## 3 Multidimensional Taylor Network Adaptive Robust Controller
### Robust Control  
It can be seen from the system expression that the system is affected by uncertain parameters, namely \(m,a,b\) and \(d(t)\). In order to reduce the impact of parameter uncertainty on the system and improve the performance by using parameter adaptive method, the following assumptions are made. **Assumption 1**. : _The uncertain parameters of the system are bounded, that is,_  
\[\left\{ \begin{array}{l}m_{\min}<m<m_{\max}\\ a_{\min}<a<a_{\max}\\ b_{\min}<b<b_{\max}\end{array} \right. \tag{2}\]  
_where \(m_{\min},a_{\min}\) and \(b_{\min}\) are the lower bounds of parameters and \(m_{\max}\), \(a_{\max}\) and \(b_{\max}\) are the upper bounds of parameters._  
_If \([\hat{m},\hat{a},\hat{b}]\) is the estimated value of the corresponding uncertain parameter \([m,a,b]\) and \([\widehat{m},\widetilde{a},\widetilde{b}]\)is the corresponding measurement error, that is, \([\widehat{m},\widetilde{a},\widetilde{b}]=[m-\hat{m},a-\hat{a},b-\hat{b}]\), the following adaptive law can be used to estimate the uncertain parameters:_  
\[[\hat{m},\hat{a},\hat{b}]^{\mathrm{T}}=-\Gamma\cdot\Omega(x_{2},h,t) \tag{3}\]  
_where \(\Gamma\) is a diagonal matrix and \(\Omega(x_{2},h,t)\) is an adaptive function; the specific composition will be given later. After the above adaptive estimation, the estimated value is also bounded, that is_  
\[\left\{ \begin{array}{l}m_{\min}<\hat{m}<m_{\max}\\ a_{\min}<\hat{a}<a_{\max}\\ b_{\min}<\hat{b}<b_{\max}\end{array} \right. \tag{4}\]  
_Set \(y_{d}(t)\) as the expected output of the system, then the system error can be obtained_  
\[e(t)=y(t)-y_{d}(t) \tag{5}\]  
_Define error function r(t)_  
\[r(t)=\dot{e}(t)+k_{1}e(t) \tag{6}\]  
_where \(k_{1}\) is the positive feedback gain. Further we can obtain_  
\[r(t)=\dot{e}(t)+k_{1}e(t)=\dot{y}(t)-\dot{y}_{d}(t)+k_{1}e(t)=\dot{x}_{1}(t)- \dot{y}_{d}(t)+k_{1}e(t)=x_{2}(t)-\dot{y}_{d}(t)+k_{1}e(t) \tag{7}\]  
_For writing convenience, set \(\widehat{x}_{2}(t)=\dot{y}_{d}(t)-k_{1}e(t)\), then the above expression can be rewritten as_  
\[r(t)=x_{2}(t)-\widehat{x}_{2}(t) \tag{8}\]  
_In summary, the original Equation (1) can be rewritten as_  
\[ \begin{array}{l}m\dot{r}(t)=g(u(t))-m\dot{\widehat{x}}_{2}(t)-ax_{2}(t)-bh (x_{2}(t))+d(t)\\ =g(u(t))+[-\widehat{x}_{2}(t),-x_{2}(t),-h(x_{2}(t))]\cdot[m,a,b]^{\mathrm{ T}}+d(t)\end{array}  \tag{9}\]  
_Let \(\Psi=[-\dot{\widehat{x}}_{2}(t),-x_{2}(t),-h(x_{2}(t))]\); then the above formula can be simplified to_  
\[m\dot{r}(t)=g(u(t))+\Psi\cdot[m,a,b]^{\mathrm{T}}+d(t) \tag{10}\]  
_where \(\dot{\widehat{x}}_{2}(t)=\ddot{y}_{d}(t)-k_{1}\dot{e}(t)\)._  
_Through Equation (8), the following control law can be given [33]:_  
\[g(u(t))=g_{a}(u(t))+g_{S}(u(t))+g_{\mathrm{MTN}}(u(t)) \tag{11}\]_The first part \(g_{a}(u(t))\) in the above formula is as follows:_  
\[g_{a}(u(t))=-\Psi\cdot[\hat{m},\hat{a},\hat{b}]^{\text{T}} \tag{12}\]  
_The role of \(g_{a}(u(t))\) is as the main component to realize the adaptive tracking of the system._  
_The second part \(g_{s}(u(t))\) in Formula (11) is as follows:_  
\[g_{s}(u(t))=-k_{2}\tau(t) \tag{13}\]  
_where \(k_{2}>0\) is the feedback gain parameter. This part is linear proportional feedback, mainly used to ensure the stability of the system._  
_The third part \(g_{\text{MTN}}(u(t))\) in Equation (11) is the compensation output of the multidimensional Taylor network, and the specific content will be described in detail in the next section._  
_Substitute Equation (11) into (10) and arrange to obtain_  
\[\dot{m}(t)+k_{2}r(t)=g_{\text{MTN}}(u(t))-\Psi\cdot[\widehat{m},\widehat{a}, \widehat{b}]^{\text{T}}+d(t) \tag{14}\]  
According to Assumption 1, namely the boundedness of disturbance and the boundedness of Equation (4), there exists an ideal \(g_{\text{MTN}}(u(t))\) optimal value denoted as \(g^{*}(u(t))\) that satisfies the following conditions  
\[\left\{