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Calibration of Measuring Systems Based on Maximum Dynamic Error

This chapter presents methods and algorithms for determining the maximum errors which can be generated by measuring systems in reference to their standards. The application of such errors in the process of calibration of measuring systems intended for measurement of dynamic signals is discussed in detail. The problem of maximum errors lies in the fact that it is impossible to analyze the full range of all possible dynamic input signals. Therefore we find out the one that represents all signals. It is the signal generating errors of maximum value.


Introduction
This chapter presents methods and algorithms for determining the maximum errors which can be generated by measuring systems in reference to their standards. The application of such errors in the process of calibration of measuring systems intended for measurement of dynamic signals is discussed in detail. The problem of maximum errors lies in the fact that it is impossible to analyze the full range of all possible dynamic input signals. Therefore we find out the one that represents all signals. It is the signal generating errors of maximum value.
The existence and availability of signals maximizing both the integral-square error and the absolute value of error are discussed, and relevant solutions are presented. Moreover, the constraints imposed on the input signal are analyzed. These constraints refer to the magnitude as well as maximum rate of signal change. The latter constraint is applied in order to match the dynamic properties of the signal to the dynamic properties of the system under test.
In the relevant literature it was proved that the signals maximizing the errors in question always achieve one of the constraints imposed on it. If only the signal amplitude is constrained, the maximizing signal is always of the 'bang-bang' type, while in the case of two constraints relating to the magnitude and the rate of change such a signal can be only of triangular or trapezoid shape with slopes resulting from the values of constraints.
For the integral-square error, no analytic solution referring to the shape of the maximizing signal has been found so far. This is why the solution of this problem presented in the chapter is based on the application of the genetic algorithm method. This method guarantees that the results are obtained in minimized calculation time which depends only on the assumed number of population, the value of the stop condition and number of assumed switches.
In the case of the absolute error, the analytical formulae and algorithm which give precise results and can be realized in a very short calculation time are considered. The signal maximizing this criterion is presented in the form of figures showing the successive steps of proceeding as well as the error shape corresponding to this signal. In this chapter respective procedures and essential mathematical operations, which allow carrying out the calibration process according to the block diagram presented above, are discussed in detail. In part two the input-output relations of measuring systems are discussed, part three is devoted to the synthesis of the mathematical model of standard and, in part four error functionals are presented and analyzed. The problem of existence and attainability of maximizing signals is discussed in part five, while parts six and seven present the procedures for determining signals maximizing the integral-square error and the absolute value of error.
The optimization of the minimax type on the basis of maximizing signals is presented in part eight. Such optimization is performed in two steps. In the first step the input signal maximizing the assumed error functional is determined, while in the second step the model parameters optimization based on the obtained signal is performed.
The identification problem of the mathematical model of measuring system is omitted in this chapter as it does not concern the theory of calibration. It was assumed that this model exist and is given in the form of transfer function.
The practical application of the presented theory and algorithms is illustrated on the example of low-pass measuring systems.
In many cases it is convenient to present the output () y t by means of the following formula

Synthesis of mathematical model of standard
The calibration procedures for measuring systems intended for measurement of static quantities are commonly known and have been used for a long time in measuring practice. They cover the hierarchy of standards and calibrations circuits.
For measuring systems intended for measurement of dynamic quantities neither legal regulations concerning hierarchy of their accuracy nor specific calibrating procedures have been worked out so far. It results from the fact that for such systems input signals are time dependent, often undetermined, and whose shapes cannot be predicted in advance. The second reason is that the problem of standards for such systems has not been solved till now, because various measuring systems fulfill different objective functions (Tomczyk&Sieja, 2006). For such a situation we expect that models of standard will satisfy mathematical notation of these functions. In the dynamic measurement selection of standard parameters can be realized by means of optimization methods, which assure the conditions of non-distortion transformation, or by means of ideal filters of a band-pass which corresponds to the range of the work of the calibrated system. For low-pass systems this standard is given by where: c -amplification coefficient, 0 t -filter delay, m  -cut-off frequency.

Error functionals
Dynamic errors applied in our calibration are defined in different function spaces by means of chosen functionals. If the values of these functionals are determined by means of maximizing signals, it means that they will be valid for any dynamic signals which can appear at the input of the calibrated system. In this way all the possible input signals are included in this special, maximizing one.
The process of maximum error determination requires a special input signal to be used, which warrants that the error determined with it will always be higher or at least equal to the value generated by any other signal. The solution of this problem needs to prove that there exists a signal maximizing the chosen error functional at all, and if so, to elaborate a suitable procedure for its determination. The maximum values of error can constitute the base to work out the hierarchy of accuracy for instruments for dynamic measurement.
The integral-square error and absolute value of error will be discussed as an example.
The choice of the functionals (14) and (15) results from the fact that they are the most common in many different engineering domains.

Existence and attainability of maximizing signals
In (Layer, 2002, Layer&Tomczyk, 2010 it was proved that signals maximizing I(x) and D(x) functionals in [0,T] always achieve one of the constraints imposed on them.
The constraints in question concern the magnitude a and rate of change .
 For systems described by means of linear differential equations, the constraint of magnitude () xt results from the measuring range of this system. Imposition on the signal of only this one constraint gives the solutions of 'bang-bang' type. As a result unexpected great values of errors are generated. It is affected by the dynamics of 'bang-bang' signals which have infinitely high derivative values in the instants of switching. Outside of these instants, they have constant values. Because of this, the 'bang-bang' signals are not matched to the dynamics of physically existing systems, since they can only transmit signals of a limited value of the rate of change. In order to match the dynamic behaviour of the input signals an additional constraint resulting from the dynamic properties of the calibrated system should be imposed.
Proper matching is obtained by restricting the maximum rate of change  of the signal.

Constraints of signal
In the case of two simultaneous constraints a and ,  the maximizing signal has a triangular shape with the slopes inclination , or a trapezoid shape with the slopes inclination   and the height . a We can analyze here two different premises referring to the rate of change  value. The first one refers to the time domain where h(t) denotes the impulse responses of the system.
In the frequency domain constraint  results from the maximum frequency of the band- where A denotes amplitude.

Procedure for determining signals maximizing the integral-square error
For the integral-square error (14)  The algorithm presented in Fig.3 includes three main operations: reproduction, crossing and mutation.
In the first step, the initial population Relations (19) and (20) allow for estimation of the usefulness of each chromosome in the population. The higher c I  value of the given chromosome gives more probability to include it in the next population.
When the difference between the obtained values of adaptation coefficients is too small, it is necessary to carry out the scaling operation of the adaptation coefficient.
The next step is the operation of reproduction. According to the probability calculated by (20) (1 ) where: 1 i d  is a descendant detector of i chromosome, and 1 j d  is a descendant detector of j chromosome.
The coefficient  in (21)  The changeability range of  is contained in the range between 0 and the third value calculated by means of (22) minus this value multiplied by  (sample interval of ). T Then the  value is selected at random from the above range, and is substituted into (21).
For the i chromosome, the mutation operation is described by

Signals constraint on magnitude
In the case of () Dx error (15) where a is the magnitude of ( Substituting (29) into (28) gives which can be computed in the very simple way.

Signals constraint on magnitude and rate of change
Using the equations (32)-(38), we can determine signal 0 () xt in the following cases In the second step, we obtain the function 2 () t  by integrating 1 () t  - Fig. 5b In the last step, we determine the function 3 () t  on the basis of 2 (): t  32 32 Finally, through integration of 3 () , t  we obtain the signal 0 () () xt x t  and this is the aim.
This operation is shown in Fig. 6a.
During the intervals in which 3 () , t     the signal shape is triangular, with the slope   .
In the intervals in which 3 () 0 , t   the signal is a constant and has the magnitude . a  For m switching moments of 3 () t  the value of error is described by the following equations: The functions 4 () t  and 5 () t  are shown in Fig. 8a, while 5 () t  and 6 () t  in Fig. 8b.

Maximum errors in optimization of models of measuring systems
In the technical domain the application of higher order models usually gives a possibility to better map the dynamic properties of real systems. On the other hand, the analysis of such models is most often difficult and time-consuming. Hence a tendency has developed to replace a higher order model by a simplified one. The class and order of such model are adopted in an arbitrary way. Its parameters can be determined by means of methods that minimize the mapping error, considering all possible input signals. Therefore it is suggested to solve the problem by using one signal which maximizes the error. In this way, the mapping error being determined is credible for any input. The application of the minimax method by means of the Levenberg-Marquardt algorithm (Tomczyk, 2009) gives good results.
This method includes two main numerical computation stages. At the first stage the 0 () xtsignal maximizing the error (Fig.10) is determined.
The notations in (56) The Levenberg-Marquardt algorithm is used for computation in the following stages: Stage 1, for k=1 1. Assume the initial values of the parameters of vector k Ξ 2. Assume the initial value of the coefficient k  (e.g. k  = 0.1) 3. Solve the matrix equation (59) The selection of coefficient values k  and  depends on the adopted programs and selected software.

Example of application
As an example let us consider the fourth order Butterworth, Tchebyhev and Bessel analog filters and integral-square error (14). These filters have the following transfer functions: As a reference let us adopt the mathematical model of ideal low-pass filter given by (12) and (13)

Conclusion
The calibration of systems based on the theory of maximum dynamic errors, makes it possible to create accuracy classes for such dynamic systems and the hierarchy of dynamic accuracy resulting from them. These hierarchies are valid regardless of the shape of the measured dynamic signal that can appear at the input of the investigated system.
The calibration methods presented in this chapter can be widely applied in many different domains, eg. in electrical metrology (transducers, strain gauge amplifiers, filters, etc), geophysics (accelerometer and vibrometer systems), medicine (electroencephalograph and electrocardiograph systems), meteorology (autocomparators, autobridges), etc.