Calculation of the modified control matrix for a selected unimorph deformable mirror to compensate the piezoelectric hysteresis effect using the inverse Bouc–Wen model

The hysteresis behavior of piezoelectric actuators degrades the positioning accuracy and bandwidth of nano-positioning systems. Therefore, considering the hysteresis of piezoelectric deformable mirrors is completely essential and also improves the modeling accuracy of adaptive optics layouts. Because of the unique adaptability and mathematical flexibility of the Bouc-Wen model it has gained popularity, and as a result, in many scientific applications, it is one of the most conventional models typically employed to describe nonlinear hysteretic systems. Among different deformable mirrors, a unimorph piezoelectric deformable mirror is a suitable choice to be used in adaptive optics systems because of its relative convenience and cost-effective production. This paper proposes a new, to the best of our knowledge, approach to determine the influence function and the voltage control matrix of a specific unimorph mirror by considering a simplified inverse Bouc-Wen hysteresis model as a frequency function. Then the results for two selected standard Zernike modes, defocus and astigmatism-x, have been simulated using Comsol Multiphysics and MATLAB at a range of 5 to 100 Hz. For a more comprehensive comparison, the root-mean-square error and the coefficients of the Zernike terms have been applied as two criteria. According to the simulation results, the hysteresis effect of piezoelectric actuators has been significantly compensated by applying the inverse Bouc-Wen model at different frequencies, especially for higher frequencies. The effectiveness of the inverse Bouc-Wen model to compensate the hysteresis has been observed in astigmatism-x mode slightly more than in the defocus mode.