## 5 Conclusions  
Aiming at the difficulty of physical parameter identification under random excitation, a novel multi-level KRRD-based polynomial frequency modulation integral analysis method was proposed. The method utilized multi-level KRRD to decompose the collected vibration acceleration signal into modal component signals corresponding to each DOF. These modal component signals contain crucial TV system information, such as stiffness and damping. In this step, the parameter identification problem of a multi-DOF system was transformed into the parameter identification problem of a single DOF system, which facilitated subsequent analysis. Moreover, the noise interference component was eliminated, and the anti-noise performance of the proposed method was improved. Additionally, the method employed the polynomial frequency modulation integral algorithm to reconstruct the velocity and displacement signals based on the single decomposed modal component signal. By combining cross-correlation theory, the dynamic differential equation of the TV system was modeled in the time domain, and the physical parameters were identified by solving the simultaneous linear equation.  
Based on numerical simulation, the identification effect of physical parameters under three different cases of variation (linear variation, quadratic variation, periodic variation) was conducted on an established three DOF TV model. The numerical results demonstrated the capability of the proposed method to effectively track various variations of physical parameters over time. Furthermore, the robustness of the proposed method was verified by adding different SNR levels of noise (20 dB, 50 dB, 100 dB) to the raw vibration acceleration signal. The identification errors of stiffness and damping are guaranteed to remain within a stable range, indicating the high identification accuracy of the proposed method even under noise interference.  
Only the vibration acceleration signal was used for parameter identification, which facilitates signal acquisition in practical engineering and improves the application po  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Error\%**} & \multicolumn{4}{c}{**SNR (dB)**} \\ \cline{2-5}  & **Noise-Free** & **20** & **50** & **100** \\ \hline \(k_{1}\) & 2.041 & 2.842 & 2.553 & 2.203 \\ \(k_{2}\) & 2.546 & 2.654 & 2.541 & 2.524 \\ \(k_{3}\) & 2.332 & 2.502 & 2.407 & 2.321 \\ \(c_{1}\) & 6.127 & 13.531 & 11.063 & 7.213 \\ \(c_{2}\) & 7.702 & 14.813 & 12.151 & 8.351 \\ \(c_{3}\) & 7.024 & 14.011 & 11.361 & 8.741 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Error analysis results under different SNR levels.  
tential of the proposed method. However, the proposed method is essentially based on the assumption of the short-term linear variation, and it has limitations in identification performance for the TV system with short-term interval non-linear variation characteristics. To enhance the adaptability and parameter identification accuracy of the proposed method, the authors will further study the non-linear fitting of the TV signal.  
Visualization, Z.S.; Writing--original draft preparation, H.L.; Writing--review and editing, H.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundations of China, grant number 12272172), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.  
Not applicable.  
The authors declare no conflict of interest.