# Dynamic Uncertainty for Compensated Second-Order Systems

^{*}

## Abstract

**:**

## 1. Introduction

_{0}, δ and ω

_{0}= 2π f

_{0}denote static gain, damping and resonance frequency, see [1–4]. When such sensors are applied for the measurement of according signals with significant frequency content near the resonance frequency the sensor output signal contains time-dependent distortions such as ringing. Analogue and digital filtering are appropriate tools to reduce these dynamic errors by compensating the dynamic response of the sensor, and techniques for the construction of compensation filters are well-known in digital signal processing (DSP), see, for instance, [1–3,5–8].

## 2. Compensation Task and Considered Digital Compensation Filters

_{S}) + ɛ[n], where f

_{S}= 1/T

_{S}denotes the chosen sampling frequency. Estimates x̂[n] of the discrete-time input signal x[n] are calculated by applying a digital deconvolution filter, see Figure 1.

## 3. Uncertainty Evaluation Methods

_{0}, Ŝ

_{0}for the system (1) with an uncertainty matrix U(δ̂, ω̂

_{0}, Ŝ

_{0}), see [14]. This uncertainty matrix can be interpreted as the covariance matrix of a joint Gaussian PDF, cf. [23]. In order to calculate the uncertainty caused by the uncertainty of the system, this uncertainty has to be propagated through the filter design. This results in the uncertainty matrix

**U**of the filter coefficient vector, where

_{θ̂}**θ**stands for the filter coefficients of the deconvolution filter, see [23]. Once the uncertainty matrix

**U**has been derived its contribution to the uncertainty of the corresponding estimate x̂[n] of the input signal can be utilized as described below.

_{θ̂}**U**, signal noise and non-perfect compensation influence the resulting uncertainty associated with x̂[n]. The contribution of signal noise is calculated by propagating the covariance of the noise through the compensation filter, see [15,17]. The non-perfect compensation due to regularization or non-perfect construction of the deconvolution filter results in remaining dynamic errors:

_{θ̂}_{comp}[n] = (g * y)[n] and the actual, unknown input of the sensor; n

_{0}denotes a possible known time sample delay. Utilizing the well-known inequality for the Fourier transform F (Ω) of a function f (t):

_{S}with f

_{S}denoting the chosen sampling frequency. The resulting bound is given by:

^{jΩ/fS}) denotes the frequency response of the compensation filter (realized by either an FIR or IIR filter), see [18,19]. Note that the upper bound Δ̄ is time-independent, and it is similar to a corresponding continuous-time result given in [13].

#### 3.1. Uncertainty evaluation for IIR filtering

#### 3.2. Uncertainty evaluation for FIR filtering

**ŷ**

_{low}[n] = (ŷ

_{low}[n],...., ŷ

_{low}[n − N

_{comp}])

^{T}; ŷ

_{low}denotes the low-pass filtered sensor output signal and

**U**

_{ylow}stands for the covariance matrix of

**ŷ**

_{low}[n]. For stationary noise only the second term on the right-hand side of (10) is time-dependent and the uncertainty evaluation can be realized at low computational costs during the measurement.

## 4. Results

^{2}= 1 e−3, σ

^{2}= 3 e−4, and σ

^{2}= 1 e−6, respectively. As sampling frequency we chose 500 kHz. According to Figure 1, the measurand of this dynamic measurement was the band-limited sensor input signal.

_{T}= 120 · πkHz.

_{0}) and damping uncertainty u(δ), can be seen especially in Figure 9 as the employed input signal has significant spectrum near the system’s resonance and thus increases. Moreover, it can be seen in Figure 9 that due to the larger cut-off frequencies of the low-pass filters the output signal noise is less attenuated than for the narrow-banded input signal shown in Figure 8. Although these characteristics of the uncertainty are similar for FIR and IIR compensation, the larger value of the error bound (4) for the IIR compensation filter causes the larger uncertainty for this filter. On the other hand, as can be seen in Figures 4 and 5, the time delay of the FIR filter result is significantly larger than that of the IIR compensation filter and hence, when speed is an issue, the IIR filter is preferable.

## 5. Conclusions

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**Figure 4.**The compensated output signals resulting from the IIR and the FIR compensation filter for the narrow-banded sensor input signal.

**Figure 5.**The compensated output signals resulting from the IIR and the FIR compensation filter for thebroad-banded sensor input signal.

**Figure 6.**Left: Frequency response of the sensor model (black) with system parameter vector (11) and the IIR compensation filter (green) designed for the available estimate (12a) of the system parameter vector for estimation of the broad-banded (25 kHz) input signal. Right: Frequency response of the actual compensated system.

**Figure 7.**Left: Frequency response of the sensor model (black) with system parameter vector (11) and the FIR compensation filter (green) designed for the available estimate (12a) of the system parameter vector for estimation of the broad-banded (25 kHz) input signal. Right: Frequency response of the actual compensated system.

**Figure 8.**Left: Uncertainty associated with the FIR compensation filter result for three different noise values obtained for the narrow-banded input. Right: Uncertainty associated with the IIR compensation filter result.

**Figure 9.**Left: Uncertainty associated with the FIR compensation filter result for three different noise values obtained for the broad-banded input. Right: Uncertainty associated with the IIR compensation filter result.

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**MDPI and ACS Style**

Eichstädt, S.; Link, A.; Elster, C. Dynamic Uncertainty for Compensated Second-Order Systems. *Sensors* **2010**, *10*, 7621-7631.
https://doi.org/10.3390/s100807621

**AMA Style**

Eichstädt S, Link A, Elster C. Dynamic Uncertainty for Compensated Second-Order Systems. *Sensors*. 2010; 10(8):7621-7631.
https://doi.org/10.3390/s100807621

**Chicago/Turabian Style**

Eichstädt, Sascha, Alfred Link, and Clemens Elster. 2010. "Dynamic Uncertainty for Compensated Second-Order Systems" *Sensors* 10, no. 8: 7621-7631.
https://doi.org/10.3390/s100807621