An Interval Type-3 Fuzzy System and a New Online Fractional-Order Learning Algorithm: Theory and Practice

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.

The Wang-Mendel Approach (WMA) focuses on combining the numerical as well as linguistic information for achieving greater explainability for inference models. The standard WMA models the linguistic information using type-1 (T1) fuzzy sets (FSs), which have a reduced capability to model the semantics of linguistic information. Therefore, we propose a novel Enhanced WMA, which models the linguistic information using the type-2 (T2) FSs. Further, our Enhanced T2 FS-based WMA can be modified to reflect the use of interval type-2 (IT2) FSs, for modelling linguistic uncertainty. IT2 FSs are suitable when better uncertainty handling capabilities are required compared to T1 FSs, however, at a computational cost lesser than the T2 FSs. Performance of Enhanced WMA is demonstrated through a real-world crop-yield prediction problem in smart agriculture and an additional exemplar application on users’ satisfaction ratings. Further, we have compared our approach with the performance obtained from the T1 FS-based WMA and the original estimations given in the original data. We found that our Enhanced WMA provides more precise estimates than the other two with 95% confidence level. To the best of our knowledge, this is the first proposal of a T2 FSs method for enhancing the modeling of linguistic uncertainty in the WMA.

In this paper, we develop the parametric graded mean integration representation (PGMIR) for the interval type 2 fuzzy sets (IT2FS). The PGMIR value can be served as the type reduced or defuzzification of an IT2FS. A set of IT2FSs can then be ordered by the corresponding PGMIR values. Firstly, the parametric representation of the embedded fuzzy sets is defined for the general IT2FS. The IT2FS is characterized by the set of embedded fuzzy sets induced from the parametric function. Then, the graded mean integration representation (GMIR) of a type 1 fuzzy set (T1FS) is extended to the IT2FS through the parametric form of an IT2FS. The GMIR of the parametric embedded T1FS is defined and is called the PGMIR. In this manner, the IT2FSs are defuzzified and can be ranked by sorting the PGMIR values. The average value of the PGMIR is used to determine IT2FS ranking. A numerical example is presented to show that the proposed approach is superior to the ranking method from IT2FS centroid.

Modeling, design and control of a 4-arm delta parallel manipulator employing type-1 and interval type-2 fuzzy logic-based techniques for precision applications

In the past several years, research results on type- 2 fuzzy control have started to emerge in the literature. None of them, however, are concerned with mathematical input-output relationship of a type-2 fuzzy controller. Establishing such relationship is important as it will deepen our understanding of type- 2 fuzzy controller structures, how they function, and their advantages and drawbacks as compared to type-1 fuzzy controllers and conventional controllers (e.g., the PID controller). The availability of such structure will also lay a foundation for more rigorous theoretical analysis and design of type-2 fuzzy control systems. In this paper, we derive the mathematical input-output relation of a type-2 fuzzy PD (or PI) controller that has one of the simplest possible configurations - only two interval type-2 fuzzy sets for each input variables, four fuzzy rules with Zadeh AND operator, and four interval type-2 output fuzzy sets. We relate the resulting structure to conventional control theory and show that it actually is a PD (or PI) controller with the proportional and derivative (or integral) gains changing as the input variable values vary.

Parallel Interval Type-2 Subsethood Neural Fuzzy Inference System

Assessment method sends messages to students to define and priorities what is important to learn and ultimately how they spend their time leaning it. Traditional grading methods are largely based on human judgments, which tend to be subjective. In addition, it is based on sharp criteria instead of fuzzy criteria and suffers from erroneous scores assigned by indifferent or inexperienced examiners, which represent a rich source of uncertainties, which might impair the credibility of the system. In an attempt to reduce uncertainties and provide more objective, reliable, and precise grading, a sophisticated assessment approach based on type-2 fuzzy set theory is developed. In this paper, interval type-2 (IT2) fuzzy sets, which are a special case of the general T2 fuzzy sets, are used. The transparency and capabilities of type-2 fuzzy sets in handling uncertainties is expected to provide an evaluation system able to justify and raise the quality and consistency of assessment judgments. A simplified implementation of interval type-2 fuzzy system using the basic knowledge of type-1 fuzzy is presented. A comparison between the use of type-1, interval type-2 fuzzy systems and the simplified IT2 fuzzy systems in reducing uncertainties and providing more transparent and fair assessment that can reflect needs of individual students and foster development is presented.

In this paper the identification and control of dynamic plants using type-2 TSK fuzzy neural system (FNS) is considered. The systems constructed on the base of type-1 fuzzy systems cannot directly handle the uncertainties associated with information or data in the knowledge base of the process. One possible way to alleviate the problem is to resort to the use of type-2 fuzzy systems. In this paper, a type-2 TSK fuzzy neural system (FNS), is proposed and its gradient learning algorithm is derived. Its performance for identification and control of time-varying plants is evaluated and compared with other approaches seen in the literature; the time-varying nature of the plants being handled as uncertainties in the plant coefficients which can be described by type-2 fuzzy sets.

A new ranking methodology for general interval type 2 fuzzy sets (IT2FS) using extended alpha cuts representation is proposed. First, the extended alpha cut representation of the IT2FS is defined. Then, the parametric representation of the type 1 fuzzy sets (T1FS) is formulated by constructing the embedded pessimistic, modal and optimistic functionals. The corresponding parametric centroid functional of the IT2FS is determined. Finally, the average centroid of the IT2FS is calculated by taking the integral average of the parametric centroid functional. In this manner, the IT2FSs are defuzzified and can be ranked by sorting the average centroids. A numerical example is presented for verifying the proposed method.

Gestures of a person symbolizing analogous pathological disorder are not always exclusive. Naturally, the gestural features of a subject suffering from the same pathological disorder vary widely over different instances. In presence of two or more gestural features, the variation of the attributes together makes the problem of pathological disorder recognition more convoluted. Wider variance in gestural features is the main source of uncertainty in the pathological disorder recognition, which has been addressed here using type-2 fuzzy sets (T2FSs). First, a type-2 fuzzy gesture space is created with the background knowledge of gestural features of different subjects for different pathological disorders. Second, the pathological disorder of an unknown gestural expression is recognized based on the consensus of the measured gestural features of the fuzzy gesture space. A fusion of interval and general type-2 Fuzzy Sets has been used to construct the fuzzy gestural space. An interval valued fuzzy set (IVFS) or interval type-2 fuzzy set (IT2FS) is constructed with primary membership functions for M gestural features obtained from N subjects, each having L instances of gestural expressions for a given pathological disorder. The corresponding general type-2 fuzzy set (GT2FS) includes the secondary memberships for individual primary upper and lower membership curves in the respective IVFS. The secondary membership of a given source, characterizing the reliability in its primary membership assignment, is determined based on the amalgamated knowledge of all the subjects’ primary membership functions. The representatives of each pathological disorder class are then evaluated by utilizing the attractive features of both IVFS and GT2FS. The pathological disorder of an unknown gestural expression is finally determined by identifying its minimal discrepancy with the existing class representatives. The adopted T2FS-based uncertainty management strategy, when applied to gender independent training and testing datasets, results in a classification accuracy of 93.41%.

This paper presents a comparative study between the firefly algorithm (FA) and the galactic swarm optimization (GSO) method, where the performance of both methods is observed and tested in the optimization of a fuzzy controller for path tracking of an autonomous mobile robot. The main contribution of this work is finding the best method that generates an optimal vector of values for the membership function optimization of the fuzzy controller. This with the goal of improving the performance of the controller and thus the trajectory generated by the autonomous robot is closer to the desired trajectory. It should be noted that the fuzzy controller that is optimized is an interval type-2 fuzzy controller, which has a greater capability for managing uncertainty than a type-1 fuzzy controller. In this case, the limiting membership functions in the interval type-2 fuzzy sets are themselves type-1 fuzzy sets that define the footprint of uncertainty. Type-2 fuzzy controllers have been shown in previous works to handle better the control of robotic systems under noisy and dynamic conditions and this is why their optimal design is very important. Simulation results show that GSO outperforms FA in the optimal design of interval type-2 fuzzy controllers.

Successful implementation of type-1 Fuzzy Systems (FS) in diverse application areas have been accomplished till date. Nonetheless type-1 FS are not able to handle significant amount of uncertainties present in dynamic real world applications. An improved performance against these uncertainties is achieved by type-2 FS. In this paper, a type-2 FS is proposed to remove variable gaussian noise from a number of images using multilevel two dimensional wavelet transform and multilevel soft threshold. We have displayed that the proposed type-2 FS achieve a good efficiency in removing the noise from the wavelet transformed images and also has a superior performance compared to type-1 FS.

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap. 3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets, type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variables, returning to linguistic variables from a numerical value of a membership function, set theoretic operations for crisp and type-1 fuzzy sets, crisp and fuzzy relations and compositionson the same or different product spaces , compositions of a type-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension Principle (which is about functions of fuzzy sets), α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cuts, multivariable membership functions and Cartesian products, crisp logic, going from crisp logic to fuzzy logic, Mamdani (engineering) implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy sets. 35 examples are used to illustrate this chapter’s important concepts.

In this paper we will present two theorems for the\njoin and meet operations for general type-2 fuzzy sets with\narbitrary secondary memberships, which can be non-convex\nand/or non-normal type-1 fuzzy sets. These results will be used\nto derive the join and meet operations of the more general\ndescriptions of interval type-2 fuzzy sets presented in [1], where\nthe secondary grades can be non-convex. Hence, this work will\nhelp to explore the potential of type-2 fuzzy logic systems which\nuse the general forms of interval type-2 fuzzy sets which are\nnot equivalent to interval valued fuzzy sets. Several examples\nfor both general type-2 and the more general forms of interval\ntype-2 fuzzy sets are presented.

In this paper, an interval type-2 Takagi-Sugeno fuzzy classification system (IT2T-SFCS) learned by particle swarm optimization (PSO) and support vector machine (SVM) for antecedent and consequent parameters optimization is proposed. The IT2T-SFCS is constructed by fuzzy if-then rules whose antecedents are interval type-2 fuzzy sets and consequents are linear state equations. The antecedents of IT2T-SFCS use the fuzzy iterative self-organizing data analysis technique (ISODATA) and PSO to learn and calculate the optimal centers and the uncertain widths of the Gaussian membership functions. Consequent parameters in IT2T-SFCS are learned through SVM for the purpose of achieving higher generalization ability. The proposed IT2T-SFCS is able to directly handle uncertainties, minimize the effects of uncertainties and get the better generalization performance, which inherits the benefits of interval type-2 T-S fuzzy system and SVM. For demonstration, IT2T-SFCS is used as a classifier in gender recognition. The experimental results show that the performance of the proposed IT2T-SFCS is superior to that of the previous mainstream classifiers.