## Appendix A Fractional Integral Approximation  
Fractional-order calculus is a generalization of the integration and differentiation from integer to non-integer order. This appendix introduces only definitions which are widely used in the area of control systems. First, gamma function \(\Gamma(z)\), which is the extension of the factorial for non-integer number \(z\), is introduced as  
\[\Gamma(z)=\int_{0}^{\infty}e^{-t}\dot{t}^{z-1}dt \tag{1}\]  
The most important property of the gamma function is  
\[z\Gamma(z)=\Gamma(z+1) \tag{2}\]Then, the definition of integral of order \(\alpha\in\mathbb{R}\) is presented. In continuous-time domain, the most often used one is the _Riemann-Liouville_ definition:  
\[{}^{\alpha}\Xi e(t)=\frac{1}{\Gamma(\alpha)}\int\limits_{0}^{t}(t-\tau)^{( \alpha-1)}e(\tau)d\tau \tag{10}\]  
At this time, the FOI is not supported in any programming language. For this reason, its numerical approximation is required to implement the FOI in any real-time control system. In a digital control system with sampling time \(T_{s}\), interval \((0,t)\) can be approximated by \(k=\frac{t}{T_{s}}\) sub-intervals. Therefore,  
\[{}^{\alpha}\Xi e(t)=\frac{1}{\Gamma(\alpha)}\sum\limits_{j=1}^{k}\int\limits _{jT_{s}}^{(j+1)T_{s}}(t-\tau)^{(\alpha-1)}e(\tau)d\tau \tag{11}\]  
Consider that \(T_{s}\) is small enough, so that \(e\) is constant in each sub-interval. Therefore,  
\[{}^{\alpha}\Xi e(t)\approx^{\alpha}\Xi_{e,k}=\frac{1}{\Gamma(\alpha)}\sum \limits_{j=1}^{k}\int\limits_{jT_{s}}^{(j+1)T_{s}}(t-\tau)^{(\alpha-1)}e(\tau )d\tau \tag{12}\]  
Following that,  
\[{}^{\alpha}\Xi_{e,k}=\sum\limits_{j=1}^{k}\left[(k-j+1)^{\alpha}-(k-j)^{ \alpha}\right]\frac{T_{s}^{\alpha}}{\alpha\Gamma(\alpha)}e_{j+1} \tag{13}\]  
From (10) and (13), we have  
\[{}^{\alpha}\Xi_{e,k}=\sum\limits_{j=1}^{k}\omega_{j}e_{j} \tag{14}\]  
with the weighting factor \(\omega_{j}\) as follows:  
\[\omega_{j}=\left[(k-j+1)^{\alpha}-(k-j)^{\alpha}\right]\frac{T_{s}^{\alpha}} {\alpha\Gamma(\alpha)}. \tag{15}\]  
Because of the infinite data in (14), the approximation of FIO cannot be directly implemented in any digital system. In this research, the recursive approximation of FIO in [31] is employed. Denote \(\Xi_{e,k-1}\) as FIO of the tracking error in the last step, and it can be computed as  
\[{}^{\alpha}\Xi_{e,k-1}=\sum\limits_{j=2}^{k}\omega_{j}e_{j-1} \tag{16}\]  
From (14) and (16), we have  
\[{}^{\alpha}\Xi_{e,k}=^{\alpha}\Xi_{e,k-1}+\sum\limits_{j=2}^{k}\omega_{j} \tilde{e}_{j-1}+\omega_{1}e_{1} \tag{17}\]  
where \(\tilde{e}_{j}=e_{j}-e_{j-1}\). We apply the short memory principle to (17) and we can consider two cases: _(a)_. If \(k<N\), where \(N=\left[\frac{L}{T_{S}}\right]\) is the number of considered data samples, then  
\[{}^{\alpha}\Xi_{e,k}=^{\alpha}\Xi_{e,k-1}+\sum\limits_{j=N-k+2}^{N}\Omega_{j} \tilde{e}_{N-k+j}+\Omega_{N-k+1}e_{1} \tag{18}\]_(b)._ If \(k\geq N\),  
\[{}^{\alpha}\Xi_{e,k}=^{\alpha}\Xi_{e,k-1}+\sum_{j=2}^{N}\Omega_{j}\xi_{k-N+j}+ \Omega_{1}e_{k-N+1} \tag{11}\]  
where  
\[\Omega_{j}=[(N-j+1)^{\alpha}-(N-j)^{\alpha}]\frac{T_{s}^{\alpha}}{\Gamma( \alpha+1)} \tag{12}\]  
The FIO is approximated by Equations (10) and (11), which can be easily implemented in any digital control system.