## 6 Simulation Examples
\right]u+\left[ \begin{array}{c}p_{1}x_{1}cos(x_ {2}^{2})+p_{3}x_{2}sin(x_{1}x_{2})\\ 0\end{array} \right] \tag{51}\]  
_where \(x=[x_{1},x_{2}]^{T}\) is the system state, \(p_{1}\in[-2,2]\), \(p_{2}\in[0,5]\), \(p_{3}\in[-1,1]\). Let \(\Delta f(x)=\left[ \begin{array}{c}p_{1}x_{1}cos(x_{2}^{2})+p_{3}x_{2}sin(x_ {1}x_{2})\\ 0\end{array} \right]\), \(\Delta g(x)=\left[ \begin{array}{c}0\\ p_{2}x_{2}^{2}\end{array} \right]\). It is easy to know that the system (51) is a mismatched system. The uncertain disturbance of the system is decomposed as_  
\[\Delta f(x)=c(x)h(x),\Delta g(x)=g(x)m(x) \tag{52}\]  
_where, \(g(x)=\left[ \begin{array}{c}0\\ 0.5\end{array} \right]\), \(c(x)=\left[ \begin{array}{c}1\\ 0\end{array} \right]\), \(h(x)=p_{1}x_{1}cos(x_{2}^{2})+p_{3}x_{2}sin(x_{1}x_{2})\), \(m(x)=p_{2}x_{2}^{2}\). Moreover, \(f_{max}(x)\) and \(g_{max}(x)\) are calculated as follows._  
\[\|g(x)^{+}c(x)h(x)\|=\|\left[ \begin{array}{cc}0&2\end{array} \right]\left[ \begin{array}{c}1\\ 0\end{array} \right]h(x)\|=0=f_{max}(x),\]  
_and_  
\[\|h(x)\|=\|p_{1}x_{1}cos(x_{2}^{2})+p_{3}x_{2}sin(x_{1}x_{2})\|\leq|2x_{1}+x_ {2}|=g_{max}(x)\]  
_Select the parameter \(\beta=1\). Then the original robust control problem is converted into solving an optimal control problem. For the auxiliary system_  
\[\dot{x}(t)=\left[ \begin{array}{cc}0&8\\ -5&1\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+\left[ \begin{array}{cc}0&1\\ 0.5&0\end{array} \right]\bar{u}, \tag{53}\]  
_find the control policy, \(\bar{u}\), such that the following performance index is minimized_  
\[ \begin{split} J(x,u)&=\int_{0}^{\infty}[f_{max}^{2}(x)+g_{max}^ {2}(x)+\beta^{2}x^{T}x+\bar{u}^{T}\bar{u}]dt\\ &=\int_{0}^{\infty}[5x_{1}^{2}+2x_{2}^{2}+4x_{1}x_{2}+\bar{u}^{T} \bar{u}]dt.\end{split}  \tag{54}\]  
_In order to obtain the obust control law by using Algorithm 2, it is assumed that the optimal cost function \(V^{*}(x)\) has a neural network structure: \(V^{*}(x)=W^{T}\phi(x)\), where \(W=[W_{1},W_{2},W_{3}]^{T}\), \(\phi(x)=[x_{1}^{2},x_{1}x_{2},x_{2}^{2}]^{T}\). The initial weight is taken as \(W_{0}=[1,-3,0.5]^{T}\), and the initial state of system is chosen as \(x_{0}=[-2,0.5]^{T}\). The neural network weights are calculated iteratively by MATLAB. In each iteration, 10 sets of data samples are collected along the nominal system trajectory to perform the batch least squares problem. After six iterations, the weight converges to \(W=\left[2.8983,-0.6859,5.2576\right]^{T}\). The optimal control of the auxiliary system is calculated as \(\bar{u}^{*}=\left[ \begin{array}{c}0.1715x_{1}-2.6288x_{2}\\ -2.8983x_{1}+0.3429x_{2}\end{array} \right]\). The robust control law of the original uncertain system is \(u^{*}=0.1715x_{1}-2.6288x_{2}\). The convergence process of neural network weight is shown in Figure 7, while the changing process of control signal is shown in Figure 8. The uncertain parameters \(p_{1}\), \(p_{2}\) and \(p_{3}\) in uncertain system (51) take different values, the state trajectories of the closed-loop system are obtained by the robust control law. Figure 9 shows the trajectory of the closed-loop system when \(p_{1}=-1\), \(p_{2}=1\) and \(p_{3}=1\). Figure 10 shows the trajectory of the closed-loop system when \(p_{1}=-1\), \(p_{2}=2\) and \(p_{3}=0\). Figure 11 shows the trajectory of the closed-loop system when \(p_{1}=0.3\), \(p_{2}=3\) and \(p_{3}=-1\). Figure 12 shows the trajectory of the closed-loop system when \(p_{1}=-2\), \(p_{2}=5\) and \(p_{3}=Figure 8: Robust control signal. Figure 7: Neural network weight. Figure 9: Closed loop system trajectory, \(p_{1}=-1,p_{2}=2,p_{3}=1\). Figure 11: Closed loop system trajectory, \(p_{1}=0.3,p_{2}=3,p_{3}=-1\). Figure 12: Closed loop system trajectory, \(p_{1}=-2,p_{2}=5,p_{3}=1\). Figure 10: Closed loop system trajectory, \(p_{1}=-1,p_{2}=2,p_{3}=0\). The nominal systems corresponding to the above two examples are linear systems. The following is an example with nonlinear nominal system. **Example 3**.