix. The output consequent matrix is reasonably approximated with a dominant consequent matrix. Also, the elements of the dominant consequent matrix is determined to minimize the approximation error function. Then the size of the dominant consequent matrix (the size of the fuzzy rule base) is reduced through the rule combination approach. The scaling factors for the fuzzy controller with the reduced rule table are adjusted to have the control system satisfy the performance indices. The effectiveness of the proposed approach is shown using simulation and experimental results.     

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     

Circuit implementation of linguistic-hedge fuzzy logic controller in current-mode approach

In this paper, a novel fuzzy logic controller called linguistic-hedge fuzzy logic controller in a mixed-signal circuit design is discussed. The linguistic-hedge fuzzy logic controller has the following advantages: 1) it needs only three simple-shape membership functions for characterizing each variable prior to the linguistic-hedge modifications; 2) it is sufficient to adopt nine rules for inference; 3) the rules are developed intuitively without heavy dependence on the endeavors of experts; 4) it performs better than conventional fuzzy logic controllers; and 5) it can be realized with a lower design complexity and a smaller hardware overhead as compared with the controllers that required more than nine rules. In this implementation, a current-mode approach is adopted in designing the signal processing portions to simplify the circuit complexity; digital circuits are adopted to implement the programmable units. This design was fabricated with a TSMC 0.35 /spl mu/m single-polysilicon-quadruple-metal CMOS process. In this chip, the LHFLC processes two input variables and one output variable. Each variable is specified using three membership functions. Nine inference rules, scheduled in a rule table with a dimension of 3 /spl times/ 3, define the relationship implications between these three variables. Under a supply voltage of 3.3 V, the measurement results show that the measured control surface and the control goal are consistent. The speed of inference operation goes up to 0.5M FLIPS that is fast enough for the control application of the cart-pole balance system. The cart-pole balance system experimental results show that this chip works with nine inference rules. Furthermore, by performing some off-chip modifications, such as shifting and scaling on the input signals and output signal of this design, according to the specifications defined by the controlled plants, this design is suitable for many control applications.     

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     

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.     

Online tuning of fuzzy PID controllers via rule weighing based on normalized acceleration

In this study, an on-line tuning method is proposed for fuzzy PID controllers via rule weighing. The rule weighing mechanism is a fuzzy rule base with two inputs namely; “error” and “normalized acceleration”. Here, the normalized acceleration provides relative information on the fastness or slowness of the system response. In deriving the fuzzy rules of the weighing mechanism, the transient phase of the unit step response of the closed loop system is to be analyzed. For this purpose, this response is assumed to be divided into certain regions, depending on the number of membership functions defined for the error input of the fuzzy logic controller. Then, the relative importance or influence of the fired fuzzy rules is determined for each region of the transient phase of the unit step response of the closed loop system. The output of the fuzzy rule weighing mechanism is charged as the tuning variable of the rule weights; and, in this manner, an on-line self-tuning rule weight assignment is accomplished. The effectiveness of the proposed on-line weight adjustment method is demonstrated on linear and non-linear systems by simulations. Moreover, a real time application of this new method is accomplished on a pH neutralization process.     

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.     

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     

Design and stability analysis of single-input fuzzy logiccontroller

In existing fuzzy logic controllers (FLCs), input variables are mostly the error and the change-of-error regardless of complexity of controlled plants. Either control input u or the change of control input Deltau is commonly used as its output variable. A rule table is then constructed on a two-dimensional (2-D) space. This scheme naturally inherits from conventional proportional-derivative (PD) or proportional-integral (PI) controller. Observing that 1) rule tables of most FLCs have skew-symmetric property and 2) the absolute magnitude of the control input |u| or |Deltau| is proportional to the distance from its main diagonal line in the normalized input space, we derive a new variable called the signed distance, which is used as a sole fuzzy input variable in our simple FLC called single-input FLC (SFLC). The SFLC has many advantages: The total number of rules is greatly reduced compared to existing FLCs, and hence, generation and tuning of control rules are much easier. The proposed SFLC is proven to be absolutely stable using Popov criterion. Furthermore, the control performance is nearly the same as that of existing FLCs, which is revealed via computer simulations using two nonlinear plants.     

基于蚁群算法的模糊控制规则的过滤简化

针对设计高维模糊控制器过程中会遇到的“规则爆炸”问题,利用蚁群算法进行控制规则的过滤简化。为了用尽量少的规则得到尽可能好的控制效果,利用蚁群算法在饵决组合优化问题中的强大优势,在已有的完备规则中优选出若干条规则嵌人模糊控制器。采用带有时间窗口的蚁群算法去克服遗传算法优选模糊控制规则时可能产生的规则不连续的问题。该文还从遗传算法和蚁群算法工作机制的角度分析了对这两种算法加入约束条件的可操作性。以单级倒立摆控制系统为对象进行仿真研究,最后的仿真结果表明该文方法可以使模糊控制规则具有更好的简化效果和鲁棒性,并能具有好的控制效果。     

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.     

Robust adaptive fuzzy controller for non‐affine nonlinear systems with dynamic rule activation

This paper describes the design of a robust adaptive fuzzy controller for an uncertain single‐input single‐output nonlinear dynamical systems. While most recent results on fuzzy controllers considers affine systems with fixed rule‐base fuzzy systems, we propose a control scheme for non‐affine nonlinear systems and a dynamic fuzzy rule activation scheme in which an appropriate number of the fuzzy rules are chosen on‐line. By using the proposed scheme, we can reduce the computation time, storage space, and dynamic order of the adaptive fuzzy system without significant performance degradation. The Lyapunov synthesis approach is used to guarantee a uniform ultimate boundedness property for the tracking error, as well as for all other signals in the closed loop. No a priori knowledge of an upper bounds on the uncertainties is required. The theoretical results are illustrated through a simulation example. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Neuro-fuzzy chip to handle complex tasks with analog performance.

This paper presents a mixed-signal neuro-fuzzy controller chip which, in terms of power consumption, input-output delay, and precision, performs as a fully analog implementation. However, it has much larger complexity than its purely analog counterparts. This combination of performance and complexity is achieved through the use of a mixed-signal architecture consisting of a programmable analog core of reduced complexity, and a strategy, and the associated mixed-signal circuitry, to cover the whole input space through the dynamic programming of this core. Since errors and delays are proportional to the reduced number of fuzzy rules included in the analog core, they are much smaller than in the case where the whole rule set is implemented by analog circuitry. Also, the area and the power consumption of the new architecture are smaller than those of its purely analog counterparts simply because most rules are implemented through programming. The paper presents a set of building blocks associated to this architecture, and gives results for an exemplary prototype. This prototype, called multiplexing fuzzy controller (MFCON), has been realized in a CMOS 0.7 /spl mu/m standard technology. It has two inputs, implements 64 rules, and features 500 ns of input to output delay with 16-mW of power consumption. Results from the chip in a control application with a dc motor are also provided.     

Structure and stability analysis of a Takagi–Sugeno fuzzy PI controller with application to tissue hyperthermia therapy

 In this paper, we first reveal the analytical structure of a simple Takagi–Sugeno (TS) fuzzy PI controller relative to the linear PI controller. The fuzzy controller consists of two linear input fuzzy sets, four TS fuzzy rules with linear consequent, Zadeh fuzzy logic AND and the centroid defuzzifier. We prove that the fuzzy controller is actually a nonlinear PI controller with the gains changing with process output. Utilizing the well-known small Gain Theorem in control theory, we then derive sufficient conditions for global stability of the fuzzy control systems involving the TS fuzzy PI controller. Finally, as an application demonstration, we apply the fuzzy PI controller to control issue temperature, in computer simulation, during hyperthermia therapy. The relationship between heat energy and tissue temperature is represented by a linear time-varying model with a time delay. The sufficient conditions for global stability are used to design a stable fuzzy control system. Our simulation results show that the fuzzy PI control system achieves satisfactory temperature control performance. The control system is robust and stable even when the model parameters are changed suddenly and significantly.     

Fuzzy classifier system architectures for mobile robotics: An experimental comparison

We present an experimental comparison between two approaches to optimization of the rules for a fuzzy controller. More specifically, the problem is autonomous acquisition of an “investigative” obstacle avoidance competency for a mobile robot. We report on results from investigating two alternative approaches to the use of a Learning Classifier System (LCS) to optimize the fuzzy rule base. One approach operates at the level of whole rule bases, the “Pittsburgh” LCS. The other approach operates at the level of individual rules, the “Michigan” LCS. In this work, both of these Fuzzy Classifier Systems were designed to operate only on the rules of fuzzy controllers, with predefined fuzzy membership functions. There are two main results from this work. First, both approaches were capable of producing fuzzy controllers with subtle interactions between rules leading to competencies exceeding that of the hand‐coded fuzzy controller presented in this article. Second, the Michigan approach suffered more seriously than the Pittsburgh approach from the well‐known LCS “cooperation/competition” problem, which is accentuated here by the structural combination of Evolutionary Computation and a fuzzy system. This problem was alleviated a little by the combination of a clustered subpopulation niche system and a fitness‐sharing scheme applied to the Michigan approach, but still remains. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 993–1019, 2007.     

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     

Learning fuzzy rules with tabu search-an application to control

We present an approach for the automatic definition of the fuzzy rules for a fuzzy controller based on the use of the tabu search (TS) scheme. We show also how the application of the TS process to the learning of a fuzzy rule base can be improved using heuristic symbolic meta rules. The paper is divided in two parts. The first part presents an introduction to TS and different learning schemes which can be used to apply it for the determination of the fuzzy control rules. The second part illustrates the application of the proposed techniques to a specific control problem-the parking of a truck and trailer. In particular, Section V illustrates the definition of a rule base for a static fuzzy controller, while Section VI presents the construction of an adaptive parking controller     

Mining the fuzzy control rules of aeration in a Submerged Biofilm Wastewater Treatment Process

This paper presents a special rule base extraction analysis for optimal design of an integrated neural-fuzzy process controller using an “impact assessment approach.” It sheds light on how to avoid some unreasonable fuzzy control rules by screening inappropriate fuzzy operators and reducing over fitting issues simultaneously when tuning parameter values for these prescribed fuzzy control rules. To mitigate the design efforts, the self-learning ability embedded in the neural networks model was emphasized for improving the rule extraction performance. An aeration unit in an Aerated Submerged Biofilm Wastewater Treatment Process (ASBWTP) was picked up to support the derivation of a solid fuzzy control rule base. Four different fuzzy operators were compared against one other in terms of their actual performance of automated knowledge acquisition in the system based on a partial or full rule base prescribed. Research findings suggest that using bounded difference fuzzy operator (O_b) in connection with back propagation neural networks (BPN) algorithm would be the best choice to build up this feedforward fuzzy controller design.