Proportional-integral-derivative improved SPGD algorithm for wavefront sensorless correction in adaptive optics

The stochastic parallel gradient descent (SPGD) algorithm is widely used for wavefront correction in wavefront sensorless adaptive optics (WFSless AO) systems. Conventional implementations typically employ a fixed gain, which slows convergence and degrades final accuracy under large aberrations. To address this limitation, we augment SPGD with a proportional-integral-derivative (PID) controller that adaptively modulates the gain during iterations. By leveraging the synergistic effects of the proportional, integral, and derivative components, the proposed method adaptively adjusts the gain coefficient during the SPGD iteration process, thereby integrating information from the current gradient, the accumulated historical error, and the gradient variation trend. This design accelerates convergence and enhances the ability of the algorithm to escape saddle points and local minima. Simulations show that, compared with SPGD, the PID-SPGD algorithm achieves approximately a 40% increase in convergence speed and about a 6% improvement in correction accuracy. Experiments verify that PID-SPGD reduces the mean radius (MR) by 23.7% and increases the intensity by roughly fourfold compared with SPGD. These results provide a theoretical rationale and practical basis for applying PID-SPGD to wavefront correction in WFSless AO systems.