of the area under the effected part of the output membership function. There are other inference methods such as averaging and sum mean square [19]. Figure 4 shows the steps involved in creating an input-output mapping using fuzzy logic [20]. 

The use of a series of fuzzy rules, and inference methods to produce a defuzzified output constitute a Fuzzy Inference System (FIS) [21]. The final manner in which the aggregation process takes place and the method of defuzzification can differ depending on the implementation of the FIS chosen. The approach discussed above is that of the Mamdani based FIS. 

![0_image_0.png](0_image_0.png)

There are several types of fuzzy inference systems which vary according to the fuzzy reasoning and the form of the if-then statements applied [16]. Another method of Fuzzy inference that is worth discussing is the Takagi-Sugeno-Kang method. It is similar to the Mamdani approach described above except that the consequent part is of a different form and as a result the defuzzification procedure is different. The if-then statement of a Sugeno fuzzy system expresses the output of each rule as a function of the input variables, and has the form [1], 

## If X Is A And Y Is B Then Z = **F(X,Y)** (26)

If the output of each rule is a linear combination of the input variables plus a constant, then it is known as a first-order Segeno fuzzy model, and has the form [1]: 

$$z=p x+q y+c$$
z = px + qy + c **(27)** 
Alternatively, the output of a rule can be a constant. The final output of the Sugeno FIS is a weighted average of the outputs from each rule [16]. 

## B. **Adaptive Neuro-Fuzzy Inference Systems**

The main difficulties with Fuzzy Inference Systems are that it is difficult to transform human knowledge into the necessary rule base, as well as to adjust the Membership functions to achieve a minimized output error for the FIS [16]. The purpose of an ANFIS (Adaptive Neuro-Fuzzy Inference System or **Adaptive Network-Based Fuzzy** 
Inference System) is to establish a set of rules along with a set of suitable membership functions that is capable of representing the input/output relationships of a given system [16]. 

An adaptive network refers a multi-layer feed-forward type structure with interconnect nodes. However, some of the nodes are adaptive, meaning that such a node's output is dependent on several parameters belonging to it [16]. The links in an adaptive network only indicates the flow of information. An adaptive network utilizes a supervised learning algorithm in order to minimize the error of the input/output mapping required, by adjusting the parameters of the adaptive nodes [16]. Therefore, a training dataset is necessary as the training process is similar to that used by neural networks except the parameters of the adaptive nodes are being adjusted instead of the weights of the links in the network. 

An ANFIS is a type of adaptive network with the adaptive nodes representing membership functions and the consequent equations along with their corresponding parameters [1]. The goal of an ANFIS is to adjust the membership functions' and consequent equations' parameters to emulate the input/output relationships of a given dataset [21]. Therefore, an ANFIS is functionally equivalent to an FIS except that it has the ability to learn and adapt through a training process using inputoutput data pairs to discover the most approximate parameters of the FIS to model the system accurately. The basic architecture of a first-order Sugeno (Takagi-SugenoKang) ANFIS with 2 inputs and 2 rules is shown in Figure 5 
[1, 16]. 

![0_image_1.png](0_image_1.png)

![0_image_2.png](0_image_2.png)

$$(27)$$

In Figure 5, Layer 1 contains a series of membership functions which determine the degree that the given input belongs to the specific fuzzy set [16]. The membership function's parameters are changed, therefore, changing the 