Resonator based non-parametric identification of linear systems

Abstract

Section 11. recalls the identification problem and the resonator based observer. Section In. introduces the resonator based identification and a detailed analysis is given. Real measurement data are provided in section IV., while section V. is the conlusion. PRELIMINARIES I. INTRODUCTIONFourier analysis is a well-known method for non-parametric frequency domain identification of linear systems [l]. IFrequency domain data are often inputs for parametric identification [2]. The utilization of multisine excitation provides the possibility of the elimination of the systematic errors like leakage and picket fence [3]. In most cases the output of the system is analyzed by the discrete Fourier transform (DFT), while the frequency domain parameters of the excitation are known in advance. The DFT is calculated via the fast Fourier transform (FFT). In order to suppress the measurement noise, averaging on the analysis results is also necessary.Resonator based observers were developed earlier to perform the recursive discrete Fourier transform (RDFT) [41, [5]. In these observers the resonators work in a common feedback loop providing zero steady-state feedback error at the resonator frequencies. The summed output of such resonators can generate any periodic signal up to the half of the sampling frequency. It is straightforward to utilize such a generator -observer pair for frequency domain non-parametric system identification: the system to be identified has to be in between the generator and the observer and the ratio of the state variables of the observer and the generator supplies the estimated transfer function. This arangement can be an alternative of the ITTbased analysis since it works as a filter bank and provides the measurement results on-line.

Analysis of Resonator-Based Harmonic Estimation in the Case of Data Loss

“…2) f r = 1/N , 0 < α < 1: The observer still performs orthogonal transforms, but the settling is not deadbeat. The RBO performs exponential averaging at the frequency of each harmonic component [7]. The poles of the observer…”

“…The settling is not deadbeat even if α = 1, but the observer is still fairly fast if all harmonic components are observed up to f s /2 [5]. Analytical results regarding noise suppression and settling time are found in [7]. A design method exists for setting arbitrary poles or even deadbeat settling [18], but usually, it is not used in real-time systems due to its computational complexity.…”

“…The resonators can be tuned with α: Small α results in good noise rejection but in slow settling. Design considerations are given in [7].…”

“…Parametersλ and η can be calculated from (25) and (26) offline. In the case of uniform resonator alignment:λ = (1 − α) 1/N [7]. Note that k counts samples, while P counts intervals.…”

“…This is particularly important when the spectrum estimation is used in real-time systems. Application examples of the RBO are, e.g., active noise and distortion control [1]- [3], analysis of power line signals [4], [5], electronic instruments [6], system identification [7], etc.…”