A Schrödinger and differential evolution framework for PID based liquid level control

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In this study, a hybrid Schrödinger optimizer with differential evolution (h-SRADE) is proposed for optimal tuning of proportional–integral–derivative (PID) controllers applied to a three-tanks liquid level control system. The original Schrödinger optimizer (SRA), inspired by wave–particle duality and quantum mechanics principles, provides a balanced exploration–exploitation mechanism; however, its convergence sensitivity in the later optimization stages motivates further enhancement. To address this limitation, differential evolution mutation and crossover operators are embedded into the SRA framework, forming the proposed h-SRADE algorithm with improved refinement capability and convergence stability. The control problem is formulated using a linearized third-order model of the three-tanks liquid level system derived from mass balance principles. PID controller parameters are optimized by minimizing a composite fitness function that simultaneously accounts for percent overshoot, steady-state error, settling time, and rise time. The optimization process is conducted within predefined practical bounds to ensure controller feasibility and robust closed-loop operation. Extensive simulation studies are carried out to evaluate the performance of the proposed method. Statistical analysis, supported by a non-parametric Wilcoxon signed-rank test, confirms the statistical significance and robustness of h-SRADE over the conventional SRA. Time-domain and frequency-domain analyses demonstrate that the h-SRADE-based PID controller achieves faster settling, reduced overshoot, lower steady-state error, and satisfactory robustness margins. Furthermore, comparative studies with several state-of-the-art optimization-based PID tuning approaches reported in the literature reveal that the proposed method provides superior dynamic performance and overall control quality. The obtained results indicate that the hybridization of the Schrödinger optimizer with differential evolution effectively enhances convergence behavior and control performance. Consequently, the proposed h-SRADE framework offers a reliable and efficient simulation-based solution for PID controller tuning in liquid level systems, providing a strong foundation for future extension toward nonlinear plant models, varying operating points, and other complex engineering control applications.

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