## 2 The Proposed Fractional Order Auto-Tuning Method  
The FO-KC auto-tuning method is used in this paper to determine the parameters of a FO-PI controller with the transfer function as indicated next:  
\[C_{PI}(s)=k_{p}\Big{(}1+k_{i}s^{-\lambda}\Big{)} \tag{1}\]  
where \(k_{p}\) and \(k_{i}\) represent the proportional and integral gains and \(\lambda\) is the fractional order, with \(\lambda_{\text{min}}<\lambda<2\). The minimum value of the fractional order can be easily computed using Reference [18]. The basic idea of the FO-KC auto-tuning method resides in determining the three parameters in (1) using three performance specifications: the gain crossover frequency \(\omega_{c}\), the phase margin \(PM\) and the iso-damping property. As mentioned in the Introduction section, the design is based on using the iso-damping property and the phase margin \(PM\) requirement to define a forbidden region in the Nyquist plane. Moreover, to ensure the iso-damping property, the optimal FO-PI controller is determined such that the difference between the slope of the loop frequency response and the slope of the forbidden region border is minimum. Figure 1 shows the forbidden region, where the center, \(C\), and its radius, \(R\), are determined using trigonometric relations based on the \(PM\) value:  
\[C=\frac{1}{cos(PM)},\ R=\sqrt{C^{2}-1} \tag{2}\]  
The _first step_ in designing the FO-PI controller using the proposed auto-tuning method consists in determining the forbidden region using (2). As it has been previously mentioned, the key point in the design consists in evaluating for the imposed gain crossover frequency, the difference between the slope of the forbidden region border and that of the loop frequency response:  
\[\left|\left|\frac{\mathrm{d}\mathrm{I}\mathrm{m}}{\mathrm{d}\mathrm{R}\mathrm{e }}-\left.\frac{\mathrm{d}\Im_{L}}{\mathrm{d}\Re_{L}}\right|_{\omega_{c}} \right|\right| \tag{3}\]  
where \(\frac{\mathrm{d}\mathrm{I}\mathrm{m}}{\mathrm{d}\mathrm{R}\mathrm{e}}\) stands for the slope of the forbidden region border and \(\left.\frac{\mathrm{d}\Im_{L}}{\mathrm{d}\Re_{L}}\right|_{\omega=\omega_{c}}\) is the slope of the loop frequency response. The _second step_ in the design consists in the computation of the slope of the forbidden region border, based on trigonometric relations in Figure 1:  
\[\frac{\mathrm{d}\mathrm{I}\mathrm{m}}{\mathrm{d}\mathrm{R}\mathrm{e}}=-\frac{ \mathrm{R}\mathrm{e}(\alpha)+C}{\mathrm{I}\mathrm{m}(\alpha)}=\frac{\cos( \alpha)}{\sin(\alpha)} \tag{4}\]  
where \(\alpha=90^{o}-PM\). The slope of the loop frequency response can be computed according to the following two equations:  
\[\left.\frac{\mathrm{d}L(j\omega)}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}} =P(j\omega_{c})\left.\frac{\mathrm{d}C(j\omega)}{\mathrm{d}\omega}\right|_{ \omega=\omega_{c}}+C(j\omega_{c})\left.\frac{\mathrm{d}P(j\omega)}{\mathrm{d} \omega}\right|_{\omega=\omega_{c}} \tag{5}\]  
\[\left.\frac{\mathrm{d}L(j\omega)}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}} =\left.\frac{\mathrm{d}\Re_{L}}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}}+ j\left.\frac{\mathrm{d}\Im_{L}}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}} \tag{6}\]  
where \(L(j\omega)=P(j\omega)C(j\omega)\) is the loop frequency response, \(P(j\omega_{c})\) is the process frequency response at the gain crossover frequency, \(\left.\frac{\mathrm{d}P(j\omega)}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}}\) is the derivative of the process frequency response, \(C(j\omega_{c})\) and \(\left.\frac{\mathrm{d}C(j\omega)}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}}\) are the controller frequency response and its derivative at the gain crossover  
Figure 1: Forbidden region for the FO-KC auto-tuning method. frequency. To determine the real \(\left.\frac{\mathrm{d}\Re_{\ell}}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}}\) and imaginary \(\left.\frac{\mathrm{d}\Im_{\ell}}{\mathrm{d}\omega}\right|_{\omega=\omega_{c}}\) parts in (6), all terms in the right hand side of (5) have to be computed.