of a **non**-**singleton** FLS. The goal of this work is not to achieve the best performance in applications such as in time series prediction, but to study and present approaches for the modelling of uncertainty in FLS inputs using solely **interval** **type**-**2** **non**-**singleton** fuzzification. The latter is valuable as the uncertainty in system inputs can directly be related to the FOU of the input FSs, while the rest of the FLS (antecedent and consequent FSs) can remain T1 FSs unless information on their respective uncertainty characteristics are known. This ap- proach is adopting an FOU creation method initially designed to determine an appropriate antecedent FS FOU and adjusted in this paper to generate appropriate FOUs for the input FSs based on input uncertainty, achieving potentially more efficient FLS design.

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where a, b and n are constant real numbers, t is the current time and τ is a **non**-negative time delay constant. The system tends to display a deterministic/periodic behaviour at τ ≤ 17 which turns chaotic when τ > 17. For comparison with other works in the literature such as [14], [20]–[23], where the target output is chosen as x(t + 6), from input vector (x(t − 18), x(t − 12), x(t − 6), x(t)) and τ = 17. For each input in this study, two intuitionistic Gaussian membership functions with uncertain standard deviation are used. Similar to [22], the 1000 learning data are generated from time series (t = 19-1018) with the first 500 data points used for training and the remaining 500 for testing. The results of applying different approaches to the prediction of Mackey-Glass are listed in Table I. As shown in Table I, IT2IFLS outperforms the modified differential evolution radial basis function neural network (MDE-RBF NN) and the **fuzzy** approaches.

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In mobile robotics, some researchers have explored the control of mobile robots using **interval** **type**-**2** **fuzzy** **logic** [21,24-29]. As Hagras states in [21] control using **type**-**2** **fuzzy** sets represents a new generation of **fuzzy** control- lers. In [25] Hagras presented an **interval** **type**-**2** **fuzzy** **logic** controller to command a robot in indoor and out- door unstructured environment. A robot was tested under different sources of **non**-systematic errors. The results showed that **type**-**2** **fuzzy** **logic** outperforms its **type**-**1** counterpart. This was shown through robot paths and control surfaces. In [27], an **interval** **type**-**2** **fuzzy** **logic** was proposed for the control of a robot tracking a mobile object in the context of robot soccer games. In this game the robot has to track a ball. To evaluate the performance of the **type**-**2** **fuzzy** **logic** against its **type**-**1** counterpart, graphical paths analysis were presented showing the way the player reaches the position of the ball. Also, an addi- tional test was made presenting the ability of **type**-**2** con- troller to track the ball with less standard deviation error than its **type**-**1** counterpart.

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Abstract—Robotics control system with leader-follower approach has a weakness in the case of formation failure if the leader robot fails. To overcome such problem, this paper proposes the formation control using **Interval** **Type**-**2**- **Fuzzy** **Logic** controller (IT2FLC). To validate the performance of the controller, simulations were performed with various environmental **systems** such as open spaces, complexes, circles and ovals with several parameters. The performance of IT2FLC will be compared with **Type**-**1** **Fuzzy** **Logic** (T1FL) and Proportional Integral and Derivative (PID) controller. As the results found using IT2FLC has advantages in environmental uncertainty, sensor imprecision and inaccurate ac- tuator. Moreover, IT2FLC produce good performance compared to T1FLC and PID controller in the above environments, in terms of small data generated in the **fuzzy** process, the rapid response of the leader robot to avoid collisions and stable movements of the follower robot to follow the leader's posture to reach the target without a crash. Especially in some situations when a leader robot crashes or stops due to hardware failure, the follower robot still continue move to the target without a collision.

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The use of convectional automatic voltage Regulator (CAVR) in synchronous generators to control the terminal voltage and reactive power has been the common phenomena in power **systems** control. Synchronous generators are nonlinear **systems** which are continuously subjected to load variations and the AVR design must cope with both normal load and fault condition of operation. Evidently, these conditions of operation result to considerable changes in the system dynamics. When the CAVR with fixed gain are used, the performance worsens and in some cases, introduces negative damping and degraded system stability. So far, a lot of work has been done in synchronous machine excitation stabilization using CAVR and controllers, all geared toward overcoming the problems enumerated above. The short comings here is that the parameters of the controllers are fixed and so if the system dynamics changes as a result of faults, the controller will be tuned manually to adjust. Modern control techniques are used extensively to achieve self-tuning (ST) control in synchronous generators. These include minimum variance (MV), generalized minimum variance (GMV), optimal predictor and pole placement (PP). In all these ST-AVR work, additional signals are used to improve robustness and are generally nonlinear. The MV generally gives very lively control and can be highly sensitive to **non** minimum phase plant. GMV, which is more robust and generalized, is vulnerable to unknown or varying plant dead time and can have difficulty with d.c offsets. PP aims to locate the closed-loop poles of the system at pre-specified locations leading to smooth controllers, but the algorithm shows numerical sensitivity when the plant model is over parameterized. Of recent, a lot of research is going on in areas of application of soft computing (**fuzzy** and neural approach) in synchronous generator controls. This work is based on **interval** **type**- **2** **fuzzy** **logic** controller (IT2FLC). (IT2FLC) in synchronous generator (SG) terminal voltage and reactive power control is designed so that it has the ability to improve the performance of **interval** **type**-**2** **fuzzy** **logic** controller. The **interval** **type**-**2** **fuzzy** **logic** controller is superior to conventional AVR controllers which continue to tune the controller parameters because it will tune and to some extent remember the values that it had tuned in the past.

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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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Atanassov [7] extended the concept of Zadeh’s **fuzzy** sets to intuitionistic **fuzzy** sets, hereafter referred to as AIFSs, which handle uncertainty by taking into account both the membership and **non**-membership degrees of an element x to a **fuzzy** set A together with extra degree of indeterminacy (hesitation). With AIFS, the **fuzzy** characteristic of “neither this or that” (neutral state) can be described, thus providing IFS the flexibility and the ability to capture more information than FS [8]. AIFSs are found to be useful for dealing with vagueness [9], [10]. Szmidt and Kacprzyk [11] state that AIFSs are useful in problem domains where the use of linguistic variable to describe the problem in terms of membership functions only seems too restrictive. According to Olej and Hajek [12], the representation of attributes by means of membership and **non**- membership functions provides a better way to express uncer- tainty. Castillo et al. [13] pointed out that the **non**-membership degrees or intuitionistic **fuzzy** indices enable the representation of imperfect knowledge and also allow adequate description of many real world problems. According to [14], when dealing with the problem of vagueness where there is insufficient information leading to an inability to satisfactorily specify the membership function, the AIFS theory becomes more suitable than **fuzzy** sets to deal with such problems. It is argued that AIFS is a tool for a more human consistent reasoning under imperfectly defined facts and imprecise knowledge [15].

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Abstract: An overview and a derivation of **interval** **type**-**2** fussy **logic** system (IT2 FLS), which can handle rule’s uncertainties on continuous domain, having good number of applications in real world. This work focused on the performance of an IT2 FLS that involves the operations of fuzzification, inference, and output processing. The out- put processing consists of **Type**-Reduction (TR) and defuzzification. This work made IT2 FLS much more accessible to FLS modelers, because it provides mathematical formulation for calculating the derivatives. Presenting extend to representation of T2 FSs on continuous domain and using it to derive formulas for operations, we developed and extended the derivation of the union of two IT2 FSs to the derivation of the intersection and union of N-IT2 FSs that is based on various concepts. The derivation of all the formulas that are related with an IT2 and these formulas de- pend on continuous domain with multiple rules. Each rule has multiple antecedents that are activated by a crisp number with T2 **singleton** fuzzification (SF). Then, we have shown how those results can be extended to T2 **non**- **singleton** fuzzification (NSF). We are derived the relationship between the consequent and the domain of uncertainty (DOU) of the T2 fired output FS. As well as, provide the derivation of the general form at continuous domain to cal- culate the different kinds of **type**-reduced. We have also applied an IT2 FLS to medical application of Heart Diseases (HDs) and an IT2 provide rather modest performance improvements over the T1 predictor. Finally, we made a com- parison of HDs result between IT2 FLS using the IT2FLS in MATLAB and the IT2 FLS in Visual C# models with T1 FISs (Mamdani, and Takagi-Sugeno).

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In the work of Nguyen et al. [16], IT2 **fuzzy** C-mean (IT2FCM) and IFS are applied for image clustering with improved performance in the clustering quality compared to FCM and IT2FCM. Nghiem et al. [17] applied intuitionis- tic T2 FS to image thresholding using Sugeno intuitionistic **fuzzy** generator. Results show improved thresholding quality compared to standard algorithms such as **type**-**1** FS and IFS. In Naim and Hagras [18], a hybrid approach using IT2 and IFS for multi-criteria group decision making (MCGDM) is proposed. In their study, IT2 FS is used to handle the linguistic uncertainty with intuitionistic evaluation used in the design of the NMF degrees. The proposed hybrid approach was evaluated on 10 candidates in a postgraduate study. Results of evaluation show better agreement with the human experts decision than IFS, FS and IT2 **fuzzy** **systems**. In the same vein, Naim et al. [19] presented a **fuzzy** **logic**-MCGDM (FL- MCGDM) for choosing appropriate and convenient lighting level to meet particular individual reading needs. The hybrid system adopted the concepts of IT2 FS and the hesitation indices of IFS with intuitionistic values used to represent the MFs of the IT2 FS for the left and right end-points. Results show that with T2 FS and IFS, the capability of FL-MDGDM is enhanced with improved capacity for decision making. In Naim and Hagras [20], FL-MCGDM is proposed for intelligent shared environment. Analysis of results indicate consistency

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Nowadays, unmanned aerial vehicles (UAVs) are widely used for various civilian and commercial applications, e.g., midair monitoring [**1**], vessel traffic management [**2**], anti- poaching patrol [3] and emergency evacuation planning [4]. In most of these applications, classical control approaches, e.g. proportional-integral-derivative control [5], sliding mode con- trol [6] and model predictive control [7], have been employed for UAVs to conduct autonomous flights. However, these well- known controllers require a precise dynamic model of the UAV and work under the assumption that significant internal as well as external uncertainties do not substantially affect the UAV **systems**. Achieving an accurate mathematical model for such complex aerial vehicles is often time-consuming and tedious [8]. In addition, the frequently-used sensors onboard the UAVs, e.g., global positioning system (GPS), inertial measurement unit (IMU) and camera, often lack precise modeling. Their measurements consist of numbers of uncertain, incomplete and possibly inaccurate information [9]. Further, the process of conducting a UAV application also contains numerous uncertain and challenging factors [10].

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shows that IT2FLC, by reducing the uncertainties of the control system, also improves the performance of the **fuzzy** controllers in reducing the structural displacement response, so that IT2FLC reduces the uncontrolled peak displacement response of stories about 10% to 30% more than the displacement response of the structure with FLC **Type**-**1**. While, the graphs indicates that none of IT2FLC and FLC **Type**-**1** couldn’t reduce the peak displacement response of stories rather than uncontrolled structure, under the influence of the acceleration of Kobe and Northridge (as near-field). It is because of this fact that, sometimes, for multi-storey buildings, a stronger ground motion may lead to earlier yielding of one floor which in turn acts as a fuse to relieve another (usually higher) floor, Especially in uncontrolled structure. Therefore, peak displacement response of stories of uncontrolled structure maybe less than structure with IT2FLC and FLC **Type**-**1**.

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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 research reported here introduces a new method for learning general **type**- **2** **fuzzy** **systems** with a unique combination of learning the footprint of uncertainty (FOU) followed by learning the secondary membership functions (SMF). In addition, we show that when using the vertical slice **type** reducer we have improvement over other approaches implemented here. Furthermore, **interval** **type**-**2** **fuzzy** **logic** **systems** were applied to answer the question of to what extent general **type**-**2** **fuzzy** sets can add more abilities and flexibilities to modeling than **interval** **type**-**2** **fuzzy** sets. A detailed analysis is carried out of the learning of general **type**-**2** **fuzzy** **systems** on a set of real- world data with and without added noise and, as such, provides significant insight into how the future of learning general **type**-**2** **fuzzy** **systems** can be carried out. These methods are applied to four benchmark problems: noise-free Mackey-Glass time series forecasting [34], noisy Mackey-Glass time series forecasting [34], and two real-world problems, namely the estimation of the low-voltage electrical line length in rural towns and the estimation of the medium-voltage electrical line maintenance cost [11].

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dataset is a time series that shows the number of lynx trapped in the Mckenzie river district per year in northern Canada and corresponds to the period 1821-1934. Similar to previous studies such as [26]–[28], the logarithms to the base 10 of the data are used in the analysis. Figures 4 and 5 show the original and the logarithmic transformed data of the Canadian lynx series respectively, with a periodicity of approximately 10 years. The series consists of 114 observations of which 100 samples are used for training and the remaining 14 are used for testing in order to validate the effectiveness of the model proposed in this study. Similar to [28], the maximum training epoch adopted is 2000. As shown in Table III, IT2IFLS outperforms the listed **non**-**fuzzy** approaches on the Canadian lynx dataset.

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In broad terms, uncertainty can be interpreted as informa- tion deficiencies in problem solving situations and it is an inseparable component of most real world applications as it depends on the variety of different circumstances [**1**]. Thus, the ability to handle uncertainties becomes an indispensable element of decision making. **Fuzzy** set (FS) theory was first introduced by Zadeh [**2**] and provided the basis for **Fuzzy** **Logic** **Systems** (FLSs) which are considered as robust **systems** to handle uncertainty in decision making [3]. FLSs have been successfully applied in a variety of areas, including data mining, pattern recognition and time series predictions [4]–[6]. FLSs processes are completed in three essential steps; fuzzi- fication, inferencing and defuzzification. In fuzzification, crisp input values are transformed into FSs and this transformation can be implemented as a **singleton** (SFLSs) or **non**-**singleton** (NSFLSs). Due to simplicity and lower computational cost of SFLSs, **singleton** fuzzification is the most commonly used

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This work, also focused on an interval type-2 non-singleton type-2 FLS IT2 NS-T2 FLS in order to determine how to assign all the parameters of the antecedent and consequent MFs using the[r]

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The simulated results show that, using a **type**-**2** FLC in real world applications can be a good option since this **type** of system is a more suitable system to manage high levels of uncertainty, as we can see in the results shown in Tables **2**-5. Simulation results indicate that the per- formance of the GA FLCT2 will better. That is mean the system will sense for change the value of IAE and ITAE. The results demonstrate that a **type**-**2** FLC can outper- form **type**-**1** FLCs that have more robustness design pa- rameters. The main advantage of the **type**-**2** FLC appears to be its ability to eliminate persistent oscillations, espe- cially when unmodelled dynamics were introduced. This ability to handle modeling error is particularly useful when FLCs are tuned offline using GA and a model as the impact of unmodelled dynamics is reduced. The sig- nificance of the work is focused to manage the uncer- tainty of the system. The nonlinear of the **systems** are big problem therefore the one of successful methods to eli- minate or reduce nonlinearity system by using **fuzzy** **type** two. It is a good option for real time applications that limited time is needed such as robot system.

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Abstract—In **non**-**singleton** **fuzzy** **logic** **systems** (NSFLSs) input uncertainties are modelled with input **fuzzy** sets in order to capture input uncertainty such as sensor noise. The performance of NSFLSs in handling such uncertainties depends both on the actual input **fuzzy** sets (and their inherent model of uncertainty) and on the way that they affect the inference process. This paper proposes a novel **type** of NSFLS by replacing the composition- based inference method of **type**-**1** **fuzzy** relations with a similarity- based inference method that makes NSFLSs more sensitive to changes in the input’s uncertainty characteristics. The proposed approach is based on using the Jaccard ratio to measure the similarity between input and antecedent **fuzzy** sets, then using the measured similarity to determine the firing strength of each individual **fuzzy** rule. The standard and novel approaches to NSFLSs are experimentally compared for the well-known problem of Mackey-Glass time series predictions, where the NSFLS’s inputs have been perturbed with different levels of Gaussian noise. The experiments are repeated for system training under both noisy and noise-free conditions. Analyses of the results show that the new method outperforms the standard approach by substantially reducing the prediction errors.

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Atanassov [3] introduced the concept of intuitionistic FS (IFS) with MF and NMF separately defined such that none is complementary to the other. In the literature, IFSs have been found to be one of the useful tools for dealing with imprecise information [4]. The reader is referred to [5] for more details on IFS. Similar to classical **type**-**1** FSs, the MFs and NMFs of IFSs may not handle the plethora of uncertainty that fraught many applications. The T2FS introduced by Zadeh is a three dimensional structure which provides the extra degrees of freedom needed to handle higher forms of uncertainty. For the generalised T2FS, the third dimension is weighted differently which makes it complex and difficult to use [6, 7]. The simpler and manageable version - **interval** **type**-**2** FS (IT2FS) - have values in the third dimension equal to **1** and this makes it easier for IT2FS to be represented on a two dimensional plane. The IFSs and IT2FSs have been widely and extensively adopted by researchers in uncertainty modeling in many applications. For example, in Nguyen et al. [8], IT2 **fuzzy** C-mean (FCM) cluster- ing using IFS is proposed. The authors show that the use of IFS with IT2FCM led to improved clustering quality particularly in the presence of noise. In Naim and Hagras [9, 10] and Naim et al. [11], IFS and IT2FS are combined to develop a multi-criteria group decision making (MCGDM) system for the assessment of post-graduate study, selection of appropriate lighting level in intelligent environ- ment and evaluation of different techniques for the choice of illumination in a shared environments respectively. The authors pointed out that the use of IFS and IT2FS in a MCGDM system provided decisions that are closer to the group decisions compared to some existing methods. However, in [9, 10, 11], only the IT2FS MFs are utilised and the intuitionistic **fuzzy** (IF) indices are evaluated on the primary MFs of the IT2FSs and no learning whatsoever is carried out on these sets.

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