## 6 Simulation Examples  
Some simulation examples are presented to verify the feasibility of the robust control design method for uncertain nonlinear systems in this section. **Example 1**. : _Consider the following uncertain nonlinear systems_  
\[\hat{x}(t)=\left[ \begin{array}{cc}0&6\\ -1&1\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+\left[ \begin{array}{c}0\\ p_{1}x_{1}cos(x_{2}^{2})\end{array} \right]+\left[ \begin{array}{c}0\\ 1+p_{2}x_{2}^{2}\end{array} \right]u \tag{47}\]  
_where \(x=[x_{1},x_{2}]^{T}\) is the system state, \(\Delta f(x)=\left[ \begin{array}{c}0\\ p_{1}x_{1}cos(x_{2}^{2})\end{array} \right]\) is the uncertain disturbance function of the system, \(\Delta g(x)=\left[ \begin{array}{c}0\\ p_{2}x_{2}^{2}\end{array} \right]\) is input uncertainty function, \(p_{1}\in[-2,2]\), \(p_{2}\in[0,10]\). Obviously,_  
\[\Delta f(x)=g(x)h(x),\Delta g(x)=g(x)m(x) \tag{48}\]  
_where \(g(x)=\left[ \begin{array}{c}0\\ 1\end{array} \right]\), \(h(x)=p_{1}x_{1}cos(x_{2}^{2})\), \(m(x)=p_{2}x_{2}^{2}\). Moreover, \(|h(x)|\leq|2x_{1}|=f_{max}(x)\). Thus, the original robust control problem is converted into calculating optimal control law. For nominal system_  
\[\dot{x}(t)=\left[ \begin{array}{cc}0&6\\ -1&1\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+\left[ \begin{array}{c}0\\ 1\end{array} \right]u, \tag{49}\]  
_find the control function \(u\), such that the performance index_  
\[ \begin{split} J(x,u)&=\int_{0}^{\infty}[f_{max}^{2}(x)+x^{T}x+u ^{T}u]dt\\ &=\int_{0}^{\infty}[5x_{1}^{2}+x_{2}^{2}+u^{2}]dt\end{split}  \tag{50}\]  
_is minimized._  
_In order to solve the robust control problem by using Algorithm 1, 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 \(W_{0}=[-1,5,1.5]^{T}\), and the initial state of system \(x_{0}=[2,-0.5]^{T}\). 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 five iterations, the weight converges to \([1.9645,2.8990,5.4038]\). The robust control law of uncertain system (47) is \(u^{*}=-1.4495x_{1}-5.4038x_{2}\). The convergence process of neural network weight is shown in Figure 1, while the changing process of control signal is shown in Figure 2. The uncertain parameter \(p_{1}\) and \(p_{2}\) in uncertain system (47) take different values, the state trajectories of the closed-loop system are obtained by the robust control law. Figure 3 shows the trajectory of the closed-loop system when \(p_{1}=-2\), \(p_{2}=1\). Figure 4 shows the trajectory of the closed-loop system when \(p_{1}=-1\), \(p_{2}=4\). Figure 5 shows the trajectory of the closed-loop system when \(p_{1}=0\), \(p_{2}=7\). Figure 6 shows the trajectory of the closed-loop system is \(p_{1}=2\), \(p_{2}=10\). From these figures, we can see that the closed-loop system is stable, which shows the effectiveness of the robust control law._  
_In this example, because of the linear property of the nominal system, MATLAB software can be used to solve LQR problem directly. With this method, the optimal control is calculated as \(u^{*}=-1.4496x_{1}-5.4038x_{2}\). It is almost the same as the result of neural network approximation, which shows the validity of Algorithm 1._Figure 1: Neural network weight. Figure 3: Closed loop system trajectory, \(p_{1}=-2\), \(p_{2}=1\). Figure 2: Robust control signal. **Example 2**. : _Consider the following uncertain nonlinear systems_  
Figure 4: Closed loop system trajectory, \(p_{1}=-1,p_{2}=4\). Figure 5: Closed loop system trajectory, \(p_{1}=0,p_{2}=7\). Figure 6: Closed loop system trajectory, \(p_{1}=2,p_{2}=10\). \[\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}{c}0\\ 0.5+p_{2}x_{2}^{2}\end{array}