Identification of Fast-rate Systems using Slow-rate Image Sensor Measurements

In this paper, we propose a method to identify the frequency response of a fast-rate motion system using measurements from a slow-rate image sensor. The integrative nature of the sensor produces a blurred image of the observed moving feature, e.g., of a laser beam deflected on a steering mirror, when the image moves much faster than the exposure period. The key concept of this paper is to extract the parameters describing the motion of the observed feature from the image blur by treating the image sensor as a nonlinear temporal-to-spatial transformation, under a known input excitation, i.e., a sinusoid. The output signal amplitude and phase characterize the frequency response of the system at the specific excitation frequency. This problem is posed as a nonlinear minimization: finding parameters to match the predicted spatial distribution with the measurement. The nonlinear mapping is shown to be noninvertible; therefore, the solution of the nonlinear minimization is nonunique. Two methods are proposed for avoiding this aliasing problem: 1) by imposing a continuity constraint and 2) by solving the minimization over two different exposure periods. The efficacy of the proposed frequency response identification approach for a multi-input/multioutput system is experimentally demonstrated by accurately obtaining the frequency response of a fast steering mirror up to 500 Hz using a 30-Hz CCD camera.