# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 9 Illustration of the Use of Basis Functions for Transient Analysis
### Commensurate Fractional-Order Sallen-Key Filter  
Figure 2 shows a simplified version of the Sallen-Key filter from [35] with an operational amplifier as the voltage buffer. Instead of classical capacitors, fractors of the impedances \(1/(s^{\alpha}\ C_{1})\) and \(1/(s^{\beta}\ C_{2})\) are used, where \(\alpha\), \(\beta\in R+\). Figure 1: (**a**) Impulse responses \(g_{F}\) and \(g\), (**b**) step responses \(h_{F}\) and \(h\) of the commensurate system with transfer function (55) and associated IOS with \(\alpha=1\). For \(\alpha=\beta\), the filter behaves like a commensurate FOS with the transfer function  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]  
where the pseudo-natural frequency and pseudo-quality factor are given by  
\[\omega_{0}=\frac{1}{\sqrt{R_{1}R_{2}C_{1}C_{2}}},Q=\frac{\sqrt{R_{1}R_{2}}}{R_ {1}+R_{2}}\sqrt{\frac{C_{1}}{C_{2}}}. \tag{60}\]  
Transfer function (59) can be rewritten in the form  
\[K_{F}(s)=\frac{1}{\left(\frac{s}{\omega_{0F}}\right)^{2\alpha}+\frac{1}{Q} \left(\frac{s}{\omega_{0F}}\right)^{\alpha}+1}, \tag{61}\]  
where  
\[\omega_{0F}=\omega_{0}^{1/\alpha} \tag{62}\]  
is the corner frequency of the asymptotic frequency characteristic of the fractional filter. Consider the Sallen-Key filter with the following parameters:  
\[\alpha=0.8,\,R_{1}=R_{2}=91\,\,\mathrm{k}\Omega,\,C_{1}=100\,\,\mathrm{nF/s^{ 1-\alpha}},\,C_{2}=1\,\,\mathrm{nF/s^{1-\alpha}}.\]  
It can be shown that \(Q=5\) and the corner frequency \(f_{0F}=\omega_{0F}/(2\pi)\) is 1 kHz. Note that the natural frequency of the associated IOS, resulting from (60), is about 174 Hz. First, the impulse response for the case \(\alpha=1\) will be found. Transfer function (59) has a pair of complex conjugate poles  
\[c=a\pm\mathrm{i}b,\,a=-109.2808,\,b=1091.4412.\]  
According to Table 3, the impulse response is  
\[g(t)=k\epsilon^{at}\sin(bt), \tag{63}\]  
\[k=\frac{\omega_{0}}{\sqrt{1-\frac{1}{4Q^{2}}}}\approx 1098.3134.\]  
Hence the impulse and step responses of the fractional filter are  
\[g_{F}(t)=k\mathrm{Im}(\varepsilon_{0}(t,c;\alpha,\alpha)), \tag{64}\]  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]0 Fractional-order Sallen–Key filter (adopted from [35]). \[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]1  
If the parameters \(C_{1}\) and \(C_{2}\) in the filter are changed to have the values \(C_{1}=C_{2}=10\) nF, then the corner and natural frequencies are preserved, but the quality factor is reduced to 0.5. This will correspond to an integer-order filter with critical damping and the two-fold real pole \(-\omega_{0}\). For critical damping, the impulse response has the form  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]2  
The corresponding responses of the fractional filter are then as follows:  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]3  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]4  
The respective waveforms computed in MATLAB are shown in \[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]5  
The obtained plots of the impulse and step responses satisfy the check for the basic features (the limit values at times zero and infinity are correct, and the local maxima and minima of the step response correspond to the impulse response passages through zero). Looking at the step response for \(Q=5\), we find a much smaller overshoot than for classical  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]6 (**a**) Impulse responses \(g_{F}\), (**b**) step responses \(h_{F}\) of the commensurate Sallen–Key filter with transfer function (59) for \(\alpha=0.8\) and for two different values of damping. second-order IOS.