# Model-Free-Based Single-Dimension Fuzzy SMC Design for Underactuated Quadrotor UAV
## 4 Control Design
### MFSDF-SMC Control Design Using Model-Free Approach  
In this sub-section, we will develop single-dimension fuzzy-based sliding mode control, embed it with model-free approach, and discuss the conventional sliding mode control that has basic architecture consisting of steering infinite quadrotor's trajectory (in our case known as sliding manifold). Thus, it will lead the system to converge all the states towards zero asymptotically. The block diagram illustrates steering at provided helical trajectory in the Figure 1. For this research work and specifically for simulations, paper suggests the tracking of underactuated quadrotor over helical trajectory. The helical trajectory is among trajectories that exhibits real-world-like scenario [2]. In addition to this, it allows one to better evaluate and study the angular and translational parameters [3]. The above conventional SMC control technique is designed by following simple steps i.e., defining the sliding surface and later ensuring that all these variables move towards sliding surface. The next step is to develop the time delay estimation-based model-free SMC control as shown in Equation (27), which in Equation (25) is the tracking error and in (26) is the sliding surface:  
\[e=Y-Y_{d} \tag{25}\]  
whereas the sliding surface is defined as:  
\[S=\dot{e}-ce \tag{26}\]  
In Equation (26), variable \(c\) represents the gain of the proposed control scheme. Considering the Equation (23), the \(u_{c}\) can be given as:  
\[u_{c}=-\frac{\hat{F}-\ddot{Y}_{d}+\dot{e}+ce}{\alpha} \tag{27}\]  
where one may see  
\[e_{est}=\ddot{e}+\dot{e}+ce \tag{28}\]  
From Equation (26) or Equation (28), one can define \(\dot{e}\),  
\[\dot{S}=-\rho s-ksign(S) \tag{29}\]  
Figure 1: Conventional block diagram of SMC. In Equation (29), \(\rho\) and \(k\) are both positive diagonal definite matrices. After embedding the model-free approach with SMC, one may introduce the fuzzy logic control design. This is because of the stability requirement during aggressive maneuvers and avoidance of chattering noise (high number of oscillations on rotors) that may harm the propellers and BLDC motors too [53; 65]. In this, the gains of SMC will be tuned by the fuzzy logic control (FLC), for which the main sliding surface and \(e\) and \(\dot{e}\) will be computed as shown in Equations (29) and (30):  
\[S=\left[ \begin{array}{cc}e&\dot{e}\end{array} \right]\left[ \begin{array}{c}k \\ 1\end{array} \right] \tag{30}\]  
Moreover, the error and rate of change of error will be derived using the vectoral distance between state trajectory and manifold mathematically as shown in Equations (29) and (30) given as \(L_{sn}\) and \(L_{o}\):  
\[L_{sn}=\frac{\left(\dot{e_{Q}}+k\dot{e_{Q}}\right)}{\sqrt{1+k^{2}}} \tag{31}\]  
\[L_{o}=\sqrt{N^{2}-{L_{sn}}^{2}} \tag{32}\]  
This vectoral distance from point \(N\) to \(L_{sn}\) can be computed in this way, as illustrated in the Equations (31) and (32). Moreover, the pictorial illustration can be seen in Figure 2. In this conventional F-SMC technique, the set of rules are designed by considering \(L_{sn}\) and \(L_{o}\) as mainly an input to fuzzy logic controller, and the output of FLC is entertained as an input to SMC design to derive the states towards this sliding surface with appropriate time and with Zeno or chattering effect. Since the fuzzy logic design is based on linguistic variable scheme, the rules are also defined in the same manner as shown in Table 2 with triangular membership functions as illustrated on next page.