## 3 Adaptive Robust Control Design
### Unknown Dynamic Estimators  
Consider an arbitrary unknown smooth multivariate function \(f(\mathbf{x})\) with \(\mathbf{x}\) as the input vector. This function can be represented in RBF NN form as [39]  
\[f(\mathbf{x})=\mathbf{W}_{f}{}^{T}\mathbf{h}_{f}(\mathbf{x})+\varepsilon_{f} \tag{29}\]  
where \(\mathbf{W}_{f}\) is the ideal neural network weights, \(\mathbf{h}_{f}=[h_{j}]^{T}\) with \(h_{j}\) represents the radial-basis function output of the \(j\)th node in the hidden layer, and \(\varepsilon_{f}\) denotes the function approximation error. **Assumption 4**. : _The weight value and the approximation error of NN are bounded as \(\|\mathbf{W}_{f}\|\leq W_{fM}\) and \(|\varepsilon_{f}|\leq\varepsilon_{fM}\) where \(W_{fM}\) and \(\varepsilon_{fM}\) are positive constants._  
The radial-basis function is usually chosen as the Gaussian function, which has the form [40]  
\[h_{j}=\exp\left(-\frac{\left\|\mathbf{x}-\mathbf{c}_{j}\right\|^{2}}{2b_{j}^{ 2}}\right) \tag{30}\]  
where \(\mathbf{c}_{j}\) is the center vector of the \(j\)th node in the hidden layer, which has the same dimension as the input vector \(\mathbf{x}\), and \(\|\bullet\|\) represents the Euclid distance of a vector. The estimation of the function \(f(\mathbf{x})\) is denoted by \(\hat{f}(\mathbf{x})\), which is the output of RBF NN as  
\[\hat{f}(\mathbf{x})=\hat{\mathbf{W}}_{f}^{T}\mathbf{h}_{f}(\mathbf{x}) \tag{31}\]  
In (11), since the system parameters are unknown, the functions \(f_{1}\), \(\hat{f}_{1}\), \(f_{2}\), and \(g_{2}\) are unknown smooth functions that are approximated by employing RBF NN-based approximators as  
\[ \begin{split} f_{1}&=\mathbf{W}_{f_{1}}^{T} \mathbf{h}_{f_{1}}(\mathbf{x})+\varepsilon_{f_{1}}\\ \hat{f}_{1}&=\mathbf{W}_{f_{1}}^{T}\mathbf{h}_{df_{1 }}(\mathbf{x})+\varepsilon_{df_{1}}\\ f_{2}&=\mathbf{W}_{f_{2}}^{T}\mathbf{h}_{f_{2}}( \mathbf{x})+\varepsilon_{f_{2}}\\ g_{2}&=\mathbf{W}_{g_{2}}^{T}\mathbf{h}_{g_{2}}( \boldsymbol{\Theta})+\varepsilon_{g_{2}}\end{split}  \tag{32}\]  
where the weight values \(\|\mathbf{W}_{f_{1}}\|\), \(\|\mathbf{W}_{df_{1}}\|\), \(\|\mathbf{W}_{g_{2}}\|\), \(\|\mathbf{W}_{g_{2}}\|\) and approximation errors \(\varepsilon_{f_{1}}\), \(\varepsilon_{df_{1}}\), \(\varepsilon_{fg_{2}}\) are bounded by \(W_{f_{1}}\), \(W_{df_{1}}\), \(W_{f_{2}}\), \(W_{g_{2}M}\) and \(\varepsilon_{f_{1}M}\), \(\varepsilon_{df_{1}M}\), \(\varepsilon_{f_{2}M}\), \(\varepsilon_{g_{2}M}\), respectively; and the input vector \(\boldsymbol{\Theta}=[x_{1},x_{2},x_{3},u]^{T}\). The estimates of \(f_{1}\), \(\hat{f}_{1}\), \(f_{2}\), and \(g_{2}\) are determined as  
\[ \begin{split}\hat{f}_{1}&=\hat{\mathbf{W}}_{f_{1}}^ {T}\mathbf{h}_{f_{1}}(\mathbf{x})\\ \hat{f}_{1}&=\hat{\mathbf{W}}_{df_{1}}^{T}\mathbf{h} _{df_{1}}(\mathbf{x})\\ \hat{f}_{2}&=\hat{\mathbf{W}}_{f_{2}}^{T}\mathbf{h} _{f_{2}}(\mathbf{x})\\ \hat{g}_{2}&=\hat{\mathbf{W}}_{g_{2}}^{T}\mathbf{h} _{g_{2}}(\boldsymbol{\Theta})\end{split}  \tag{33}\]  
The approximation errors are determined by  
\[ \begin{split}\tilde{f}_{1}&\overset{\Delta}{=}f_{1} -\hat{f}_{1}=\tilde{\mathbf{W}}_{f_{1}}^{T}\mathbf{h}_{f_{1}}+\varepsilon_{f_{ 1}}\\ \tilde{f}_{1}&\overset{\Delta}{=}\hat{f}_{1}-\hat{f}_ {1}=\tilde{\mathbf{W}}_{df_{1}}^{T}\mathbf{h}_{df_{1}}+\varepsilon_{df_{1}}\\ \tilde{f}_{2}&\overset{\Delta}{=}f_{2}-\hat{f}_{2}= \tilde{\mathbf{W}}_{f_{2}}^{T}\mathbf{h}_{f_{2}}+\varepsilon_{f_{2}}\\ \tilde{g}_{2}&\overset{\Delta}{=}g_{2}-\hat{g}_{2}= \tilde{\mathbf{W}}_{g_{2}}^{T}\mathbf{h}_{g_{2}}+\varepsilon_{g_{2}}\end{split}  \tag{34}\]  
where  
\[ \begin{split}\tilde{\mathbf{W}}_{f_{1}}&=\mathbf{W}_{ f_{1}}-\hat{\mathbf{W}}_{f_{1}}\\ \tilde{\mathbf{W}}_{df_{1}}&=\mathbf{W}_{df_{1}}- \hat{\mathbf{W}}_{df_{1}}\\ \tilde{\mathbf{W}}_{f_{2}}&=\mathbf{W}_{f_{2}}- \hat{\mathbf{W}}_{f_{2}}\\ \tilde{\mathbf{W}}_{g_{2}}&=\mathbf{W}_{g_{2}}- \hat{\mathbf{W}}_{g_{2}}\end{split}  \tag{35}\]  
The adaption mechanisms are determined as  
\[