Abstract — In this paper, we describe a method for edge detection in gray scale images based on the Sobel operator and **fuzzy** **logic**. The goal is to improve a standard method for edge detection in order to obtain better results. The tests were made with an efficient **type**-1fuzzy inference system (T1FIS) and the results shows that the edges obtained with the **fuzzy** **logic** are better and more precise than the basic edge detection method. For defuzzification process, centroid method is used. The proposed **type**-**1** **fuzzy** **logic** edge detection method was tested with the benchmark images and synthetic images. We used the merit of pratt measure to illustrate the performance of using **type**-**1** **fuzzy** **logic**.

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Abstract—Most applications of both **type**-**1** and **type**-2 **fuzzy** **logic** systems are employing singleton fuzzification due to its simplicity and reduction in its computational speed. However, using singleton fuzzification assumes that the input data (i.e., measurements) are precise with no uncertainty associated with them. This paper explores the potential of combining the uncer- tainty modelling capacity of interval **type**-2 **fuzzy** sets with the simplicity of **type**-**1** **fuzzy** **logic** systems (FLSs) by using interval **type**-2 **fuzzy** sets solely as part of the non-singleton input fuzzifier. This paper builds on previous work and uses the methodological design of the footprint of uncertainty (FOU) of interval **type**-2 **fuzzy** sets for given levels of uncertainty. We provide a detailed investigation into the ability of both types of **fuzzy** sets (**type**- **1** and interval **type**-2) to capture and model different levels of uncertainty/noise through varying the size of the FOU of the underlying input **fuzzy** sets from **type**-**1** **fuzzy** sets to very “wide” interval **type**-2 **fuzzy** sets as part of **type**-**1** non-singleton FLSs using interval **type**-2 input **fuzzy** sets. By applying the study in the context of chaotic time-series prediction, we show how, as uncertainty/noise increases, interval **type**-2 input **fuzzy** sets with FOUs of increasing size become more and more viable.

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Navigation of autonomous mobile robots in dynamic and unknown environments needs to take into account different kinds of uncertainties. **Type**-**1** **fuzzy** **logic** research has been largely used in the control of mobile robots. However, **type**-**1** **fuzzy** control presents limitations in handling those uncertainties as it uses precise **fuzzy** sets. Indeed **type**-**1** **fuzzy** sets cannot deal with linguistic and numerical uncertainties associated with either the mechanical aspect of robots, or with dynamic changing environment or with knowledge used in the phase of conception of a **fuzzy** system. Recently many researchers have applied **type**-2 **fuzzy** **logic** to improve performance. As control using **type**-2 **fuzzy** sets represents a new generation of **fuzzy** controllers in mobile robotic issue, it is interesting to present the performances that can offer **type**-2 **fuzzy** sets by regards to **type**-**1** **fuzzy** sets. The paper presented deep and new comparisons between the two sides of **fuzzy** **logic** and demonstrated the great interest in controlling mobile robot using **type**-2 **fuzzy** **logic**. We deal with the design of new controllers for mobile robots using **type**-2 **fuzzy** **logic** in the navigation process in unknown and dy- namic environments. The dynamicity of the environment is depicted by the presence of other dynamic robots. The per- formances of the proposed controllers are represented by both simulations and experimental results, and discussed over graphical paths and numerical analysis.

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The simulation results of the edge detectors implemented in MATLABR2010a software [20] are shown. This simulation is done on the test image lena and two synthetic images, square and polar shown in figure 11[21] ,created in MATLABR2010a. The results of Morphological gradient, Integrated **Type** **1** **fuzzy** **logic** system and Integrated interval T2FS and integrated generalized T2FS are shown in fig 7, 8, 9, 10 respectively..

This paper aimed to compare **type**-**1** and **type**-2 **fuzzy** **logic** performance in controlling swarm robot as tools for complex problem modeling, especially for path navigation. Each has its advantages and disadvantages with some **type** of **fuzzy** **logic** system. In general, the **Type** 2 **Fuzzy** **Logic** System (T2FLS) has better performance rather than the **Type**-**1** **Fuzzy** **Logic** System (T1FLS). T1FLS is much faster than others, particularly when considering Real Time apps and easier to design when compared to T2FLS, but has no resistance to interference, and does not support some degree of uncertainty. At T2FLS though it is more complex computation than T1FLS but effective in handling uncertainty. In resolving the problem of uncertainty, **Type**-2 **Fuzzy** **Logic** System (T2FLS) has been used in modeling uncertainties in solving complex problems as well as improving accuracy.

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T2FLSs are more complex than **type**-**1** ones. The major difference being the present of **type**-2 is their antecedent and consequent sets. T2FLSs result in better performance than **type**-**1** **fuzzy** **logic** systems (T1FLSs) on the applications of function approximation, modeling, and control. Besides, neural networks have found numerous practical applications, especially in the areas of prediction, classification, and control [18, 23]. The main aspect of neural networks lies in the connection weights which are obtained by training process. Based on the advantages of T2FLSs and neural networks, the **type**-2 **fuzzy** neural network (T2FNN) systems

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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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The EKF has been used to learn the parameters of some traditional **fuzzy** **logic** systems [44], [45] and intuitionistic **fuzzy** systems of **type**-**1** [46], [47]. However, because of the high dimensionality of the **fuzzy** system parameters, using the standard EKF can be more complicated [44], [48] especially for larger problem domains. In order to alleviate this computa- tional burden, the EKF is used in a decoupled form - DEKF - because it is faster and easier to implement [48] with the most useful properties of the EKF still preserved [49]. The DEKF algorithm has been used previously in [44] to train a T2 FLS where the parameters of both the antecedent and consequent parts of the T2 FLS were gathered into two separate vectors (antecedent and consequent parameter vectors). Similar to [38], we adopt a hybrid learning methodology (KF-based and GD) to adjust the antecedent and consequent parameters of the proposed model. While GD is also adopted for the update of the antecedent parameters, our model utilises the DEKF approach, different from [38], to adjust the consequent parameters of the new and extended framework of IT2 FLS, otherwise known as IT2 IFLS [34] for the first time in this study with the aim of achieving improved system performance in terms of error minimisation and faster convergence. To the best knowledge of the authors, there is previously no work in the literature where DEKF and GD is used for the optimisation of IT2 IFLS-TSK parameters.

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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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Step (4): The interface mechanism of the FLC is represented by a 7 7 decision table. The set of decision rules relating all possible combinations of input to outputs is based on previous experience in the field. This set is made up of 49 rules expressed using the same linguistic variables as those of the inputs and is stored in the form of a decision table shown in Table **1**.

part of Table 4. The system responses for Scenario 5 and 7 are illustrated in Fig.10c and Fig.10d, respectively. It can clearly observe that the T2 **fuzzy** moon landing system was able to pilot the spaceship to the dock without a crash for all testing scenarios while T1 and conventional PD controller structures crashed the spaceship in three of them. The PD structure was not able to handle the uncertainty and thus, in the first three testing scenarios, the crash occurred for several reasons. Scenario 5 was failed because the 4 condition of the successful landing (presented in Section 4.1) was violated. PD structure has also violated **1** condition at Scenario 6 and 7. The T1 **fuzzy** structure crashed the spaceship in Scenario Numbers 6 and 7 since it hit/touched the terrain (**1** condition). The last crash (Scenario 8) of the T1-FLC structure occurred due to the fact the required angle condition (5 condition) for a successful landing could not be satisfied as it resulted with oscillating system response. The handled scenarios clearly show that the proposed T2 **fuzzy** moon landing system can handle uncertainties and various operating points when compared to its T1 and conventional PD controller counterparts.

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T IME series forecasting is an important application area that has been extensively researched. It involves the sequential collection of observations over time with the purpose of developing a model that captures the underlying dependencies among attributes of the data. A wide range of approaches have been employed in the analysis of time series data. More recently, the use of soft computing methodologies such as **fuzzy** **logic** (**type**-**1** and **type**-2), neural networks, simulated annealing and genetic algorithms have been reported in the literature for time series forecasting [**1**]–[4]. These latter approaches have shown significant improvements over the traditional statistical methods because they are non-linear and are able to approximate any complex dynamical systems better than linear statistical models [5].

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ABSTRACT: A low power programmable frequency divider is proposed in this paper which is appropriate for WLAN applications. Multi- modulus architecture in dynamic **logic** with the minimum number of transistors is designed in 0.18µm CMOS technology. By using mixer, bandpass filter and switches, the divide ratios improved to 18. A technique is implemented in the dynamic 2-3 programmable divider cell for decreasing the glitches which leads to low power consumption. Based on simulation results it works up to 5GHz, with the average power about 37nW. Under a supply voltage of 1.8V, the total chip area of the multi- modulus programmable divider is 3100µm 2 .

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Recent work had proposed the use of neural based systems to learn the **type**-2 FLC parameters. However, these approaches require existing data to optimize the **type**-2 FLC. Thus, they are not suitable for applications where there is no or not sufficient data available to represent the various situations faced by the IT2FLC controller. Genetic Algorithms (GAs) do not require a priori knowledge such as a model or data but perform a search through the solution space based on natural selection, using a specified fitness function. We did not evolve the interval **type**-2 FLC rule base as it will remain the same as the interval **type**-**1** FLC rule base. However, the FLC antecedents and consequents will be represented by interval **type**-2 MFs rather than **type**-**1** MFs.

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Conventional PID controllers have been well developed and applied for about half a century [**1**], and are extensively used for industrial automation and process control today. The main reason is due to their simplicity of operation, ease of design, in expensive maintenance, low cost, and effectiveness for most linear systems. Recently, motivated by the rapidly dev eloped advanced micro-electronics and digital processors, conventional PID controllers have gone through a technological evolution, from pneumatic controllers via analog electronics to micro-processors via digital circuits [**1**, 5]. However, it has been known that conventional PID controllers generally do not work well for nonlinear systems, higher-order and time-delayed linear systems, and particularly complex and vague systems that have no precise mathematical models. To overcome these difficulties, various types of modified conventional PID controllers sue h as auto-tuning and adaptive controllers were developed lately [5]. Also, a class of non-conventional **type** of PID controllers employing **fuzzy** **logic** have been designed and simulated for this purpose [4, 5, and 12]. Stability of these **fuzzy** PID controllers are analyzed and guaranteed [4, 5, and 12]. Many simulation examples have been given to show the superior performance of this class of **fuzzy** PID controllers. Yet, despite the significant improvement of the **fuzzy** PID controllers over their classical counterparts, the constant control gains of these controllers are tuned manually; so generally do not achieve their best performance due to the lack of optimization.

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The use of a large number of rules in a **fuzzy** **logic** controller makes the control system more accurate and precise, providing a high performance, but increases the computational load of the processor. The reduction of the rule number of adaptive interval **type**-2 **fuzzy** controllers is possible through the ANFIS optimization technique that uses as inputs a **type**-**1** **fuzzy** controller with a large number of rules and the error and the integral of the error. In the proposed optimization method, the inputs and the outputs of a **type**-**1** **fuzzy** controller with a 49 rule base constitute the training data for the adaptive network-based **fuzzy** inference system (ANFIS). The training paradigm uses a gradient descent and a least squares algorithm to optimize the antecedent and the consequent parameters respectively. This allows obtaining a new **fuzzy** system with a rule base made up of only three rules Fig. 4(a) but with the same high control performance of the original **fuzzy** controller. The optimized **type**-**1** **fuzzy** system, with first order Sugeno inference, represents the new **type**-**1** **fuzzy** controller and takes the place of the previous **type**-**1** **fuzzy** controller with 49 rules.

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The proposed EKF-based learning IT2IFLS model is evalu- ated using a real world datasets from the Australia’s National Electricity Market (NEM) namely New South Wales (NSW) electricity market. Similar to [29], the NSW electricity market for the year 2008 is used for the analysis. The dataset was downloaded from [30] and consists of 17568 instances with attributes of regional reference price (RRP) as the input. The price data are treated as time series data and are partitioned into four separate datasets according to [29] as representatives of the four seasons in Australia. The input data for analysis is generated from four previous values [x(t − 4), x(t − 3), x(t − 2), x(t − **1**)] with x(t + **1**) as the output. There are a total of 336 data samples for each season which reduces to 331 after input generation. The first 231 data points are used for training while the remaining 100 data samples are used for testing in each season. There are 16 rules generated with a total of 6(4) + 2*16(4+**1**) = 184 parameters.

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As previously stated, GIT2FS is basically a **fuzzy** system augmented by a learning process based on a genetic algorithm (GA). In GIT2FS, GAs operates to search an appropriate Knowledge Base (KB) of a **fuzzy** system for a particular problem and to make sure those parameter values that are optimal with respect to the design criteria. The KB parameters constitute the optimization space, which is transformed into suitable genetic representation on which the search process operates. The KB is composed by interval **type**-2 membership functions (IT2MF), shortly (MF), and **fuzzy** rule base (RB), as mentioned before. So, 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. When tuning membership functions, an individual population represents parameters of the membership function shapes at which **fuzzy** rule base is predefined in advance. In contrast, if be desired to tune **fuzzy** rules base, the population represents all of **fuzzy** rules possibility that membership functions is assumed before. Fig.**1** shows these conceptions.

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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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