## 5 Conclusions and Future Works  
Nonlinear equations effectively represented the proposed nonlinear model of the interconnected tank system. To facilitate control using a fractional PID controller, it was necessary to linearize these equations around an equilibrium point. This was accomplished by developing the Taylor Series and applying the Laplace transform, resulting in a transfer function suitable for fractional PID control. The linearization process enabled the accurate modeling and subsequent control of the system's dynamics, ensuring that the complex behavior of the nonlinear system could be managed effectively using advanced control techniques. The implementation of fractional PID control for the system's transfer function involved tuning the controller constants and fractional orders using meta-heuristic algorithms: Ant Colony Optimization (ACO), Gray Wolf Optimizer (GWO), and Flower Pollination Algorithm (FPA). Each algorithm was applied with different performance error criteria. The system's responses to a step input were simulated on the Matlab platform, demonstrating the feasibility and robustness of these algorithms in tuning the FOPID controllers. This approach allowed for the precise adjustment of the controller parameters to achieve desired performance characteristics in a nonlinear environment. The system's transient response to step inputs was thoroughly analyzed for each tuning strategy and performance index. Key metrics such as overshoot, settling time, and rise time were extracted from the simulation results. For instance, the GWO algorithm resulted in an overshoot of 10.0831% and a settling time of 5.1937 s, while the ACO and FPA algorithms exhibited lower overshoots and shorter settling times. Additionally, the system's ability to reject disturbances was evaluated, revealing that the ACO and FPA algorithms provided superior disturbance rejection capabilities, maintaining system stability and minimizing deviation from the setpoint more effectively than GWO. These response curves provided a comprehensive understanding of how each tuning method influenced the system's dynamic behavior, highlighting the strengths and weaknesses of each algorithm in achieving optimal control performance and robust disturbance rejection. A comparative validation using performance metrics and graphical analysis of the output responses established the efficiency of each tuning method. The GWO algorithm showed higher overshoot and settling times than ACO and FPA. Specifically, the FPA algorithm demonstrated superior performance with an ISE criterion value of 297.83, lower than the ACO's 389.536. Furthermore, the FPA algorithm required fewer iterations to converge to the optimal proportional, integral, and derivative constants (three iterations for constants and four iterations for fractional orders) compared to the ACO algorithm  
Figure 8: Algorithm iterations to find fractional orders \(\lambda\), \(\mu\). (four iterations for constants and eight iterations for fractional orders). These findings underscore the FPA algorithm's effectiveness in achieving faster convergence and better performance, making it a preferable choice for tuning fractional PID controllers in nonlinear systems. Conceptualization: R.P. and S.S.; Methodology: W.P. and M.A. ; Software: R.P. and W.P. ; Validation: S.S. and M.A. ; Formal Analysis: R.P. ; Investigation: W.P. ; Resources: M.A. ; Data Curation: W.P. ; Writing--Original Draft Preparation: R.P. and W.P. ; Writing--Review and Editing: M.A. and S.S.; Visualization: W.P. ; Supervision: R.P. and S.S.; Project Administration: W.P. ; Funding Acquisition: W.P. All authors have read and agreed to the published version of the manuscript. This research was funded by Universidad Politecnica Salesiana (UPS) also supports the Master's program in Electronics and Automation and covers the costs associated with publishing in Algorithms Journals. Additionally, UPS funded my Ph.D. studies at Ferrara University.