1. 1 Filtering Professor Asgeir J. Sørensen, Department of Marine Technology, Norwegian University of Science and Technology, Otto Nielsens Vei 10, NO-7491 Trondheim, Norway E-mail: Asgeir.Sorensen@ntnu.no

2. 2 Outline Motivation Analog filtering –Ideal filters –Nonideal low pass filter –Nonideal high pass filter –Bandstop filter - Notch

3. 3 Filtering of measurement noise. Most sensor signals are contaminated by some noise that may have negative impact on the controller performance. By filtering we achieve a change in the relative amplitudes of the frequency components in a signal or even elimination of some frequency components. Reconstruction of non-measured data. For many applications important process states are not measured. Model based filtering techniques - state estimation - can be applied to reconstruct unmeasured signals and perform filtering before the signals are used in a feedback control system. Dead reckogning. All kind of equipment will fail according to some failure rate. Model based filters replace the measured signal by a mathematical model prediction for some period of time. Motivation

4. 4 Shaping Filtering. The power spectral density of the output of a linear system satisfies: Motivation (cont.) Then, the filter can be designed to Simulate time series of a signal with a particular power spectral density by filtering white noise. Example Simulation of Ship Roll Motion Fourier representation (multi sine)Filtered white noise (shaping filter)

5. 5 Conventional filters used in marine control systems Low pass filter suppressing e.g. noise High pass filter Band stop filters, e.g. notch filter Band pass filter Cascaded low pass and notch filter for wave filtering

6. 6 Analog low pass filter - Tolerances

7. 7 Low pass filter - Butterworth filter n = 1:b Ý s Þ = g c s +g c x % f +g c x f =g c x n = 2:b Ý s Þ = g c 2 s 2 + 2 g c s +g c 2 x 6 f + 2 g c x % f +g c 2 x f =g c 2 x n = 3:b Ý s Þ = g c 3 s 2 +g c s +g c 2 Ý s +g c Þ x 4 f + 2 g c x 6 f + 2 g c 2 x % f +g c 3 =g c 3 x

8. 8 High pass filter High pass filter may be designed by substituting in the equation describing low pass filter, e.g. The corresponding high pass filter becomes

9. 9 Band stop filter - Notch filter

10. 10 Examples Filter examples by Finn Haugen, KYBSIM http://techteach.no/kybsim/filters/