## 3 Results and Discussion
### Real-Time Adaptation Performance  
The real-time adaptation capabilities of the LSTM-enhanced model identification system were evaluated through extensive testing under various operating conditions. The analysis focuses on two key aspects: parameter convergence characteristics and control quality metrics.  
The LSTM-enhanced system demonstrates significantly improved parameter convergence characteristics compared to conventional methods. The convergence rate shows a 65% improvement, which can be expressed mathematically as follows:  
\[\tau_{c}=0.35\tau_{n} \tag{24}\]  
where \(\tau_{c}\) represents the convergence time of the LSTM-enhanced system and \(\tau_{n}\) represents that of the conventional system.  
Parameter estimation accuracy improved by 45%, as quantified by the following equation:  
\[\left|\left|\theta_{est}-\theta_{true}\right|\right|^{2}=0.55||\theta_{conv}- \theta_{true}||^{2} \tag{25}\]  
where \(\theta_{est}\), \(\theta_{true}\), and \(\theta_{conv}\) represent the estimated, true, and conventional parameter vectors, respectively.  
These results demonstrate that the LSTM-enhanced system provides significant improvements in both parameter convergence and control quality, making it particularly suitable for applications requiring robust real-time adaptation and precise control under varying conditions, as confirmed by the comparative analysis shown in Table 6.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Metric** & **Conventional** & **LSTM Enhanced** & **Improvement** \\ \hline Convergence Time (ms) & 250 & 87.5 & 65\% \\ Estimation Accuracy (\%) & 85 & 98.3 & 45\% \\ Tracking Error (\%) & 12.5 & 4.8 & 61.6\% \\ Estimation Lag (ms) & 45 & 12 & 73.3\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Parameter convergence metrics.  
A comprehensive analysis of the limitations and applicability of the proposed SOPTD model across various industrial processes reveals several key challenges. The model exhibits limitations in approximating complex systems, including accuracy degradation in high-order systems, performance reduction in nonlinear systems, and difficulties in parameter estimation for time-varying systems. Additionally, challenges exist in representing system dynamics, particularly in the simplification of complex dynamics and handling of multiple time-delay systems.  
Our analysis of industrial process applicability shows that chemical processes and fluid control systems demonstrate high applicability, while heat treatment and reactor control processes require additional considerations for nonlinearity compensation. To address these limitations, we propose enhancement strategies, including nonlinearity compensation algorithms, adaptive parameter estimation methods, and hybrid modeling techniques. The control structure improvements incorporate robust control methods and predictive control elements.  
Simulation results confirm that the SOPTD model, despite its limitations, can be effectively applied across various industrial processes when appropriate compensation strategies are implemented. The proposed hybrid approach offers a balanced solution between model accuracy and practical implementation, while future research should focus on developing improved algorithms and empirical validation across diverse industrial settings.