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To gather information about physical parameters from electric or electronic methods, the signal to noise ratio (SNR) plays a decisive role. It can be used as a measure of goodness for the current data acquisition. With a laser Doppler vibrometer a contactless measurement of oscillating surfaces is available. It uses the Doppler Effect caused by the local deviation of an oscillating surface, which can be analyzed interferometrically. Depending on the roughness of the surface, interference phenomena can occur and are usually known as speckle effects. The coherence behavior of the light in a laser Doppler vibrometer can lead to destructive interference, with the result that the signal to noise ratio is too low to perform a sufficient measurement. This interference related phenomenon is also called speckle-dropout. To counteract this effect, the vibrometer was equipped with an adaptive optics, which can modify specifically the phase front of the coherent wave. In a first approach, the potential of signal optimization was investigated. Based on superposed Zernike polynomials, special phase pattern were calculated and written into that adaptive optics. Such polynomials are the common method to describe wave fronts in optical systems and, accordingly, are sufficiently precise analyzed. Each Zernike polynomial has a related coefficient to weight it individually in a superposition. These coefficients are the decision variables in an optimization algorithm. Correlated with a loop- back control, the coefficients can be interpreted as regulating variables. With the assumption that the system states are close enough to the optimal states, an extremum seeking control was developed to track and hold the system at that optimum. The algorithm depends on the successive parabolic interpolation, which was developed by Heath for one-dimensional problems. It was extended for solving a multi-dimensional problem definition and, furthermore, embedded into a loop- back control.
This paper presents the current design of the extremum seeking control and discusses the benefits with some results of the improved measurements and is structured as follows. The investigated system will be introduced in section 1. Also, the measurement concept will be shown in this section. It is followed by the mathmatical framework in section 2. It gives an overview over the developed concepts based on the successive parabolic interpolation. In addition, the Newton method, an approach to solve a non-linear optimization problem, is described. The next section 3 contains the nucleus of the work. It deals with the derivation of the developed control law of the extremum seeking control. The paper is completed with a results section and the conclusion of this work.
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