On the identification of fast dynamics using slow rate camera measurements

This paper addresses the identification of the dynamics of a high bandwidth motion system using measurements from a slow rate integrative sensor. The motivation of this work is the need to characterize the dynamics of adaptive optics elements, such as deformable mirrors, using image array based sensors, such as CCD or CMOS wavefront sensors, which sample at a significantly lower rate than the actuator bandwidth. The integrative nature of the sensor produces a blurred image when the image moves much faster than the exposure period. The key concept of this paper is to extract system dynamics from the image blur. We consider the image blur as a nonlinear temporal-to-spatial transformation under a known input excitation, such as a sinusoid. The output signal parameters, such as amplitude and phase, characterize the frequency response of the system at the specific excitation frequency. This problem may be posed as a nonlinear minimization: finding output signal parameters to match the predicted spatial distribution with the measurement. However, the nonlinear mapping is not one-to-one, or, equivalently, the solution of the nonlinear minimization is non-unique. We propose two methods for avoiding this aliasing problem: by imposing a continuity constraint or by solving the minimization over two different exposure periods. The efficacy of the proposed identification approach for a single-input/single-output system is experimentally demonstrated by accurately obtaining the frequency response of a fast steering mirror up to 500Hz using a 30 Hz CCD camera.