# Model-Free Adaptive Control Based on Fractional Input-Output Data Model
## 3 Fractional Input-Output Equivalent Model  
The discrete-time fractional difference of the sequence \(x(k)\in\mathbb{R}\), with \(k\in\mathbb{N}_{0}\), is defined as  
\[\Delta_{N}^{\alpha}x(k)=\sum_{i=0}^{N}\omega_{\alpha}(i)x(k-i), \tag{2}\]  
where \(\alpha\in(0,1]\) is the order, and the fractional binomial coefficient \(\omega_{\alpha}(i)\) is determined recursively as  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]  
for \(i\geq 1\), and initial condition \(\omega_{\alpha}(0)=1\). Accordingly, the equivalent model based on the FPPD (fractional pseudo partial derivative) \(\theta(k)\) is proposed as  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\theta(k)\Delta^{\alpha}_{N_{u}}u(k). \tag{4}\]  
Relying on the definition in (2), we have  
\[\Delta^{\alpha}_{N_{y}}y(k+1)=\omega_{\alpha}(0)y(k+1)+\cdots+\omega_{\alpha}( N_{y})y(k-N_{y}+1), \tag{5}\]  
and  
\[\Delta^{\alpha}_{N_{u}}u(k)=\omega_{\alpha}(0)u(k)+\cdots+\omega_{\alpha}(N_{ u})u(k-N_{u}). \tag{6}\]  
Simplifying \(\omega(i)=\omega_{\alpha}(i)\) and using (3), we obtain  
\[y(k+1) = \theta(k)\Delta^{\alpha}_{N_{u}}u(k) \tag{7}\] \[-\omega(1)y(k)-\omega(2)y(k-1)-\cdots-\omega(N_{y})y(k-N_{y}+1).\]  
Therefore, the equivalent model is formulated as  
\[\hat{y}(k+1) = \hat{\theta}(k)\Delta^{\alpha}_{N_{u}}u(k) \tag{8}\] \[-\omega(1)y(k)-\omega(2)y(k-1)-\cdots-\omega(N_{y})y(k-N_{y}+1),\]  
where \(\hat{y}(k+1)\) is the output of the equivalent model and \(\hat{\theta}(k)\) is the estimated FPPD. With the aim of utilizing the fractional input-output equivalent model proposed by (8), the adaptive network multiple-input fuzzy rules emulating network (MiFREN) is employed to establish \(\hat{\theta}(k)\). The fuzzy network architecture relies on the IF-THEN rules:  
\(\circ\) Rule\({}_{i=1,\cdots N_{f}}\):  
\(\left\{\text{IF }u(k-1)\text{ is }\mu_{u_{i}}\text{ and }y(k)\text{ is }\mu_{y_{i}}\text{ THEN }\hat{\theta}_{i}(k)=\beta_{i}(k)\right\}\),  
where \(N_{f}\) is the number of IF-THEN rules, \(\mu_{u_{i}}\) and \(\mu_{y_{i}}\) are membership functions of \(u(k-1)\) and \(y(k)\), respectively, and \(\beta(k)\in\mathbb{R}^{N^{f}}\) is the weight-vector of output consequences. In addition, considering \(\varphi_{i}(k)=\mu_{u_{i}}\wedge\mu_{y_{i}}\), with the product as the t-norm, as the degree of validity of Rule\({}_{i}\). Then, the estimated FPPD, or \(\hat{\theta}(k)\), is obtained as  
\[\hat{\theta}(k)=\sum_{i=1}^{N_{f}}\varphi_{i}(k)\beta_{i}(k)=\varphi^{T}(k) \beta(k). \tag{9}\]  
An alternative for the membership functions is given by  
* \(x\) is POSITIVE: \(\mu_{P}(x)=0.5[1+\tanh(\sigma x-1)]\);
* \(x\) is ZERO: \(\mu_{N}(x)=1-\tanh^{2}(\sigma x)\); and
* \(x\) is NEGATIVE \(\mu_{Z}(x)=0.5[1-\tanh(\sigma x+1)]\),  
for agglomeration parameter \(\sigma>0\). Note that, if the range of argument \(x\) is small, the parameter \(\sigma\) should be large, and vice-versa. Moreover, the adaptation allows us to reduce the number of fuzzy sets, and in turns the number of rules, to be considered. The learning law of the weight parameters \(\beta(k)\) in (9) is derived from minimizing the cost function  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]0  
where \(\hat{\varrho}(k+1)\) is the model error given by  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]1By employing the gradient descent approach to minimize the cost function (10), the learning law for \(\beta(k+1)\) is derived as  
\[\omega_{\alpha}(i)=\Big{(}1-\frac{1+\alpha}{i}\Big{)}\omega_{\alpha}(i-1), \tag{3}\]2  
where \(\eta>0\) is the learning rate.