## 4 Numerical Simulations  
Consider the Furuta pendulum-like system reported in [17], which serves as a useful underactuated platform for studying control theory problems. As shown in Figure 2, the system consists of two couple beams: \(\phi\) is the angle measured from the \(X\) axis towards the first link, and \(\theta\) is the angle measured from the vertical \(Z\) axis towards the second link, both contained in the vector of generalized coordinates \(q=[\phi\ \ \theta]^{\top}\). Thus, its dynamic equations are in the Lagrange-Euler form [30,31]  
\[M(q)\ddot{q}(t)+C(q,\dot{q})\dot{q}(t)+G(q)=\bar{B}u(t), \tag{19}\]  
where  
\[M(q) =\left[ \begin{array}{cc}J_{arm}+m_{p}r^{2}+m_{p}l_{p}^{2}\sin^{2} \theta&m_{p}rl_{p}\cos\theta\\ m_{p}rl_{p}\cos\theta&J_{p}+m_{p}l_{p}^{2}\end{array} \right],\] \[C(q,\dot{q}) =\left[ \begin{array}{cc}\frac{1}{2}m_{p}l_{p}^{2}\dot{\theta} \sin 2\theta&\frac{1}{2}m_{p}l_{p}^{2}\dot{\theta}\sin 2\theta-m_{p}rl_{p} \dot{\theta}\sin\theta\\ -\frac{1}{2}m_{p}l_{p}^{2}\sin\theta&0\end{array} \right],\] \[G(q) =\left[ \begin{array}{cc}0\\ -m_{p}sl_{p}\sin\theta\end{array} \right],\] \[\bar{B} =\left[ \begin{array}{c}1\\ 0\end{array} \right],\quad u=\frac{K_{t}(V_{m}-K_{m}\phi)}{R_{m}}.\]  
Consider the mapping \(x=[x_{1}\ x_{2}\ x_{3}\ x_{4}]^{\top}=[\phi\ \theta\ \dot{\phi}\ \dot{\theta}]^{\top}\), its nonlinear model from the Lagrange-Euler equations is  
\[\dot{x}= \begin{bmatrix}\dot{q}\\ \dot{q}\end{bmatrix} = \begin{bmatrix}\dot{q}\\ M(q)^{-1}(\bar{B}u-C(q,\dot{q})\dot{q}-G(q))\end{bmatrix} . \tag{20}\]  
Considering the values reported in Table 1, the linearization of the nonlinear system (20) around the origin in the form of (3), and consider \(x_{1}\) as the output to be regulated, with the matrices  
\[A = \begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ 0&76.2884&-0.5689&0\\ 0&82.2655&-0.2399&0\end{bmatrix} ,\quad B= \begin{bmatrix}0\\ 0\\ 17.084\\ 7.2054\end{bmatrix} , \tag{21}\] \[C = \begin{bmatrix}1&0&0&0\end{bmatrix} .\]  
Figure 1: Block-diagram of the controller scheme. Consequently and by assuming \(\rho=2\), \(\alpha_{1}=1,\alpha_{2}=1\) and \(\epsilon=0.05\), the block matrices for (5), are:  
\[ \begin{split} S=\begin{bmatrix}-200&1\\ -40000&0\end{bmatrix},\qquad S_{H}=\begin{bmatrix}200\\ 40000\end{bmatrix},\\ Q_{w}=Q_{ref}=\begin{bmatrix}1&0\end{bmatrix}.\end{split}  \tag{22}\]  
Then, remembering the linear central manifolds mappings \(x_{ss}=\Pi\omega\) and \(u_{ss}=\Gamma\omega\), its gains are computed by (7) resulting in  
\[\Pi= \begin{bmatrix}1&0\\ 0&0\\ 0&1\\ 0&0\end{bmatrix} ,\quad\Gamma= \begin{bmatrix}0&0.0333\end{bmatrix} .\]  
Therefore, for the stabilization by the LMIs (9) and (10), \(\varphi=5.9\), \(\beta=22\) and \(x(0)= \begin{bmatrix}5^{\circ}&3^{\circ}&0^{\circ}/s&0^{\circ}/s\end{bmatrix} ^{\top}\) are set, respectively, resulting in  
\[K= \begin{bmatrix}8.8650&-90.7956&3.3963&-12.2266\end{bmatrix} . \tag{23}\]  
Next, by using the LMI (16) a fault actuator observer of the fifth order was designed, which means that \(p_{a}=5\left(f_{a}^{(5)}(t)\approx 0\right)\), including a decay rate of 1, resulting in the increased observer gain \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Description** & **Value** \\ \hline \(M_{p}\) & Pendulum mass & 0.027 Kg \\ \(L_{p}\) & Large pendulum & 0.153 m \\ \(r\) & Large arm & 0.0826 m \\ \(I_{p}\) & Pendulum inertia & 0.000177 m\({}^{4}\) \\ \(J_{arm}\) & Arm inertia & 0.00018374 m\({}^{4}\) \\ \(k_{t}\) & Armor constant & 0.0333 \\ \(K_{m}\) & Motor constant & 0.0333 \\ \(R_{m}\) & Motor resistance & 8.7 \(\Omega\) \\ \(g\) & Gravity & 9.81 \(\frac{m}{s^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the Furuta pendulum. Figure 2: Furuta pendulum. \[\bar{L}= \begin{bmatrix}35.8\\ 293.8\\ 1286.2\\ 2707.9\\ -416.6\\ -1008.2\\ -1543.4\\ -1349.1\\ -512.2\end{bmatrix} .\]  
It is worth mentioning that the higher the order of the derivative of \(f_{a}(t)\), the better the approximation is, at the cost of increasing the computational burden [17].