Output Feedback Displacement Control for Piezoelectric Deformable Mirrors with Unknown Hysteresis and Dynamics Coupling

Deformable mirrors (DMs) are key components of adaptive optics (AO) systems, enabling precise wavefront control in a wide range of applications. When actuated by piezoelectric actuators (PAs), the resulting PA-DM can exhibit nonlinear behaviors, including hysteresis and dynamic coupling between actuators, which degrade performance. This paper presents a displacement-control strategy for PA-DMs that explicitly accounts for these unknown nonlinearities and inter-actuator couplings. The proposed approach combines state feedback control for accurate displacement tracking with an extended high-gain observer (EHGO) to estimate the unknown nonlinear effects. A 41-actuator PA-DM is used as a case study to evaluate the controller's performance. Simulation results show that the proposed controller significantly improves performance, achieving the desired Zernike-mode aberration reconstruction with a peak residual of 0.01 μm, compared to 1.2 μm obtained using a conventional control approach. The controller is further tested under realistic atmospheric conditions: a fast-changing wavefront with a Greenwood frequency of 35 Hz representing mid-level turbulence, and a slow-changing wavefront with a Greenwood frequency of 3.8 Hz representing low-level turbulence. The results demonstrate consistent performance across both scenarios, with an average RMSE of 0.23 μm for the fast-changing wavefront and 0.021 μm for the low-changing wavefront.