network with **fuzzy** adaptive resonance theory (GRNNFA) for the analysis of this first set of data. Similar to [65], we also study the performance of IT2AIFLS when the output of the nonlinear Friedman equation is noise free. In this second case, 1000 test samples are generated with ˆ n = 0 (this we refer to test dataset **2**). Similar to [65] we adopt self-constructing neural **fuzzy** inference network (SONFIN) and support vector based **fuzzy** model (SVR-FM) for **type**-1 comparison with our model. The parameters of SONFIN are learned using training-error minimisation through the com- bination of Kalman filtering and GDA. For **type**-**2** systems, we adopt **type**-**2** models such as **type**-**2** FLS, self-evolving **interval** **type**-**2** **fuzzy** neural network (SEIT2FNN) and **interval** **type**-**2** **fuzzy** neural network with support vector **regression** (IT2FNN-SVR). T2FLS employs GDA for parameter learning referred to as T2FLS-G. SEIT2FNN is designed with structure learning and utilises rule-ordered Kalman filter together with GDA for parameter learning. SEIT2FNN has IT2FS in the antecedents trained with GDA with TSK **interval** **type**-1 sets in the consequent. Two flavors of IT2FNN-SVR are proposed in [65] namely IT2FNN-SVR(N) and IT2FNN-SVR(F). The difference between the two is in the representation of the input nodes. The former consists of input nodes with numerical values with **interval** output nodes while the latter consists of input nodes with **fuzzy** numbers and **interval** output nodes. SONFIN and SEIT2FNN are previous studies involving the first author in [65]. We compare our results with these models already reported in the literature as shown in Table III. The results in Table III indicate the RMSE and standard deviation for AIFLS, IT2AIFLS and similar works in the literature. It is shown that IT2AIFLS exhibits lower RMSE compared to its **type**-1 counterpart, the non-**fuzzy**, the two T1FLSs and the T2FLSs. For 30 Monte-Carlo realisations, the average RMSE and standard deviation for IT2AIFLS on Friedman#**2** with additive noise are 1.5057 and 0.1022 respectively.

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The DEKF has been used to train recurrent neural networks [18] and radial basis neural networks [16] with application to three different control applications and iris classification problem respectively and has been shown to provide good performance. In [20], the DEKF is used to train multilayer perceptron (MLP) networks for forecasting zonal locational marginal price (LMP) of electric energy of the Pennsylvania- New Jersey-Maryland (PJM) electricity market. Khanesar et al [21] used the decoupled EKF to train IT2FLS and applied the system to identification and control **problems**. To the best knowledge of the authors, this is the first study where a decoupled EKF is used to update the parameters of **interval** **type**-**2** **intuitionistic** **fuzzy** **logic** system (IT2IFLS).

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Abstract—This paper presents a novel application of a hybrid learning approach to the optimisation of membership and non-membership functions of a newly developed **interval** **type**-**2** **intuitionistic** **fuzzy** **logic** system (IT2 IFLS) of a Takagi- Sugeno-Kang (TSK) **fuzzy** inference system with neural network learning capability. The hybrid algorithms consisting of decou- pled extended Kalman filter (DEKF) and gradient descent (GD) are used to tune the parameters of the IT2 IFLS for the first time. The DEKF is used to tune the consequent parameters in the forward pass while the GD method is used to tune the antecedents parts during the backward pass of the hybrid learning. The hybrid algorithm is described and evaluated, prediction and identification results together with the runtime are compared with similar existing studies in the literature. Performance comparison is made between the proposed hybrid learning model of IT2 IFLS, a TSK-**type**-1 **intuitionistic** **fuzzy** **logic** system (IFLS-TSK) and a TSK-**type** **interval** **type**-**2** **fuzzy** **logic** system (IT2 FLS-TSK) on two instances of the datasets under investigation. The empirical comparison is made on the designed systems using three artificially generated datasets and three real world datasets. Analysis of results reveal that IT2 IFLS outperforms its **type**-1 variants, IT2 FLS and most of the existing models in the literature. Moreover, the minimal run time of the proposed hybrid learning model for IT2 IFLS also puts this model forward as a good candidate for application in real time systems.

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Abstract—Conventional **fuzzy** time series approaches make use of **type**-1 or **type**-**2** **fuzzy** models. **Type**-1 models with one index (membership grade) cannot fully handle the level of uncertainty inherent in many real world applications. The **type**-**2** models with upper and lower membership functions do handle uncertainties in many applications better than its **type**-1 counterparts. This study proposes the use of **interval** **type**-**2** **intuitionistic** **fuzzy** **logic** system of Takagi-Sugeno-Kang (IT2IFLS-TSK) **fuzzy** inference that utilises more parameters than **type**-**2** **fuzzy** models in time series forecasting. The IT2IFLS utilises more indexes namely upper and lower non-membership functions. These additional parameters of IT2IFLS serve to refine the **fuzzy** relationships obtained from **type**-**2** **fuzzy** models and ultimately improve the forecasting performance. Evaluation is made on the proposed system using three real world benchmark time series **problems** namely: Santa Fe, tree ring and Canadian lynx datasets. The empirical analyses show improvements of prediction of IT2IFLS over other approaches on these datasets.

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Abstract. Several **fuzzy** modeling techniques have been employed for handling uncertainties in data. This study presents a comparative eval- uation of a new class of **interval** **type**-**2** **fuzzy** **logic** system (IT2FLS) namely: **interval** **type**-**2** **intuitionistic** **fuzzy** **logic** system (IT2IFLS) of Takagi-Sugeno-Kang (TSK)-**type** with classical IT2FLS and its **type**-1 variant (IFLS). Simulations are conducted using a real-world gas com- pression system (GCS) dataset. Study shows that the performance of the proposed framework with membership functions (MFs) and non- membership functions (NMFs) that are each intervals is superior to clas- sical IT2FLS with only MFs (upper and lower) and IFLS with MFs and NMFs that are not intervals in this problem domain.

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Abstract—This paper presents an approach to prediction based on a new **interval** **type**-**2** **intuitionistic** **fuzzy** **logic** system (IT2IFLS) of Takagi-Sugeno-Kang (TSK) **fuzzy** inference. The gradient descent algorithm (GDA) is used to adapt the parame- ters of the IT2IFLS. The empirical comparison is made on the designed system using two synthetic datasets. Analysis of our results reveal that the presence of additional degrees of freedom in terms of non-membership functions and hesitation indexes in IT2IFLS tend to reduce the root mean square error (RMSE) of the system compared to a **type**-1 **fuzzy** **logic** approach and some **interval** **type**-**2** **fuzzy** systems.

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In this study, a novel variable impedance control for a lower-limb rehabilitation robotic system using voltage control strategy is presented. The majority of existing control approaches are based on control torque strategy, which require the knowledge of robot dynamics as well as dynamic of patients. This requires the controller to overcome complex **problems** such as uncertainties and nonlinearities involved in the dynamic of the system, robot and patients. On the other hand, how impedance parameters must be selected is a serious question in control system design for rehabilitation robots. To resolve these **problems** this paper, presents a variable impedance control based on the voltage control strategy. In contrast to the usual current-based (torque mode) the use of motor dynamics lees to a computationally faster and more realistic voltage-base controller. The most important advantage of the proposed control strategy is that the nonlinear dynamic of rehabilitation robot is handled as an external load, hence the control law is free from robot dynamic and the impedance controller is computationally simpler, faster and more robust with negligible tracking error. Moreover, variable impedance parameters based on **Interval** **Type**-**2** **Fuzzy** **Logic** (IT2Fl) is proposed to evaluate impedance parameters. The proposed control is verified by a stability analysis. To illustrate the effectiveness of the control approach, a 1-DOF lower-limb rehabilitation robot is designed. Voltage- based impedance control are simulated through a therapeutic exercise consist of Isometric and Isotonic exercises. Simulation results show that the proposed voltage- based variable impedance control is superior to voltage-based impedance control in therapeutic exercises.

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Quality of web service (QoWS) monitoring is an important component in web service as it evaluates web service delivery performance and detects **problems**. Our previous work proposed a **fuzzy** model for QoWS monitoring due to uncertain nature of web service environment. However, **fuzzy** models are computationally costly. In this work, we propose a parallelization implementation of the models. The objective of this paper is to compare the performance between Mamdani- and Sugeno-based **fuzzy** inference systems (FIS) when they are applied to the QoWS monitoring models. The results suggested that Sugeno models produced less processing time than that of Mamdani models. However, Mamdani models benefited from parallelization more than that of Sugeno models by recoding higher percentage of improvement in terms of average processing time. This work will be expanded to investigate the implementation of the models in cluster computers and using a higher **type** of **fuzzy** **logic**, namely **interval** **type**-**2** **fuzzy**.

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The **interval** **type**-**2** FLC works as follows: the crisp inputs from the input sensors are first fuzzified into input **type**-**2** **fuzzy** sets; singleton fuzzification is usually used in **interval** **type**-**2** FLC applications due to its simplicity and suitability for embedded processors and real time applications. The input **type**-**2** **fuzzy** sets then activate the inference engine and the rule base to produce output **type**-**2** **fuzzy** sets. The **type**-**2** FLC rules will remain the same as in a **type**-1 FLC but the antecedents and/or the consequents will be represented by **interval** **type**-**2** **fuzzy** sets. The inference engine combines the fired rules and gives a mapping from input **type**-**2** **fuzzy** sets to output **type**-**2** **fuzzy** sets. The **type**-**2** **fuzzy** outputs of the inference engine are then processed by the **type**-reducer which combines the output sets and performs a centroid calculation which leads to **type**-1 **fuzzy** sets called the **type**-reduced sets. There are different types of **type**-reduction methods. In this paper we will be using the Center of Sets **type**-reduction as it has reasonable computational complexity that lies between the computationally expensive centroid **type**-reduction and the simple height and modified height **type**-reductions which have **problems** when only one rule fires . After the **type**-reduction process, the **type**-reduced sets are defuzzified (by taking the average of the **type**-reduced set) to obtain crisp outputs that are sent to the actuators.

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GIT2FS is basically a **fuzzy** system **type**-**2** augmented by a learning process based on a genetic algorithm (GA). Used to store complex unstructured information and structured used by a computer system is called as knowledge base (KB) technology. In GIT2FS, GAs operates to search an appropriate Knowledge Base (KB) of a **fuzzy** system for a particular problem. The optimization space is developed by Knowledge Base (KB) parameters. The Knowledge Base (KB) parameter is transformed into suitable genetic representation on which the search process runs. The KB is composed by **interval** **type**-**2** membership functions (IT2MF), denoted (MF), and **fuzzy** rule base (RB), as mentioned before. There are some options to design Genetic IT2 **Fuzzy** System, e.g. tuning or learning membership functions, or **fuzzy** rule base or both of them, sequentially or concurrently. An individual population represents parameters of the membership function shapes at which **fuzzy** rule base is predefined in advance and then tuning membership function take place. The population represents all of **fuzzy** rules possibility that membership functions is assumed before, if be desired to tune **fuzzy** rules base. Figure 5.1 shows these conceptions. Recently, there are some successful applications of GIT2FS to real world **problems**, e.g. modeling and decision making, control, robotics, manufacturing, consumer products.

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The **interval** **type**-**2** FLC works as follows: the crisp inputs from the input sensors are first fuzzified into input **type**-**2** **fuzzy** sets; singleton fuzzification is usually used in **interval** **type**-**2** FLC applications due to its simplicity and suitability for embedded processors and real time applications. The input **type**-**2** **fuzzy** sets then activate the inference engine and the rule base to produce output **type**-**2** **fuzzy** sets. The **type**-**2** FLC rules will remain the same as in a **type**-1 FLC but the antecedents and/or the consequents will be represented by **interval** **type**-**2** **fuzzy** sets. The inference engine combines the fired rules and gives a mapping from input **type**-**2** **fuzzy** sets to output **type**-**2** **fuzzy** sets. The **type**-**2** **fuzzy** outputs of the inference engine are then processed by the **type**-reducer which combines the output sets and performs a centroid calculation which leads to **type**-1 **fuzzy** sets called the **type**-reduced sets. There are different types of **type**-reduction methods. In this paper we will be using the Center of Sets **type**-reduction as it has reasonable computational complexity that lies between the computationally expensive centroid **type**-reduction and the simple height and modified height **type**-reductions which have **problems** when only one rule fires. After the **type**-reduction process, the **type**-reduced sets are defuzzified (by taking the average of the **type**-reduced set) to obtain crisp outputs that are sent to the actuators.

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I N the last three decades process control and automation area had a tremendous improvement due to advances on computational tools. Many of regulatory control actions that were performed by human operators are now performed automatically with aid of computers. Nonetheless, in a pro- cess with hundreds of variables, instruments and actuators it is impossible that a person or a group can manage every and any alarm triggered by an abnormal event. Therefore the Fault Detection and Diagnosis (FDD) field had received extensive attention. According to [1], the current challenge for control engineers is the automation of Abnormal Event Management (AEM) using intelligent control systems. Inside this field, Instrument Fault Detection and Diagnosis is a potential tool to prevent process performance degradation, false alarms, missing actions, process shutdown and even safety **problems**. A well-known strategy related to this pro- blem is preventive maintenance. In that, periodical tests and calibration are made in instruments. This is a cumbersome task where instruments are dismantled, cleaned, reassembled and calibrated. Even so, this is not a guarantee that faults will not occur [**2**]. This paper presents an **Interval** **Type**- **2** **Fuzzy** **Logic** (IT2FL) classifier to detect and diagnose temperature sensor faults in an alternative compressor, named Sales Gas Compressor (SGC), operating in a Gas Processing Unit (GPU).

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The concept of **fuzzy** set was introduced by Zadeh[7] in 1965 for defining uncertainty. In 1975, Zadeh[8] introduced the notion of **interval**-valued **fuzzy** sets as an extension of **fuzzy** sets, in which the values of the membership degrees are intervals of numbers instead of the numbers. In 1986, Atanassov introduced **Intuitionistic** **Fuzzy** Sets [4] which provides the opportunity to model the problem precisely based on the existing information and observations. After three years Atanassov and Gargov[5] proposed **Interval**- Valued **Intuitionistic** **Fuzzy** set (IVIFS) which is helpful to model the problem more accurately. The **fuzzy** graph theory was first introduced by Rosenfeld [10] in 1975. Yeh and Bang [12] also introduced **fuzzy** graphs independently. **Fuzzy** graphs are useful to represent relationships which deal with uncertainty. It has numerous applications to **problems** in various fields. **Interval**-Valued **Fuzzy** Graphs (IVFG) are defined by Akram and Dudek[**2**] in 2011. Atanassov[6] introduced the concept of **intuitionistic** **fuzzy** relations and **Intuitionistic** **Fuzzy** Graph (IFG). ShovanDogra [11] introduced different types of products of **fuzzy** graphs. S.N.Mishra and A.Pal[9] introduced the product of **interval** valued **intuitionistic** **fuzzy** graph. Akram and BijanDavvaz[1] introduced Strong **Intuitionistic** **Fuzzy** Graphs (SIFG). The notions of Strong **Interval**-Valued **Intuitionistic** **Fuzzy** Graphs (SIVIFG) are introduced by A.MohamedIsmayil andA.Mohamed Ali [3]. This paper has been organized as follows. Preliminaries required for this study are given in section **2**. In section 3 and 4, Boxdot and Star product on **interval**-valued **intuitionistic** **fuzzy** graphs has been defined and some of its properties are discussed.

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office equipment, spacecraft power systems, laptop computers, and telecommunications equipment, as well as DC motor drives [4]. Several control techniques for DC–DC converters have been reported in the literature, such as linear based control techniques, sliding mode control technique, and **fuzzy** **logic** control technique. Although the structure and design of linear based control techniques are simple, their performance usually depends on the working conditions of the controlled system. Sliding mode control technique needs a system model to be designed. One of the most important **problems** in design of this controller is control chattering [5]. Traditional **fuzzy** techniques provide for the output voltage regulation against input voltage However, the performance of this controller depends on the experience and knowledge of human experts. In general, trial-and-error tuning procedure is used to adjust parameters of the rule base and membership sets [6]. This means that these parameters will be change from one expert to another expert. The controlled system performance may be undesirably affected from these uncertainty conditions. Thus, a **type**-**2** **fuzzy** controller will be highly suitable to tackle the uncertainty which occurs in traditional **fuzzy** **logic** controllers. Karnik and Mendel [7], [8] established a complete **Type**-**2** FLS theory to handle uncertainties in FLS parameters.

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The concept of **fuzzy** sets was introduced by Zadeh [11] and later Atanassov [1] generalized this idea to **intuitionistic** **fuzzy** sets using the notion of **fuzzy** sets. On the other hand Coker [4] introduced **intuitionistic** **fuzzy** topological spaces using the notion of **intuitionistic** **fuzzy** sets. In this paper, we introduced **intuitionistic** **fuzzy** 𝜷 generalized continuous mappings and studied some of their basic properties. We arrived at some characterizations of **intuitionistic** **fuzzy** 𝜷 generalized continuous mappings.

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An IFS A is called **intuitionistic** **fuzzy** b open set, **intuitionistic** **fuzzy** -open set, **intuitionistic** **fuzzy** pre-open set, **intuitionistic** **fuzzy** regular open set, **intuitionistic** **fuzzy** semi open set, **intuitionistic** **fuzzy** generalized open set, **intuitionistic** **fuzzy** generalized pre open set, **intuitionistic** **fuzzy** generalized open set and **intuitionistic** **fuzzy** weakly generalized open set (IFbOS, IFOS, IFPOS, IFROS, IFSOS, IFGOS, IFGPOS, IFGOS and IFWGOS) if the complement of A c is an IFbCS, IFCS, IFPCS, IFRCS, IFSCS, IFGCS,

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Genetic operators such as crossover and mutation are ap- plied to the parents in order to produce a new generation of candidate solutions. As a result of this evolutionary cycle of selection, crossover and mutation, more and more suitable solutions to the optimization problem emerge with- in the population. Increasingly, GA is used to facilitate FLSs design [9]. However, most of the works discuss **type**-1 FLC design. This paper focuses on genetic algo- rithm of **type**-**2** FLCs. There are two very different ap- proaches for selecting the parameters of a **type**-**2** FLS [4]. **Type**-**2** FLCs designed via the partially dependent ap- proach are able to outperform the corresponding **type**-1 FLCs [9], The **type**-**2** FLC has a larger number of de- grees of freedom because the **fuzzy** set is more complex. The additional mathematical dimension provided by the **type**-**2** **fuzzy** set enables a **type**-**2** FLS to produce more complex input-output map without the need to increase the resolution. To address this issue, a comparative study involving **type**-**2** and **type**-1 FLCs with similar number of degrees of freedom is performed. The totally independent approach is adopted so that the **type**-**2** FLC evolved using GA has maximum design flexibility.

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This section describes the complete algorithm using rough set based on **intuitionistic** **type**-**2** **fuzzy** c- means clustering for robust and fast segmentation, which is a bottleneck to restrict the application of magnetic resonance imaging in clinic, and the segmentation of brain MRI now is confronted with presence of uncertainty and noise, many various kinds of algorithms have been proposed to handle this problem. In this paper, a hybrid clustering algorithm combined with a new **intuitionistic** **fuzzy** factor and local spatial information is proposed, where randomness is handled by **type**-**2** **fuzzy** **logic**, vagueness could be dealt with the rough set, and the **intuitionistic** **fuzzy** **logic** can address the external noises. The proposed algorithm is listed in the following three subsections:

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otherwise; the **type**-1 FLC performance might deteriorate (Mendel, 2001). As a consequence, research has started to focus on the possibilities of higher order FLCs, such as **type**-**2** FLCs that use **type**-**2** **fuzzy** sets. A **type**-**2** **fuzzy** set is characterized by a **fuzzy** MF, that is, the membership value (or membership grade) for each element of this set is a **fuzzy** set in [0, 1], unlike a **type**-1 **fuzzy** set where the membership grade is a crisp number in [0,1] (Hagras, 2004). The MF of a **type**-**2** **fuzzy** set is three dimensional and includes a footprint of uncertainty. It is the third dimension of the **type**-**2** **fuzzy** sets and the footprint of uncertainty that provide additional degrees of freedom making it possible to better model and handle uncertainties as compared to **type**-1 **fuzzy** sets. In this paper, adaptive network based **fuzzy** inference system (ANFIS) was used as **interval** **type**-**2** **fuzzy** **logic** controller (IT-2FL) in control strategies of the Heat Exchanger. **Interval** type2 **fuzzy** **logic** control was not taken into consideration by this approach in most of the cited investigations, despite some of its advantages indicated in this study. Proposed **type**-**2** **fuzzy** **logic** controller combines two different control techniques which are adaptive network based **fuzzy** **logic** inference system control and **interval** **type**-**2** **fuzzy** **logic** control, and uses their control performances together. Adaptive network based **fuzzy** inference system (ANFIS) uses a hybrid learning algorithm to identify parameters of Sugeno-**type** **fuzzy** inference systems. A combination of the least squares method and the back propagation gradient descent method is used in training **interval** type2 **fuzzy** inference system (IT2FIS) membership function parameters to emulate a given training data set. Moreover MATLAB/Sim-Mechanics toolbox and computer aided design program (Solid Works) was used together for visual simulations.. Also MATLAB/ANFIS toolbox was used to create adaptive network based **fuzzy** **logic** inference system controllers.

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convergence of powers of **fuzzy** matrix. Ragab et al. [33,34] presented some properties of the min-max composition of **fuzzy** matrices. Hashimoto [18,19] studied the canonical form of a transitive **fuzzy** matrix. Hemashina et al. [20] Investigated iterates of **fuzzy** circulant matrices. Dererminant theory, powers and nilpotent conditions of matrices over a distributive lattice are consider by Zhang [43] and Tan [41]. After that Pal, Bhowmik, Adak, Shyamal, Mondal have done lot of works on **fuzzy**, **intuitionistic** **fuzzy**, **interval**-valued **fuzzy**, etc. matrices [1-12,24,25,27-32,35-39].

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