Deep Neural Network Based Modeling Method of Piezoelectric Fast Steering Mirrors Using Koopman Operator Theory

Because the piezoelectric fast steering mirror (PFSM) typically has fast response speed, high resolution, low energy consumption, and low electrical noise, it plays a crucial role in precision tracking systems for laser communication. However, accurately describing the dynamical behaviour of the PFSM is challenging due to the existence of hysteresis. Therefore, this study explores modeling algorithms for PFSM system based on the Koopman operator. We primarily construct the Deep-Koopman architecture, utilizing a two-layer deep neural network (DNN) instead of manually selecting basis functions. This maps the system state to a high-dimensional space and represents the evolution trajectory of PFSM system based on the linear representation of the Koopman operator, which is more precise compared to the model solved based on extended dynamic mode decomposition (EDMD). By incorporating time-delay embedding to process the original sampled sequences, the prediction accuracy is further improved, leading to the delay-deep Koopmam model. Its effectiveness and advantage have been verified through experiments.