# Frequency Response Matrix Estimation From Partially Missing Data – for Periodic Inputs

Time-variant dynamics are present in quite some engineering applications. Think of, for example, pit corrosion of metals where the time-variation is induced by the changing surface; thermal drift in power electronics and lithium-ion batteries whose characteristics are temperature and charge dependent; mortification in ex vivo experiments; muscle fatigue causing time-dependent relationships between, for instance, the hand-grip force and electro-myographic signals; and airplane dynamics which depend on the changing flight speed and height. In all these applications the dynamics evolve in a smooth and non-periodic manner.

Similar to the frequency response function (FRF) for linear time-invariant systems, the time-variant frequency response function (TV-FRF) gives quick insight into the time-variant dynamics of complex systems. A non-parametric procedure for estimating the TV-FRF from noisy observations of an arbitrarily time-variant system excited by an arbitrary excitation is presented in [1].

In some cases data is missing due to sensor failure, sensor saturation and/or data transmission errors. The best solution to this problem would be to repeat the measurements. However, the measurements can be lengthy or expensive and repeating them is not always an option, especially in case of arbitrarily time-variant systems. In this project a nonparametric estimation procedure is proposed that can estimated the TV-FRF from noisy observation is the presence of missing data. The proposed procedure is an original extension of the missing data algorithm for FRFs [2], to TV-FRFs. In the sequel the approach is illustrated on real measurements.

*Figure 1: Second order time-variant bandpass filter with input u(t), output y(t), and gate voltage p(t). The electronic circuit consists of a high gain operational amplifier (CA741CE), a JFET transistor (BF245B), three resistors (R_1=R_2=10 "k" Ω and R_3=470 Ω), and two capacitors (C_1=C_2=10 "nF" ). During the experiment the gate voltage p(t) varies cubically between -860 mV and -1.02 V*

A time-variant bandpass electronic circuit (see Figure 1) is excited by coloured Gaussian noise exciting the band [208 Hz, 11 kHz]. The input and output rms-values are, respectively, 46.5 mV and 83.7 mV, and the time-variation originates from the cubically varying gate voltage of the JFET transistor (see Figure 1).

**Figure 2: Comparison between the estimated (o) and measured (x) missing output samples – Electronic circuit, 50% missing data. Top row: missing output samples. Bottom row: difference between the measured and the estimated missing samples (red lines) and the predicted standard deviation (black lines). Left column: random missing data pattern. Right column: clipping**

Starting from N=15001 input-output samples three TV-FRFs are estimated nonparametrically: (i) using all data (no missing values), (ii) 50% of the output samples are missing at random (see Figure 2), and (iii) 50% of the output samples are missing by saturating the output signal (see Figure 2). Figure 2–Figure 4 show the results.

Figure 2 compares the estimated to the measured missing output samples. For both missing data patterns the estimation error – except at the borders – is at the level of ± 20 µV, while the missing output samples vary between ± 20 mV.

**Figure 3: Estimated TV-FRF – Electronic circuit, 50% missing data. Top row: TV-FRF estimate using all data (red surface) and its predicted standard deviation (green surface), and magnitude of the complex difference between the estimates using all data and the 50% missing data set respectively (blue surface). Bottom row: frequency-time plot of the TV-FRF estimate using the 50% missing data set. Left column: random missing data pattern. Right column: clipping.**

Figure 3 compares the TV-FRF estimates based on the missing data sets with that obtained using all data (no missing values). It can be seen that the estimation error is at the level of the estimation uncertainty for both missing data patterns.

**Figure 4: Simulation error of the nonparametric TV-FRF estimate – Electronic circuit, 50% missing data. Output DFT spectrum using all data (red lines); simulation error of the TV-FRF estimate using all data (green lines) and its predicted standard deviation (black lines); and magnitude of the complex difference between the simulated outputs of the TV-FRF estimates using all data and the 50% missing data set respectively (blue lines). Left column: random missing data pattern. Right column: clipping**

Finally, Figure 4 compares the simulation error of the non-parametric TV-FRF estimates with and without missing output values. It can be seen that the simulation error of the TV-FRF estimates using the missing data sets is at the same level of that using all data (the blue lines in Figure 5 are below the green lines).

It can be concluded that it is possible to estimate nonparametrically the TV-FRF in the presence of missing data; even for large fractions of missing values.

#### References

- R. Pintelon, E. Louarroudi, and J. Lataire (2015). Nonparametric time-variant frequency response function estimates using arbitrary excitations, Automatica, vol. 51, no. 1, pp. 308-317.
- D. Ugryumova, D., R. Pintelon, and G. Vandersteen (2015). Frequency response function estimation in the presence of missing output data, IEEE Trans. Instrum. and Meas., vol. 64, no. 2, pp. 541-553.
- Ugryumova, D., R. Pintelon, and G. Vandersteen (2015). Frequency response matrix estimation from missing input-output data, IEEE Trans. Instrum. and Meas., vol. 64, no. 11, pp. 3124-3136