## 2 Main Results
### Normal Model
: _[_40_]_ _The first order sliding mode differentiator is expressed as_  
\[\left\{ \begin{array}{l}\dot{\omega}_{1}=\gamma_{0}=-\epsilon_{1}|\omega_{1} -f(t)|^{\frac{1}{2}}sign(\omega_{1}-f(t))+\omega_{2}\\ \dot{\omega}_{2}=\epsilon_{2}sign(\omega_{2}-\gamma_{0})\end{array} \right. \tag{11}\]  
_where \(\omega_{1},\gamma_{0},\text{and }\omega_{2}\) denote states of the first order sliding mode differentiator, \(\epsilon_{1}\) and \(\epsilon_{2}\) are the designed parameters, and \(f(t)\) is an unknown function. Consequently, \(\gamma_{0}\) can approximate the differential term \(f(t)\) with any arbitrary precision if the initial errors \(\omega_{1}(t_{0})-f(t_{0}),\omega_{2}(t_{0})-f(t_{0})\) are bounded._  
**Lemma 5**. : _[_41_]_ _If there exists a continuously positive function \(V(x,t):\mathbf{R}^{n}\times\mathbf{R}^{+}\rightarrow\mathbf{R}^{+}\), with two scalars \(c_{1}>0\) and \(c_{2}\geq 0\) and where \(\mu_{1}\) and \(\mu_{2}\) are class \(K_{\infty}\)-functions, then \(V(x,t)\) satisfies Equations (12) and (13)_  
\[\mu_{1}(|x|)\leq V(x,t)\leq\mu_{2}(|x|) \tag{12}\]  
\[\dot{V}\leq-c_{1}V+c_{2} \tag{13}\]  
_with regard to \(x\in\mathbf{R}^{n}\) and \(t>0\). Then, there is the following solution for arbitrarily initial value \(x(0)\in\mathbf{R}^{n}\) and satisfies_  
\[V(x,t)\leq V(0)e^{-c_{1}t}+\frac{c_{2}}{c_{1}},\quad\forall t>0 \tag{14}\]  
**Lemma 6**. : _[_42_]_ _Let the unknown function \(f(Z)\) be defined over a compact set \(\Omega_{2}\); then for any approximation accuracy \(l^{*}>0\), there exists an RBFNN such that_  
\[f(Z)=W^{*T}\Psi(Z)+l(Z) \tag{15}\]  
_where \(Z\in\Omega_{2}\subset\mathbf{R}^{n}\) is the input vector of the neural networks with \(n\) being the input dimension. \(W=[w_{1},w_{2},\cdots,w_{m}]^{T}\in\mathbf{R}^{m}\) is the weight vector with the neural network node number \(m\) and \(W^{*}\) is the ideal constant weight vector. \(l(Z)\) denotes the approximation error with \(\|l(Z)\|\leq l^{*}\). \(\Psi(Z)\) is the smooth basis vector \(\Psi(Z)=[\Psi_{1}(Z),\Psi_{2}(Z),\cdots,\Psi_{m}(Z)]^{\mathsf{T}}\in\mathbf{R}^{m}\) with \(\Psi_{i}(Z)\) selected from the commonly used Gaussian functions_  
\[\Psi_{i}(Z)=exp\bigg{[}\frac{-(Z-c_{i})^{\mathsf{T}}(Z-c_{i})}{b_{i}^{2}} \bigg{]},\ \ \ i=1,2,\cdots,m \tag{16}\]  
_where \(c_{i}=[c_{i1},c_{i2},\cdots,c_{in}]^{\mathsf{T}}\) and \(b_{i}\) are the center and width of the Gaussian functions, respectively. The ideal constant weight vector is defined such as_  
\[W^{*}:=\arg\min_{W\in\mathbf{R}^{m}}\Bigg{\{}\sup_{Z\in\Omega_{z}}|f(Z)-\hat{W}^{ \mathsf{T}}\Psi(Z)|)\Bigg{\}} \tag{17}\]  
_It's noteworthy that the ideal weight vector \(W^{*}\) is unknown and its elements need to be estimated by designed adaptive law. Nevertheless, in this paper, minimum parameter learning theory is used instead of a direct estimate with the variable \(\theta=\hat{G}^{-1}\|W^{*}\|^{2}\)._  
**Remark 2**. : _The ideal weight \(W^{*}\in\mathbf{R}^{m}\) realizes \(m\) unknown elements to be estimated. More estimators will result in a good many of adjustment parameters, thus directly increasing the computational burden. By updating the estimation values of the norm for the unknown neural network weight vectors but not their weights only the unknown parameter \(\theta\) needs to be estimated in this paper, hence the computational complexity can be reduced substantially._  
For the system of Equation (4), it is necessary for the system transformation and the reference trajectory to satisfy the following assumptions:  
**Assumption 2**. : _There exist two positive constants \(\beta_{1}\) and \(\beta_{2}\) such that the desired reference trajectory signal \(y_{d}\) and its time derivative \(\dot{y}_{d}\) have corresponding boundedness, i.e., \(|y_{d}|\leq\beta_{1}\), \(|\dot{y}_{d}|<\beta_{2}\)._  
**Assumption 3**. : _For a time-varying unknown compound disturbance \(d\), there exists an unknown positive constant \(d_{m}\), i.e., \(|d|<d_{m}\)._  
**Assumption 4**.