A GL(n, ℝ) differential algebraic equation method for numerical differentiation of noisy signal

We show that the problem "real-time numerical differentiation" of a given noisy signal in time, by supplementing a compensated controller in the second-order robust exact differentiator, the tracking differentiator or the continuous hybrid differentiator, can be viewed as a set of differential algebraic equations (DAEs) to enhance a precise tracking of the given noisy signal. Thus, we are able to solve the highly ill-posed problem of numerical differentiation of noisy signal by using the Lie-group differential algebraic differentiators (LGDADs) based on the Lie-group GL(n, R), whose accuracy and tracking performance are better than before. The "index-two" differentiators (ITDs), which do not need any parameter in the governing equations, can efficiently depress the chattering, and also improve the dynamical performance under noise. The tracking results obtained by the LGDADs and the ITDs are quite promising.