## 5 Conclusions and Future Plans  
In this study, a multi-staged adaptive LMS (MSA-LMS) algorithm, which is an extension of the least mean squares (LMS) algorithm commonly used in the adaptive filtering system for active vibration and noise control, was developed and validated through simulation. Following are the three major contributions of this paper:  
(1) Introduction of a new concept for adaptive step size: The introduction of exponential function reduced the size of error as time increased, thereby improving system stability with decreasing step size. To prevent the divergence phenomenon through normalization using the input signal power in the initial state with respect to the error value, an energy autocorrelation function of instantaneous error was introduced, thereby ensuring that a misadjustment in the steady state was not affected. (2) Proposed MSA-LMS algorithm with multi-stage structure: According to the characteristics of the LMS algorithm, the signal after the control has a different spectrum to that of the original signal, which means that the number of frequency elements continues to increase. Furthermore, the conventional adaptive filtering system has a structure composed of the same reference signal and \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Control Algorithm**} & \multirow{2}{*}{**Error Analysis**} & \multicolumn{3}{c}{**Time**} \\ \cline{3-5}  & & **2.0–3.0 s** & **3.0–3.5 s** & **3.5–4.0 s** \\ \hline \multirow{2}{*}{**LMS Algorithm**} & Maximum & 0.1712 & 0.1701 & 0.1712 \\  & Minimum & \(-\)0.1835 & \(-\)0.1839 & \(-\)0.1858 \\  & Peak-to-Peak & 0.3547 & 0.3540 & 0.3876 \\ \hline \multirow{2}{*}{**MSA-LMS Algorithm**} & Maximum & 0.1305 & 0.0453 & 0.0651 \\  & Minimum & \(-\)0.0590 & \(-\)0.0178 & \(-\)0.0199 \\ \multirow{2}{*}{**Algorithm**} & Peak-to-Peak & 0.1895 (47\%\(\downarrow\)) & 0.0631 (82\%\(\downarrow\)) & 0.0850 (78\%\(\downarrow\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Peak-to-Peak value analysis for the estimation errors with LMS and MSA-LMS algorithms (“Incommensurate Sinusoids” case). Figure 13: Comparison of predicted frequency spectra. Key: controlled signal with least mean squares algorithm; controlled signal with multi-staged adaptive least mean squares algorithm. control system regardless of the time when one reference signal is determined, and consequently, the spillover phenomenon caused by the control deteriorates. To minimize this phenomenon, we proposed a new algorithm with several stages at regular intervals. This control strategy was also implemented by applying it to active periodic structures. The multi-stage architecture with multi-channel algorithms has the potential to actively attenuate vibration and noise signals with complex spectra in motors and decelerators used in next-generation powertrains for automobiles, aircrafts, and ships. (3) Numerical results of \"Incommensurate Sinusoids\" and \"Periodic Sinusoids\": It was demonstrated that the signal tracking result of the MSA-LMS algorithm was excellent. In particular, it can be confirmed that the misadjustments were properly adjusted for a signal including an unexpected high-frequency component, due to a special signal pattern or a spillover in the time domain generated by the variation of the frequency interval. Thus, the tracking performance was excellent. The observed peak-to-peak value of the estimation error in the case of the proposed MSA-LMS algorithm having the adaptability of step size could be further reduced by 90% or more, compared to the LMS algorithm with fixed step size. In addition, it was confirmed that the input signal was significantly reduced with respect to the size of the dominant spectrum in the existing frequency components, compared with the results of the LMS algorithm. Overall, we confirmed that the proposed MSA-LMS algorithm was superior to the existing LMS algorithm with respect to the ability to cope with unexpected frequency components in addition to the existing ones, owing to the characteristics of the LMS algorithm.