# A Shadowed Type-2 Fuzzy Approach for Crossover Parameter Adaptation in Differential Evolution
## 1 Introduction  
Fuzzy logic is a modern tool, and one of its most important applications is process control. It has shown rapid growth as it can solve problems related to information or knowledge uncertainty, providing a formal method for expressing knowledge in a way that is understandable for humans. It uses linguistic variables for this purpose.  
The success of applying fuzzy logic is primarily due to its ability to utilize ambiguous concept models, which helps in reducing the intuitive complexity of a process. This allows for control operations to be carried out, albeit in an approximate or heuristic manner, on nonlinear or time-varying processes [1; 2; 3; 4; 5].  
Fuzzy logic has achieved significant success in industrial control. While there are various versions of controllers that employ fuzzy logic, the term \"fuzzy controller\" typically refers to a control system with an internal structure that includes a fuzzy system to govern the controller. One of the frequently mentioned advantages of fuzzy controllers, in comparison to other controller types, is their ability to be designed even in the absence of an exact model of the plant to be controlled, as they are rule-based [6; 7; 8; 9; 10; 11; 12; 13; 14; 15].  
A new type of fuzzy sets of higher order, known as type-2 fuzzy sets, was proposed by [PERSON] in 1975 as an enhancement to fuzzy sets. These sets can handle system uncertainties by considering degrees of membership. The literature demonstrates the development of numerous works that utilize type-2 fuzzy controllers, which exhibit improved results [16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. Therefore, in this work, an interval type-2 fuzzy controller is implemented for this reason.  
Similarly, this work also presents the utilization of shadowed type-2 fuzzy systems, which represent an alternative branch in fuzzy logic. The concept of shadowed sets was originally introduced to enhance the interpretability of type-1 fuzzy sets and address the issues of excessive precision [26; 27; 28; 29; 30]. In this work, ST2-FSs will be employed to dynamically adapt the crossover parameter (CR) of differential evolution, with the primary objective of improving the results.  
The aforementioned aims to demonstrate the significance of this development, wherein the CR parameter is dynamically utilized in DE. Few works in the literature showcase the performance of the dynamic CR parameter, hence emphasizing the main reason for its adoption. Furthermore, we have an interval type-2 fuzzy system controller for the motor, which is commonly employed in the state-of-the-art technologies. Moreover, this work presents the integration of these two concepts to validate the potential of their combined application and assess their effectiveness.  
The contribution of this article is to utilize a novel approach, shadowed type-2 fuzzy systems (ST2-FS), for dynamically adapting the crossover parameter of differential evolution (DE), in this way obtaining a new variant of DE. This adaptation aims to enhance the performance of the DE algorithm when applied to the optimization of the motor position control problem with interval type-2 fuzzy logic.  
This paper provides the following content: Section 2 presents the definition of ST2-FSs, Section 3 introduces the DE theory, Section 4 describes the utilization of the DE algorithm with shadowed fuzzy, Section 5 presents the controller, Section 6 outlines the Results, and Section 7 concludes the paper.