## 3 Sliding Surface and Control Design
### RADP-Based Sliding Surface Design  
In what follows, the RADP with NN approximation is used to find this SS through obtaining a numerical solution for the above HJB equation, the analytical solution of which is rather difficult to calculate. Based on the universal approximation property of NN, approximate \(J^{*}(x_{1})\) by a NN with only one hidden layer as  
\[J^{*}(x_{1})=W_{c}^{T}\sigma_{c}(x_{1})+\varepsilon_{c}(x_{1}), \tag{14}\]  
where \(W_{c}\in\mathbb{R}^{p}\) is the ideal weight, \(\sigma_{c}(x_{1})\in\mathbb{R}^{p}\) is the activation function, \(p\) is the number of neurons, and \(\varepsilon_{c}(x_{1})\) is the unknown approximation error of this NN. Next, partial derivative of the OCF is written as  
\[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\] \[=(\
abla\sigma_{c}(x_{1}))^{T}W_{c}+\
abla\varepsilon_{c}(x_{1}). \tag{15}\]  
However, the ideal NN weights are unknown, the estimation of NN weight vector \(\hat{W}_{c}\) approximates the OCF. \[\hat{J}^{*}(x_{1})=\hat{W}_{c}^{T}\sigma_{c}(x_{1}). \tag{16}\]  
The partial derivative of the OCF by approximation is  
\[\
abla\hat{J}^{*}(x_{1})=\frac{\partial\hat{J}^{*}(x_{1})}{\partial x_{1}}=( \
abla\sigma_{c}(x_{1}))^{T}\hat{W}_{c}. \tag{17}\]From (15), the optimal control can be given as  
\[x_{2}^{*} = -\frac{1}{2}R^{-1}g_{1}^{T}(x_{1})((\
abla\sigma_{c}(x_{1}))^{T}W_{c }+\
abla\epsilon_{c}(x_{1})). \tag{18}\]  
From (17), the approximate optimal control policy can be expressed as  
\[\hat{x}_{2}^{*}=-\frac{1}{2}R^{-1}g_{1}^{T}(x_{1})(\
abla\sigma_{c}(x_{1}))^{T} \hat{W}_{c}. \tag{19}\]  
For the following derivation, two non-negative matrices are denoted as follows  
\[\mathcal{A}(x_{1})=\
abla\sigma_{c}(x_{1})g_{1}(x_{1})R^{-1}g_{1}^{T}(x_{1})(( \
abla\sigma_{c}(x_{1}))^{T},\]  
\[\mathcal{B}(x_{1})=\
abla\sigma_{c}(x_{1})((\
abla\sigma_{c}(x_{1}))^{T}.\]  
By using the cost function and control policy denoted by the NN, the Hamiltonian function could be given by  
\[H(x_{1},W_{c}) = x_{1}^{T}Qx_{1}+W_{c}^{T}\
abla\sigma_{c}(x_{1})f_{1}(x_{1}) \tag{20}\] \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]0 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]1  
where the term \(e_{CH}\) is the residual error of the NN expression. \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]2 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]3 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]4 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]5 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]6  
The approximate Hamiltonian by the estimated weight is described as  
\[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]7 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]8 \[\
abla J^{*}(x_{1}) =\frac{\partial J^{*}(x_{1})}{\partial x_{1}}\]9  
letting \(e_{c}=\hat{H}(x_{1},\hat{W}_{c})-H(x_{1},W_{c})\), and \(\hat{W}_{c}=W_{c}-\hat{W}_{c}\). Based on (20) and (22), the \(e_{c}\) with \(\hat{W}_{c}\) is formulated by  
\[=(\
abla\sigma_{c}(x_{1}))^{T}W_{c}+\
abla\varepsilon_{c}(x_{1}). \tag{15}\]0 \[=(\
abla\sigma_{c}(x_{1}))^{T}W_{c}+\
abla\varepsilon_{c}(x_{1}). \tag{15}\]1 \[=(\
abla\sigma_{c}(x_{1}))^{T}W_{c}+\
abla\varepsilon_{c}(x_{1}). \tag{15}\]2 \[=(\
abla\sigma_{c}(x_{1}))^{T}W_{c}+\
abla\varepsilon_{c}(x_{1}). \tag{15}\]3  
Now, the next step is to train the CNN and obtain an update law of weights by minimizing the simple cost criterion \(E_{c}=(1/2)e_{c}^{2}\). For avoiding the difficulty of finding an admissible initial control policy, the improved weight updating rule [30] is also selected. The following general assumption [2; 32] are given before the training process and the stability analysis. **Assumption 4**. : _Considering the nominal subsystem (3) with the cost function (5) and substituting the optimal feedback control (11) into the subsystem (3). Let \(J_{s}(x_{1})\) be a continuously differentiable function and simultaneously a Lyapunov function candidate.