A softsign-enhanced PIDA controller for automatic voltage regulation with robustness analysis

Automatic voltage regulator (AVR) systems require fast, accurate, and robust voltage regulation under parameter variations, external disturbances, measurement noise, and actuator limitations. This study proposes a softsign-enhanced proportional-integral-derivative-acceleration (SSE-PIDA) controller for AVR applications and optimally tunes its eight parameters using the Schrödinger-based optimization algorithm (SRA). The proposed controller combines filtered derivative and acceleration actions with a bounded softsign nonlinear branch, aiming to improve transient damping, reduce excessive control action, and enhance robustness under non-ideal operating conditions. To distinguish the contribution of the optimizer from that of the controller structure, the same SSE-PIDA architecture is tuned using SRA, TOC, ALA, GA, and WOA under identical optimization settings. The SRA-tuned SSE-PIDA controller achieves a rise time of 0.0321 s, a settling time of 0.0609 s, and zero overshoot in the nominal unit-step response. It also provides the lowest best, average, worst, and standard deviation fitness values among the compared optimizers. Further comparisons with recent PID-, FOPID-, and PIDD 2 -based AVR controllers show faster transient response and improved damping performance. Robustness is evaluated under reference changes, disturbance, measurement noise, input-limiter effects, and parametric uncertainty. The simulation results indicate that the proposed SRA-based SSE-PIDA controller offers a promising high-speed and robust AVR control strategy.