## 5 Simulation and Experimental Results
### Study Example 2  
A third order uncertain unstable system having an internal minimum-phase zero is considered for study [14]:  
\[\left\{\begin{aligned} \dot{x}_{1}&=3x_{1}-1.5x_{2}+0.5x_{3}+2u(t), \\ \dot{x}_{2}&=2x_{1},\\ \dot{x}_{3}&=x_{2}+\tanh(x_{3})+w(t)\\ y&=0.25x_{2}+0.75x_{3}\end{aligned}\right. \tag{36}\]  
being \(d(x,w(t))\triangleq\tanh(x_{3})+w(t)\), with \(w(t)=0\), \(0\leq t<5\) s and \(w(t)=5(t-5)e^{-\frac{(t-5)}{2}}\). The system matrices are:  
\[A=\begin{bmatrix}3&-1.5&0.5\\ 2&0&0\\ 0&1&0\end{bmatrix},B_{u}=\begin{bmatrix}2\\ 0\\ 0\end{bmatrix},B_{f}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix},C=\begin{bmatrix}0&0.25&0.75\end{bmatrix}.\]  
The proposed sliding surface designed with reaching phase elimination for this case is obtained as  
\[\begin{array}{ll}\overset{\star}{\sigma}_{m}=&\sigma-\sigma(0)e^{-\alpha_{s }t}\\ =&c_{1}y+\dot{y}-0.75(d)e^{-\alpha_{s}t}\end{array} \tag{37}\]  
where \(d\) can be replaced by estimated state \(\hat{x}_{4}\). Equivalent control law \(u_{eq}\) and disturbance caring term \(u_{n}\) is obtained as  
\[u_{eq}= -[(0.25c_{1}+0.75)\cdot 2\hat{x}_{2}+0.75c_{1}\hat{x}_{2}+0.5(3\hat{x}_ {1}-1.5\hat{x}_{2}+0.5\hat{x}_{3})]-k_{s}\overset{\star}{\sigma}_{m} \tag{38}\] \[u_{n}= -[0.75c_{1}\hat{x}_{4}+0.75\hat{x}_{4}] \tag{39}\]  
where the sliding surface coefficient is \(k_{s}=20\), control gain is \(c_{1}=50\), observer poles are kept at \(s=-5\) (same as placed in [14]), \(\alpha_{s}=2\), step size of simulation is \(0.001\) s, the solver  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Controller**} & \multirow{2}{*}{
\begin{tabular}{c} **Set-Point (\(r\))** \\ **Output (\(y\))** \\ \end{tabular} } & \multicolumn{2}{c}{**Error Indices**} & & & & \\ \cline{3-8}  & & **IAE** & **ISE** & & & & **ITSE** & **Control Efforts (\(u\))** \\ \cline{3-8}  & & **(10\({}^{-2}\))** & **(10\({}^{-2}\))** & **(10\({}^{-2}\))** & **ITAE** & **(10\({}^{-2}\))** & **in ITAE** \\ \hline GESOBC & error = y \(-\) & 311.5 & 256.1 & 13.47 & 1030 & 280.5 \\ EGESOBC & error = y \(-\) & 175.5 & 103.9 & 5.827 & 254.4 & 284.4 \\ LESOMC & error = y \(-\) & 179.4 & 101.8 & 6.910 & 322.3 & 355.5 \\ Proposed scheme & error = y \(-\) & 96.66 & 47.43 & 2.736 & 71.81 & 259.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of performance indices for Example 1.  
Figure 4: Error dynamics of proposed controller with reaching phase elimination for Example 1.  
used in simulation is ode4 (Runge-Kutta), and saturation is not considered in control signal for this example. The efficacy of the proposed controller is better as compared to others as shown in Figure 5. The sliding surface response is shown in Figure 6 and it is observed that sliding surface starts exactly from 0 value.  
The error dynamics of actual and observed states are shown in Figure 7, and it is verified that it is approaching towards zero, whereas real disturbance and its estimation are also presented in Figure 7. The performance indices for this example are shown in Table 2. It is clear from the performance indices that less error is provided by the proposed controller than by other compared techniques in all aspects.  
Figure 5: Response of output and control signal for Example 2.  
Figure 6: Sliding surface response for Example 2.