## 5 Conclusions  
This paper presented a control scheme that integrates hybrid systems, exponential functions, and a PID controller to achieve smooth trajectory transitions of the process variable to the setpoint. The hybrid system enables dynamic resetting of the exponential function parameters, ensuring that even in the presence of discontinuous setpoint changes, the process variable smoothly converges to the new target.  
Through both numerical simulations and real-time experiments, it was demonstrated that the proposed approach effectively reduces transient oscillations typically caused by poorly tuned PID controllers. This results in reduced actuator effort and, consequently, energy savings. Notably, the same PID gain values were used for both the classical and hybrid PID controllers, highlighting the improved performance of the proposed method without additional tuning requirements.  
It is important to highlight a potential limitation of the proposed hybrid control strategy. Specifically, the method's performance may be compromised if the decay rate \(\alpha\) is selected too aggressively, that is, when the resulting exponential trajectory demands a faster closed-loop response than the original PID controller can stably achieve. In such cases, the hybrid mechanism may impose excessive control effort or even destabilize the system. Therefore, to preserve the stability guaranteed by the original PID controller, it is suggested to select \(\alpha\) such that the exponential reference evolves more slowly than the system under the influence of the original PID. This consideration is crucial for practical implementation and underscores the need for careful parameter selection, as explained in more detail in Remarks 2 and 3.  
As future work, it is proposed to explore alternative functions to replace the exponential function, providing different trajectory shaping strategies for setpoint tracking.  
\begin{table}
\begin{tabular}{c c} \hline
**Decay Rate \(\alpha\)** & **Relative Energy Savings (\%)** \\ \hline
0.5 & 84.55 \\
1.0 & 75.56 \\
2.0 & 60.26 \\ \hline \end{tabular}
\end{table}
Table 1: Relative energy savings of the hybrid PID controller for different values of \(\alpha\).  
Additionally, extending the approach to other control applications and refining the hybrid switching mechanism could further enhance its adaptability and effectiveness.  
Conceptualization, I.LL.-P. and J.A.M.-C.; formal analysis, J.C.G.-H., I.I.L.-P., J.O.E.-A. and L.A.P.-C.; investigation, J.O.E.-A., R.T.-H. and J.A.M.-C.; methodology, J.O.E.-A., R.T.-H., L.A.P.-C. and J.A.M.-C.; project administration, J.A.M.-C.; writing--original draft, J.A.M.-C.; writing--review and editing, I.I.L.-P., J.O.E.-A., J.C.G.-H., R.T.-H. and J.A.M.-C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data Availability StatementData is contained within the article.  
The authors gratefully acknowledge the partial support provided for this research by the Secretaria de Ciencia, Humanidades, Tecnologia e Innovacion (Secichit) through the SNII (Sistema Nacional de Investigadoras e Investigadores) scholarship and project CBF-2025-I-641. Additionally, the Instituto Politecnico Nacional contributed to this work through research project 20253657, as well as by providing scholarships as part of the programs EDI (Estimulo al Desempeno de los Investigadores), COFAA (Comision de Operacion y Fomento de Actividades Academicas), and BEIFI (Beca de Estimulo Institucional de Formacion de Investigadores).  
The authors declare no conflicts of interest.