# Model-Free Adaptive Control Based on Fractional Input-Output Data Model
## 4 Adaptive Controller  
By utilizing the fractional equivalent model proposed in (8), the adaptive control law is derived in this section. First, the filtered tracking error \(s(k)\) is defined as  
\[s(k)=e(k)+\gamma_{e}e(k-1), \tag{28}\]  
where \(\gamma_{e}<1\) is a parameter of design, and \(e(k)=r(k)-y(k)\) is the tracking error for the smooth reference \(r(k)\). The reaching condition is designed as  
\[s(k+1)=-\gamma_{s}s(k), \tag{29}\]  
where the constant parameter \(\gamma_{s}<1\) is the reaching rate. Combining (28) and (29) results in  
\[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]  
The dynamics in (7) can be rewritten as  
\[y(k+1)=\theta(k)[u(k)+u^{N_{u}}_{\Delta}(k-1)]-y^{N_{y}}_{\Delta}(k), \tag{31}\]  
where  
\[u^{N_{u}}_{\Delta}(k-1)=\omega(1)u(k-1)+\cdots+\omega(N_{u})u(k-N_{u}). \tag{32}\]  
Thus, the error dynamics in (30) can be processed as  
\[r(k+1)-y(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k), \tag{33}\]or  
\[y(k+1)=r(k+1)+\gamma_{s}s(k)+\gamma_{e}e(k), \tag{34}\]  
and consequently  
\[r(k+1)+\gamma_{s}s(k)+\gamma_{e}e(k)=\theta(k)[u(k)+u_{\Delta}^{N_{\eta}}(k-1)] -y_{\Delta}^{N_{y}}(k). \tag{35}\]  
Rearranging (35), the ideal control law \(u^{*}(k)\) can be obtained as  
\[u^{*}(k)=\frac{1}{\theta(k)}\Big{[}r(k+1)+\gamma_{s}s(k)+\gamma_{e}e(k)+y_{ \Delta}^{N_{y}}(k)\Big{]}-u_{\Delta}^{N_{u}}(k-1). \tag{36}\]  
Nevertheless, since \(\theta(k)\) is unknown, the control law \(u(k)\) is established by \(\hat{\theta}(k)\) in (9) as  
\[u(k) = \frac{1}{\hat{\theta}(k)+\varepsilon_{\theta}\text{sign}(\hat{ \theta}(k))}\Big{[}r(k+1)+\gamma_{s}s(k)+\gamma_{e}e(k)+y_{\Delta}^{N_{y}}(k) \Big{]} \tag{37}\] \[s(k+1)=-\gamma_{s}s(k), \tag{29}\]0  
where \(\varepsilon_{\theta}\) is a positive constant. Theorem 1: _The sliding surface \(s(k)\) and the tracking error \(e(k)\) remain as bounded sequences for small enough positive gains \(\gamma_{s}\) and \(\gamma_{e}\in(0,1)\)._  
Proof: Dynamics (31) with control law (37) leads to  
\[s(k+1)=-\gamma_{s}s(k), \tag{29}\]1 \[s(k+1)=-\gamma_{s}s(k), \tag{29}\]2 \[s(k+1)=-\gamma_{s}s(k), \tag{29}\]3  
where  
\[s(k+1)=-\gamma_{s}s(k), \tag{29}\]4  
Thereafter, the tracking error is derived as  
\[s(k+1)=-\gamma_{s}s(k), \tag{29}\]5 \[s(k+1)=-\gamma_{s}s(k), \tag{29}\]6 \[s(k+1)=-\gamma_{s}s(k), \tag{29}\]7  
Referring to the definition of \(s(k)\) in (28), one obtains  
\[s(k+1)=-\gamma_{s}s(k), \tag{29}\]8 \[s(k+1)=-\gamma_{s}s(k), \tag{29}\]9 \[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]0  
In addition, consider  
\[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]1\[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]2  
and  
\[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]3  
Thus, the dynamics in (40) and (41) can be rearranged as  
\[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]4  
and  
\[e(k+1)=-\gamma_{s}s(k)-\gamma_{e}e(k). \tag{30}\]5  
respectively. In view of (45) and (46), one obtains the closed-loop dynamics  
\[ \begin{bmatrix}s(k+1)\\ e(k+1)\end{bmatrix} =\underbrace{ \begin{bmatrix}A_{1,1}(k)&A_{1,2}(k)\\ A_{2,1}(k)&A_{2,2}(k)\end{bmatrix} }_{\lambda_{\theta}} \begin{bmatrix}s(k)\\ e(k)\end{bmatrix} + \begin{bmatrix}1\\ 1\end{bmatrix} [\xi_{r}(k)+\xi_{y}(k)], \tag{47}\]  
where  
\[A_{1,1}(k)=\gamma_{s}\xi_{\theta}(k), \tag{48}\]  
\[A_{1,2}(k)=\gamma_{\epsilon}[1+\xi_{\theta}(k)], \tag{49}\]  
\[A_{2,1}(k)=A_{1,1}(k), \tag{50}\]  
and  
\[A_{2,2}(k)=\gamma_{\epsilon}\xi_{\theta}(k). \tag{51}\]  
Note that \(\xi_{\theta}(k)\) is a bounded sequence; then, it is possible to find small enough values \(\bar{\gamma}_{\epsilon}\) and \(\bar{\gamma}_{s}\), such that \(A_{\theta}\) is a convergent matrix, whenever \(0<\gamma_{\epsilon}<\bar{\gamma}_{\epsilon}\) and \(0<\gamma_{s}<\bar{\gamma}_{s}\). One way to determine conservative bounds for \(\bar{\gamma}_{\epsilon}\) and \(\bar{\gamma}_{s}\) relies on utilizing the Gershgorin Circle Theorem, which states that all the proper values of \(A_{\theta}\) lie inside of at least one of the circles of Gershgorin.