## 4 Proposed Shadowed Fuzzy System Using HS Algorithm  
In this section, we provide a description of the structure of the fuzzy system employed for the experimental analysis. This fuzzy system, referred to as SFHSA (shadowed fuzzy harmony search algorithm), is depicted in more detail in Figure 3, providing a visual representation of its components and structure. SFHSA serves as the framework for integrating the ST2-FHS approach into the optimal type-2 fuzzy controller design. It effectively manages the harmonization process, enabling efficient search space exploration and exploitation. The utilization of shadow sets and the  
Figure 2: Representation of the HS algorithm. incorporation of two alpha planes play crucial roles in enhancing the performance of the algorithm, which can be viewed as the key contribution of this work. Figure 3 offers a comprehensive illustration of the SFHSA structure, allowing for a clear understanding of its key components and their interconnections. This diagram serves as a visual aid to facilitate the comprehension of the algorithm's implementation and its information flow. By employing the SFHSA methodology, this article aimed to address the challenges associated with optimizing the parameters of the membership functions in an interval type-2 fuzzy controller. The proposed approach not only enhances the computational efficiency but also ensures the attainment of more effective and accurate control outcomes. The subsequent sections of this article will delve into the experimental results obtained using SFHSA and present a comprehensive analysis of its performance. These findings will shed light on the effectiveness and reliability of the proposed methodology in addressing the control challenges associated with type-2 fuzzy systems. Equations (11) and (12) represent the calculation of the input and output of the SFHSA. \[\text{Iteration}=\frac{\text{Current Iteration}}{\text{Maximum of iterations}} \tag{11}\]  
\[\text{HMR}=\frac{\sum_{i=1}^{n_{\text{hmr}}}\mu_{\text{i}}^{\text{hmr}}( \text{hmr}_{\text{i1}})}{\sum_{i=1}^{n_{\text{hmr}}}\mu_{\text{i}}^{\text{hmr}}} \tag{12}\]  
In these equations, HMR represents the memory consideration, r\({}_{\text{hmr}}\) denotes the number of rules in the SFHSA corresponding to HMR, hmr\({}_{\text{i1}}\) represents the output result for rule i corresponding to HMR, and u\({}_{\text{i}}^{\text{hmr}}\) represents the membership function of rule i corresponding to HMR. Figure 3: Diagram of the main proposal. Algorithm 2 depicts the pseudocode of the enhanced SFHSA with parameter adaptation and Figure 4 shows the SFHSA flowchart. ```
1. Define objective function \(f(x),x=\left(X_{1},\dots,X_{n}\right)^{T}\)
2. Initially generate harmonics (matrix of real numbers)
3. Establish the pitch adjustment rate (PArate) and limits of tone
4. Establish the harmony memory accepting (HMR)
5. While (t < Maximum number of iterations)
6. Calculate iteration using Equation (11)
7. Calculate new HMR parameter using a SFHSA fuzzy system of Equation (12)
8. Produce a new harmony and accept the best system
9. Setting the tone for new harmonies (solutions)
10. If (rand > HMR)
11. Randomly select an existing harmony
12. Else if (rand > PArate)
13. Setting the tone at random within a bandwidth
14. Else
15. Produce a new harmony through a randomization
16. End if
17. Accepting new harmonies (solutions) best
18. End while
19. Find the best solutions
```  
**Algorithm 2:** SFHSA  
The design of SFHSA consists of a Mamdani-type system, which includes an input named \"Iteration.\" This input is characterized by three membership functions (MFs) called \"Small\", \"Medium\", and \"Big\". The output of the SFHSA system is represented by the HMR parameter, which is defined by three MFs: \"Small\", \"Medium\", and \"Big\". Figure 5 illustrates the structure of SFHSA. The SFHSA rules were developed incrementally, and they were created based on previous experimentation.