# Model-Free Robotic Architecture with Task-Multiplexer and Discrete-Time Adaptive Control
## 3 Identification of an Equivalent Model  
The affine equivalent model of the controlled plant in (2) is defined by  
\[q(k+1)=f_{a}(w_{k})+f_{b}(w_{k})u(k), \tag{4}\]  
where \(f_{a}(w_{k})\) and \(f_{b}(w_{k})\) are unknown functions. According to \(f_{u}(k)\) in (3) and the model in (4), we have  
\[f_{u}(k)=\frac{\partial q(k+1)}{\partial u(k)}=f_{b}(w_{k}). \tag{5}\]  
The two functions \(f_{a}(w_{k})\) and \(f_{b}(w_{k})\) in (4) are identified by the adaptive network MiFREN, as illustrated in Figure 2. The estimated output of \(\hat{q}(k+1)\) is obtained as  
\[\hat{q}(k+1)=\hat{f}_{a}(k)+\hat{f}_{b}(k)u(k), \tag{6}\]  
where \(\hat{f}_{a}(k)\) and \(\hat{f}_{b}(k)\) are the estimated functions of \(f_{a}(w_{k})\) and \(f_{b}(w_{k})\), respectively. By using the general relation of MiFREN, the functions \(\hat{f}_{a}(k)\) and \(\hat{f}_{b}(k)\) are utilized as  
\[\hat{f}_{a}(k)=\beta_{a}^{\mathrm{T}}(k)\hat{\sigma}(k), \tag{7}\]  
and  
\[\hat{f}_{b}(k)=\beta_{b}^{\mathrm{T}}(k)\hat{\sigma}(k), \tag{8}\]  
where \(\beta_{a}(k)\) and \(\beta_{b}(k)\) are weight parameter vectors, and \(\hat{\sigma}(k)\) is the membership vector, as depicted in Figure 2. The network architecture in Figure 2 is designed by a set of IF-THEN rules, such as:  
_IF \(q(k)\) is positive and \(q(k+1)\) is positive THEN \(\hat{f}_{a}(k)\) should be positive and \(\hat{f}_{b}(k)\) should be positive._  
In this case, two inputs \(q(k)\) and \(q(k+1)\) have three linguistic variables, as the following membership functions: Positive (p), Zero (z), and Negative (n). The design of membership functions will be demonstrated by the experimental system in Section 5. Next, the online learning scheme for adjustable parameters \(\beta_{a}(k)\) and \(\beta_{b}(k)\) is established with the convergence analysis by the following Theorem. Figure 1: Discrete-time robotic control system block diagram. VFD, variable-frequency drive. **Lemma 1**. : _For the equivalent model given by (6)-(8) with the idea weight parameters \(\beta_{a}^{*}\) and \(\beta_{b}^{*}\), the convergence of the internal signals \(\tilde{\beta}_{a}(k)=\beta_{a}^{*}-\beta_{a}(k)\) and \(\tilde{\beta}_{b}(k)=\beta_{b}^{*}-\beta_{b}(k)\), and the approximation error \(\hat{e}(k+1)=q(k+1)-\hat{q}(k+1)\) can be guaranteed by the tuning laws given by_  
\[\beta_{a}(k+1)=\beta_{a}(k)+\eta_{h}(k)\hat{e}(k+1)\hat{\sigma}(k), \tag{9}\]  
_and_  
\[\beta_{b}(k+1)=\beta_{b}(k)+\eta_{h}(k)\hat{e}(k+1)u(k)\hat{\sigma}(k), \tag{10}\]  
_where the learning rate \(\eta_{h}(k)\) is determined by_  
\[f_{u}(k)=\frac{\partial q(k+1)}{\partial u(k)}=f_{b}(w_{k}). \tag{5}\]0  
_with \(0<\gamma_{h}<1\)._  
Proof. : By combining the tuning laws (9) and (10), we get  
\[f_{u}(k)=\frac{\partial q(k+1)}{\partial u(k)}=f_{b}(w_{k}). \tag{5}\]1  
where \(\beta_{t}(k)=[\beta_{a}(k)\ \beta_{b}(k)]^{T}\) and \(\hat{\sigma}_{t}(k)=[\hat{\sigma}(k)\ \hat{\sigma}(k)u(k)]^{T}\). There exists an idea weight parameter \(\beta_{t}^{*}=[\beta_{a}^{*}\ \beta_{b}^{*}]^{T}\) for MiFREN such that  
\[f_{u}(k)=\frac{\partial q(k+1)}{\partial u(k)}=f_{b}(w_{k}). \tag{5}\]2  
where \(\varepsilon_{h}(k)\) is the residue error and \(|\varepsilon_{h}(k)|\leq\varepsilon_{H}\) where \(\varepsilon_{H}\) a positive constant. The approximation error dynamic \(\hat{e}(k+1)\) can be expressed as  
\[ \begin{array}{rcl}\hat{e}(k+1)&=&q(k+1)-\hat{q}(k+1),\\ &=&[\beta_{t}^{*}-\beta_{t}(k)]^{T}\hat{\sigma}_{t}(k)+\varepsilon_{h}(k),\\ &=&\tilde{\beta}_{t}^{T}(k)\hat{\sigma}_{t}(k)+\varepsilon_{h}(k),\end{array}  \tag{14}\]  
where \(\tilde{\beta}_{t}(k)=\beta_{t}^{*}-\beta_{t}(k)\). Subtraction of \(\beta_{t}^{*}\) from both sides of the tuning law (12) leads to  
Figure 2: MiFREN identification model. \[\tilde{\beta}_{t}(k+1)=\tilde{\beta}_{t}(k)-\eta_{h}(k)\vartheta_{t}(k)\hat{e}(k+1). \tag{15}\]  
Substitution of (14) into (15) gives  
\[