# A Robust Intelligent Controller for Autonomous Ground Vehicle Longitudinal Dynamics
## 3 Controller Development
### Robust Adaptive RBNN-SMC
#### 3.3.3 Robust Law
This leads to the differentiation of Equation (20) as:  
\[\Delta\hat{\mathfrak{M}}_{j}(\mathcal{Z})=\Delta\hat{\omega}_{j}\mathcal{H}_{ j}(\mathcal{Z})+\hat{\omega}_{j}\frac{\partial\mathcal{H}_{j}}{\partial\hat{ \mathfrak{h}}_{j}}\Delta\hat{\mathfrak{h}}_{j}+\hat{\omega}_{j}\frac{\partial \mathcal{H}_{j}}{\partial\hat{c}_{j}}\Delta\hat{c}_{j}+\mathbb{J}_{2j} \tag{23}\]  
The linearization neglects the high-order terms of the [PERSON] expansion of \(\hat{\mathfrak{M}}(\mathcal{Z})\), which causes a \(\mathbb{J}_{2}\) error quantity of \(\mathbb{J}_{2}=\sum\limits_{j=1}^{n}\mathbb{J}_{2j}\), which is assumed to be constrained by the positive constant \(\mathbb{J}_{n}\) as \(|\mathbb{J}_{2}|<\mathbb{J}_{2n}\)[46]. The vectors of the partial derivative of \(\mathcal{H}\) with respect to \(\hat{\mathfrak{h}}\) and \(\hat{c}\), \(\frac{\partial\mathcal{H}_{j}}{\partial\hat{\mathfrak{h}}_{j}}\) and \(\frac{\partial\mathcal{H}_{j}}{\partial\hat{c}_{j}}\), are expressed as  
\[\mathcal{H}^{\prime}_{\hat{c}}= \begin{bmatrix}\frac{\partial\mathcal{H}_{1}}{ \partial\hat{c}_{1}}&0&\dots&0\\ 0&\frac{\partial\mathcal{H}_{2}}{\partial\hat{c}_{2}}&\dots&0\\ \vdots&0&\ddots&\vdots\\ 0&0&\dots&\frac{\partial\mathcal{H}_{n}}{\partial\hat{c}_{n}}\end{bmatrix}  \in\mathbb{R}^{n\times n}\]  
\[\mathcal{H}^{\prime}_{\hat{\mathfrak{h}}}= \begin{bmatrix}\frac{\partial \mathcal{H}_{1}}{\partial\hat{\mathfrak{h}}_{1}}&0&\dots&0\\ 0&\frac{\partial\mathcal{H}_{2}}{\partial\hat{\mathfrak{h}}_{2}}&\dots&0\\ \vdots&0&\ddots&\vdots\\ 0&0&\dots&\frac{\partial\mathcal{H}_{n}}{\partial\hat{\mathfrak{h}}_{n}}\end{bmatrix}  \in\mathbb{R}^{n\times n}\]  
\(\widetilde{\omega}\), \(\widetilde{\mathfrak{h}}\), and \(\widetilde{c}\) denote the estimation error as  
\[\left\{ \begin{array}{l}\widetilde{\mathfrak{M}}=\mathfrak{M}^{*}-\mathfrak{ M}\\ \widetilde{\omega}=\omega^{*}-\hat{\omega}\\ \widetilde{\mathfrak{h}}=\mathfrak{h}^{*}-\hat{\mathfrak{h}}\\ \widetilde{c}=c^{*}-\hat{c}\end{array} \right. \tag{24}\]  
Subsequently, \(\Delta\widetilde{\mathfrak{M}}\) can be written as  
\[\Delta\widetilde{\mathfrak{M}}=\mathfrak{M}^{*}-\mathfrak{M}+\mathbb{J}_{1}= \Delta\mathfrak{M}+\mathbb{J}_{1} \tag{25}\]  
where \(\Delta\mathfrak{M}=\sum\limits_{j=1}^{n}\Delta\mathfrak{M}(\mathcal{Z})\). Then, based on Equations (23) and (25), we can obtain the following:  
\[\widetilde{\mathfrak{M}}(\mathcal{Z})=\widetilde{\omega}^{T}\mathcal{H}( \mathcal{Z})+\omega^{T}\mathcal{HUnder the condition \(|\mathfrak{J}_{1}+\mathfrak{J}_{2}|\leq o\), the parameters \(\hat{\omega}\), \(\hat{\mathfrak{h}}\), and \(\hat{\mathfrak{c}}\) can be adjusted by following adaptive laws:  
\[ \begin{cases}\hat{\omega}&=-a_{1}|\mathcal{Z}|H(\mathcal{Z})\\ \hat{\mathfrak{h}}&=-a_{2}|\mathcal{Z}|\mathcal{H}_{\hat{\mathfrak{h}}}^{ \prime}\omega\\ \hat{\mathfrak{c}}&=-a_{3}|\mathcal{Z}|\mathcal{H}_{\hat{\mathfrak{c}}}^{ \prime}\omega\end{cases}\begin{cases}\hat{\omega}_{j}&=-a_{1}|\mathcal{Z}| \mathcal{H}_{\hat{\mathfrak{h}}}(\mathcal{Z})\\ \hat{\mathfrak{h}}_{j}&=-a_{2}|\mathcal{Z}|\frac{\partial\hat{H}_{\hat{ \mathfrak{h}}}}{\partial\hat{\mathfrak{h}}_{j}}\omega_{j}\\ \hat{\mathfrak{c}}&=-a_{3}|\mathcal{Z}|\frac{\partial\hat{H}_{\hat{ \mathfrak{h}}}}{\partial\hat{\mathfrak{c}}_{j}}\omega_{j}\end{cases}  \tag{27}\]  
where \(a_{k},k=\{1,2,3\}\) are non-zero-positive constants. **Proposition 1**. : _Considering the system expressed by Equation (13), the lumped uncertainties are coherent with the inequality \(\mathfrak{O}(x,t)\leq\mathfrak{M}\)._  
If the sliding surface is set as  
\[\mathcal{Z}(t)=\mathfrak{p}e+\dot{e} \tag{28}\]  
the control law is defined as  
\[\mathfrak{U}=\mathfrak{U}_{eq}+\mathfrak{U}_{sw}+\mathfrak{U}_{bnn} \tag{29}\]  
with  
\[  
From Equations (25) and (26), we can obtain the following:  
\[