## 3 The Design of the Adaptive Fuzzy Sliding-Mode Controller  
In this section, a sliding-mode controller will be designed to improve the robustness of MIMO nonlinear fractional-order systems, which contain uncertain disturbances and unknown nonlinear functions. The key to designing this controller is to discover the input control laws \(u_{i}(t),i=1,2,\ldots,n\) and the effective sliding-mode surfaces \(s_{i}(t),i=1,2,\ldots,n\). The system can be written as  
\[\left\{\begin{array}{l} \begin{array}{l}C\,D_{1}^{a}x_{1}=f_{1}(x)+g_{1}(x) +d_{1}(t)+u_{1}(t)\\ \,0\,D_{1}^{a}x_{2}=f_{2}(x)+g_{2}(x)+d_{2}(t)+u_{2}(t)\\ \,\vdots\,\vdots\\ \,0\,D_{1}^{a}x_{n}=f_{n}(x)+g_{n}(x)+d_{n}(t)+u_{n}(t)\end{array} \right. \tag{9}\]  
where \(d_{i}(t),i=1,2,\ldots\ldots,n\) are bounded uncertain disturbances, and \(\|d_{i}(t)\|\leq d_{i}\), \(i=1,2,\ldots\cdots\), where \(d_{i}>0\). \(x^{T}=[x_{1},x_{2},\cdots,x_{n}]\in R^{n}\) are vector variables of the system. \(f_{i}(x),i=1,2,\ldots,n\) denote known terms of the system, and \(g_{i}(x),i=1,2,\ldots,n\) are the unknown internal structure functions of the system, which are difficult to accurately obtain. The following flowchart in Figure 1 effectively illustrates the intricate design and implementation of AFMSC. By leveraging the inherent capabilities of the integral SMC, significant enhancements are achieved in improving the switching function, thereby optimizing overall system performance. In scenarios characterized by uncertain interference and unknown nonlinear factors, it is strongly necessary to employ adaptive control strategies and leverage FLS for designing and implementing an equivalent control. These approaches enable robustness against unpredictable disturbances while ensuring precise regulation in dynamic environments. Furthermore, they facilitate adaptation to varying operating conditions, ultimately enhancing system stability and reliability. Figure 1: The flowchart of the fuzzy sliding mode. Supposing that the reference state signals are continuously differentiable functions, the tracking errors are defined as  
\[e_{i}=x_{i}-x_{id} \tag{10}\]  
The designed integral sliding-mode surfaces with \(e_{i}\) are defined as follows:  
\[s_{i}(t)=e_{i}(t)+{m_{i}}^{C}I_{t}^{a}e_{i}(t)-h_{i}(t),i=1,2,\ldots,n \tag{11}\]  
where \(m_{i}>0,i=1,2,\ldots,n\), \({}^{C}I_{t}^{a}\) is the fractional-integral operator. To make sure the sliding-mode surfaces pass through the coordinate origin, the functions \(h_{i}(t)\) satisfy  
1. \(h_{i}(0)=e_{i}(0)+{m_{i}}^{C}I_{t}^{a}e_{i}(0)\);
2. \(h_{i}(t)\to 0\) as \(t\rightarrow\infty\);
3. \(h_{i}(t)\) have a continuous derivative. In this paper, let \(h_{i}(t)=h_{i}(0)e^{-pt},i=1,2,\ldots,n\), where \(p>0\). The positive constants \(m_{i},i=1,2,\ldots,n\) determine the decay rate and velocity reaching the sliding-mode surfaces. Unlike traditional SMC techniques that use discontinuous switching functions with fixed-form switching functions, like exponential or sign switching functions, this study suggests a novel approach in which FLSs are methodically used to produce adaptive switching signals. The proposed fuzzy switching functions effectively emulate sliding-mode dynamics through real-time fuzzy reasoning algorithms that process multiple-input variables, including sliding surface variables and their derivatives. Specifically, the FLSs establish a nonlinear mapping between the system state errors and the required control compensation through carefully designed membership functions and rule bases derived from Lyapunov stability criteria.