Multi-dictionary matching pursuit for servo error analysis applied to iterative learning control

Wavelets and associated multiresolution analysis has had a major impact on signal processing, data compression, computer vision, telecommunication and a variety of other engineering and scientific disciplines. However, all of the aforementioned applications are limited to either wavelet analysis or wavelet characterization. The use of wavelets as basis in the direct control of the system dynamics has not been exploited [G.W. Wei et al., 2003]. In this paper we show how we can better control the time-varying nonlinear behavior of a motion system, i.e. a wafer stage apparatus, by integrating an adaptive iterative learning control (ILC) technique with the time-frequency analysis of the servo error when acceleration feed-forward is applied. The performance of the presented learning control technique relies on an accurate identification of time-varying nonlinear and stochastic effects present in the servo error signal. The identification of these effects is performed by means of time-frequency analysis of the servo error and therefore, our goal is to obtain a high-resolution time-frequency energy distribution of the analyzed signal. In this paper we present a comparative analysis of the servo error energy density by four means: Wigner distribution; piecewise-linear Wigner distribution; adaptive signal decomposition over one dictionary of modulated versions of orthonormal bases of compactly supported wavelets having a fixed number of vanishing moments (which we call simple atomic dictionary); and by means of combining several simple atomic dictionaries into a complex atomic dictionary. We show that the latter approach leads to an improved time-frequency energy distribution.