# Adaptive Model-Free Coupling Controller Design for Multi-Axis Motion Systems
## 3 Adaptive Model-Free Coupling Controller Design Using RFNN
The adaptive law of RFNN is derived by the well-known back-propagation algorithm, as  
\[W(t_{k+1})=W(t_{k})+\Delta W(t_{k})=W(t_{k})+\eta(-\frac{\partial E(t_{k})}{ \partial W}) \tag{15}\]  
where \(\eta\) and \(W=[\ \theta,\ w_{r}]^{T}\) represent the learning rate and tuning parameters of the RFNN. From Equation (15), the update rule of \(w_{\bar{y}}\) is  
\[w_{ij}(t_{k+1})=w_{ij}(t_{k})-\eta^{w}\frac{\partial E(t_{k})}{\partial w_{ ij}} \tag{16}\]  
where  
\[\frac{\partial E(t_{k})}{\partial w_{ij}}=e(t_{k})\frac{\partial y(t_{k})}{ \partial w_{ij}}=e(k)\frac{\partial y(t_{k})}{\partial u(t_{k})}\frac{ \partial u(t_{k})}{\partial w_{ij}} \tag{17}\]  
where \(y_{u}(t_{k})=\partial y(t_{k})/\partial u(t_{k})\) denotes the system sensitivity [21]. By recursive applications of the chain rule, the error term for each layer is first calculated, and then the corresponding parameters layers are adjusted. Similarly, the update law of \(\theta_{ij}\) is  
\[\theta_{ij}(t_{k+1})=\theta_{ij}(t_{k})-\eta^{\theta}\frac{\partial E(t_{k})} {\partial\theta_{ij}} \tag{18}\]where  
Figure 3:0 Figure 3:1  
The remaining work is to develop the convergence theorem for selecting appropriate learning rates. For adaptive control, the learning rate selection affects the stability result. In the following, we introduce the stability condition by the Lyapunov approach for simplified one-dimensional case. Recall the error cost function as Lyapunov candidate as  
Figure 3:2  
The error difference due to the learning can be represented by [21; 22; 23]  
Figure 3:3  
thus,  
\[\Delta W\equiv-[ \begin{array}{cc}\eta^{w}&\eta^{\theta}\end{array} ]e(t_{k}) \left[ \begin{array}{cc}\frac{\partial e(t_{k})}{\partial\eta^{\theta}}\\ \frac{\partial e(t_{k})}{\partial\theta}\end{array} \right]=[ \begin{array}{ cc}\eta^{w}&\eta^{\theta}\end{array} ]e(t_{k})\left[ \begin{array}{ cc}y_{u}(t_{k})\frac{\partial O^{(4)}(t_{k})}{\partial\eta}\\ y_{u}(t_{k})\frac{\partial O^{(4)}(t_{k})}{\partial\theta}\end{array} \right] \tag{22}\]  
Finally, we summarize the stability theorem, as follows. **Theorem 1**. : _Consider the adaptive control problem of (4), with control scheme Figure 4. Let \(\eta^{w}\) and \(\eta^{\theta}\)be the learning rates of parameters \(w\) and \(\theta\), respectively. The asymptotically convergence of closed-loop system is guaranteed if_  
\[0<\left\{\eta^{w}\|y_{u}(t_{k})\cdot\frac{\partial O^{(4)}(t_{k})}{\partial w }\|^{2}+\eta^{\theta}\|y_{u}(t_{k})\cdot\frac{\partial O^{(4)}(t_{k})}{ \partial w}\|^{2}\right\}<2 \tag{23}\]  
This guarantees the stability of closed-loop system. Proof. : As in Equations (11) and (12), our goal is to have the stability condition that Equation (12) satisfies. Herein, the discrete-time approach is utilized to finish it. Therefore, the change of the Lyapunov candidate is  
We establish time varying contour error coordinate from machine tool coordinate through transform matrix \(F(t)\), which includes normal vector, tangent vector, and bi-normal in order to reduce contour error; therefore, contour error is defined as error of normal vector, tangent vector, and bi-normal; the concept map is shown in Figure 2. Recall Section 2, the instant contour error is defined as  
\[e(t)=F^{T}(t)x(t)-F^{T}(t)x_{d}(t). \tag{26}\]  
herein,  
\[\|e(t)\|=\|F^{T}(t)x(t)-F^{T}(t)x_{d}(t)\|. \tag{27}\]  
Figure 5 introduces the illustration of on-line contour error. From Figure 5a, we can observe that there exists time delay problem in control, such that \(F(t)\) is not same in current actual trajectory and current desired trajectory, however, Equation (26) adopts the same one to treat it. In other words, there exists error between calculation and actual contour errors, and this is unavoidable.