## 2 Theories  
**Definition 1**. : _The \(\alpha\)-order Caputo fractional derivative of the function is defined as the following form:_  
\[ \begin{array}{l}\mathbb{C}D_{t}^{\alpha}f(t)=\left\{\begin{array}{l}\frac{1 }{\Gamma(m-\alpha)}\int_{t_{0}}^{t}\frac{f^{m}(\tau)}{(t-\tau)^{\alpha-m+1}}d \tau,m-1<\alpha<m\\ \frac{d^{m}f(t)}{d^{2m}},\alpha=m\end{array}\right.\end{array}  \tag{1}\]  
\(m\) _is a positive integer number and represents the integer part of \(\alpha\). \(\Gamma\) is the Gamma function, which satisfies \(\Gamma(1)=1\)._  
If \(m\) is equal to \(\alpha\) in the above definition, fractional-order algorithms will be equivalent to integer-order algorithms. Fractional-order calculus is homogeneous; for any constant, there exists  
\[ \begin{array}{l}\mathbb{C}D_{t}^{\alpha}[af(t)+bg(t)]=a_{t_{0}}^{C}D_{t}^{ \alpha}f(t)+b_{t_{0}}^{C}D_{t}^{\alpha}g(t)\end{array}  \tag{2}\]  
The relationship between fractional-integral and fractional-differential is  
\[ \begin{array}{l}\mathbb{C}D_{t}^{\alpha}\mathbb{C}_{t_{0}}^{\alpha}f(t)=f(t )\end{array}  \tag{3}\]  
**Definition 2**. : _Global Stability: If there exists a positive defined, continuously differentiable, radially unbounded function \(V(x)\) defined in \(R^{n}\), such that \(\dot{V}(x)\) is negative definite in \(R^{n}\), then the origin is globally asymptotically stable._  
The integer-order differential operation can be regarded as a special case of the fractional-order operation; thus, because of the function \(\dot{V}=D_{t}^{1-\varepsilon}D_{t}^{\alpha}V\), if there exists a positive define function \(V(x)\) and a negative define function \(\mathop{\circ}\limits_{0}^{\infty}D_{t}^{\alpha}V(x)\), the system will be proved to be stable. **Definition 3**. : _Sliding-mode control: The primary concept of the SMC involves designing the switching sliding-mode surface of the system in accordance with the anticipated dynamic characteristics of the system. The system state is then closed from outside the hyperplane to the switching sliding-mode surface through the use of a sliding-mode controller. Once the system state reaches this surface, the controller ensures that the system reaches its origin along this path. This process offers strong robustness as it depends solely on the characteristics and parameters of the designed switching sliding-mode surface and is unaffected by external interference. To ensure that the system state can be reached and maintained on this surface, it must satisfy a specific formula: \(\mathop{\lim}\limits_{s\to 0}\mathop{\rm s}\limits\mathop{\rm s}\limits<0\)._  
**Definition 4**. : _Fuzzy logic systems:The FLSs consist of three main components: the fuzzy rule base, fuzzification, and defuzzification operators. The fuzzy rule base contains inference rules for multiple-input and single-output systems:_  
\(R^{l}\)_: If_ \(x_{1}\) _is_ \(A_{1}^{l}\) _and_ \(x_{2}\) _is_ \(A_{2}^{l}\cdot\cdot\cdot\) _and_ \(x_{n}\) _is_ \(A_{n}^{l}\)_, then_ \(y\) _is_ \(B^{l}\)_,_ \(l=1,2,\ldots,N\)_._  
_By utilizing singleton function, center average defuzzification, and product inference, FLS can be effectively expressed as_  
\[\hat{f}(x)=\frac{\sum\limits_{l=1}^{N}\mathop{\circ}\limits_{i=1}^{N}\mathop{ \circ}\limits_{i=1}^{n}\mu A_{i}^{l}(x_{i})}{\sum\limits_{l=1}^{N}\left(\prod \limits_{i=1}^{n}\mu A_{i}^{l}(x_{i})\right)} \tag{4}\]  
_Define the fuzzy basis functions as_  
\[\xi_{l}(x)=\frac{\prod\limits_{i=1}^{n}\mu A_{i}^{l}(x_{i})}{\sum\limits_{l=1} ^{N}\left(\prod\limits_{i=1}^{n}\mu A_{i}^{l}(x_{i})\right)} \tag{5}\]  
_where the function_ \(\mu A_{i}^{l}(x_{i})\) _is the membership function.