Sensor Measurement Errors Detection Methods Septimiu Pop, Dan Pitica, and Ioan Ciascai Applied Electronics Department, Technical University of Cluj apoca, 28 Memorandumului, Cluj apoca, Romania Septimiu.pop@ael.utcluj.ro Abstract: This paper studies the methods of gross errors identification of transducer measurements from hydro energetic building especially from dams. The transducer used in dams monitoring is read every day during the year. The obtained information with one transducer can be seen as the discreet answers of a system. The erroneous measurements appear as a large variation from normal characteristics. The correction methods are mainly based on digital signal processing and approximation function. This paper proposes three method of correction: digital low pass filter, adaptive filter and approximation function. For low pas filtering is used a FIR (Finite Impulse Response) filter with Hamming window. An FIR filter is always stable, realizable, and provides a linear phase response under specific condition. These characteristics recommend FIR filter many filter design. Ambient temperature is the cause of the physical parameter variation detected with transducer. For error correction is applied an adaptive filtering, using the desired response. The approximation method function is less exhaustive and is based on the velocity of variation of result data from day to day. The data which is not respecting these conditions is removed. These three methods are used to determine a maxim correlation between results signals and original signal.. ITRODUCTIO The major motivation of this study results from the importance of hydro-energetic buildings in any country s economy. Safety analysis of a dam consists in analyzing a large amount of information obtained with transducers placed in the building structure. For the right decision over building behavior is required that the data analyzed is error free. Manual error detection is a too difficult job. The requirement is to develop automatic correction methods. The Fig. shows the data from transducer measurement over three year. For data processing the transducer measurements over time are used like discrete signal. Digital techniques used are from digital signal processing. The method proposed in this paper is based on the measurement estimation through signal processing. The correction consists in eliminating the erroneous measurements witch do not respect estimation sequences. The block diagram of proposed correction method is show in Fig. 2. T[ Fig.. Temperature transducer measurements. Measurements estimation Ť[ Measurement filter T true [ Fig. 2. Block diagram of correction method. 978-4577 22 0/20/$ 26.00 20 IEEE 44 34th Int. Spring Seminar on Electronics Technology
The purpose of this article is to introduce the digital filtering technique and numerical approximation in measurement analyze. The digital signal processing is very interested in measurement analyze from numerous reasons: allows to manipulate the discrete values using the design filter, they make it possible to test signal quality, are easy to implement. 2. GEERAL DESCRIPTIO From fig. is obviously that the errors that affect the transducer measurements appear like a high noise signal. The errors occur only from time to time, it is not a present phenomenon. The measurement model is described by the next equation. T[ = Ttrue [ + e[ () T true [ true measurement value e[ measurement error; n = : In the next processing stage are used a test signal with known signal noise. T true [ = non if : [ T ~ [ ± ΔT ]; e [ = T[ T ~ [ ] T[ out n (2) t t T[n + ] - T[ Δ = T t n t (3) n= Δ + Δ n Δ T is the dynamic range of temperature variation. Measurements specification: finite length, sample frequency of measurement is one measurement on day during the six 2.5 years. 2.. FIR low pas filter To design the digital low pass filter the window technique by Hamming type [] is used. The filter output depends on input samples. T ~ FIR (n) = - k= h ( k) T (n - k). (4) Where: T FIR (n) filter output data; h(k) is the filter coefficients; T(n-k) shifter value of input signal T(n), M is the filter length. The filter response is described by the linear shift-invariant difference equation [2]. The filter design specifications are: Sample frequency = /24*3600 [Hz] Cutoff frequency = (Sample frequency /0) Hz M = 32 Fig. 3. Test signal. The following section describes the digital techniques used for measurements estimation with input data (Fig. 3). ~ The output, T [ n ] estimate the input signal T [. The measurement error is present on test signal Fig.3, and is detected with measurement filter. The measurement filter declares a value to be erroneous if his value is outside of dynamic range. The dynamic range is described with follow equation. T true [ = T[ if [ T ~ [ ± ΔT ]; e [ 0; T out [ = : Fig. 4. Filter low pass repose. Low pas digital filter is designed using Matlab [3]. The filter response at test input signal T[ is show in the next figure. 978-4577 22 0/20/$ 26.00 20 IEEE 45 34th Int. Spring Seminar on Electronics Technology
Fig. 5. Measurements estimation using low pas digital filter response. The measurements errors are shown in fig.6. 2.2. Adaptive filter The principle of an adaptive filter is shown in fig. 8. The adaptive filter [3] uses two input signals; the first one is the erroneous signal T[. The past values of the same signal are applied on the reference input. T[n-k]. In figure 8 the estimate signal Ť[ is the digital filter output. The difference between past input and the actual output of the digital filter is the error signal e F lms [. The error e F LMS [ is used to calculate the best value of filter coefficients. The most popular method for filter coefficient adjusted to obtain the desired response is adaptive LMS (least mean squares) algorithm. The filter coefficients are determinant with equation. w + [ k] = w [ k] + μ e [ T[ ] (6) n n F lms n μ is the step of adaptive algorithm. Fig. 6. Detected errors with digital filter response. To estimate the probability distribution of data e[ (known input error) and e FIR [ (detected error) a statistic analyze of errors through histogram is used. Fig. 8. Principle of an adaptive filter. The estimate signal with adaptive filter is show in the next figure. Fig. 7. Histogram of measurements error obtained with FIR estimation. From figure 7 results that the distribution of detected error is the same with input error. Another way to compare those two signals is to compare their power. P = e[ e[ (5) n= 0 C 2 The errors powers are: P e = 2.8 [ ] 0 C 2 and Pe FIR = 9.8 [ ]. Fig. 9. Estimate measurement with adaptive filter. The first step to estimate the signal with adaptive filter is affected by digital filter coefficient converges. To eliminate this disadvantage in data processing is recommended to use sequences with length more that. This error is present also at FIR filter if the initial conditions are 0. The detected errors with estimate signal from fig. 9 are show in next figure. 978-4577 22 0/20/$ 26.00 20 IEEE 46 34th Int. Spring Seminar on Electronics Technology
Fig. 0. Detected errors with digital filter response. The error from fig. 0 contains also the estimation error with adaptive filter. 2.3. Methods of mathematical approximation The mathematical approximation method supposes to estimate the measurements evolution with a polynomial equation using data fitting [5]. Approximation is an alternative technique to signal processing used in measurement estimation. Approximation can be polynomial or trigonometrically. The trigonometric equation is accepted because the physical phenomena from hydro-energetic building like dams are cyclic phenomena. The mathematical function for approximation is described by the next equation: P T ~ (n) = a n (7) p= 0 P 0 + p p= p p [ cos( c n ] p T ~ (n) = a a ) (8) Fig. 2. Detected errors with approximation function. 2.4. Comparative analysis of processing methods Processing method applied in measurement analyze realize a linear combination of some coefficient with measurements value. These are evaluated with errors histogram and the noise power (error). A statistic analyze of a result data (after measurements correction) is based on correlation function between free error data T F [ and the true measurements T true [ described with () [6]. That can show whether and how strongly T true [ and T F [ are related [4]. The main result of a correlation is called the correlation coefficient (or "ρ"). It ranges from -.0 to +.0. The closer ρ is to + or -, the more closely the two variables T true [ and T F [ are related. rt,ttrue [d] ρ T,Ttrue = (9) r [0] r [0] T,Ttrue Ttrue,Ttrue rt, Ttrue [d] = T [ Ttrue[ n d] n= (0) The histogram of detected errors through estimate techniques is show in next figure. Fig.. Estimate with approximation function. The approximation functions depend by the length of estimated signal. The signal from figure is interpolate with an eleven order function. Fig. 3. Histogram of measurements errors. Between histograms there is not a significant difference. However the LMS filter response, estimate more bathers in the measurement evolution. The 978-4577 22 0/20/$ 26.00 20 IEEE 47 34th Int. Spring Seminar on Electronics Technology
number of measurements that respects the condition ~ T [ [ T [ ± ΔT ] is greater that in others methods. This result is confirmed also by errors power. Pe 2.8 [ºC 2 ] Pe FIR 20.05 [ºC 2 ] Pe lms 20.3 [ºC 2 ] Pe f 8.8 [ºC 2 ] From statistical analyze of results the optimal estimation of measurements (from fig. 3) is obtained with adaptive filter. The performance of correction method depends of detected errors. Method errors are obtained with difference of input errors (know error for test model) and detected error. e method [ = e[ e [ () lms For test signal the correlation between input errors and detected errors is high 0.9 (--). This value shows the reliability of correction method. A maxim correlation (ρ[d] = ) are obtains just if estimation is precisely; e method [ = e[ e lms [ = 0. 3. COCLUSIO The obtained results shows that the processing method from signals analyzes may be applied in measurements analyze. Using one of them depends of data statistics. The measurements are a building response to a physical stimulus. For resistive transducers used in dams for temperature monitoring the stimulus is ambient temperature. Measurement estimations can be done based on digital signal processing methods presented in this paper. The erroneous measurements are eliminated by applying a measurement filter. The digital processing allows applying data filtering online. ACKOWLEDGEMET Fig. 4. Methods error. The method errors depend by the accuracy of the estimator. With adaptive filter response the method error is ±4ºC. The correlation between true measurement and filtered data is shows in fig. 5. These results were obtained on research grants with SC. Hidroelectrica SA between 2009 and 200. REFERECES [] Les Thede; Practical Analog and Digital Filter Design 2004. [2] Behrouz Farhang-Boroujeny Adaptive Filters: Theory and Applications ISB 0-47-98337-3. [3] Septimiu Pop, Dan Pitica, Ioan Ciascai, A Correlation Method for Improving Temperature Sensors Measurements ISSE 2007 Cluj apoca Romania. [4] Jaan Kiusalaas umerical Methods in Engineering with MATLAB ISB-3 978-0-5-28-0. [5] B. A. Shenoi Introduction to Digital Signal Processing and Filter Design ISB-0 0-47- 46482-. [6] Hoang Pham Engineering - Statistical methods, Rutgers the State University of ew Jersey ISB-0: 852338067. Fig. 5. Correlation function between free error data T true [ and T F [. 978-4577 22 0/20/$ 26.00 20 IEEE 48 34th Int. Spring Seminar on Electronics Technology