# Neural Robust Control for a Mobile Agent Leader-Follower System
## 3 Neuro-Robust Control
### Controller Design  
Applying the results presented in the previous section, we obtain the relative degree matrix given by  
\[A(x)=\left[ \begin{matrix}\frac{m_{1}}{R(m_{1}^{2}-m_{2}^{2})}&-\frac{m_{2}}{R( m_{1}^{2}-m_{2}^{2})}\\ -\frac{m_{2}}{R(m_{1}^{2}-m_{2}^{2})}&\frac{m_{1}}{R(m_{1}^{2}-m_{2}^{2})}\\ \end{matrix} \right], \tag{18}\]  
where the \(rank(A(x))=2\) and \(det(A(x))\
eq 0\), and  
\[b(x)=\left[ \begin{matrix}\frac{m_{1}\omega_{L}(\omega_{L}-\omega_{R})}{m_{1}^ {2}-m_{2}^{2}}+\frac{m_{2}\omega_{R}(\omega_{L}-\omega_{R})}{m_{1}^{2}-m_{2}^ {2}}\\ -\frac{m_{2}\omega_{L}(\omega_{L}-\omega_{R})}{m_{1}^{2}-m_{2}^{2}}-\frac{m_{ 1}\omega_{R}(\omega_{L}-\omega_{R})}{m_{1}^{2}-m_{2}^{2}}\\ \end{matrix} \right]. \tag{19}\]  
Then, we have the next local coordinate transformation as described in Section 2; Equations (3)-(5):  
\[ \begin{split} z_{1}^{1}&=h_{1}(x)\\ z_{2}^{1}&=b_{1}+a_{11}u_{1}+a_{12}u_{2}\\ z_{1}^{2}&=h_{2}(x)\\ z_{2}^{2}&=b_{2}+a_{21}u_{1}+a_{22}u_{2},\\ \end{split}  \tag{20}\]  
where the linearized states are defined as \(z_{1}^{1}=w_{R}\) and\(z_{1}^{2}=w_{L}\) in Equation (16). Finally, the transformed system is as follows:  
\[ \begin{split}\dot{z}_{1}^{1}&=\underbrace{\frac{m_{ 1}vz_{1}^{2}(z_{1}^{2}-z_{1}^{1})}{m_{1}^{2}-m_{2}^{2}}+\frac{m_{2}vz_{1}^{1}( z_{1}^{2}-z_{1}^{1})}{m_{1}^{2}-m_{2}^{2}}}_{b_{1}}\\ &\underbrace{+\frac{m_{1}}{R(m_{1}^{2}-m_{2}^{2})}}_{a_{11}}u_{1} -\underbrace{\frac{m_{2}}{R(m_{1}^{2}-m_{2}^{2})}}_{b_{12}}u_{2},\\ \end{split}  \tag{21}\]  
Figure 2: Differential mobile robot model [27]. \[\dot{z}_{1}^{2}= \underbrace{-\frac{m_{2}vz_{1}^{2}(z_{1}^{2}-z_{1}^{1})}{m_{1}^{2}-m_ {2}^{2}}-\frac{m_{1}vz_{1}^{1}(z_{1}^{2}-z_{1}^{1})}{m_{1}^{2}-m_{2}^{2}}}_{b_{2}} \tag{22}\] \[\underbrace{-\frac{m_{2}}{R(m_{1}^{2}-m_{2}^{2})}}_{a_{21}}u_{1} \underbrace{+\frac{m_{1}}{R(m_{1}^{2}-m_{2}^{2})}u_{2}}_{a_{22}}u_{2}.\]  
The control law for asymptotic trajectory tracking is given by Equation (6), where \(y_{R}^{(r)}\) represents the \(r\)-th derivative of the reference angular velocity for each wheel. The nonlinear functions denoted as \(b_{1}(z)\) and \(b_{2}(z)\) in (21) and (22) are uncertain and need to be estimated for the controller to compensate them. An RHONN is proposed to estimate these functions. The system takes the following form:  
\[\dot{z}_{1}^{1}= \overbrace{b_{1}(z)}^{estimated}+a_{11}(z)u_{1}+a_{12}(z)u_{2}, \tag{23}\] \[\dot{z}_{1}^{2}= \overbrace{b_{2}(z)}^{estimated}+a_{21}(z)u_{1}+a_{22}(z)u_{2}.\]  
In Equations (21) and (22), the terms \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\) do not depend on the states and are therefore constant. Additionally, note that these terms depend on the system parameters, but it is sufficient to know their signs and a nominal value. In actuality, there is no need to estimate them; it is sufficient for the controller to know their nominal value. Hence, the matrix \(A\) takes the following constant form:  
\[A= \begin{bmatrix}\delta_{1}&-\delta_{2}\\ -\delta_{3}&\delta_{4}\end{bmatrix} , \tag{24}\]  
where all \(\delta_{i}\) are positive. Then, the following control law using the estimated parameters by RHONN is obtained:  
\[u_{1}=-\frac{\delta_{2}(\hat{b_{2}}-\omega_{Ld}^{(r)}+e\omega_{L}k_{2})}{ \delta_{1}\delta_{4}-\delta_{2}\delta_{3}}-\frac{\delta_{4}(\hat{b_{1}}- \omega_{Rd}^{(r)}+e\omega_{R}k_{1})}{\delta_{1}\delta_{4}-\delta_{2}\delta_{3}}, \tag{25}\]  
\[u_{2}=-\frac{\delta_{1}(\hat{b_{2}}-\omega_{Ld}^{(r)}+e\omega_{L}k_{2})}{ \delta_{1}\delta_{4}-\delta_{2}\delta_{3}}-\frac{\delta_{3}(\hat{b_{1}}- \omega_{Rd}^{(r)}+e\omega_{R}k_{1})}{\delta_{1}\delta_{4}-\delta_{2}\delta_{3 }}, \tag{26}\]  
where \(e_{\omega_{L}}=\omega_{L}-\omega_{Ld}\) and \(e_{\omega_{R}}=\omega_{R}-\omega_{Rd}\). The previous process was only described for a single mobile agent; for the leader-follower system, it is necessary to duplicate the procedure for the follower agent. By estimating the parameter \(b\), we eliminate certain restrictions of the exact linearization method, such as the requirement of knowing the complete state.