General Type-2 Fuzzy Logic Systems Research Articles

Study on sensible beginning divided-search enhanced Karnik-Mendel algorithms for centroid type-reduction of general type-2 fuzzy logic systems

<abstract> <p>General type-2 fuzzy logic systems (GT2 FLSs) on the basis of alpha-plane representation of GT2 fuzzy sets (FSs) have attracted considerable attention in recent years. For the kernel type-reduction (TR) block of GT2 FLSs, the enhanced Karnik-Mendel (EKM) algorithm is the most popular approach. This paper proposes the sensible beginning divided-search EKM (SBDEKM) algorithms for completing the centroid TR of GT2 FLSs. Computer simulations are provided to show the performances of the SBDEKM algorithms. Compared with EKM algorithms and sensible beginning EKM (SBEKM) algorithms, the SBDEKM algorithms have almost the same accuracies and better computational efficiency.</p> </abstract>

The Enhanced Wagner–Hagras OLS–BP Hybrid Algorithm for Training IT3 NSFLS-1 for Temperature Prediction in HSM Processes

This paper presents (a) a novel hybrid learning method to train interval type-1 non-singleton type-3 fuzzy logic systems (IT3 NSFLS-1), (b) a novel method, named enhanced Wagner–Hagras (EWH) applied to IT3 NSFLS-1 fuzzy systems, which includes the level alpha 0 output to calculate the output y alpha using the average of the outputs y alpha k instead of their weighted average, and (c) the novel application of the proposed methodology to solve the problem of transfer bar surface temperature prediction in a hot strip mill. The development of the proposed methodology uses the orthogonal least square (OLS) method to train the consequent parameters and the backpropagation (BP) method to train the antecedent parameters. This methodology dynamically changes the parameters of only the level alpha 0, minimizing some criterion functions as new information becomes available to each level alpha k. The precursor sets are type-2 fuzzy sets, the consequent sets are fuzzy centroids, the inputs are type-1 non-singleton fuzzy numbers with uncertain standard deviations, and the secondary membership functions are modeled as two Gaussians with uncertain standard deviation and the same mean. Based on the firing set of the level alpha 0, the proposed methodology calculates each firing set of each level alpha k to dynamically construct and update the proposed EWH IT3 NSFLS-1 (OLS–BP) system. The proposed enhanced fuzzy system and the proposed hybrid learning algorithm were applied in a hot strip mill facility to predict the transfer bar surface temperature at the finishing mill entry zone using, as inputs, (1) the surface temperature measured by the pyrometer located at the roughing mill exit and (2) the time taken to translate the transfer bar from the exit of the roughing mill to the entry of the descale breaker of the finishing mill. Several fuzzy tools were used to make the benchmarking compositions: type-1 singleton fuzzy logic systems (T1 SFLS), type-1 adaptive network fuzzy inference systems (T1 ANFIS), type-1 radial basis function neural networks (T1 RBFNN), interval singleton type-2 fuzzy logic systems (IT2 SFLS), interval type-1 non-singleton type-2 fuzzy logic systems (IT2 NSFLS-1), type-2 ANFIS (IT2 ANFIS), IT2 RBFNN, general singleton type-2 fuzzy logic systems (GT2 SFLS), general type-1 non-singleton type-2 fuzzy logic systems (GT2 NSFLS-1), interval singleton type-3 fuzzy logic systems (IT3 SFLS), and interval type-1 non-singleton type-3 fuzzy systems (IT3 NSFLS-1). The experiments show that the proposed EWH IT3 NSFLS-1 (OLS–BP) system presented superior capability to learn the knowledge and to predict the surface temperature with the lower prediction error.

Study on centroid type-reduction of general type-2 fuzzy logic systems with sensible beginning weighted enhanced Karnik–Mendel algorithms

General type-2 fuzzy logic systems have received wide concerns in current academic subject. Type-reduction is the kernel module for the systems. This paper shows the interpretations for the beginning of Karnik–Mendel (KM) algorithms. According to the famous numerical integration technique, the weighting approaches of enhanced Karnik–Mendel (EKM) algorithms are put forward. Then, the sensible beginning weighted enhanced Karnik–Mendel (SBWEKM) algorithms are put forward to perform the centroid type-reduction. Compared with the EKM algorithms, WEKM algorithms and SBEKM algorithms, this approach helps to improve both the absolute errors and convergence speeds as shown in four computer simulation experiments.

Study on non-iterative algorithms for center-of-sets type-reduction of Takagi–Sugeno–Kang type general type-2 fuzzy logic systems

The paper performs the center-of-sets (COS) type-reduction (TR) and de-fuzzification for Takagi–Sugeno–Kang (TSK) type general type-2 fuzzy logic systems (GT2 FLSs) on the basis of the alpha-planes expression of general type-2 fuzzy sets. Actually, comparing the popular Karnik–Mendel (KM) algorithms with other non-iterative algorithms is an important question in T2 society. Here the modules of fuzzy inference, COS TR, and de-fuzzification for TSK type GT2 FLSs are discussed by means of non-iterative Nagar–Bardini (NB) algorithms, Nie–Tan (NT) algorithms, and Begian–Melek–Mendel (BMM) algorithms. Simulation instances are constructed to illustrate the performances of three types of non-iterative algorithms compared with the KM algorithms. It is proved that, the proposed non-iterative algorithms can enhance the computational efficiencies significantly, which afford the potential application value for designers of GT2 FLSs.

An Approach Towards the Design of Interval Type-3 T–S Fuzzy System

This article providesa systematic approach for the design of an interval type-3 (IT3) Takagi–Sugeno (T–S) fuzzy logic system (FLS) using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha $</tex-math></inline-formula> - plane representation. An IT3 FLS is designed with the baseline of the general type-2 (GT2) FLS in a similar manner as an IT2 FLS was designed from the baseline of type-1 FLS. Hence, IT3 FLS evolved as a successor of GT2 FLS, where secondary membership function is an interval type-2 fuzzy set (FS), and values of tertiary membership are unity over the footprint of uncertainty of secondary membership. This extra degree of freedom in IT3 FLS provides better modeling capability as compared to GT2 FLS in the presence of a high degree of uncertainty and vagueness. The proposed system will be more appealing while dealing with uncertain information or data, which is supposed to be generated from uncertain sources; i.e., there exist uncertainties even in the presence of uncertainty. The computations needed for the design of IT3 FLS are derived using IT2 FS and GT2 FS mathematics. The design algorithms adopted for the baseline IT2 T–S fuzzy system are as per the modified interval type-2 fuzzy c-regression model clustering algorithm and hyper-plane-shaped membership function. The proposed methodology is applied to several benchmark examples and obtained results are compared with recently developed fuzzy modeling methods having a comparable number of rule bases. The proposed IT3 T–S FLS shows good performance in terms of accuracy when data is corrupted by noise and uncertainties related to missing or unvarying data exist. The computational cost is linear with design parameters and by optimum choice of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha $</tex-math></inline-formula> -planes, it is still bearable considering advantages and nature of applications.

Design of sampling-based noniterative algorithms for centroid type-reduction of general type-2 fuzzy logic systems

General type-2 (GT2) fuzzy logic systems (FLSs) become a popular research topic for the past few years. Usually the Karnik–Mendel algorithms are the most prevalent approach to complete the type-reduction. Nonetheless, the iterative quality of these types of computational intensive algorithms might impede applying them. For the improved types of algorithms, some noniterative algorithms can enhance the calculation efficiencies greatly, while it is still an open problem for comparing the relation between the discrete TR algorithms and corresponding continuous TR algorithms. First, the sum and integral operations in discrete and continuous noniterative algorithms are compared. Then, three kinds of noniterative algorithms originate from the type-reduction of interval type-2 FLSs are extended to complete the centroid type-reduction of general T2 FLSs. Four computer simulations prove that while changing the number of samples suitably, the calculational results of discrete types of algorithms may accurately gain on the related continuous types of algorithms, and the calculational times of discrete types of algorithms are obviously less than the continuous types of algorithms, this may offer the possible meaning for designing and applying T2 FLSs.

Design of Discrete Noniterative Algorithms for Center-of-Sets Type Reduction of General Type-2 Fuzzy Logic Systems

Design of Discrete Noniterative Algorithms for Center-of-Sets Type Reduction of General Type-2 Fuzzy Logic Systems

Comparison Study of Iterative Algorithms for Center-of-Sets Type-Reduction of Takagi Sugeno Kang Type General Type-2 Fuzzy Logic Systems

Studying the block of center-of-sets (COS) type-reduction (TR) for Takagi Sugeno Kang (TSK) inference-based general type-2 fuzzy logic systems (GT2 FLSs) is meaningful for applying the systems. Blocks of fuzzy reasoning, COS type-reduction and defuzzification for TSK type GT2 FLSs are first given. According to three kinds of iterative algorithms for computing the centroids of interval type-2 fuzzy sets (IT2 FSs), the paper extends these types of algorithms for studying the COS type-reduction of TSK type GT2 FLSs. Six computer simulation experiments show the computational costs of proposed three kinds of iterative algorithms by computing the outputs of GT2 FLSs, which affords the potential guidance value designers and users of T2 FLSs.

Fixed-time adaptive general type-2 fuzzy logic control for air-breathing hypersonic vehicle

This paper proposes a novel fixed-time adaptive general type-2 fuzzy logical control (FAGT2FLC) scheme for an air-breathing hypersonic vehicle (AHV) with uncertainties. Firstly, the AHV dynamic model is transformed into a strict feedback form. Then, the FAGT2FLC is designed based on the transformed model to improve robustness and guarantee fixed-time convergence of the closed-loop system. The general type-2 fuzzy logic system (GT2FLS) is utilized to approximate the model uncertainties; for the purpose of designing adaptive laws, the [Formula: see text]-plane method is employed to represent the GT2FLS. A parameter projection operator is used to solve the possible singularity problem of parameter adaption. Besides, a fixed-time differentiator is used to deal with the “explosion of terms” inherent in backstepping method. Theoretical analysis based on relevant lemmas shows that the closed-loop system will converge into a small error band in fixed time. Lastly, detailed simulations are carried out to demonstrate the effectiveness and superiority of the proposed control scheme.

Discrete Non-iterative Centroid Type-Reduction Algorithms on General Type-2 Fuzzy Logic Systems

Since the alpha-planes expressions of general type-2 fuzzy sets (GT2 FSs) have been proposed, general type-2 fuzzy logic systems (GT2 FLSs) that are dependent on GT2 FSs are becoming quite popular to fuzzy logic researchers. Usually enhanced Karnik–Mendel (EKM) algorithms are adopted for performing the kernel block of type-reduction (TR). However, the essence of EKM-based TR process probably hinders the GT2 FLSs from real-world applications. It is an intriguing as well as unsolved problem for comparing EKM algorithms with other non-iterative algorithms. This paper provides a framework encompassing fuzzy reasoning, defuzzification, as well as type-reduction. Furthermore, the continuous of NT (CNT) algorithm is shown to be a precise approach when it is used to execute the centroid TR of GT2 fuzzy logic systems. Four computer tests display the characteristics of discrete Nagar-Bardini (NB) and Nie-Tan (NT) non-iterative algorithms. Compared with EKM approach, the developed one exhibits some superiorities in terms of guaranteeing high computation accuracy and low computation burdens, which broadens the application ranges for the proposed method.

A dynamic general type-2 fuzzy system with optimized secondary membership for online frequency regulation

This study suggests a new control system for frequency regulation in AC microgrids. Unlike to the most studies, challenging conditions such as variation of wind speed, multiple load disturbance, unknown dynamics and variable solar radiation are taken to account. To cope with uncertainties, a novel dynamic general type-2 (GT2) fuzzy logic system (FLS) by an optimized secondary membership is suggested. The secondary membership and rule parameters of dynamic GT2-FLS are online tuned through the adaptive optimization rules. The optimization rules are determined such that the robustness and stability to be guaranteed. Also, a new compensator is presented to tackle with estimation error and perturbations. The simulations verify that schemed controller outperforms than conventional methods.

Triangular Fuzzy-Rough Set Based Fuzzification of Fuzzy Rule-Based Systems

Abstract In real-world approximation problems, precise input data are economically expensive. Therefore, fuzzy methods devoted to uncertain data are in the focus of current research. Consequently, a method based on fuzzy-rough sets for fuzzification of inputs in a rule-based fuzzy system is discussed in this paper. A triangular membership function is applied to describe the nature of imprecision in data. Firstly, triangular fuzzy partitions are introduced to approximate common antecedent fuzzy rule sets. As a consequence of the proposed method, we obtain a structure of a general (non-interval) type-2 fuzzy logic system in which secondary membership functions are cropped triangular. Then, the possibility of applying so-called regular triangular norms is discussed. Finally, an experimental system constructed on precise data, which is then transformed and verified for uncertain data, is provided to demonstrate its basic properties.

A Fractional Order General Type-2 Fuzzy PID Controller Design Algorithm

Fractional order PID controller was received attentions in control problems for it had 2 freedom adjustable parameters. Practices had proved that better results could be obtained by introduction of fractional order PID in control problems. And in recently years, fractional order PID combined with fuzzy logic system was gradually got attention, the typical of which was fractional order type-1 fuzzy PID controller. For fractional order PID and fractional order type-1 fuzzy PID controller couldn't deal with the uncertainty of system, so fractional order interval type-2 fuzzy PID controller was applied to solve this problem. But interval type-2 fuzzy sets were a simplification of type-2 fuzzy sets, and there may be have the defect of information loss. A fractional order general type-2 fuzzy PID controller was proposed in this article. The proposed controller based on general type-2 fuzzy logic system, which can make full use of the advantages of general type-2 fuzzy logic system in describing the uncertainty of the system. The fractional order general type-2 fuzzy PID controller utilized a simplified type reduction called NT type reduction algorithm. The NT type reduction algorithm can get the defuzzification result directly and avoided iterative process as KM type reduction commonly used in interval type-2 fuzzy controller. The simulations of 3 processes and a practical inverted pendulum system show that fractional order general type-2 fuzzy PID controller can reduce overshoot, improve system response speed and accelerate system stability time in comparing with other controllers. Especially, when the system has disturbance, parameters uncertainty or structure uncertainty, the fractional order general type-2 fuzzy PID controller has better control effects than other compared controllers.

Study on weighted Nagar-Bardini algorithms for centroid type-reduction of general type-2 fuzzy logic systems

Study on weighted Nagar-Bardini algorithms for centroid type-reduction of general type-2 fuzzy logic systems

General 3-D Type-II Fuzzy Logic Systems in the Polar Frame: Concept and Practice

This paper deals with the concept and application of general three-dimensional (3-D) type-2 fuzzy logic systems in the polar frame (G3DT2FLS). 1 1 Because many acronyms are used in this paper, they are summarized in Table I for the convenience of the reader. We focused on the automatic membership function (MF) Generator, general type-2 polar fuzzy membership, new geometric operators in polar frame, inference consisting of fuzzy 3-D polar rules and antecedents/consequents $\theta$ -slice and $\alpha$ -planes. The cubic smoothing spline is introduced to generate the upper and lower MFs according to the information theory. Three indices of compactness, smoothness, and entropy are employed for tuning of MFs. A measure of ultrafuzziness, min, max, equal, and reduced ultrafuzziness are suggested for the polar method. Additionally, the set theoretic operations of 3-D type-2 fuzzy sets in the polar frame are discussed. We also prove the join, meet, centroid, and type-reduction operations in the polar frame. Several rule sets are given to show the usefulness and complexity of the proposed method. The performance of the partial general 3D polar type-2 fuzzy logic system showed linear growth as the number of rules was increased. Computation time tests showed that the algorithm reduces the computation time by a maximum of 67% compared with discrete MF and is shown by an extreme 98% for the geometric procedure. These results indicate significant improvements in computation time for the spline interpolation over the existing methods.

Forecasting by designing Mamdani general type-2 fuzzy logic systems optimized with quantum particle swarm optimization algorithms

Much more attention has been focused on studying and applying general type-2 fuzzy logic systems (GT2 FLSs) in recent years. The paper designs a type of Mamdani GT2 FLS for studying forecasting problems based on the data of permanent magnetic drive (PMD) loss. During the system design process, we choose the primary membership functions (MFs) of antecedent, consequent and input measurement general type-2 fuzzy sets (GT2 FSs) as Gaussian type MFs with uncertain standard deviations. The corresponding vertical slices (secondary MFs) are chosen as the triangle MFs. All the parameters of Mamdani GT2 FLSs are optimized by the quantum particle swarm optimization (QPSO) algorithms. Noisy data of PMD loss are adopted for both training and testing the proposed FLSs forecasting approaches. Simulation studies and convergence analysis are employed to show the effectiveness and feasibility of the proposed GT2 FLSs forecasting methods compared with their T1 and IT2 counterparts.

Study on Sampling Based Discrete Nie-Tan Algorithms for Computing the Centroids of General Type-2 Fuzzy Sets

General type-2 fuzzy logic systems (GT2 FLSs) have become a hot topic in current academic field. Computing the centroids of general type-2 fuzzy sets (also called type-reduction) is a central block in GT2 FLSs. Recent studies prove the continuous Nie-Tan (CNT) algorithms to be actually an accurate approach to calculate the centroids of interval type-2 fuzzy sets (IT2 FSs). This paper compares the sum operation in discrete NT algorithms and the integral operation in CNT algorithms. According to the alpha-planes representation theory of general type-2 fuzzy sets (GT2 FSs), both the discrete and continuous NT algorithms can be extended to compute the centroids of GT2 FSs. Four computer simulation experiments indicate that, when the centroid type-reduced sets and defuzzified values of GT2 FSs are solved, to properly increase the number of sampling points of primary variable can make the results of discrete NT algorithms exactly approach to the accurate benchmark CNT algorithms. Furthermore, the computation efficiency of sampling based discrete NT algorithms is much higher than the CNT algorithms.

Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted Nie–Tan algorithms

In recent years, researching on general type-2 fuzzy logic systems (GT2 FLSs) has become a hot topic as the development of alpha-planes representation of general type-2 fuzzy sets. The iterative Karnik–Mendel (KM) algorithms are used to perform the key block of type-reduction (TR) of GT2 FLSs. However, the KM algorithms are computationally intensive and time-consuming, which is not adapted to real-time applications. In the enhanced types of algorithms, the noniterative Nie–Tan (NT) algorithms decrease the computational cost greatly. Moreover, the closed-form Nie–Tan algorithms which calculate the outputs by averaging the lower and upper bounds of the membership functions have been proved to be actually an accurate algorithm for performing TR. The paper expands the NT algorithms to three different forms of weighted NT (WNT) algorithms according to the Newton–Cotes quadrature formulas of numerical integration techniques. Four computer simulation examples are adopted to analyze the performances of WNT algorithms when solving the type-reduction of general type-2 fuzzy logic systems. The proposed WNT algorithms have smaller absolute errors and faster convergence speed compared with NT algorithms, which provide the potential value for designers and adopters of GT2 FLSs.

Forecasting by general type-2 fuzzy logic systems optimized with QPSO algorithms

With the development of α-planes representation result of general type-2 fuzzy sets, the optimization and application of general type-2 fuzzy logic systems (GT2 FLSs) based on general type-2 fuzzy sets (GT2 FSs) has become a hot topic in current academic research. The efficient and energy conserving permanent magnetic drive (PMD) presents relatively high uncertainty as an emerging technology. The paper studies on forecasting problems based the data of torque and revolutions per minute (rpm) of PMD. In the proposed GT2 FLSs design, the antecedent, input measurement primary membership functions of GT2 FSs are chosen as Gaussian type-2 membership functions with uncertain standard deviation. While the consequent parameters are selected as deterministic values. Quantum particle swarm optimization (QPSO) algorithms are used to optimize all the parameters of the suggested GT2 FLSs. The torque and rpm data of PMD are used to train and test the proposed advanced FLSs forecasting methods. Simulation studies and convergence analysis show the effectiveness of the proposed GT2 FLSs methods compared with their type-1 (T1) and interval type-2 (IT2) methods for forecasting.

Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted enhanced Karnik–Mendel algorithms

With the development of $$\alpha $$ -planes representation of general type-2 fuzzy sets (GT2 FSs), general type-2 fuzzy logic systems (GT2 FLSs) based on GT2 FSs have become a hot topic in academic field. While type-reduction (TR) is most critical block for a T2 FLS, generally speaking, the most popular Karnik–Mendel (KM) or enhanced KM (EKM) algorithms are used to perform the TR. The paper connects the EKM and the continuous version of EKM algorithms together and expands the EKM algorithms to three different forms of weighted EKM (WEKM) algorithms resort to the Newton–Cotes quadrature formulas of numerical integration techniques, while the EKM algorithms just become a special case of the WEKM algorithms. Four computer simulation examples are used to illustrate the performances of the WEKM algorithms. Compared with the EKM algorithms, the WEKM algorithms have smaller absolute error and faster convergence speed to compute the centroid defuzzified value of GT2 FLSs in general, which make them potentially applicable for T2 FLSs designers and adopters.