# Model-Free Robotic Architecture with Task-Multiplexer and Discrete-Time Adaptive Control
## 4 MiFREN Adaptive Control and Closed-Loop Analysis
### Adaptive Algorithm and Closed Loop Analysis  
An adaptive algorithm will be established to tune the adjustable parameters \(\beta(k)\) under the consideration of closed-loop performance and stability analysis. By using the dead-zone function, the adaptive algorithm is proposed as  
\[\beta(k+1)=\beta(k)+\frac{\eta(k)}{\tilde{\gamma}_{u}||\varphi(k)||^{2}}e_{k+1} \varphi(k), \tag{30}\]  
where \(\eta(k)\) is the time-varying learning rate and \(\tilde{\gamma}_{u}\) is a constant, which will be discussed below.  
The learning rate plays an importance role of the controller performance. The following theorem manifests the selection of the learning rate, regarding the closed loop analysis.  
**Theorem 1**.: _For the control law (28) with the adaptive algorithm (30) applied to the discrete-time systems (1) and (2), the convergence of tracking error and internal signals can be guaranteed when the learning rate is given as_  
\[\eta(k)=\frac{\tilde{\gamma}_{u}}{\tilde{f}_{b}(k)+\varepsilon_{o}}<\frac{ \tilde{\gamma}_{u}}{\varepsilon_{o}}, \tag{31}\]  
_where_  
\[0<\tilde{\gamma}_{u}\leq\varepsilon_{o}, \tag{32}\]  
_and_  
\[0<\tilde{f}_{u}\leq\tilde{\gamma}_{u}. \tag{33}\]  
**Remark 2**.: _Practically, \(\tilde{\gamma}_{u}\) is unknown and, thus, it can be approximated by \(\tilde{\gamma}_{u}=C_{u}\max\{\hat{f}_{b}(k)\}\), where \(1\leq C_{u}\)._  
Figure 3: Multi-input Fuzzy Rule Emulated Network (MiFREN) Controller with an IF-THEN rule.  
Proof.: The proof is conducted with two parts. The first part is to prove the convergence of the internal signals \(\tilde{\beta}^{T}(k)\) when \(\tilde{\beta}(k)=\beta^{*}-\beta(k)\). Subtraction of \(\beta^{*}\) from both sides of (30) leads to  
\[\tilde{\beta}(k+1)=\tilde{\beta}(k)-\frac{\eta(k)}{\gamma_{u}||\varphi(k)||^{2} }\epsilon_{k+1}\varphi(k). \tag{34}\]  
By using (27) with (26) and (28), the error dynamic can be rearranged as  
\[e(k+1) = r(k+1)+q(k+1),\] (35) \[= f(w(k),u^{*}(k))-f(w(k),u(k)),\] \[= \frac{\partial f(w(k),\
u)}{\partial\
u}\Big{|}_{\
u=u_{c}(k)} \Big{[}u^{*}(  
Referring to the adaptive algorithm in (30), it is clear that  
\[\Delta\beta(k) = \beta(k+1)-\beta(k), \tag{45}\] \[= \frac{\eta(k)}{\gamma_{u}||\varphi(k)||^{2}}\varepsilon_{k+1} \varphi(k).\]  
Substitution of (45) into (44) gives  
\[\Delta e(k)=-\eta(k)\frac{\hat{f}_{b}(k)}{\hat{\gamma}_{u}}e(k+1). \tag{46}\]  
Thereafter, the relation in (46) can be rearranged as  
\[e(k+1)-e(k) = -\eta(k)\frac{\hat{f}_{b}(k)}{\hat{\gamma}_{u}}e(k+1),\] \[e(k+1) = \frac{1}{1+\eta(k)\frac{\hat{f}_{b}(k)}{\hat{\gamma}_{u}}}e(k). \tag{47}\]  
For the convergence of the sequence in (47), it is required that  
\[\Big{|}\frac{1}{1+\eta(k)\frac{\hat{f}_{b}(k)}{\hat{\gamma}_{u}}}\Big{|}<1. \tag{48}\]  
By using the learning rate in (31), it's clear that the condition in (48) always holds. The convergence proof of the tracking error is completed.  
The validation of the proposed control scheme and the setting of the designed parameters and membership functions will be addressed in the following section.