[Paper Review] Parameter identification using the Hilbert transform

Andrew Allison, Derek Abbott|arXiv (Cornell University)|Oct 27, 2002

Advanced Measurement and Metrology Techniques6 references4 citations

TL;DR

This paper proposes a fast, approximate method for estimating the damping coefficient α and damped natural frequency ωd of a second-order system using the Hilbert transform to reconstruct the analytic signal from time-domain data. By applying the logarithm and differencing to the analytic signal, the method estimates α and ωd via the slope of log(y₂(t)), achieving −0.4% error in ωd and −5.5% error in α in simulations, with robustness enhanced by using median-based slope estimation.

ABSTRACT

Many physical systems can be adequately modelled using a second order approximation. The problem of plant identification reduces to the problem of estimating the position of a single pair of complex conjugate poles. One approach to the problem is to apply the method of least squares to the time domain data. This type of computation is best carried out in "batch" mode and applies to an entire data set. Another approach would be to design an adaptive filter and to use autoregressive, AR, techniques. This would be well suited to continuous real-time data and could track slow changes on the underlying plant. I this paper we present a very fast but approximate technique for the estimation of the position of a single pair of complex conjugate poles, using the Hilbert transform to reconstruct the analytic signal.

Motivation & Objective

• To develop a computationally efficient method for identifying the parameters of a damped oscillatory system from time-domain data.
• To estimate the damping coefficient α and damped natural frequency ωd of a second-order system using only sampled response data after a step or impulse excitation.
• To provide a fast, approximate alternative to iterative non-linear least squares or AR methods for real-time or batch parameter estimation.
• To reduce sensitivity to initial conditions and measurement noise by leveraging the analytic signal and median-based slope estimation.

Proposed method

• Reconstruct the analytic signal y₂(t) from real-time sampled data using the discrete Hilbert transform via FFT.
• Apply the natural logarithm to the analytic signal to obtain log(y₂(t)) = log(C) − jϕ + (−α + jωd)t.
• Compute the time difference of log(y₂(t)) to estimate the complex slope (−α + jωd) at each time step.
• Use the median of the real and imaginary parts of the differences to robustly estimate the slope, minimizing the impact of outliers from initial transients.
• Extract α and ωd from the median slope estimates: α = −median(real(D)), ωd = median(imag(D)).
• Implement the algorithm in MATLAB using simple operations: hilbert(), log(), unwrap(), diff(), and median().

Experimental results

Research questions

• RQ1 Can the Hilbert transform be used to efficiently reconstruct the analytic signal from sampled time-domain data of a damped oscillatory system?
• RQ2 To what extent can the complex slope of the logarithm of the analytic signal accurately estimate the damping coefficient α and damped frequency ωd?
• RQ3 How does median-based estimation improve robustness against noise and transient artifacts in the data?
• RQ4 What are the limitations of this method in terms of numerical stability and boundary effects?

Key findings

• The method estimated the damped natural frequency ωd with a −0.4% error, achieving high accuracy despite being approximate.
• The damping coefficient α was estimated with a −5.5% error, indicating moderate accuracy under the tested conditions.
• The use of median-based slope estimation significantly reduced the influence of outliers caused by initial impulse or step transients.
• Boundary effects due to the Gibbs phenomenon were observed in the Hilbert transform reconstruction, particularly near data edges.
• Numerical instability occurred for large time values due to underflow in exp(−αt), limiting applicability for long-duration signals.
• The algorithm is computationally efficient and suitable for real-time or batch processing, offering a fast alternative to iterative methods.