# Model-Free Adaptive Control Based on Fractional Input-Output Data Model
## 3 Fractional Input-Output Equivalent Model
Applying the chain rule with (10) through (9) and (8), \(\frac{\partial\hat{E}(k+1)}{\partial\beta(k)}\) in (12) can be obtained as  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]3 \[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]4  
Substitution of (13) into (12) leads to the learning law  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]5  
Furthermore, the appropriate learning rate \(\eta\) is established by the following result to guarantee an acceptable performance of the equivalent model. **Lemma 1**. : _Consider the fractional-order dynamics in (5), whose equivalent model is given in (8) and (9). Then, the boundedness of the internal signals and the model error is guaranteed if the learning rate \(\eta\) is selected with a positive constant \(\mu_{\eta}\) as_  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]6  
_where_  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]7  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]8  
_and_  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]9  
Proof. : According to the universal approximation property of the MiFREN method [32], with the estimated FPPD in (9), and the ideal weight vector \(\beta^{*}\), one has that \(\theta(k)\) in (9) can be rewritten as  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]0  
where \(|\varepsilon_{\theta}(k)|\leq\varepsilon_{\theta}^{M}\) is the bounded residue error. By using (19) with (7), one gets  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]1 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]2 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]3  
where  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]4  
Next, using (8) and (9) produces  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]5By utilizing the model error (11) with (20) and (22) yields  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]6 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]7 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]8  
where  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]9  
Applying one-step forward with (24) and using (14) leads to  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]0 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]1 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]2 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]3  
Note that, for \(\beta(k)\perp\varphi(k)\), one has that \(\beta(k)^{T}\varphi(k)=\beta^{*}\varphi(k)\). Thereby, only the case of \(\beta(k)\parallel\varphi(k)\) is considered, and consequently,  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]4 \[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]5  
Therefore, the learning rate \(\eta\) in (15) renders  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]6  
Thus, \(\tilde{\beta}(k)\) is a bounded sequence for a control effort \(u(k)\) with bounded fractional difference \(\Delta^{\mathrm{g}}_{N_{u}}u(k)\). Thereafter, the model error \(\hat{e}(k)\) is also a bounded sequence according to (23) and the boundedness of \(\tilde{\beta}(k)\).