# Adaptive Model-Free Coupling Controller Design for Multi-Axis Motion Systems
## 3 Adaptive Model-Free Coupling Controller Design Using RFNN  
In this study, our objective is to develop an adaptive model-free coupling controller for multi-axis systems to minimize the contour error, based on the time varying coordinate. Therefore, we transform the system to contour error coordinate by matrix \(F(t)\). The corresponding adaptive control scheme is introduced in Figure 4, the RFNN generates the suitable control signal to drive system output to follows desired trajectory, the input of RFNN is \(s\), \(s=\dot{e}+\alpha e\), \(\alpha>0\), and the RFNN is updated by the stable learning mechanism. The detailed description of learning mechanism for selecting stable learning rate is introduced, as follows. Recall system (4) and define \(e=x_{f}-x_{fd},s=\dot{e}+\alpha e\), the dynamic model of multi-axis system in time-varying coordinate space is  
\[\overline{M}\dot{x}_{f}+\overline{V}_{m}\dot{x}_{f}+\overline{B}\dot{x}_{f}+Nx _{f}=\overline{F}_{c}-\overline{F}_{r}-\overline{F}_{f} \tag{8}\]  
There exists an ideal stabilizing controller by nonlinear cancellation and linear state feedback  
\[\overline{F}_{c}=\overline{M}\dot{s}+(\overline{B}+\overline{V}_{m})s+ \overline{M}(\dot{x}_{fd}-\alpha\dot{e})+\overline{B}(\dot{x}_{fd}-\alpha e)+ \overline{V}_{m}(\dot{x}_{fd}-\alpha e)+N(e+x_{fd})+\overline{F}_{r}+\overline {F}_{f}-\dot{e}-\beta e,\;\beta>0 \tag{9}\]  
such that the following closed-loop system is stable  
\[ \begin{array}{l}\overline{M}\dot{x}_{f}+\overline{V}_{m}\dot{x}_{f}+ \overline{B}\dot{x}_{f}+Nx_{f}+\overline{F}_{r}+\overline{F}_{f}\\ =\overline{M}\dot{s}+(\overline{B}+\overline{V}_{m})s+\overline{M}(\dot{x}_{ fd}-\alpha\dot{e})+\overline{B}(\dot{x}_{fd}-\alpha e)+\overline{V}_{m}(\dot{x}_{fd}- \alpha e)+N(e+x_{fd})+\overline{F}_{r}+\overline{F}_{f}-\dot{e}-\beta e.\end{array}  \tag{10}\]  
Figure 4: The proposed adaptive model-free coupling control scheme. Figure 3: The configuration of the proposed recurrent fuzzy neural network (RFNN) [21]. Based on the Lyapunov stability theorem, the Lyapunov candidate function is  
\[V=\frac{1}{2}e^{T}Pe \tag{11}\]  
where \(P\) is positive definite. The ideal control (9) results \(\dot{V}=-e^{T}\beta e\leq 0\) guaranteeing the stability of the closed-loop system. However, the nonlinear cancellation will not be perfect when the system is experiencing uncertainties. In addition, the discrete-time approach is utilized here; therefore, our goal is to derive the RFNN controller such that the variation of (11) satisfies  
\[\Delta V(t_{k})=V(t_{k+1})-V(t_{k})\leq 0 \tag{12}\]  
where \(t_{k}\) denotes discrete-time instant. This implies that the contour error approaches to zero, i.e., the state \(x_{f}\) will follow the desired \(x_{\theta}\), the control system is stable in the sense of Lyapunov. According above description, multi-axis linear motors system exists an ideal controller and we can use adaptive model-free coupling controller based on RFNN to make system stability; the RFNN can be expressed as \(w^{T}\Psi_{r}(e,s,\theta)\), where \(w\) is the weighting matrix of RFNN and \(\Psi\) denotes the firing strength of the corresponding fuzzy rules. By the universal approximation property of the RFNN [21], the adaptive coupling controller is \(\overline{F}\), i.e.,  
\[\overline{F}=w^{*T}\Psi_{r} \tag{13}\]  
where \(w^{*}\in\Re^{PQ}\) is optimal weighting matrices and \(Q\) represents the rule number of RFNN. From the control architecture shown in Figure 4, we must pay attention to the training of the RFNN for guaranteeing the stability of closed-loop system. Note that our goal is to minimize the contour error; thus, let us define the cost function as  
\[E(t_{k})=\frac{1}{2}\|x_{f}(t_{k})-x_{fd}(t_{k})\|^{2}=\frac{1}{2}\|F^{T}(t_{k })x(t_{k})-F^{T}(t_{k})x_{d}(t_{k})\|^{2}\equiv\frac{1}{2}\|e(t_{k})\|^{2} \tag{14}\]  
where \(\|.\|\) is the Euclidean distance and \(t_{k}\) denotes the discrete-time instant.