Designing a cascaded exponential PID controller via starfish optimizer for DC motor and liquid level systems.

In this study, a novel cascaded exponential proportional-integral-derivative (exp-PID) controller tuned by the starfish optimization algorithm (SFOA) is proposed for enhancing the transient and steady-state performance of nonlinear dynamic systems. The design objective is to achieve improved adaptability, robustness, and precision under varying operating conditions and external disturbances. The exponential PID structure introduces nonlinear modulation in the proportional and derivative components, enabling smoother control action and superior damping characteristics compared to conventional PID and fractional-order PID designs. The proposed SFOA-based exp-PID controller is validated on two benchmark systems: a DC motor speed control system and a three-tank liquid-level process. Across multiple independent trials, the controller achieved outstanding results, with the DC motor system attaining a rise time of 0.0039 s, settling time of 0.0083 s, and zero overshoot, while the three-tank system reached a rise time of 1.72 s, settling time of 2.47 s, overshoot of 1.5%, and steady-state error of 9.22 × 10⁻⁵%. Comparative analyses with recently developed algorithms (including the flood algorithm, greater cane rat algorithm, mantis search algorithm, and dandelion optimizer) as well as previously reported methods demonstrate the superior convergence behavior, stability, and accuracy of the proposed controller. Statistical evaluations further confirm the method's robustness and consistent performance across repeated runs.