We study the asymptotic stability of Lur'e systems with butterfly hysteresis nonlinearities modeled by the Preisach operators with respect to a set of equilibrium points. We present the input-output rate property of the Preisach operators that exhibit butterfly hysteresis behavior. Based on this characterization, we present sufficient conditions on the linear systems that guarantee the asymptotic stability of the closed-loop system to a set of equilibrium points via circle criterion. Finally numerical simulations are presented to demonstrate these results.

We present the conceptual design and initial development of the hysteretic deformable mirror (HDM). The HDM is a completely new approach to the design and operation of deformable mirrors (DMs) for wavefront correction in advanced imaging systems. The key technology breakthrough is the application of highly hysteretic piezoelectric material in combination with a simple electrode layout to efficiently define single actuator pixels. The set-and-forget nature of the HDM, which is based on the large remnant deformation of the newly developed piezomaterial, facilitates the use of time division multiplexing to address the single pixels without the need for high update frequencies to avoid pixel drift. This, in combination with the simple electrode layout, paves the way for upscaling to extremely high pixel numbers (≥128 × 128) and pixel density (100∕mm 2 ) DMs, which is of great importance for high spatial frequency wavefront correction in some of the most advanced imaging systems in the world.

We present modeling and analysis of a hysteretic deformable mirror where the facesheet interacts with a continuous layer of piezoelectric material that can be actuated distributively by a matrix of electrodes through multiplexing. Moreover, a method to calculate the actuator influence functions is described considering the particular arrangement of electrodes. The results are presented in a semi-analytical model to describe the facesheet's deformation caused by a high-density array of actuators, and validated in a simulation. The proposed modeling of an interconnection layout of electrodes is used to determine the optimal pressures the actuators must exert to achieve a desired surface deformation.

We study in this paper the control of hysteresisbased actuator systems where its remanence behavior (e.g., the remaining memory when the actuation signal is set to zero) must follow a desired reference point. We present a recursive algorithm for the output regulation of the hysteresis remnant behavior described by Preisach operators. Under some mild conditions, we prove that our proposed algorithm guarantees that the output remnant converges to a desired value. Simulation result shows the efficacy of our proposed algorithm.

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