Wavelets in control engineering

In this report a for control engineering new analysis and filter technique is presented: the wavelet transform. Basically the wavelet transform can be described as a time-frequency technique. It presents the signal on both a time as frequency axis, comparable with a windowed Fourier transformation that is shifted along the time axis. The basic difference with Fourier techniques is that the window width is changed as function of analyzing frequency. Furthermore there is much freedom in choosing the analysis function, which let the wavelet transform do more than only discover frequency information. Besides as analysis technique the wavelet transform can be used for filtering purposes. With the discrete version of the wavelet transform (DWT) a decomposition in frequency-dependent coefficients is possible, from which the original signal can be reconstructed. The coefficients can be processed in several ways, giving the DWT exotic properties compared with linear filters. To understand this, an introduction in filter banks is presented which forms the basis for the discrete wavelet transform. For both analysis and filter technique general applications are presented. Next is examined if wavelets can be interesting in control engineering. Since the origins of the DWT lie in the field of signal processing, the available algorithms are efficient but not optimal with respect to the delay times. Therefore a real-time wavelet filter algorithm is derived, which can also be used for analyzing purposes. Expressions for the delay time show that such a filter cannot be used as an online controller. However, for off-line filtering or in supervisory loops wavelets can be interesting. Two application are worked out: 1. Wavelet filters for encoder quantization denoising Encoders, widely used in motion systems, always generate noise which is especially annoying when derivatives of the measurement have to be calculated. Often lowpass filtering is applied, but if the amplitudes and frequencies of a signal are spread over a broad range, this approach fails. After an study on quantization noise, the properties of the DWT seems very suitable to minimize quantization effects. Some design rules for a dedicated wavelet filter are proposed and the technique is tested on several artificial and real-life signals. 2. Online feature detection on a CD-player setup Sometimes it is desirable to adapt the controller if certain unwanted events disturb the closed-loop. If these events are the result of external disturbances, often the only way to detect them is using a measurement which is part of the closed-loop. Fast detection can then be beneficial to minimize performance loss. Wavelets show good results in isolating features (time-patterns in signals), especially short-living events. Building dedicated waveforms can improve the results and the real-time algorithm makes fast detection possible. The experiences are validated on a servo-loop of a CD-player setup, on which external disturbances are presented in the form of shocks and disc-faults. For some disturbances very early detection is possible: the wavelet filter already isolates features where the measurement still moves within the noiselevel.