# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 9 Illustration of the Use of Basis Functions for Transient Analysis
### Minimum-Phase Fractional-Order System  
A commensurate minimum-phase system is analyzed in [34], with the transfer function  
\[K_{\rm F}(s)=\frac{\sum\limits_{i=0}^{4}b_{i}s^{ika}}{\frac{10}{10}},\alpha=1.2, \tag{55}\]  
where  
\(a_{0}=b_{0}=750\),000, \(a_{1}=3\),563,000, \(a_{2}=6\),433,000, \(a_{3}=6\),032,000, \(a_{4}=3\),357,000, \(a_{5}=1\),167,000, \(a_{6}=251\),620,  
\(a_{7}=31\),914, \(a_{8}=2193\), \(a_{9}=75\), \(a_{10}=1\), \(b_{1}=690\),000, \(b_{2}=240\),000, \(b_{3}=-26\),000 and \(b_{4}=-4000\).  
For system (55), let us find the basis functions of its natural response and use them to construct the formulas for the impulse and step responses.  
First, the basis functions and the impulse response of the associated IOS will be found. Substituting \(\alpha=1\) into (55) yields the transfer function \(K(s)\) (see Equation (2)). The decomposition of \(K(s)\) into partial fractions and the Laplace inversion lead to the impulse response  
\[g(t)=\sum\limits_{k=1}^{6}P_{k}e^{\alpha_{k}t}+2e^{\sigma_{k}t}(P_{7},\cos b _{7}t-P_{7i}\sin b_{7}t)+2e^{\alpha_{9}t}(P_{9r}\cos b_{9}t-P_{9i}\sin b_{9}t). \tag{56}\]  
Equation (56) implies that the transfer function \(K(s)\) has six real poles \(a_{k}\), \(k=1...\) 6, and two pairs of complex conjugate poles of type \(c_{k}=a_{k}\pm{\rm i}b_{k}\). The corresponding residues are denoted by the symbols \(P_{k}=P_{kr}+{\rm i}P_{ki}\). The poles and residues were calculated with a high precision using the MATLAB symbolic toolbox, and, after rounding, they are summarized in Table 6.  
A similar decomposition of the function \(K(s)/s\) can be used to obtain the formula for the step response.  
Based on (56) and Table 6, the formula of the impulse response of the associated commensurate system with transfer function \(K_{F}\) (\(s\)) (55) can be directly written as  
\[g_{F}(t)=\sum\limits_{k=1}^{6}P_{k}\epsilon_{0}(t,a_{k};\alpha,\alpha)+2(P_{7r }{\rm Re}(\epsilon_{0}(t,c_{7};\alpha,\alpha))-P_{7i}{\rm Im}(\epsilon_{0}(t,c _{7};\alpha,\alpha))+P_{9i}{\rm Re}(\epsilon_{0}(t,c_{9};\alpha,\alpha))-P_{9 i}{\rm Im}(\epsilon_{0}(t,c_{9};\alpha,\alpha))) \tag{57}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(k\) & \(a_{k}\) & \(b_{k}\) & \(P_{kr}\) & \(P_{kk}\) \\ \hline
1 & \(-22.4369\) & & 0.0244 & \\
2 & \(-4.8430\) & & 0.9733 & \\
3 & \(-2.7791\) & & \(-9.9797\) & \\
4 & \(-2.1809\) & & 10.0304 & \\
5 & \(-1.0017\) & & \(-3.1207\) & \\
6 & \(-0.4993\) & & 1.5711 & \\
7 & \(-18.8014\) & 2.1491 & \(-0.0086\) & \(-0.0386\) \\
8 & \(-18.8014\) & \(-2.1491\) & \(-0.0086\) & \(0.0386\) \\
9 & \(-1.8282\) & 1.7367 & 0.2592 & \(0.7145\) \\
10 & \(-1.8282\) & \(-1.7367\) & 0.2592 & \(-0.7145\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Parameters of the partial fraction expansion of transfer function (55).  
The step response is easily determined from (57) using (54):  
\[\begin{array}{l}h_{F}(t)=\sum\limits_{k=1}^{6}P_{k}\varepsilon_{0}(t,a_{k}; \alpha,\alpha+1)\\ +2(P_{\gamma_{T}}\text{Re}(\varepsilon_{0}(t,c_{\gamma_{T}};\alpha,\alpha+1))-P _{\gamma_{1}}\text{Im}(\varepsilon_{0}(t,c_{\gamma};\alpha,\alpha+1))+P_{ \gamma_{T}}\text{Re}(\varepsilon_{0}(t,c_{\phi};\alpha,\alpha+1))-P_{\psi_{1}} \text{Im}(\varepsilon_{0}(t,c_{\phi};\alpha,\alpha+1)))\end{array} \tag{58}\]  
Figure 1 shows the plots of the corresponding impulse and step responses in MATLAB. For illustration, the responses of the IOS (\(\alpha=1\)) were added. Since the same results were obtained in Micro-Cap, they were not added in Figure 1.  
It follows from the comparison of the IOS and FOS responses in Figure 1 that the FOS responses exhibit less damping due to the increase in the \(\alpha\) parameter from 1 to 1.2. In [35], only the step response \(h_{F}\) is published, and its exact comparison with the analysis results of Figure 1 cannot be performed with the lack of numerical data. Therefore, the correctness of the results from Figure 1 was verified with the FFT test.