# Model-Free-Based Single-Dimension Fuzzy SMC Design for Underactuated Quadrotor UAV
## 4 Control Design
### MFSDF-SMC Control Design Using Model-Free Approach
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(L_{sn}\) & **NB** & **NS** & **NS** & **Z** & **PS** & **NS** & **PB** \\ \hline PB & Z & PS & PM & PB & PB & PB \\ NS & NS & Z & PS & PM & PB & PB \\ PS & NM & NS & Z & PS & PM & PB \\ Z & NB & NM & NS & Z & PS & PM & PB \\ NS & NB & NB & NM & NS & Z & PS & PM \\ NS & NB & NB & NB & NM & NS & Z & PS \\ NB & NB & NB & NB & NB & NM & NS & Z \\ \hline \hline \end{tabular}  
* PB: Positive Big; NB: Negative Big; PM: Positive Medium; NM: Negative Medium; PS: Positive Small; NS: Negative Small; Z: Zero.  
\end{table} 
Table 2: Rule base for bounded-layer fuzzy-based SMC. The Figure 2 is the graphical illustration for the variables \(L_{sn}\) and \(L_{o}\) on the basis of which the rule base for bounder-layer fuzzy-based SMC is computed. One may see these rules in Table 2 given below:  
Furthermore, it is share that the output of this FLC design is based on singleton utilizing the concept of center of gravity (CoG) presented in the Figure 3. The entire range for all output and input membership functions as shown in Figures 3 and 4 respectively ranges in between \(-1\) to \(1\), whereas the display ranges in between \(-0.6\) and \(+0.6\). The block diagram of this fuzzy-based sliding mode control is also shown in Figure 5.  
In this paper, two issues are discussed: the tackling of unmodelled dynamic factors using model-free approach and the suggestion of single-input- or dimension-based fuzzy sliding mode controller. It has been observed that fuzzy logic control design involves several steps: fuzzification, inference, and defuzzification. These individual steps consume some computation time that is computed and discussed in the results. This computation time and control input energy derived using F-SMC control will not be suitable for aggressive maneuvers. Thus, paper recommends single input or dimension-based fuzzy sliding mode control. Single-dimension-based fuzzy sliding mode control (SDF-SMC) is designed using signed distance method [3; 65] where one may convert the two-dimensional set of rules into single-dimension as shown in Table 3. This can only be  
Figure 4: Input membership functions. Figure 5: Block diagram for fuzzy-based sliding mode control (FSMC). Figure 3: Output membership functions based on singleton (CoG). deviates at 66.51 s and remains away from reference track till 70.51 s. One may see this impact of unmodelled dynamic factor during its maneuvering in Figure 9.  
In the results shown in Figure 10 are position trajectories on \(x,y\) and \(z\) axis. The unmodelled dynamic factors and unknown external disturbances are introduced in the simulation via a random function generator. One may see the underactuated quadrotor deviates at 25 th second in all positional vectors. Moreover, the angular velocities such as roll, pitch and yaw responses are illustrated in Figure 11. This is the same with the angular velocities, where after introducing the unmodelled dynamic factors and external disturbance one may see the high number of oscillations at 25 th second known as chattering or Zeno effect. This Zeno phenomenon is available in roll, pitch and even in yaw response. Figure 10: Position responses derived using SMC control technique. Figure 3.0 Quadrotor deviates from trajectory due to unmodelled dynamic factors. Due to the high number of oscillations known as Zeno phenomenon, paper suggests the amalgamation of fuzzy logic design with sliding mode control. This constitutes a kind of underactuated quadrotor craft. The quadrotor deviates from the path at 59.51 s but again follows the same path at 63.51 s as one may remove the unmodelled dynamic factors and external disturbances. In addition to this, one may also see the chattering effect.