## 4 Simulation
### Incommensurate Sinusoids: \(\omega_{2}\
eq n\omega_{1},\omega_{3}\
eq n\omega_{1}\)  
For incommensurate sinusoids, \(A_{1}=0.029\), \(A_{2}=0.053\), \(A_{3}=0.071\), \(\omega_{1}=10\) Hz, \(\omega_{2}=10\) Hz, and \(\omega_{2}=10\) Hz were used as controls. It was also assumed that there was no phase difference in each term. Figure 9 is a graphical comparison between the LMS algorithm and the proposed MSA-LMS algorithm. In the case of \"Incommensurate Sinusoids\" that were relatively less related to the frequencies involved than the \"Periodic Sinusoids\" discussed earlier, the signal pattern in the time domain was found to be irregular, as shown in Figure 10. Nevertheless, it can be seen that the signal tracking performance of the proposed MSA-LMS algorithm was much better than that of the LMS algorithm with fixed step size. Figure 8: Comparison of predicted frequency spectra. Key: controlled signal with least mean squares algorithm; controlled signal with multi-staged adaptive least mean squares algorithm. To compare the proposed MSA-LMS algorithm with the LMS algorithm in detail, the range from 3.5 to 4 s was expanded, as shown in Figure 10. The MSA-LMS algorithm was adaptive with respect to the frequency and error of the filtered signal. On the other hand, the LMS algorithm that had a fixed step size exhibited a relatively low tracking performance, owing to its failure to detect such changes. Figure 11 compares the estimation error of the active vibration control between the two algorithms. It can be observed that there was no significant difference up to 2 s. However, between 2 and 3 s, the maximum and minimum values of the control error were \(0.1712\) and \(-0.1835\), respectively, for the LMS algorithm, while the corresponding values were \(0.1305\) and \(-0.059\) for the MSA-LMS algorithm. Similarly, between 3 and 3.5 s, the respective values were \(0.1701\) and \(-0.1839\) for LMS, while \(0.0453\) and \(-0.0178\) for MSA-LMS. Finally, between 3.5 and 4 s, the respective values were \(0.1712\) and \(-0.1858\) for LMS, while \(0.0651\) and \(-0.0199\) for MSA-LMS. Figure 10: Comparison of desired signal and filter output controlled with LMS and MSA-LMS in the case of the control of “Incommensurate Sinusoids” in the time domain, \(\omega_{2}\
eq n\omega_{1},\omega_{3}\
eq n\omega_{1}\). Key:  
It can be observed from the above results that the peak-to-peak value of the proposed MSA-LMS algorithm, which is more adaptive than the existing LMS algorithm, decreased by 47% between 2 and 3 s, by 82% between 3 and 3.5 s, and by 78% between 3.5 and 4 s. Similar to the previous case (Period Sinusoids), the peak-to-peak value of the error did not decrease over time in the case of the LMS algorithm, whereas it was reduced by about two times in the case of the MSA-LMS algorithm, owing to its multi-stage concept. Because of the complexity of the incommensurate sinusoids, the tracking performance of the LMS algorithm was found to be worse than that observed in the case of the periodic sinusoids, whereas the MSA-LMS algorithm had a similar tracking performance with a higher percentage of decrease in the error value. Figure 12 is a graph comparing the error between the LMS algorithm and the proposed MSA-LMS algorithm. It can be observed that the MSA-LMS algorithm exhibited slightly better tracking performance than that observed in the previous case (Periodic Sinusoids) between 2 and 3 s, by virtue of its energy autocorrelation function of exponential function and instantaneous error. The above results are summarized in Table 2 below. Figure 11: Predicted estimation error with LMS and MSA-LMS algorithms. Figure 12: Comparison of predicted estimation errors. Key: controlled signal with least mean squares method; controlled signal with multi-staged adaptive least mean squares method. Figure 13 is a graph comparing the frequency spectra between the LMS algorithm and the proposed MSA-LMS algorithm.