## 7 Conclusions and Future Work  
An enhanced LMS algorithm has been proposed and applied to an adaptive digital filtering system combined with sliding mode control. The method benefits from the nonlinear control technique inside the control system structure for better adaptability and tracking performance. This adaptive filtering system was used for the vibration reduction and control of a continuous smart structure with piezoelectric patches attached to it. The SM-LMS method allows for significant and robust vibration  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Peak Control**} & \multicolumn{3}{c}{**Amplitude**} \\ \cline{2-4}  & **Open-Loop** & **With LMS** & **With SM-LMS** \\ \hline
**1 st Peak** (15 Hz) & 0.0022920 & 0.0007087 (**10.2 dB\(\downarrow\)**) & 0.0004849 (**13.5 dB\(\downarrow\)**) \\
**2 nd Peak** (33 Hz) & 0.0013280 & 0.0016330 (**1.8 dB\(\uparrow\)**) & 0.0011170 (**1.5 dB\(\downarrow\)**) \\
**3 rd Peak** (117 Hz) & 0.0045400 & 0.0037110 (**1.8 dB\(\downarrow\)**) & 0.0025390 (**5.0 dB\(\downarrow\)**) \\
**4 th Peak** (312 Hz) & 0.0028750 & 0.0046440 (**4.2 dB\(\uparrow\)**) & 0.0031780 (**0.9 dB\(\uparrow\)**) \\
**Overall RMS** & \(3.0888\times 10^{-4}\) & \(3.2784\times 10^{-4}\) (**0.5 dB\(\uparrow\)**) & \(2.2431\times 10^{-4}\) (**2.8 dB\(\downarrow\)**) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Vibrational amplitudes of each case for the original beam structure.  
Figure 20: Experimental result for the modified structure: (**a**) time domain; and (**b**) frequency domain.  
reduction for not only the original structure, but also for a beam with modified dynamics. Thus, it can be a good solution for reducing the vibration of structures with unknown dynamics.  
The contributions of this research are summarized as: First, the proposed control system has improved stability, robustness, and capability to manage signals with multiple spectra with increasing complexity and frequency of components to be controlled in the output of the disturbance plant. Numerical analyses demonstrated that the method has adequate tracking performance. Second, a smart structure with piezoelectric patches was constructed and its dynamic model has been identified with frequency domain identification method; third, a conventional control method has been validated numerically in the case of active vibration control and showed limitations. Third, the method was validated for the original structure and a modified structure. The controller had acceptable tracking performance in both cases.  
This work could be validated with several real-world applications in active vibration control, such as vehicle vibration reduction and vibration attenuation for engineering structures. In addition, a hybrid controller could be proposed to control two or more directions simultaneously for realistic applications by effectively managing an extensive range of multi-spectral signals. Self-sensing actuator could be implemented with one PZT material, which acts as sensor and actuator simultaneously. Intelligent control methods such as neural networks could also be more adaptive and robust to changeable system dynamics.  
This paper was supported by the National Research Foundation of Korea (NRF) grant funded by the government of Korea (Ministry of Science, ICT & Future Planning) (No. 2016R1C1B1010888) as well as by the 2017 Yeungnam University Research Grant (217A380020).  
[PERSON] and [PERSON] initiated and developed the ideas related to this research work. [PERSON] and [PERSON] developed novel methods, derived relevant formulations, and carried out performance analyses and numerical analyses. [PERSON] wrote the paper draft under [PERSON]'s guidance and [PERSON] finalized the paper.  
The authors declare no conflict of interest.