## 5 Simulation and Experimental Results
### Experimental Model
#### 5.3.2 Simulation Results for SLFM  
The proposed EGESO based sliding mode controller is applied to 4 th order uncertain system with matched uncertainty considered in 3 rd channel as given in (42). Indeed, it is found that the disturbance does not affect the SLFM system in the same channel as the control action, which demands a complex closed-loop control [40]. State \(x_{3}\) with matched disturbance added into it becomes \(\dot{x}_{3}=936.39x_{2}-41.19x_{3}+72.4593u+d(t)\) and \(\dot{x}_{1}=x_{3}\), where \(d(t)\triangleq 9\deg\) from \(11\leq t<12\) s and \(d(t)\triangleq 0\deg\) from \(12\leq t<14\) s and the disturbance matrix \(B_{f}\) becomes \([0\ 0\ 1\ 0]^{T}\). EGESO is used to estimate all four states and extended lumped disturbance. The sliding surface design for this system, considering all initial conditions of \(\hat{x}(0)\) are zero, is obtained as  
\[ \begin{array}{lcl}\dot{\sigma}_{m}=&\sigma-\sigma(0)e^{-\alpha_{s}t}\\ =&c_{s}(\hat{x}-\mathbf{r})+\hat{x}_{5}-(c_{s}\hat{x}(0)-c_{s}\cdot\mathbf{r}+ \hat{x}_{5})e^{-\alpha_{s}t}\end{array}  \tag{43}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Description** & **Unit** \\ \hline \(\alpha\) & Arm end-point deflection & degree \\ \(\theta\) & Servo motor gear angle & degree \\ \(L\) & Flexible link length & cm \\ \(T_{l}\) & Load torque & Nm \\ \(D\) & Link end-point deflection (Arc length) & cm \\ \(B_{eq}\) & Viscous damping coefficient & \\ \(K_{stiff}\) & Total stiffness of model & Nm/deg \\ \(J_{link}\) & Moment of inertia of link & Kg-m\({}^{2}\) \\ \(J_{eq}\) & Equivalent moment of inertia of the model & Kg-m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Nomenclature used for parameter. where \(\hat{x}_{5}\) is an estimation of disturbance \(d(t)\), \(\hat{x}=[\hat{x}_{1}\ \hat{x}_{2}\ \hat{x}_{3}\ \hat{x}_{4}]^{T}\), r is a set point for servo motor gear angle \(\theta\) and gain matrix \(c_{s}=[5-24.3847\ 1.0-0.2]\). Equivalent control law \(u_{eq}\) and disturbance caring term \(u_{n}\) is obtained as  
\[u_{eq} =-\left(c_{s}B\right)^{-1}\cdot\left(c_{s}\cdot A\hat{x}-\alpha_{s}c _{s}re^{-\alpha_{s}t}+\hat{x}_{5}e^{-\alpha_{s}t}+k_{s}\cdot(\vec{\sigma}_{m}) +\hat{x}_{5}\right) \tag{44}\] \[u_{n} =-\left(c_{s}B\right)^{-1}\cdot\left(c_{s}\cdot B_{f}\hat{x}_{5}- \alpha_{s}\hat{x}_{5}e^{-\alpha_{s}t}\right) \tag{45}\]  
where the sliding surface coefficient \(k_{s}=30\), observer gain obtained as [51.81 \(-27.2945\)\(-146.3584\)\(-245.1937\)\(234.7045\)\(743.9036\)], \(\alpha_{s}=4.5\), step size of simulation = 0.001 s, solver used in simulation is ode4 (Runge-Kutta). The proposed controller response is compared with other methods, and set-point tracking of \(\theta\) and arm end-point deflection angle (\(\alpha\)) are observed as shown in Figure 10. It is found that the proposed controller is not giving any overshoot as compared to the other two methods, so its effect on \(\alpha\) seems to be very less. The proposed controller provides less control signal at the initial stage to provide smooth output \(\theta\). It works better after external disturbances are added into the system at time \(11\leq t<12\) s. Figure 11 shows the control input for simulation results. The sliding surface response is shown in Figure 12 and it is observed that it is approaching zero. The actual and estimated states are \(\theta\), \(\dot{\theta}\), \(\dot{\alpha}\), \(\dot{\alpha}\) given in Figure 13. It is observed that GESO properly measures the estimated states, whereas the extended state estimates disturbance acts on the system immediately. The estimation of disturbance and error in disturbance estimation (error dynamics) are shown in Figure 14.  
From the performance indices shown in Table 4, it is clarified that the proposed controller provides a minor error compared to other techniques in all cases.