# Switched Observer Design for a Class of Non-Linear Systems
## 4 Simulation Results  
In this section, we illustrate the proposed observer design procedure through a numerical example. Consider a one link manipulator with revolute joints actuated by a DC motor. The elasticity of the joint may be well-modeled by a linear tensional spring. The elastic coupling of the motor shaft to the link introduces an additional degree of freedom. The states of this system are motor position and velocity, and the link position and velocity. The corresponding state-space model is  
\[\dot{\theta}_{m} =\omega_{m} \tag{53}\] \[\dot{\omega}_{m} =\frac{k}{J_{m}}(\theta_{1}-\theta_{m})-\frac{B}{J_{m}}\omega_{m} +\frac{K_{\tau}}{J_{m}}\tau\] \[\dot{\theta}_{1} =\omega_{1}\] \[\dot{\omega}_{1} =-\frac{k}{J_{1}}(\theta_{1}-\theta_{m})-\frac{mgh}{J_{1}}sin( \theta_{1})\]  
where \(\theta_{m}\), \(\omega_{m}\), \(\theta_{l}\) and \(\omega_{l}\) are the motor and link position and velocities, respectively. \(J_{m}\) and \(J_{l}\) are the inertia of the motor and link, respectively, \(h\), and \(M\) represent the length and mass of the link, \(b\) is the viscous friction, and \(K_{\tau}\) is the amplifier gain. The torque due to the stiffening spring is \(\tau\). The system modeled by (53) takes the form of (3) by considering  
\[x(t) = \begin{bmatrix}\theta_{m}(t)&\omega_{m}(t)&\theta_{l}(t)&\omega_ {l}(t)\end{bmatrix} ^{T} \tag{54}\] \[A = \begin{bmatrix}0&1&0&0\\ -\frac{k}{J_{m}}&-\frac{B}{J_{m}}&\frac{k}{J_{m}}&0\\ 0&0&0&1\\ -\frac{k}{J_{1}}&0&\frac{k}{J_{1}}&0\end{bmatrix} ^{T},\] \[B_{f} = \begin{bmatrix}0&0&0&-\frac{mgh}{J_{1}}\end{bmatrix} ^{T},\;B_{u}= \begin{bmatrix}0&\frac{K_{\tau}}{J_{m}}&0&0\end{bmatrix} ^{T}\] \[f(x(t)) =sin(x_{3}(t)).\]  
The adopted numerical values for the process parameters are taken from Table 1.  
Applying our approach, we obtain  
\[h_{1,j} =0,\;j=1,2,3 \tag{55}\] \[h_{1,3} =-3.33\cos(\hat{x}_{3}(t))\]then  
\[\overline{h}_{1,3} =-3.33\] \[\underline{h}_{1,3} =3.33\]  
which defines the set of matrix vertices of the uncertain polytopic matrix \(A(\alpha)\) in (11) as follows  
\[A_{1} =\left( \begin{array}{cccc}0&1.0&0&0\\ -270.3&-12.43&270.3&0\\ 0&0&0&1.0\\ -107.5&0&218.1&0\\ \end{array} \right)\] \[A_{2} =\left( \begin{array}{cccc}0&1.0&0&0\\ -270.3&-12.43&270.3&0\\ 0&0&0&1.0\\ -107.5&0&-3.008&0\\ \end{array} \right)\]  
On the other side, \(\tilde{X}\) defined in (43) is normalized considering the following polytopic set  
\[\tilde{X}=\Big{\{}\tilde{\epsilon}\in\mathbb{R}^{4}:|\tilde{x}_{1}|\leq 0.5,\ |\tilde{x}_{2}|\leq 0.25,|\tilde{x}_{3}|\leq 0.5,\ |\tilde{x}_{4}|\leq 0.5 \Big{\}}.\]  
The procedure defined in Algorithm 1 is used to determine the appropriate selection of the design parameters. Thus, Figure 2 shows the feasibility region of the optimization problem (52) subject to the LMI conditions in (23), (48), and (51) with respect to \(\mu\) and the upper bound of the decay rate \(\gamma\), considering the cases given by \(\chi=1.0\times 10^{5}\), \(\chi=5.0\times 10^{5}\), \(\chi=1.0\times 10^{6}\), and \(\epsilon=0.01\) as parameters of design which allow the numerical implementability of the state observer. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Numerical Values** \\ \hline \(J_{m}\) & \(3.7\times 10^{-3}\) kg m\({}^{2}\) \\ \hline \(J_{l}\) & \(9.3\times 10^{-3}\) kg m\({}^{2}\) \\ \hline \(h\) & \(1.5\times 10^{-1}\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values. Figure 2: Feasibility region for guaranteed decay rate \(\gamma\) with \(\epsilon=0.01\). Similarly, Figure 3 shows the feasibility region of the optimization problem (52) subject to the LMI conditions in (35), (48), and (51) under the same selection of design parameters. As expected, Figures 2 and 3 show that whenever the upper bound of the prescribed decay rate \(\gamma\) and the guaranteed \(H_{\infty}\) cost are set to be higher, respectively, the feasibility region associated with the solution of the corresponding convex optimization problem decreases for required low upper bound of the observer gain norm \(\chi\).