# A Robust Intelligent Controller for Autonomous Ground Vehicle Longitudinal Dynamics
## 3 Controller Development
### Robust Adaptive RBNN-SMC
#### 3.3.3 Robust Law  
To accommodate disturbances and parametric uncertainties, a robust law is given in terms of the upper limit of lumped uncertainty, as follows:  
\[\mathfrak{U}_{r}=\frac{1}{\mathfrak{S}(x,t)}(-q\mathcal{Z}-\mathcal{K}sign( \mathcal{Z})) \tag{17}\]  
where \(q\) and \(\mathcal{K}\) indicate the switching gains and \(\mathcal{K}\) is chosen as [43; 44]\(\mathcal{K}=\mathfrak{M}+o\), where \(\mathfrak{M}\) is the upper limit of the lumped uncertainty, given in [43], and \(o\) is a positive constant. The idea of designing a control law considering the upper limits of disturbances and uncertainties is insufficient to guarantee high performance against real non-static disturbances. To solve this problem, a neuronal architecture was adopted to approximate the upper limit (\(\mathfrak{M}\)) depending on the variation in these critical factors. Hence, the subdivision of the robust law (\(\mathfrak{U}_{r}\)) into two terms, \(\mathfrak{U}_{sw}\) and \(\mathfrak{U}_{bmn}\), was as follows:  
\[\left\{ \begin{array}{l}\mathfrak{U}_{r}=\mathfrak{U}_{sw}+\mathfrak{U}_{bmn }\\ \mathfrak{U}_{sw}=\frac{1}{\mathfrak{S}(x,t)}(-q\mathcal{Z}-osign(\mathcal{Z}) )\\ \mathfrak{U}_{bmn}=\frac{-\mathfrak{M}sign(\mathcal{Z})}{\mathfrak{S}(x,t)} \end{array} \right. \tag{18}\]  
As it is expressed in the Equation (18), the term \(\mathfrak{U}_{bmn}\) contains the approximation of the lumped uncertainty upper bound by the RBNN, as described in Figure 4.  
\(\hat{\mathfrak{M}}\) is the output of RBNN and presents the estimation of \(\mathfrak{M}\) according to the expression  
\[\hat{\mathfrak{M}}(\mathcal{Z},\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{        \mathfrak{      }}}}}}}}}}}}}} )\mathcal{H}(\mathcal{Z}) \tag{19}\]  
where \(\mathcal{H}\) is the Gaussian activation function constituted by the radial basis function defined as  
\[\mathcal{H}_{j}(\mathcal{Z})=e^{\left(-\frac{(\mathcal{Z}-\mathfrak{\mathfrak{ \mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{ \mathfrak{        \mathfrak{     }}}}}}}}}}}}} {j}})\right. \tag{20}\]  
The input of the RBNN structure is the sliding variable \(\mathcal{Z}\). Each hidden node has a center parameter vector and an input vector of the same dimension. On the other hand, \(\mathfrak{M}\) can be expressed using RBNN's ideal parameters \(\omega^{*}\), \(\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{\mathfrak{ \mathfrak{\mathfrak{ \mathfrak{     \mathfrak{      }}}}}}}}}}}}}}\), and \(c^{*}\), given as [42; 45]:  
\[\mathfrak{M}=\mathfrak{M}^{*}+\mathfrak{J}_{1}=\omega^{*T}\mathcal{H}^{*}( \mathcal{Z})+\mathfrak{J}_{2} \tag{21}\]  
Figure 4: The RBNN structure. \(\mathbb{J}_{1}\) denotes the approximation error of the RBNN, which is assumed to be limited by the non-zero positive constant \(\mathbb{J}_{n}\) as \(|\mathbb{J}_{1}|<\mathbb{J}_{n}\)[42; 45]. The estimated weight vector (\(\hat{\omega}\)), the estimated center vector (\(\hat{\mathfrak{h}}\)), and the estimated width vector (\(\hat{c}\)) are defined as  
\[\hat{\omega}= \begin{bmatrix}\hat{\omega}_{1}\\ \hat{\omega}_{2}\\..\\ \hat{\omega}_{n}\end{bmatrix} ;\mathcal{H}= \begin{bmatrix}\mathcal{H}_{1}\\ \mathcal{H}_{2}\\..\\ \mathcal{H}_{n}\end{bmatrix} ;\hat{c}= \begin{bmatrix}\hat{c}_{1}\\ \hat{c}_{2}\\..\\ \hat{c}_{n}\end{bmatrix}  \tag{22}\]  
where \(n\) is the hidden-layer node number. Obtaining the expression allows the adaptation of the parameters \(\hat{\omega}\), \(\hat{\mathfrak{h}}\), and \(\hat{c}\), which provides the adjustment of \(\mathfrak{M}\) according to external factors and uncertainties. To achieve this, the Gaussian function is linearized as in [46].