## 1 Introduction  
Proportional-Integral-Derivative (PID) controllers are widely used in the process industry because they are easy to understand, simple in structure, and easy to implement. While traditional PID controllers remain fundamental, modern control challenges require more sophisticated approaches that can adapt to complex dynamic systems and uncertain environments. The most important aspect when designing a PID controller is determining its parameters, which has led to extensive research on parameter determination methods. Among them, Ziegler-Nichols Rule, Cohen-Coon Method, Integral Absolute Error (IAE), Integral Square Error (ISE), Integral Time Absolute Error (ITAE), and Internal Model Control (IMC) methods are widely used [1, 2, 3, 4, 5, 6]. However, most of these methods are easy to apply for primary and secondary models but are not easily applicable for higher-order models. Recent advances in artificial intelligence and machine learning have opened new possibilities for control system design. In particular, Long Short-Term Memory (LSTM) networks have shown promising results in time-series prediction and system identification tasks [7, 8]. Combining these modern techniques with traditional control methods creates a powerful hybrid approach that can leverage the benefits of both worlds. This hybrid methodology not only improves control performance but also enhances system adaptability and robustness. Research is actively being conducted on universal tuning methods for PID controllers that can achieve good performance for high-order models or processes with large delay times. A recent study by [PERSON] et al. demonstrated the effectiveness of deep learning in PID control optimization, achieving significant improvements in control accuracy. However, their approach focused primarily on parameter optimization without considering real-time adaptation [9, 10, 11]. Our proposed method extends this work by incorporating real-time model identification and adaptive control strategies, offering improved performance in dynamic environments. The research on reduction models is performed in both time domain and frequency domain and has achieved significant progress through the integration of deep learning approaches [12, 13]. Comparing model reduction in the time domain with model reduction in the frequency domain, the reduced model obtained in the time domain is comparable to the response of the real model in the time domain but not in the frequency domain. The model reduction method in the frequency domain is more complex than the model reduction method in the time domain, but it shows that the reduced model is comparable to the response of the real model in both time domain and frequency domain. In the time domain, typical model reduction methods include Pade approximation and Routh stability approximation [12]. [PERSON]'s model reduction method is a typical method in the frequency domain [12, 13, 14, 15, 16]. However, [PERSON]'s method cannot be applied when the real model does not pass the entire region of the Nyquist line, and there is a case where the steady-state response between the high-order model and the reduced model is inconsistent in the model reduction process. To address these limitations, our proposed hybrid approach incorporates LSTM networks for dynamic system identification and Genetic Algorithms for parameter optimization [17]. The LSTM component enables better capture of temporal dependencies and system dynamics, while the Genetic Algorithm optimizes the control parameters for enhanced performance [18]. This combination allows the system to do the following:  
1. Adapt more quickly to changes in system dynamics;
2. Handle nonlinearities more effectively;
3. Provide more robust performance under uncertainty;
4. Reduce computational overhead through efficient model reduction;  
To solve previous problems, we employ a hybrid methodology combining numerical computation, Genetic Algorithms, and LSTM networks to obtain the coefficients of the reduced model [19].