Data-driven modeling and control of an X-ray bimorph adaptive mirror

Abstract

Adaptive X-ray mirrors are being adopted on high-coherent-flux synchrotron and X-ray free-electron laser beamlines where dynamic phase control and aberration compensation are necessary to preserve wavefront quality from source to sample, yet challenging to achieve. Additional difficulties arise from the inability to continuously probe the wavefront in this context, which demands methods of control that require little to no feedback. In this work, a data-driven approach to the control of adaptive X-ray optics with piezo-bimorph actuators is demonstrated. This approach approximates the non-linear system dynamics with a discrete-time model using random mirror shapes and interferometric measurements as training data. For mirrors of this type, prior states and voltage inputs affect the shape-change trajectory, and therefore must be included in the model. Without the need for assumed physical models of the mirror's behavior, the generality of the neural network structure accommodates drift, creep and hysteresis, and enables a control algorithm that achieves shape control and stability below 2 nm RMS. Using a prototype mirror and ex situ metrology, it is shown that the accuracy of our trained model enables open-loop shape control across a diverse set of states and that the control algorithm achieves shape error magnitudes that fall within diffraction-limited performance.

A real-time machine-learning-driven control system of a deformable mirror for achieving aberration-free hard X-ray wavefronts

“…Then, the voltage applied to the bender actuator was tuned to 170 V to bend the mirror surface and form the focus in the desired position. In this study, as in [11], we limited the voltage change applied to the 18 local actuators to within a range of [-100 V, 100 V] for the training data collection.…”

“…The controller is built by extending the system described in [11]. The system is a two-part framework: (i) an ML-based predictive model of the non-linear dynamics of the adaptive optics using a feedforward neural network and (ii) a controller to drive the mirror to obtain a desired spatial distribution of a chosen wavefront property using non-linear quadratic cost regulation over a finite horizon.…”

“…The meaning of 𝑤𝑝 𝑡+1 is that the model predicts the wavefront property spatial distribution one time step in the future given a finite history of wavefront property shapes and voltages, starting with its current shape, and applied set of voltages, going back to 𝑛 time steps. In this study, we kept ∆𝑡 = 2s, as used in [11], for comparison purposes and to safely have time to change the input voltages and collect the wavefront measurement. The chosen wavefront property is detected as a 2D spatial distribution and mapped to 1D profiles along vertical and horizontal coordinates.…”

“…The controller algorithm solves a quadratic cost function with an expression similar to the iterative linear quadratic regulator [23] that penalizes the prediction error at each intermediate step. As described in [11], the controller provides the succession of voltages to be applied to the actuators by directly minimizing the non-convex objective function using Adam [21]. We explicitly chose to maintain the conditions of our previous study for comparison purposes, even though it does not theoretically guarantee the optimality of the converged solution.…”

“…Traditional iterative control methods based on the linear response model [9,10] can take tens of minutes to converge to the targeted solution and are, therefore, unsuitable for real-time AO control. Recent work demonstrated the possibility of controlling a bimorph mirror using machine learning (ML) based on a neural network (NN) trained on a collection of surface shapes measured by a Fizeau interferometer [11]. A closed-loop, fast-shaping control of a bimorph mirror was recently reported for the surface shape feedback system of an array of interferometric sensors [12].…”