## 5 Numerical Example  
Several examples are presented to explain the feasibility of the algorithm. **Example 1**. : _Consider plant (1) and give the system parameters in the following form to illustrate the effectiveness of our results._  
\[A_{1} =\left[ \begin{array}{cccc}-9.9477&-0.7476&0.2632&5.0337\\ 52.1659&2.7452&5.5532&-24.4221\\ 26.0922&2.6363&-4.1975&-19.2774\\ 0&0&1&0\\ \end{array} \right],B_{1}=\left[ \begin{array}{cccc}0&0\\ 0&1\\ 1&0\\ 0&0\\ \end{array} \right],\] \[A_{2} =\left[ \begin{array}{cccc}-4.501&-0.7102&0.3011&5.0005\\ 52&2.9021&5.5504&-24\\ 20.0028&2.6&-4.1905&-17.2203\\ 0&0&1&-2\\ \end{array} \right],B_{2}=\left[ \begin{array}{cccc}0.2&0.15\\ 3.5&-7.6\\ -6&4.5\\ 0&0&0\\ \end{array} \right],\] \[C_{1} =\left[ \begin{array}{cccc}-0.01&0&0&0\\ 0&-0.01&0&0\\ 0&0&0&-0.01\\ \end{array} \right],C_{2}=\left[ \begin{array}{cccc}-0.01&0&0&0\\ 0&-0.01&0&0\\ 0&0&0&0\\ \end{array} \right].\]  
_The control input \(u(t)=-\left[cos(t);cos(t)\right]^{T}\). The transition rate matrix \(\Pi\) is chosen in \(\left[ \begin{array}{cccc}-0.6&0.6\\ -0.8&0.8\\ \end{array} \right]\) and the transition rate is shown in Figure 1. Following the observer design process, the simulation results are presented in Figure 2 with the initial value \(x(0)=[1;-2;-1;0]^{T}\). Figure 1(a)-d are the trajectories of the system states and their estimated states. It can be seen that the achieved observer performance is ideal. From the error image--after about 1.6 s, the error between the estimated value and the actual value is very small. However, the matrix \(\left[\left(sI-A_{2}\right)\left(I-C_{2}^{-}C_{2}\right),B_{2}\right]\) is not a full rank. So this model is unable to complete the model reference tracking control; whatever the reference model is, the controller does not exist. This example shows that the effect of the observer is ideal, but the algorithm is not suitable for all systems._  
Figure 1: Transition rates. **Example 2**. : _Consider the following satellite tracking system with three modes._  
\[A_{1} =\left[ \begin{array}{cccccc}0&0&0&0.94786&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&1\\ 1.6829793\times 10^{-8}&0&0&0&1.45842\times 10^{-4}&0\\ 0&0&0&-1.45842\times 10^{-4}&0&0\\ 0&0&-5.31747\times 10^{-9}&0&0&0\\ \end{array} \right],\] \[A_{2} =\left[ \begin{array}{cccccc}0&0&0&1.24649664327&0&0\\ 0&0&0&0&0.886083509114&0\\ 0&0&0&0&0&0.1\\ 1.2797796\times 10^{-8}&0&0&0&1.45842\times 10^{-4}&0\\ 0&0&0&-1.45842\times 10^{-4}&0&0\\ 0&0&-5.31747\times 10^{-9}&0&0&0\\ \end{array} \right],\] \[A_{3} =\left[ \begin{array}{cccccc}0&0&0&1.0153803503256841&0&0\\ 0&0&0&0&1.1889863481182&0\\ 0&0&0&0&0&0.1\\ 1.5710731\times 10^{-8}&0&0&0&1.45842\times 10^{-4}&0\\ 0&0&0&-1.45842\times 10^{-4}&0&0\\ 0&0&-5.31747\times 10^{-9}&0&0&0\\ \end{array} \right],\] \[B_{1} =\left[ \begin{array}{cccccc}0&0&0\\ 0&0&0\\ 0&0&0\\ 0.8&0&0\\ 0&0&1\end{array} \right],\] \[B_{2} =\left[ \begin{array}{cccccc}0&0&0\\ 0&0&0\\ 0&0&0\\ 1&0&0\\ 0&0.8&0\\ \end{array} \right],\]  
_where the transition rate \(\Pi=\left[ \begin{array}{cccc}?&?&0.6\\ 0.5&-0.8&0.3\\?&?&-0.8\end{array} \right]\). The matrices \(A_{m}\) and \(C_{m}\) in the reference model are designed as \(A_{m}=\left[ \begin{array}{cccccc}0&\omega_{1}&0\\ -\omega_{1}&0&0\\ 0&0&-2\end{array} \right]\), \(C_{m}=\)\(I_{3}.\omega_{1}=0.008\pi\). \(x_{m0}=\left[100/\sqrt{2};100/\sqrt{2};0\right]\). If the output of the satellite model can track the circular signal generated by the reference model, the orbiting motion can be completed. Set the simulation time at 5000 s._  
Figure 2: Results of the observer under initial conditions \([1,-2,1,0]\). _After computations by the LMI Toolbox in Matlab,_  
\[K_{1} =\left[ \begin{array}{ccccc}-0.0045365&0.0007&-0.0004&-0.0306&0.012 2&-0.0081\\ 0.002321&-0.0034&0.001&0.0169&-0.0646&0.0124\\ -0.0015825&0.001&-0.0026&-0.0112&0.0124&-0.0542\\ \end{array} \right].\] \[K_{2} =\left[