A new SIMO filter for the estimation of higher order derivatives

Abstract

This research proposes a new differentiator for estimating higher order derivatives of an input signal. The main reason why higher order derivatives are necessary is that Active Inference makes use of generalized coordinates. This means that it keeps internally track of higher order temporal derivatives of states, inputs and measurements. The difficult part of generalized coordinates are the generalized measurements, because these should be measured from a physical system. However, it is not always possible to measure all states of a system and it is definitely not possible to measure all higher order derivatives. A solution to this is to estimate the derivatives of states by using real time differentiators.

A short literature study about differentiators is conducted and found that the state of the art differentiator is the algebraic estimation approach differentiator (AEAD). However, this and other differentiators have the problem that when they are converted to discrete time, they cannot track polynomial signals anymore. For that reason, this thesis proposes a new differentiator: the recurrent
low pass algebraic differentiator (RLPAD).

The proposed differentiator is compared with the (AEAD) in two experiments. The first experiment evaluates the performance based on three analytical inputs with known higher order derivatives. The three inputs are: a polynomial, a sine and a combination of the two. Additionally these three inputs are corrupted by Gaussian white noise. This experiment concludes that the proposed method outperforms the (AEAD) significantly when a polynomial or polynomial combined with sine input is used. They perform similar for sine inputs, although (RLPAD) is considered slightly better.

The second experiment evaluates how the two differentiators perform when real sensor data is used. In order to conduct the experiment, the raw data is smoothed by a Savitzky-Golay filter to create a ground truth about the derivatives. This experiment concludes that the differentiators have an optimal set of parameters, where either the one or the other performs better. The proposed method performs a lot better when few derivatives are estimated. On the other hand (AEAD) performs better when there are more derivatives estimated. However, this is not true when the noise gets stronger. (AEAD) is more sensitive to noise than the proposed method.

Based on the experiments, the proposed method (RLPAD) is the better differentiator for creating generalized measurements in Active Inference for three reasons. The first is that it can track polynomial signals. The second is that it has stronger noise attenuation. And the third reason is that it is computationally faster.