Bio-Inspired Controller Design via Dholes-Inspired Optimization: A Novel Gompertz Function-Augmented PID Strategy for Electro-Hydraulic Actuator Control

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Electro-hydraulic actuator systems are widely used in precision motion-control applications; however, their displacement regulation remains challenging because fast response, low overshoot, and high steady-state accuracy must be achieved simultaneously under strongly dynamic operating conditions. In this study, a proportional-integral-derivative (PID) controller augmented with a Gompertz function (PID-G) is proposed for the position control of a four-way valve-controlled linear actuator, and its parameters are tuned by the recently introduced dholes-inspired optimizer (DIO). First, a control-oriented mathematical model of the electro-hydraulic actuator system is established by combining the valve and actuator dynamics. Then, the PID-G structure is formulated by incorporating a nonlinear Gompertz-based term into the conventional PID framework, and the resulting seven-parameter tuning problem is cast as an optimization task using a composite objective function that accounts for overshoot, steady-state error, rise time, and settling time. The effectiveness of DIO is evaluated comparatively against flood algorithm (FLA), covariance matrix adaptation evolution strategy (CMA-ES), and particle swarm optimization (PSO) under identical simulation conditions. The results show that DIO provides the best optimization performance, yielding the lowest best, average, and standard-deviation values of the objective function among the compared algorithms. In the time domain, the DIO-based PID-G controller achieves the most favorable overall response with a rise time of 0.079511 s, a settling time of 0.099326 s, an overshoot of 0.15110%, and a steady-state error of 0.089317%. The superiority of the DIO-based design is further confirmed by lower values of error based performance metrics (IAE, ISE, ITAE, and ITSE), improved convergence characteristics, and statistically significant advantages in the Wilcoxon test. Additional comparisons with different (PI, PID, 2DOF-PID, and FOPID) controllers also demonstrate that the proposed PID-G structure provides markedly better transient and error-based performance when tuned by DIO. Frequency-domain and varying-setpoint results further indicate satisfactory stability margins, robust tracking ability, and bounded control effort. Overall, the study shows that combining DIO with a Gompertz-augmented PID structure constitutes an effective strategy for high-performance electro-hydraulic actuator displacement control.

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