ааааааааааааааааааааааааааааа аPRECISE MEASUREMENT BY NON-LINEAR SENSOR

аааааааааааааааааааааааааааааааааааааааа WITH STAIRCASE REFERENCE SIGNALа аааааааааааааааааааааааааааааааааааааа

 

ааааааааааа Sergey Sibeykin

 

аа аAbstract. We use a staircase reference signal for measurement of electrical and non-electrical values by a nonlinear sensor. A sensor with a frequency output signal is preferable. We can measure the input signal itself (e.g. acceleration) or integral from it (e.g. velocity). Signals are considered as random variables.а The result was found that the error of measurement is minimum if reference signal has the same duration and raw moments as the input signal. The reference signal can cover only a part of the input signal. A concept of negative probability was introduced to describe the composition of signal and the method of its transformation. The physical interpretation of the positive and negative probabilities is subsequent direct or inverse integration of signal.

 

ааа Index Terms: Measurement; Measurement errors; Acceletation measurement; Velocity measurement; Voltmeters; Probability.ааааааааааааааааааааааааааааа

 

ааа I. Introduction

ааа This method of measurement can be applied in various fields (in trains, missiles, aviation, automobiles, submarines, and robotics) to measure acceleration, velocity, vibrations, direct/alternating voltage, etc.

ааа A task is to measure physical values by a nonlinear sensor with minimal error. The output signal of sensor is a frequency which permits to calculate parameters of signal, including moments. This sensor may be piezo-element, since its frequency depends on the pressure of inertial mass. Measured values may be:а pressure, acceleration, vibration, etc. We can measure also direct or alternating voltage that applies to varicap. The piezo-element (or varicap) is part of the oscillation circuit. A measured value may also be an integral from the input signal (velocity, if the input signal is acceleration).

ааа Let's consider the measurement of velocity (Fig. 1). Our input signal is acceleration x(t) that is measured by a piezo-accelerometer. Counter of impulses measures its output signal (frequency). The code of Counter is proportional the reading W(t) of a measured velocity V(t).а

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


In the simplest case we find the measured velocity V(t) that has got some value Vr. This task may be solved as follows:

ааа We put the reading Wr of the value Vr in Register by integration of a constant reference signal. This acceleration is due to EarthТs gravity g = 9.81m/sec2. Therefore, time of integration Tr = Vr/g. In the process of measuring, the Digital Comparator compares the codes of Counter and Register. When W (t) = Wr, Comparator gives the signal that the velocity V(t) has a value Vr. Lets denote this moment of time by T.а However, since the sensor may not be a perfect machine, the real velocity V(T) may be different from Vr by a value of DV. Error of measurement DV = V(T) - Vr .

ааа We can decrease the error DV by replacing the constant reference signal g by the staircase reference signal Xr, which approximates the input signal x(t) (Fig. 2a, here the input signal x(t) is linear). Then we have the mutual compensation of errors for W(T) from the input signal x(t) and Wr from the reference signal Xr [1]. Therefore, when W(T) = Wr, V(T) Vr, the error DV is at a minimum. The staircase reference signal is preferable because it is easier to provide parameters of stairs Xi and Ti with higher accuracy.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


ааа Usually, we know approximately an acceleration that we will measure.а Therefore, we know the expected velocity Ve at a time T, and can assume that Vr = Ve.а We receive the values Xi from -g to +g by rotation of accelerometer relative to the local vertical line.а We integrate Xi for a time Ti in order to obtain Vr = X1T1 + X2T2 + X3T3+..., and put the reading Wr of a reference velocity Vr in Register.

ааа Our task is to find the optimal reference signals, i.e. what values Xi and Ti minimize the error of measurement DV.

 

аа II. Assumptions

ааа a) We will consider the input x(t) and the reference Xr signals as random variables. The main characteristic of random variable X is probability function p(x). It has two properties [2] :

ааа p(x) is non-negative:аааааааааааааа p(x)0;ааааааааааааааа ааааааааааааааааааааааааааааааааааааааааааааааааааааааааа(1)

ааа

 

 

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а For the linear input signal x(t), depicted on Fig. 2, probability function p(x) has an even distribution. Then we can derive from (2) p(x) = 1/Xmax.

ааа The probability function p(Xr) of staircase reference signal with n stairs is a composition of functions di, which is proportional to the delta-function d [3]:

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Probability functions p(x) and p(Xr) are depicted on Fig. 2b.аааааааааааааа

ааа We will also use moments for describing a random variable X. The raw moment Mi of i-th order is:

 

 


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We can also define the moments Mi as an expectation for variable x(t) of degree iа :

 

 

 


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аb) We describe the transfer function of the sensor as a Maclaurin series:

y = a0 +(1 + a1)x +a2x2 +a3x3+ ...= x + a0а +a1x +a2x2 +a3x3 +...ааааа (6)аааааааа

 
 


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Here a0 is the bias, a1 is the error of scale factor, and a2, аa3 ... describe the nonlinear error.

ааа An ideal sensor has the transfer function y = x.

ааа c) We suppose the parameters of the sensor have short term stability, i.e. values a0, a1, a2... do not change from calibration till measurement.

ааа d) Sensor does not have hysteresis.

 

ааа III. Algorithm for an optimal reference signal

ааа LetТs try to minimize the error of measurement. To do this, we need to find and equate the reading of velocity W(t) at a time T and a value Wr.

ааа Taking into account (6), the reading W(T) is:

 

 

 


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ааа Using (5), following is what we get from (7):

W(T) = TM1 + a0T + a1TM1 + a2TM2 + a3TM3 + ...аааааааааааааааааааааааааааааааааааааааааааааааааааааа (8)

The reading of an integral from the reference signal Xr, i.e. the reading of a reference velocity Vr:

 

 

 

 


ааа

The values Yi of the staircase reference signal Xr are constant, therefore:

Wr = T1Y1+ T2Y2+ T3Y3+ ...аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (9)ааа

Taking into account (6), we obtain:

 

Wr = T1(X1 + a0 + a1X1 + a2X12 + a3X13 + ...) +

аааа T2(X2 + a0 + a1X2 + a2X22 + a3X23 + ...) + T3(X3 + a0 + a1X3 + a2X32 + a3X33 + ...) +...

or

Wr = T1X1 + T2X2 + T3X3 +...+a0(T1 + T2 + T3 +...) + a1(T1X1 + T2X2 + T3X3 +...) +

аааа + a2(T1X12 + T2X22 + T3X32 +...) + a3(T1X13 + T2X23 + T3X33 +...) + ...ааааааааааааааааааа (10)

 

Equating (8) and (10), put the terms with aiа to the right side of the equation:

TM1 - (T1X1 + T2X2 + T3X3 +...) =

а = a0 (T1 + T2 + T3 ... -T) + a1 (T1X1 + T2X2 + T3X3 ... -TM1) +

а + a2 (T1X12 + T2X22 + T3X32 ... -TM2) + + a3 (T1X13 + T2X23 + T3X33 ... -TM3) +...

 

Here, value TM1 is the velocity V(T), and the value T1X1+ T2X2+ T3X3+... is Vr. Therefore, the left part of this equation is the error of measurement DV=TM1 -(T1X1 + T2X2 + T3X3 +...) = V(T) - Vr. Then

а

аа DV = a0 (T1 + T2 + T3 ... -T) + a1 (T1X1 + T2X2 + T3X3 ... -TM1) +

+ a2 (T1X12 + T2X22 + T3X32 ... -TM2) + + a3 (T1X13 + T2X23 + T3X33 ... -TM3) +...аааааа (11)

 

аа Error DV is equal to zero if the terms in parentheses on the right part of (11) are zero. Let us denote the terms of order >n as insignificant, i.e ai = 0 if i > n. Then conditions for DV = 0 are:

 

 

 

 

 

 

 

 

 

 

 

 


аа Let us multiply and divide the right parts of (12) by the durations of reference signal

Tr = T1 + T2 + T3 а+...+ Tn. We obtain in these right parts the products of Tr and moments Mr,i of the staircase reference signal Xr. We can write (12) in short form as:

 

аT Mi = Tr M r, i , i = 0, 1, 2, ... n-1,ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (13)

 

here n is the quantity of stairs of the reference signal Xr.

ааа Therefore, the error of measurement is minimized if the durations and the moments of input and reference signals are equal.

ааа The staircase reference signal with n stairs can decrease the error DV from the first n terms of transfer function (6) to zero, if we precisely evaluate the moments Mi of input signal.

ааа To find the optimal reference signal, we have to decide on the task of momentsТ problem, i.e. find the parameters of the reference signal on its moments, which are equal to moments of input signal.

ааа For example, let the reference signal Xr has four stairs (n=4), and the input signal x(t) be linear. In accordance with (4), or (5), the first, second, and third raw moments of this signal are:

ааа M1 = Xmax/2, M2 = X2max/3, M3 = X3max/4.

ааа Let the values Xi of the staircase reference signal Xr be uniformly distributed along the measurement's scale: X1 = 0; X2 = Xmax/3; X3 = 2Xmax/3; X4 = Xmax.

ааа We know the values Mi and Xi, and therefore we can find the values Ti by solving a system of equations (12) via determinants. Then T1 = T/8; T2 = 3T/8; T3 = 3T/8; T4 = T/8.

ааа

IV. Error of measurement

ааа The resulting relative error of measurement is.

ааааааа

 

 

 

 


ааа dmeth аis a methodical error, because it is the error of evaluation for moments Mi of input signal x(t).

ааа dmeth ddde,ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (14)

Here dd is a relative error of direct measurement (without application of reference signals);а

ааааааа de is a relative error of evaluation of the first raw moment M1.

ааа The proof for (14) is omitted.

ааа dres is a residual error because of (n+1)-th, (n+2)-th, ...order terms of transfer function (6).ааа

ааа dr аis an error of the reading Wr of the reference velocity Vr.ааа

аа

V. Case when reference signal covers part of input signal

ааа Our reference signal is limited by a value g, but we often measure much more acceleration. The new method works properly in this case as well.

ааа Let acceleration changes itself in the range from 0 to 10g linearly, Xmax=10g. Let the values

X1 = -g;а X2 = -0.5g;а X3 = 0.5g;а X4 = g.

ааа Then by solving (12) we obtain the next durations of stairs:

T1 = -143.7778T, T2 = 305.1111T, T3 = -348.2222T, T4 = 187.8889T.

ааа Values T1 and T3 are negative. This means Counter (Integrator) must subtract impulses for times T1 and T3 (Inverse integration).

ааа We see that the reference signals may cover only a part of the input signal, or otherwise go out of its range. Then in accordance with (9), the reading Wr of reference velocity Vr is the difference of great numbers YiTi. This limits the accuracy of Wr and, therefore, the accuracy of measurement. The range of the reference signal may be only in several times less than the range of the input signal.

 

ааа VI. A continuous measurement

ааа We can introduce various values Wr at many points of the trajectory. Therefore, the measurement of velocity V(t) becomes continuous. We have the next sequence of operations:

ааа 6.1. Calibration. We find the readings Y1, Y3, Y3,ЕYn for chosen values X1, X2, X3,ЕXn of the reference signal and save them in Register.

ааа 6.2. Measurement.

ааа a) Evaluation of moments M1, M2, M3, ...Mn-1 for the next point of measurement. For this we can find the expected velocity Ve by extrapolation from measured values V(t).

ааа b) Calculation of the values T1, T2, T3, ...Tn by solving (12) and the expected reading Wr of the reference velocity Vr according to (9):а Wr = T1 Y1 + T2 Y2 + T3 Y3 + . . . + Tn Yn .

аа Here, we can calculate Wr by multiplication and addition (subtraction, if Ti is negative) instead of integration. This significantly decreases the time of preparation for measurement (i.e. the time between the next and previous points of measurement is very short).

ааа c) comparing W(t) and Wr. When W(t)=Wr, Comparator gives a signal that V(t) has got Vr.

ааа If we find moments M1, M2, M3, ... Mn-1а in the process of measurement, the difference between expected values of these moments and their real values is minimized. According to (14), accuracy of measurement is high. A computer calculation showed that the relative error of measurement may get 10-5 if nonlinear error of sensor is a few percentage points.

 

ааа VII. Measurement of voltage

ааа We can measure the average value of input signal x(t), for example, voltage, too. The measured voltages apply to a varicap , which is part of the LC circuit. Voltage converts to a frequency:.

 

 

 

 

 

 

 


The nonlinear error is minimized by this method. The staircase reference signal Xrа is created by connecting the stable sources of voltage X1, X2, ... Xn for times T1, T2, ... Tn to varicap. In comparison with the voltmeter of dual slope integration, this voltmeter has a low level of flicker noise (1/f noise). The resolution of this voltmeter can get a few microvolts by a reference signal with four stairs.

 

ааа VIII. One Mathematical Consequence

ааа The proposed method has one interesting consequence.

ааа Let us find the probability functions p(Xr) of the staircase reference signal Xr covering only a part of the input signal thatТs described in part V. Using (3), we receive the probability functionа p(Xr), depicted in Fig.3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


а Because T1 and T3 are negative, d1 and d3 are also negative. As we can see from Fig. 3, the

а4

х di = d

i =1

 

property (1) is not satisfied, but property (2) is satisfied because

 

and the integral from delta-function d is 1. How can we interpret this result?

ааа The sign "plus" or "minus" for the value of probability function p(Xr) determines direct or inverse integrations for stairs of reference signal Xr. Therefore, the probability functions p(Xr) describe not only the random variable Xr, but also the composition of this random variable and the method of its transformation. Let us call this composition a Уtransformer process.Ф For some of its values, the probability function p(Xr) can be negative. The negative probability function may be useful for researching nonlinear systems. Here, the results of direct integration for a positive variable X and inverse integration for a negative variable -X do not coincide.

ааа There are papers that focus on the use of negative probabilities, including works of Nobel prize-winners Paul Dirac andа Richard Feynman. The best and last review of these papers is in [4]. But negative probabilities did not win recognition yet, partly because ofа contradictions with Kolmogorov probability theory. But letТs use a quotation from [4]: УThere were no experiments where we can say that there were directly Уdetected negative probabilitiesФ. In this research we found the interpretation of positive and negative probabilities as direct and inverse integration of signals considered as random variables.

 

ааа IX. Conclusions

ааа We consider the input x(t) and the reference Xr signals as random variables. This permits us to use the apparatus of probability theory and receive new results.

ааа The error of measurement DV is minimized when the duration and the moments of input signal x(t) are the same as the duration and the moments for staircase reference signals Xr. When applying these conditions to the chosen values of stairs Xi, we find the durations Ti of these stairs by solving a system of equations (12).

ааа If the reference signal Xr covers the most part of the input signal x(t), the probability function p(Xr) describes the reference signal Xr. We compare the reading Wr of the integral from the reference signal Xr to the reading W(t) of the integral from the input signal x(t) for the determination of x(t) or V(t).

ааа If the reference signal Xr covers only a part of the input signal x(t), the probability function p(Xr) describes the composition of the reference signal and the method of its treatment, i.e. the transformer process. Then some values Ti are negative and we can obtain the reading Wr only by combining the direct and inverse integrations. The transformer process may be useful for further research of nonlinear systems.

ааа Integration of the staircase reference signal can be substituted by operations of multiplication and addition (and subtraction, in case of inverse integration). Then the measurement may be continuous.

ааа Error of measurement is lesser, nearer the moments of the real input signal to their expected values. The exact evaluation of the first moment is especially important. Error of measurement can be reduced to 10-5 if the moments Mi are determined at time of measurement.

 

References

ааа [1]. Sergey Sibeykin. Method and device of determination of an input signal which changes in time. USA, Patent US No. 6,240,435 b1, Date of Patent May 29, 2001.

ааа [2] Ash, Robert B. Probability and Measure theory. 2000, pp.174Ц176.

ааа [3] Papoulis, A. Probability, Random Variables, and Stochastic Processes.1965, pp.88Ц103.

ааа [4] Espen Gaarder Haug. Derivatives Models on Models. 2007, pp. 317-333.ааааа



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