Sascha Mayer

An interval arithmetic approach for the estimation of the domain of attraction

Since the analysis of asymptotic stability is not sufficient for safety-critical nonlinear systems, the analysis of the domain of attraction is the focus of current research. While several approaches for polynomial systems were presented in the last years, based on sums of squares/linear matrix inequalities (SOS/LMIs) techniques or using a sampling method, only a few approaches are available for non-polynomial systems. In this paper we present a new branch-and-bound-method using the Lyapunov stability theory. It is based on interval arithmetic and delivers lower and upper bounds for the maximum contour line of a given Lyapunov function, which bounds a subset of the domain of attraction. Our approach can applied for polynomial systems as well as for non-polynomial systems.

Parallel Computation of Domains of Attraction for Nonlinear Dynamic Systems

In this paper a parallel version of a new approach for investigation of the stability of nonlinear dynamic system swill be presented. It is an extension of an interval arithmetic approach which computes tight approximations for domains of attraction using the Lyapunov stability theory. Unfortunately, the execution time of the serial algorithm grows exponentially with an increase in the number of state space variables, which makes it time-consuming to solve tasks of higher dimensions. We will show that parallelization is a good strategy to overcome this drawback.

A new approach for signal loss compensation in a vibrometer

This paper presents an approach for signal loss compensation in vibrometry, a contactless measurement technique. Since a new scanning method based on computer generated holograms was developed, now signal loss caused by speckle effects can be reduced and in the best case significantly compensated. For this, multiple Zernike polynomials are used in a Zernike expansion mapped to a computer generated hologram, which modifies the wave front of a coherent laser beam. The coefficients of each Zernike polynomial are used as parameters in a global optimization. Solving this optimization problem with a suitable cost function leads to an increased signal level.

Extremum seeking control to avoid speckle-dropouts in a vibrometer

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.

Advanced techniques in vibrometry by using spatial light modulators

Vibrometry is a modern contactless measurement technique to analyze vibrations in different applications. It is extremely important for industry and research. Since spatial light modulators (SLM) have been developed, new techniques in vibrometry are possible. This paper presents two applications which use the SLM as actor instead of a galvanic mirror. Both techniques modify the phase of the wave front of a laser beam by using an SLM. On the one hand, it allows varying the focus plane and repositioning of a spot without any mechanical device. For that, special patterns have to be written to the SLM. On the other hand, overlay patterns can be used to reduce speckle drop-outs nearly without changing position and focus of the spot. For both applications an optimized signal level is the key to get sufficient results in a rougher environment.