Dynamic measuring methods: a review


ACTA IMEKO 
ISSN: 2221-870X 
March 2019, Volume 8, Number 1, 64 - 76 

 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 64 

Dynamic measuring methods: a review 

Aleksandr L. Shestakov1  

1 South Ural State University, Lenina, 76, Chelyabinsk, Russia 

 

 

Section: RESEARCH PAPER  

Keywords: Dynamic measurements; dynamic error analysis; restoration of a dynamically distorted signal; regularisation; evaluation of the dynamic 
characteristics of measuring equipment 

Citation: Aleksandr L. Shestakov, Dynamic measuring methods: a review, Acta IMEKO, vol. 8, no. 1, article 10, March 2019, identifier: IMEKO-ACTA-08 
(2019)-01-10 

Section Editor: Franco Pavese, Italy 

Received March 23, 2018; In final form May 31, 2018; Published March 2019 

Copyright: © 2019 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 

Funding: The work was supported by Act 211, Government of the Russian Federation, contract № 02.A03.21.0011. 

Corresponding author: Aleksandr Shestakov, e-mail: a.l.shestakov@susu.ru 

1. THE CONCEPT OF DYNAMIC MEASUREMENTS  

The term ‘dynamic measurements’ refers to measurements 
whereby the output signal is significantly different to the 
measuring input signal due to the inertia lag of the measuring 
equipment or measuring system. Such situations occur when 
impulse or other quick-changing actions are measured. A 
sensor, or the whole test facility, can act as measuring 
equipment. A common example of such measurements is an 

impulse pressure measurement 𝑃(𝑡) (Figure 1). 

 
As a rule, the value of the dynamic error is a time function. 

It is significantly different to the measurement uncertainty in 
static measurements. In dynamic measurements, errors of tens 
of per cents are common. 

2. SHORT HISTORICAL BACKGROUND 

The pioneering work on dynamic measurement is the study 
of the French physicist Cornu [1], published in 1887. In this 
work, he studied the motion of a galvanometer and associated 
the value of the damping factor with the physical parameters of 
the device.  

Later on, the inventor of an oscilloscope, French engineer 
Blondel [2], considered dynamic models of some of its modules 
(1893). Also, the work of D. I. Mendeleev [3], published in 
1897 and focused on accurate weighing on the unsteady 
laboratory balance should be referred to the first academic 
papers. In this work the great Russian chemist without using 
differential equations found formulas for accurate weighing in 
the context of the swing of laboratory balance.  

The work of A. N. Krylov [4], wherein a device for 
measuring mine explosion pressure was examined, can be 
considered as a starting point in the development of the theory 
of dynamic measurement. Measuring error was studied as a 
time function for the first time. V. A. Granovskii had a key role 
in the development of the theory of dynamic measurement, and 

 

Figure 1. Dynamic measuring of impulse pressure: Pi = measuring input 
signal, P0 = observed output signal 

ABSTRACT 
The paper presents an overview of the directions in the development of dynamic measurement methods. First, the concept of dynamic measurements 
and a short historical background are outlined. The author describes several types of dynamic characteristics of measuring equipment, which fully 
characterise their dynamic properties. The next part is devoted to an analysis of the types of dynamic errors and their corrections. In conclusion, an 
overview of the identification of dynamic characteristics is given. 

mailto:a.l.shestakov@susu.ru


 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 65 

in his book [5], he presented all the main parts of this theory: 
the determination of the dynamic characteristics of measuring 
equipment, analysis of dynamic errors, dynamic error 
correction, and issues of dynamic measurement uniformity. 

3. DYNAMIC CHARACTERISTICS OF MEASURING 
EQUIPMENT  

There are several types of the dynamic characteristics of 
measuring equipment, which fully characterise its dynamic 
properties. The transfer function of the measuring equipment is  

𝑊(𝑝) =
𝑦(𝑝)

𝑢(𝑝)
=

𝑝𝑚 + 𝑏𝑚−1𝑝
𝑚−1 + ⋯ + 𝑏1𝑝 + 𝑏0

𝑝𝑛 + 𝑎𝑛−1𝑝
𝑛−1 + ⋯ + 𝑎1𝑝 + 𝑎0

, (1) 

where 𝑢(𝑝) and 𝑦(𝑝) are Laplace transforms of input and 
output signals 𝑚 < 𝑛, and its discrete analogue is: 

𝑊(𝑧) =
𝑦(𝑧)

𝑢(𝑧)
=

𝑧𝑙 + 𝑟𝑙−1𝑧
𝑙−1 + ⋯ + 𝑟1𝑧 + 𝑟0

𝑧𝑘 + 𝑞𝑘−1𝑧
𝑘−1 + ⋯ + 𝑞1𝑧 + 𝑞0

 (2) 

𝑧 is a delay operator. The frequency characteristics of the 
measuring equipment, amplitude-frequency and phase-
frequency characteristics are illustrated in Figure 2. 

The impulse transient function ℎ(𝑡) (Figure 3) is a response 
of the measuring equipment to the 𝛿-impulse of the infinitely 
small duration, in terms of time and an infinitely large 
amplitude  

∫ 𝛿
∞

0

(𝑡)𝑑𝑡 = 1  

The transient function 𝑘(𝑡) (Figure 4) is a response of the 
measuring equipment to the unit step input.  

All these characteristics fully characterise the dynamic 
properties of the measuring equipment, and from this 
perspective, they are equivalent. Furthermore, there are 
methods for converting one characteristic into another. 
Theoretical methods for defining dynamic characteristics have 
been quite fully developed in terms of identifying automatic 
control systems. Moreover, there is no differential difference 
between the automatic control system and the dynamic 
measuring system. However, the dynamic characterisation of 
the measuring equipment is a complex problem. In the passport 
data provided by the sensor manufacturers, the presence of 
dynamic characteristics is still the exception rather than the 
norm. For complex dynamic measuring facilities, this process 
requires special experiments.  

4. ANALYSIS OF DYNAMIC MEASURING ERRORS  

After the information on dynamic characteristics of 
measuring equipment is obtained, the next question is: What 
error occurs when taking measurements in dynamic mode? We 
can single out several approaches to the analysis of dynamic 
measuring errors. For the first, the deterministic approach, a 
structural diagram of the dynamic error analysis can be 
represented in the following way (Figure 5): 

The Laplace transforms of dynamic errors take the form 

(𝑝) = 𝑢(𝑝)[𝑊(𝑝) − 1] (3) 

In fact, the dynamic error through the inverse Laplace 
transform is measured as follows 

 

Figure 2. Amplitude- and phase-frequency characteristics of measuring 
equipment 

 

Figure 3. Impulse transient function of the measuring equipment. 

 

Figure 4. Transient function of the measuring equipment. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 66 

(𝑡) = 𝐿−1{𝑢(𝑝)[𝑊(𝑝) − 1]}, (4) 

and through the solution of an integral equation 

(𝑡) = ∫ ℎ
∞

0

(𝑡 − 𝜏)𝑢(𝜏)𝑑𝑡 − 𝑢(𝑡), (5) 

where 𝑊(𝑝) and ℎ(𝑡 − 𝜏) are the transfer function and the 
impulse transient function of measuring equipment respectively.  

Equation (4) and equation (5) show that the dynamic error 
of measurements is the time function and depends both on the 
dynamic properties of the measuring equipment and on the 
measuring signal. 

The analytical expression for the dynamic errors of the 
measuring step signal for a measuring transducer of arbitrary 
order was outlined by V. A. Granovskii [5]. In a general case in 
which the input of the measuring system is affected by the 
input signal of the Laplace transform, which can be represented 

as 𝑢(𝑝) = 𝑢1(𝑝) 𝑢2(𝑝)⁄ . The measuring system has the 
transfer function 𝑊(𝑝) = 𝑀(𝑝) 𝐷(𝑝)⁄ , where 𝑀(𝑝) and 𝐷(𝑝) 
are polynomials of 𝑚 and 𝑛 order, respectively, and 𝐾(0) =
𝑀(0) 𝐷(0)⁄ . The dynamic error, in steady-state conditions, 
takes the form 

(𝑡) = ∑
𝑢1(𝜌𝑘 )

𝑢′2(𝜌𝑘 )

𝜈

𝑘=1

[
∏ (1 −

𝜌𝑘
𝛾𝑖

)𝑚𝑖=1

∏ (1 −
𝜌𝑘
𝜆𝑖

)𝑛𝑖=1

− 1] 𝑒 𝜌𝑘(𝑡) (6) 

where 𝜆1, … 𝜆𝑛 and 𝛾1, … 𝛾𝑚 are transfer function poles and 
zeros. 𝜌𝑘  is a pole of the Laplace transform of the input signal 
𝑢 [6]. In the same work, dependencies for determining the 
poles and zeros of the measuring system, providing the required 
maximum allowable errors to measure linear, parabolic, and 
exponential signals, are obtained.  

More complex signals can be approximated by a 
combination of simpler signals and taking into account non-
zero initial conditions when they are joined. However, analytical 
dependencies are lengthy. The mathematical simulation method 
is more preferable for analysing dynamic errors in such a case. 
This approach is useful for the analysis of dynamic errors when 
signals of the known form are measured. 

[7] presents the expressions of the dynamic errors of the 
measuring system used when periodic square impulse and 
trapezoidal wave signals are measured. The transition function 

ℎ(𝑡) is used here as a characteristic of the dynamic properties 
of the measuring system.  

There is also an approach whereby a dynamic error is 
determined as an integral in some period of observing an 
integral square error. Thus, [8] analyses the method for 
decreasing an order of the dynamic system based on the 
minimisation of a quadratic error criterion for a simplified 
system with regards to a high-order system.  

The error formation scheme is shown in Figure 6. 

The method for determining a signal 𝑢0(𝑡) switching from 
+1 to –1 at the moments 𝑡1, 𝑡2, … 𝑡𝑟  (Figure 7), maximising 
the square error on the interval [0, 𝑇], was suggested.  

The results of this work can be considered as a method for 
finding a signal that generates the maximum dynamic error of 
the measuring system.  

[9] considers a method for obtaining a switching input 

signal, limited in the value |𝑢(𝑡)| ≤ 𝑎, whose speed is also 
limited. |�̇�(𝑡)| ≤ 𝜃,  𝑡 ∈ [0, 𝑇] maximises the integral square 
error of the difference between the signals of the measuring 
system and its simplified dynamic model. The method requires 
the solution of a set of complicated integral-convolution 
equations. As the author notes, applying this method, it is 
difficult to receive a signal with more than 25 switching 
operations.  

[10] suggests a method for generating a switching signal on 
the basis of wavelet transform. The measured input signal has 
limitations in amplitude and the input change rate (Figure 8). 

The method generates a switching signal that maximises the 
integral square error of the dynamic measuring system. The 
switching points are specified by the genetic algorithm.  

Based on the approach by which the measured input signal 
(which provides the maximum integral square error of the 
dynamic measuring system) is generated, an attempt to create a 
hierarchy of dynamic measuring systems according to their 
dynamic accuracy was made in [11]. The diagram for 
determining dynamic accuracy is suggested in Figure 9.  

Here, 𝐾𝑥 (𝑝)and 𝐾𝑠 (𝑝) are the transfer functions of the 

 

Figure 5. Structural diagram of forming dynamic errors in the deterministic 
case. 

 

Figure 6. Block diagram for determining the integral square error. 

 

Figure 7. Input signal maximising the integral square error. 

 

Figure 8. Input signal maximising the integral square error. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 67 

measuring system and a standard system with the desired 
dynamic characteristics. The system error has two components. 

The coefficient 

𝑐 = lim
𝑝→∞

[𝐾𝑥 (𝑝) − 𝐾𝑠 (𝑝)] (7) 

specifies a steady-state component of the error 𝑐𝑢(𝑡), and  

𝑦(𝑡) = ∫ 𝑘(𝑡 − 𝜏)𝑢(𝜏)
𝑡

0

𝑑𝑡 (8) 

where 𝑘(𝑡 − 𝜏) is an impulse transient function of the 
difference between the systems 𝐾𝑥 (𝑝) − 𝐾𝑠 (𝑝). 

A criterion for the dynamic accuracy of the measuring 
system is proposed on the basis of the quadratic criterion 

𝐼 = ∫ [𝑐𝑢(𝑡) + 𝑦(𝑡)]
𝑇

0

2

𝑑𝑡 (9) 

to use its maximum value 

max 𝐼 [ (𝑡)] = max
𝑡∈[0,𝑇]

max
𝑢∈[𝑈]

𝐼{ (𝑡)[𝑢(𝑡), 𝑐, 𝑘(𝑡), 𝑇]} (10) 

The attempt to classify (create a hierarchy) dynamic 
measuring systems by dynamic errors is very useful. However, it 
has not been as successful as the establishment of measuring 
equipment in the case of static measurements by measurement 
uncertainty and accuracy classes. It is not clear what is 
considered as a standard measuring system. The situation is 

logical. As shown in [9], the measured signal 𝑢(𝑡), maximising 
the integral square error, is disruptive. Furthermore, finding it is 
highly complicated, especially for high-order systems. The 
interval of integration should also be compared with the 
dynamic properties of the measuring system. Nevertheless, the 
problem statement is promising.  

The third approach to the analysis of dynamic errors is 
based on consideration of the measured signal as random with 
the known spectral density. Then, the structural diagram for 
estimating the measurement errors can be presented in the 
following form (Figure 10). 

In Figure 10, 𝑊(𝑗𝜔) is a frequency characteristic of the 
dynamic measuring system. 𝑆𝑢 (𝜔),   𝑆𝑛 (𝜔),   𝑆𝑤 (𝜔),   𝑆𝑦 (𝜔),  

𝑆𝜀 (𝜔) are spectral densities of the input signal 𝑢(𝑡); input and 
output noise 𝑛(𝑡) and 𝑤(𝑡); the output signal 𝑦(𝑡), and the 
error (𝑡). The spectral densities, even the function in the form 

of fractional rational functions, can be represented as [12], [13]  

𝑆𝑢 (𝜔) =
𝑏0 + 𝑏1𝜔

2 + ⋯ + 𝑏𝑚 𝜔
2𝑚

𝑎0 + 𝑎1 𝜔
2 + ⋯ + 𝑎𝑛 𝜔

2𝑚
.  

The typical characteristic of the spectral density of the 
measured input signal indicates that this signal always has a 
restricted frequency spectrum (Figure 11). Taking into account 

the mutual uncorrelatedness of signals 𝑢(𝑡), 𝑛(𝑡), and 𝑤(𝑡), 
the spectral density of the error takes the form 

𝑆𝜀 (𝜔) = |1 − 𝑊(𝑗𝜔)|
2𝑆𝑢(𝜔) + |𝑊(𝑗𝜔)|

2𝑆𝑢(𝜔) + 𝑆𝑤(𝑗𝜔) (11) 

The average value of the squared error is calculated by the 
spectral density 

̄2 =
1

2𝜋
∫ 𝑆𝜀 (𝜔)

∞

−∞

𝑑𝜔 .  

5. DYNAMIC ERROR CORRECTION  

By this term, we refer to the decrease in dynamic measuring 
errors by mathematical processing of the measurement results 
using the knowledge of dynamic characteristics of the 
measuring system. There is also another name for this process: 
the restoration of a dynamically distorted signal. 

The first approach, which results in a number of methods 
for solving this problem, is based on the description of the 
process of dynamic measurements by the Volterra equation of 

the first kind. The impulse transient function ℎ(𝑡 − 𝜏) is used 
as a dynamic characteristic of the measuring equipment (Figure 
12).  

The measured input signal 𝑢(𝑡) is connected with the 
observed output signal 𝑦(𝑡) by the integral equation 

𝑦(𝑡) = ∫ ℎ(𝑡 − 𝜏)
𝑡

𝑇1

𝑢(𝜏)𝑑𝜏, (12) 

where 𝑇1 ≤ 𝑡 ≤ 𝑇2. In this case, 𝑇1 can be equal to 0 or –∞, 
and 𝑇2 = ∞. ℎ(𝑡 – 𝜏) is an impulse transient function of the 
measuring equipment. As the variable 𝑡 is time, the output 
signal cannot appear before the input one. That is why a 
requirement of physical realisability should be met: 

ℎ(𝑡 − 𝜏) = 0 in case of 𝑡 < 𝜏. 

 

Figure 9. Block diagram for determining dynamic errors of the system 

 

Figure 10. Structural diagram of the formation of dynamic errors in the case 
of a random signal. 

 

Figure 11. Typical characteristic of the spectral density of the measured 
signal. 

 

Figure 12. Block diagram of the measuring equipment represented by the 
impulse transient function. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 68 

To restore the measured signal 𝑢(𝑡), it is necessary to solve 
equation (12). The direct method for solving equation (12) is 
based on replacement of the integral by the sum [14]: 

∑ 𝐴𝑗 ℎ𝑖𝑗 𝑢𝑗 = 𝑦𝑖 ,

𝑖

𝑗=1

 (13) 

are coefficients of the quadrature formula. There are many such 
formulas [15], [16]. However, a general feature of this approach 

is the impossibility of the direct determination of 𝑢1 for further 
development of a recurrent algorithm. To find it, it is necessary 

to differentiate 𝑦(𝑡) at the starting point 

𝑢1 =
𝑦′(𝑇1)

ℎ11
 . (14) 

In the presence of noise in the output signal, this leads to a 
significant error.  

There are many possible methods of solving equation (12). 

In [17], it is suggested to approximate the output signal 𝑦(𝑡) 
and the impulse transient function ℎ(𝑡 – 𝜏) by a certain type of 
function. Taking into account a damped oscillating nature of 
the studied processes, we can approximate 

𝑦(𝑡) = 𝑒 −𝛽|𝑡| ⋅ cos 𝜔 𝑡 (15) 

ℎ(𝑡 – 𝜏) = 𝑒 −𝛼|𝑡−𝜏| ⋅ cos(𝜈(𝑡 − 𝜏)), (16) 

where 𝛽, 𝛼, 𝜔, 𝑣 are approximation parameters. Then, equation 
(12) can be solved by simple numerical or even analytical 
methods.  

One method for solving equation (12) is based on the series 

expansion of the desired signal 𝑢(𝑡) by eigenfunctions 𝜑𝑖 (𝑡)of 
the impulse transient function ℎ(𝑡 – 𝜏) [18], which satisfies the 
equation 

∫ ℎ(𝑡 − 𝜏)
𝑎

−𝑎

𝜑𝑖 (𝜏)𝑑𝜏 = 𝜆𝑖 𝜑𝑖 (𝑡), (17) 

where 𝜆𝑖 is an eigenvalue of the function ℎ(𝑡 – 𝜏). Then, 𝑢(𝑡) is 
expressed in terms of the series sum, as follows: 

𝑢(𝑡) = lim
𝑁→∞

∑ 𝑆𝑖  𝜑𝑖 (𝑡) .

𝑁

𝑖=0

 (18) 

The recurrent algorithm for calculating the measured signal 
is suggested in [19]: 

𝑢𝑛+1(𝑡) = 𝑦(𝑡) + 𝑢𝑛 (𝑡) − ∫ ℎ(𝑡 − 𝜏)
𝑡

𝑇1

𝑢𝑛(𝜏)𝑑𝜏 , (19) 

where 𝑢0(𝑡) = 0; 𝑢1(𝑡) = 𝑦(𝑡). 
The common feature of the abovementioned methods is the 

high sensitivity of solving the Volterra integral equation to 
noise, which is always present in the output signal of the 
measuring equipment. This is due to the incorrectness of the 
task in restoring the dynamically distorted signal. One of the 
first works, which has been devoted to the peculiar features of 
dynamic measurement, is the work of G. N. Solopchenko [20]. 

In [21], the integral equation is solved taking into account 
the output noise of the measuring equipment. The integral 
equation (12) is approximated by a discrete equation, which in 

matrix form, is written as �̄� = 𝐻�̄� + �̄�, where 

�̄� = [𝑦(0), 𝑦(1), … , 𝑦(𝑀 – 1)]𝑇   

�̄� = [𝑢(0), 𝑢(1), … , 𝑢(𝑁 – 1)]𝑇  

�̄� = [𝑧(0), 𝑧(1), … , 𝑧(𝑀 – 1)]𝑇 . 

It is assumed that the output noise z of the measuring 
equipment is a Gaussian probability distribution with the 

average 𝐸{�̄�} = 0 and the known covariance matrix 𝑅𝑧 =
𝐸{�̄��̄� 𝑇 }. Minimising the mean-square error by the least squares 
method, the values of the measured signal takes the form 

�̄� = (𝐻𝑇 𝑅𝑧
– 1𝐻)– 1 𝐻𝑇 𝑅𝑧

– 1�̄� .  

The method is quite simple and takes into account the noise 
assuming it is Gaussian. However, we note that the 
measurement error increases with time, and it does not quite 
solve the problem of noise control.  

When the dynamically distorted signal is restored, the noise 
present in the output signal is amplified. The desire to weaken 
the influence of this phenomenon has led to the development 
of methods for regularising the solution to equation (12). The 
work of A. N. Tikhonov [22] has been highly valuable in the 
development of these methods. The principle of the method 
for producing regularised solutions of equation (12) is as 
follows. Let the output signal of the measuring equipment be 
represented in the following form: 

𝑦(𝑡) = 𝑦𝑇 (𝑡) + 𝜈(𝑡), (20) 

where 𝜈(𝑡) is measurement noise; 𝑦𝑇 (𝑡) is an output signal of 
the measuring equipment without noise at its output. Applying 
the Fourier transform to equation (12), reduced to the infinite 
limits of integration, we can write 

𝑢(𝜔) ⋅ 𝐻(𝜔) = 𝑦𝑇 (𝜔) + 𝜈(𝜔), (21) 

where 𝑢(𝜔), 𝑦Т(𝜔), 𝜈(𝜔) is the Fourier transform of the 
corresponding signals, and 𝐻(𝜔) is the Fourier transform of 
the impulse transient function ℎ(𝑡 – 𝜏). 𝐻(𝜔) characterises the 
frequency properties of the measuring equipment. Then, 

𝑢(𝜔) = 𝑢𝑇 (𝜔) +
𝜈(𝜔)

𝐻(𝜔)
 (22) 

where 𝑢𝑇 (𝜔) is the Fourier transform of the accurate measured 
signal. 

The application of the inverse Fourier transform to equation 
(22) gives the formula 

𝑢(𝑡) = 𝑢𝑇 (𝑡) +
1

2𝜋
∫

𝜈(𝜔)

𝐻(𝜔)
𝑒 −𝑗𝜔𝑡

+∞

−∞

𝑑𝜔, (23) 

in which the integral diverges, as a rule. The reason for that is 

the presence of high frequencies in the function 𝜈(𝜔) and the 
excess of the degree of a denominator under the degree of a 

numerator in Н(𝜔). The idea of the regularisation method is 
the suppression of the influence of high frequencies 𝜔, 

multiplying the function 
𝑦(𝜔)

𝐻(𝜔)
 by a stabilising factor𝑓(𝜔, 𝛼), 

which depends on the parameter 𝛼 and frequency 𝜔. A. N. 
Tikhonov [22] proposes determining the stabilising factor as 

𝑓(𝜔, 𝛼) =
|𝐻(𝜔)|2

|𝐻(𝜔)|2 + 𝛼𝑀(𝜔)
 (24) 

where 𝑀(𝜔) is a given even function of 𝜔. With this in mind, 
we obtain the classes of the regularised solutions of integral 
equation (12): 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 69 

𝑢𝛼 (𝑡) =
1

2𝜋
∫

𝐻(−𝜔) ⋅ 𝑦(𝜔)

𝐻(𝜔) ⋅ 𝐻(−𝜔) + 𝛼 ⋅ 𝑀(𝜔)

+∞

−∞

 𝑒 −𝑗𝜔𝑡 𝑑𝜔 . (25) 

In this case, one of the possible variants of the polynomial 

𝑀(𝜔) takes the form 

𝑀(𝜔) = ∑ 𝑞𝑛 𝜔
2𝑛

𝑝

𝑛=0

 (26) 

where 𝑞𝑛  is a given non-negative constant.  
It is suggested that the value of parameter 𝛼 should be 

found according to ‘discrepancy’. Let us identify the limiting 

error 𝛿 of determining the output signal of the measuring 
equipment 𝑦(𝑡). Let us define the ‘discrepancy’ of the 
approximate solution: 

𝜌(𝛼) = ∫ [∫ ℎ(𝑡 − 𝜏)𝑢𝛼 (𝜏)𝑑𝜏 − 𝑦(𝑡)
𝑡

𝑜

]
+∞

−∞

2

𝑑𝑡 . (27) 

We choose a monotonic sequence of numbers 

𝛼0, 𝛼1, . . . , 𝛼𝑛 , e.g. a geometric progression. For each value of 
𝛼𝑖 , the ‘discrepancy’ 𝜌(𝛼𝑖 ) is calculated. As the desired value of 
𝛼, we take such 𝛼𝑘, for which the equality 𝜌(𝛼𝑘 ) = 𝛿 is 
fulfilled with the required accuracy. The approximate solution 
of this equation can be found with the help of other numerical 
methods; for example, Newton methods [23], [24]. 

In [24], an optimal regularised solution of integral equation 
(12) is obtained when the additional information on the spectral 
densities of the output noise of the measuring equipment 

𝑆𝜈 (𝜔) and the input signal 𝑈(𝜔) is available. This solution 

does not depend on parameter 𝛼   and has the form 

𝑈𝑜𝑝𝑡 (𝑡) =
1

2𝜋
∫

𝐻(−𝜔) ⋅ 𝑦(𝜔)

𝐻(𝜔) ⋅ 𝐻(−𝜔) +
𝑆𝜈 (𝜔)
𝑈(𝜔)

⋅ 𝑒 −𝑗𝜔𝑡
+∞

−∞

𝑑𝜔 . 

This result coincides with the result of applying the 
optimum filtration according to Wiener [25], [26]. 

In many practical cases, the spectral density of noise of the 

output signal 𝑦(𝑡) can be determined according to the 
observation results. It is more difficult to specify the spectral 

density of the input signal 𝑢(𝑡). Its evaluation can be obtained 
on the basis of a priori information about the measured signal. 
This complicates the task of the optimal restoration of the 
input signal of the measuring equipment. 

The impulse transient function used above is not the only 
characteristic that reflects the dynamic properties of the 
measuring equipment. The dynamic properties can also be 
defined by a differential equation with the corresponding 
coefficients; a transfer function; and a set of amplitude-
frequency and phase-frequency characteristics [5]. The use of 

these characteristics enables the development of relevant 
methods for the restoration of a dynamically distorted signal or 
to correct dynamic errors.  

The use of differential equations enables the development of 
methods of dynamic measurements based on the results of 
automatic control theory. [27] presents a review of automatic 
control theory and the prospects of its application in 
measurement technology. In [28], the term ‘dynamics’ is 
considered from different angles in respect to the problems of 
metrology. The importance of studying measuring signals is 
noted. A classic description of dynamic systems on the basis of 
differential equations is given. 

The author pays attention to the related concepts of the 
properties and behaviours of systems. The concepts of the 
stability, stationary state, and drift of dynamic systems are also 
analysed in the work. The basis for applying the methods of 
automatic control theory to the correction of dynamic 
measuring errors is the similarity of structures of automatic 
control systems and dynamic measuring systems [32]. The 
classic structure of the automatic control system is illustrated in 
Figure 13, which is a graphical representation of the system of 
differential equations. 

In Figure 13, 𝑢 is an input signal of the system; 𝑥 is a state 
vector of the system; 𝑦 is an observation vector; and 𝐴, 𝐵, 𝐶, 𝐷 
are the corresponding matrices. The required dynamic 
properties are provided by setting the coefficients of the 

feedback matrix 𝐾. 
In dynamic measuring systems, it is impossible to establish 

the feedback from the output to the input, since the input’s 
measured action is not available. That is why, in such systems, 
the possibility of reducing measurement errors is realised by 
means of a special filter at the output of a primary measuring 
transducer (sensor). In this case, the structural diagram of the 
dynamic measuring system is shown in Figure 14. 

In Figure 14, u is an input measured vector of the system; 

𝑥𝑆, 𝑦𝑆  are state and observation vectors; 𝐴𝑆, 𝐵𝑆 , 𝐶𝑆, 𝐷𝑆 are 
corresponding matrices of the sensor; 𝑥𝑓 , 𝑦𝑓  are vectors of the 

state and observation of the filter; 𝐴𝑓 , 𝐵𝑓 , 𝐶𝑓 , 𝐷𝑓  are filter 

matrices of the corresponding dimension; and 𝐾 is a matrix for 
setting the coefficients to provide the required dynamic error. 
In this system, based on the similarity of the sensor and filter 
structures, the latter can be considered as a dynamic sensor 
model. Therefore, dynamic measuring systems have been 
developed based on the models for creating automatic control 
systems outlined in [30].  

A dynamic measuring system with modal control of dynamic 
characteristics has the following structure, as illustrated in 
Figure 15. 

At this time, the transfer functions of sensor 𝑊𝑆 and a 
model of sensor 𝑊𝑀𝑆 are equal. If the output signals 𝑦𝑆  and 
𝑦𝑀𝑆  are close to each other, then the input signals 𝑢 and 𝑢𝑀𝑆 
will differ from each other slightly. 

Consequently, the input signal of the model 𝑢𝑀𝑆 can be used 
to judge the input signal of the sensor 𝑢. This is the basic idea 

 

Figure 13. Structural diagram of the automatic control system. 

 

Figure 14. Structural diagram of the dynamic measuring system. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 70 

of the correcting the dynamic measuring error. In this system, 

the transfer function 𝑊𝐷𝑆(𝑝) = 𝑢𝑀𝑆(𝑝) 𝑢(𝑝)⁄  can be obtained 
with any given zero and poles. If 

𝑊𝑆 (𝑝) =
𝑏𝑚 𝑝

𝑚 + 𝑏𝑚−1𝑝
𝑚−1 + ⋯ + 𝑏1𝑝 + 𝑏0

𝑝𝑛 + 𝑎𝑛−1𝑝
𝑛−1 + ⋯ + 𝑎1𝑝 + 𝑎0

, 

then the transfer function of the dynamic measuring system can 
be obtained by  

𝑊𝐷𝑆 (𝑝) =
(𝑏𝑚 – 𝑑𝑚 )𝑝

𝑚 + (𝑏𝑚−1– 𝑑𝑚−1)𝑝
𝑚−1 + ⋯ + (𝑏0– 𝑑0)

𝑝𝑛 + (𝑎𝑛−1– 𝐾𝑛−1)𝑝
𝑛−1 + ⋯ + (𝑎1– 𝐾1)𝑝 + (𝑎0– 𝐾0)

 

where 𝑑0, 𝑑1, …, 𝑑𝑚 are configurable filter parameters 𝑊𝑓 , and 

𝐾0, 𝐾1, . . . , 𝐾𝑛−1are configurable feedback vector parameters of 
𝐾 [31]. 

The development of this method for the observed state 
coordinate vector is outlined in [31], [34]. If the spectral density 

of the measuring signal 𝑢, and the observed coordinates are 
known, then the structural diagram of this measuring system is 
shown in Figure 16. 

If the spectral densities of the measuring signal 𝑆𝑢 (𝜔) and 
noises 𝑆𝑣2

1 (𝜔), … , 𝑆𝑣2
𝑚 (𝜔) in the channels of the observed 

state coordinates are known, then, in [33], the optimum 
individual characteristics of the correcting device that reduces 
the measuring error are obtained. 

Dynamic measuring systems based on the iterative principle 
of the measuring signal recovery are given in [32], [34]. In the 
structure of such a measuring system, there are several sensor 
models. Every subsequent model processes the measuring 
errors of the previous ones. The main property of such 
measuring systems is their low sensitivity to sensor noise and 
their ability to reduce dynamic error without the significant 
increase of the system bandwidth.  

The structural diagram of this measuring system is shown in 
Figure 17 [29]. 

In [32], [35], dynamic measuring systems in which a sliding 
mode is implemented are observed and explained. The 
structures ensuring the stability of such systems are analysed. A 
combination of high dynamic accuracy and low sensitivity to 
sensor noise is shown. A structural diagram of a dynamic 
measuring system in sliding mode is given in Figure 18. 

The presence of a relay element for a high-order system may 
cause instability of the system. To ensure stability a cascading 
structure of the measuring system is given in reference [32]. 
Every subsequent cascade restores the measuring signal more 
accurately than the previous one. The results of the system 
modelling in a sliding mode with three cascades are shown in 
Figure 19. 

In [32], [36], dynamic measuring systems are observed on 
the basis of neural network structures. The stability conditions 
of such systems are investigated therein. Special means of 
ensuring the suppression of sensor noise are introduced to the 
system structures. The unique feature of neural network 
dynamic measuring systems is their ability to adapt to the type 
of such measuring signals and to reduce the measuring error in 
the multiple measuring of such signals.  

A feature of dynamic measuring systems developed on the 
basis of methods of automatic control theory is the possibility 
to obtain an additional channel for dynamic error estimation 
[32]. An error in an estimate of a dynamic error is comparable 
with a measuring error of the input signal of the measuring 
system. 

A structural diagram of the channel for dynamic error 
estimation is shown in Figure 20. 

In [32], [37], intelligent dynamic measuring systems that can 
adapt their parameters to dynamic measuring errors are 

 

Figure 15. Structural diagram of a dynamic measuring system with modal 
control of dynamic characteristics. 

 

Figure 16. Structural diagram of a measuring system with the observed 
state coordinate vector. 

 

Figure 17. Structural diagram of an iterative dynamic measuring system. 

 

Figure 18. Structural diagram of a dynamic measuring system in sliding 
mode. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 71 

considered. Measuring systems with parameter adaptation after 
the performance of measurements during the processing of 
their results are considered. Measuring systems with adaptation 
of dynamic parameters in the process of measuring are also 
given. 

Based on ideas of automatic control theory, [38] considers 
dynamic measuring systems from the standpoint of 
simultaneous measurement and observation. A structural 
diagram of this measuring system is shown in Figure 21. Here, 

𝑃 is a dynamic process in the system; 𝑀𝑈 and 𝑀𝑌 are 
measurement processes 𝑢(𝑡) and 𝑦(𝑡) correspondingly; and 
𝑥𝑜𝑏𝑠 (𝑡) is the observation process.  

The use of the observation process allows us to refine the 
model of the process P. This approach is mainly focused on 
multi-connected dynamic measuring systems, where u is the 
vector of the measured coordinates. Its implementation is based 
on the use of the methods and results of automatic control 
theory, which are well developed at present [39]. This approach 
is promising both from the standpoint of the development of 
the general theory of dynamic measurements and in 
applications to dynamic measurements of the specific systems. 

Another approach in dynamic measurements that considers 
the uncertainty of the dynamic model is based on the use of 
chaos theory. Thus, in [40], compensation for the dynamic error 
is considered in the presence of uncertainty in the coefficients 
of the sensor dynamic model. The uncertainty of these 
coefficients is expressed on the basis of the method of 
polynomial chaos theory. The method also assumes the 
description of the input measured signal in an analytical way. 
This is its essential limitation. This method gives an estimate of 
the measurement error and takes into account noise at the 
sensor output. 

6. IDENTIFICATION OF THE DYNAMIC CHARACTERISTICS OF 
A MEASURING EQUIPMENT 

Concerning the identification of dynamic characteristics and 
methods of solving this problem, measuring equipment does 
not differ from automatic control systems. For automatic 
control systems, a great many identification methods have been 

developed [41]. However, these methods do not pay much 
attention to the problem of accuracy in the identification of 
dynamic characteristics. However, they are an important feature 
of measuring equipment. The approach of determining dynamic 
characteristics with an estimate of their accuracy was proposed 
by V. A. Granovskii [5]. The determination of dynamic 
characteristics requires the formation of test signals (Figure 22). 

On a number of test signals, he identified errors in the 
determination of the transient characteristics, impulse response, 
and frequency characteristics. He proposed a method for the 
multiple integration of differential equations of the measuring 
equipment, which helps to determine the coefficients of 
differential equations and errors in their determination for the 
input test signals of an arbitrary shape. He developed an 
adaptive method for the identification of dynamic 
characteristics. This method determines the transfer function by 
the subsequent complication of the transfer function, step by 
step, and identifies optimal coefficients at every step. Based on 
the minimisation of integral quadratic error, this method makes 
use of five types of test signals: stepwise; exponential; and three 
impulse signals of triangular, cosine, and sinusoidal forms. 

The determination of impulse and transfer functions by 
means of statistical regularity is outlined in [42], [43]. In fact, 
determining the dynamic characteristics of measuring 
equipment is not an easy task, especially for multi-axis sensors 
and multichannel measuring systems. Quite often, the test 
equipment is a complicated facility. In the vast majority of 
cases, the specification of requirements for the test equipment 
is made by means of the expert method, and the accuracy of the 
identification of dynamic characteristics is not estimated. There 
are several specific examples of the determination of dynamic 
characteristics, given below. 

In [44], the technology for the determination of dynamic 
characteristics for a six-component force-and-moment sensor is 
considered. Eigen frequencies for each of the six coordinates 
are considered to be such characteristics. The study offers two 
methods of obtaining these frequencies. The computational 
method uses an ANSYS SOFTWARE simulation. The 
experimental method measures the frequency of natural 
oscillations after impulse force. 

In [45], the method used for the dynamic calibration of the 
force measurement devices is considered. In this case, a 

 

Figure 20. Structural diagram of a channel for dynamic error estimation.  

 

Figure 19. Results of modelling a tri-cascade dynamic measuring system in 
sliding mode. 

 

Figure 21. Structural diagram of the measuring process and observation.  

 

Figure 22. Structural diagram of the equipment used to identify dynamic 
characteristics.  



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 72 

standard force dynamometer and force dynamometer to be 
tested connected in a series through the load frame in the 
general structure are used. By exciting the oscillations of this 
structure, the frequency response of the force measuring device 
is defined, including the natural oscillation frequency.  

In [46], the experimental determination of the dynamic 
characteristics of the mechanical system is considered. Based on 
measurements with accelerometers and interferometers, the 
amplitude-frequency and phase-frequency characteristics of the 
system along the axes are determined. Frequency characteristics 
of noise in a mechanical system are also estimated.  

In [47], the influence of a dynamic effect on the process and 
accuracy of a flowmeter calibration is considered. The dynamic 
properties of flowmeters are given in the form of a delay in the 
signal for converting the measuring flow signal into an electrical 
signal. The study proposes that the model of dynamics is 
described by the equation of the first order. Its parameters are 
determined in the process of calibration. This reduces the 
measurement uncertainty of the flowmeter. 

In [48], the calibration method of a torque measuring 
transducer operating in dynamic modes is considered. This 
method is based on an accurate estimate of mass angular 
acceleration with a known momentum of inertia. 

The researchers in [49], [50] consider the determination of 
the frequency characteristics of the kernel of the Volterra 
equations for single- and multi-channel dynamic systems. This 
method is based on the use of the Fourier transform when 
applying known harmonic (sinusoidal) signals to the input of 
the dynamic system. This method identifies the frequency 
characteristics of a dynamic system with non-linearities. This 
method is pointed to the application in dynamic measuring 
systems to measure the electrical signals in a broad band. 

The method of dynamic calibration of a high-pressure 

sensor based on a quasi- function is considered in [51]. This 
signal is given in the form of a half sine wave with the duration 
of several microseconds, µs. Using the fast Fourier transform of 

the input quasi -signal f(t) and the output signal y(t): 

𝐹(𝜔) = fft[𝑓(𝑡)], 𝑌(𝜔) = fft[𝑦(𝑡)]  

and the normalised frequency response of a pressure sensor is 
determined experimentally 

𝐻(𝜔) = |

𝑌(𝜔)
𝑌(0)

𝐹(𝜔)
𝐹(0)

| . 

The quasi 𝛿-impulse is created by the bullet going through 
the Hopkinson bar. 

The researchers in [52] consider the behaviour of a dynamic 
measuring system in the neighbourhood of a user-specified 
frequency. A method of approximation of a frequency response 
in the neighbourhood of the frequency by polynomials is 
proposed. The polynomial approximation takes into account 
the non-zero initial conditions of a dynamic system. However, 
in this case, it does not take into account disturbances and noise 
in the system, considering that at frequency response 
approximation, the input and output signals of the measuring 
system are known. The error of the polynomial approximation 
is also studied. This method is pointed to the analysis of the 
behaviour of a dynamic measuring system concerning the 
measurement of harmonic (sinusoidal) signals in the 
neighbourhood of a certain frequency. 

The creation of mathematical models of measuring 
equipment that adequately reflects the properties has made it 
possible to develop measuring equipment with optimal 
characteristics and to obtain higher measurement accuracy 
when processing measuring results. This approach is 
implemented in mathematical models, and software is created in 
NPL. Thus, in [53], one can see the application of the method 
of mathematical simulation to analyse dynamic measurements 
executed in NPL. The first example demonstrates the 
mathematical model of a lock-in amplifier. The comparison 
between the operation of a lock-in amplifier in a real 
experiment and the results of the simulation reveals sources of 
additional error. The second example considers simulation of 
an underwater acoustics network of three hydrophones. The 
case is considered when one of the sensors has a smaller 
bandwidth than the others. The software helps to simulate the 
operation of this network under various conditions. The third 
example considers an imperfect clock with a quartz oscillator. 
The accuracy of the temporal data fit has a direct influence on 
dynamic error. Based on the Monte Carlo method, this software 
is used in the simulation of the long-term behaviour of the 
timing device, taking into account long-term drift and jitter 
within the measurement data. The fourth example presents a 
shock tube as a means of generation of calibration pressure 
signals with a broad bandwidth. The effects in a shock tube are 
simulated on the basis of wave propagation in a shock tube. 
The simulation enables the determination of the optimal 
parameters of a shock tube. 

[81] is devoted to the development and application of a new 
open-source software package for the analysis of dynamic 
measurements. 

At the same time, in many publications, the question of the 
determination of dynamic characteristics and measurements in 
different problems is analysed. 

In [54], [55], the method of autoregressive analysis of the 
spectrum of the output signal is used to determine the dynamic 
characteristics of the pressure transducer (the transfer function 
in the frequency domain and the transient function in the time 
domain). In this case, the spectral analysis is undertaken on the 
basis of the Fourier transform of the transducer response to a 
step change in the pressure signal. The results of the 
experimental studies show good correspondence between the 
characteristics of the obtained dynamic models and real 
transducers. 

A generator of dynamic pressure with a rotating valve is 
proposed in [56]. The methods of frequency response 
estimation and the speed of the pressure transducer are 
developed on the basis of an analysis of the transducer’s 
reaction to the generated signal as a series of rectangular 
pressure impulses with variable frequencies and amplitudes. 
Experimental studies have shown good repeatability of the 
amplitude spectrum of the generated signal and the possibility 
to compare the frequency responses of two pressure 
transducers while simultaneously observing their output signals. 

In [57], a vibration pressure transducer with a frequency 
output is considered as the object of analysis. To determine the 
dynamic characteristics of a transducer, the method of 
parametric identification is used with a harmonic input action 
and a controlled average value and amplitude. When modelling 
the measuring system, the method of computational 
hydrodynamics is used. The results of the modelling allow us to 
reveal the effects during the operation of the given pressure 
transducer, which influence dynamic characteristics a great deal. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 73 

In [58]-[64], methods of developing the variable effects of 
different types of dynamic calibration of pressure transducers 
are considered. In [58] and [59], generators of impulse pressure 
in the form of a short sinusoid half-wave are presented. The 
principle of operation of the first thereof is the use of the 
system with drop mass, and the second is based on the use of 
the system with a Hopkinson bar and a piston gauge. In [60] 
and [61], the system of calibration with the generator of a 
periodically varying pressure in the form of a double-acting 
pneumatic actuator is considered. Methods of dynamic 
calibration that are based on the use of a shock tube as a 
generator of step-change pressure are studied in [62]-[64]. 
Dynamic calibration with the use of one or another calibration 
action involves the determination of the dynamic characteristics 
of the resulting impulsive, amplitude-frequency, or transient 
functions of pressure transducers. 

An experimental method of the determination frequency 
response of a pressure sensor of the second order is considered 
in [64]. The experimental installation generates the input step 
signal. According to the transient process, the parameters of 
transfer function are determined, and the frequency 
characteristics are determined from them. 

The methods used for the dynamic calibration of different 
types of conditioning amplifiers are presented in [82]. 

The necessity for the dynamic measurement of pressure 
arises at low [65] and high [66] temperatures of gases and 
liquids [67]. 

There are also examples of the use of the Kalman filter to 
determine the parameters of a dynamic measuring system. 

The researchers in [68] consider the method of dynamic 
weighing with letter scales. For the model of scales of the 
second order, the coefficients of the difference equations of 
weights and the weight of an object are determined with the 
Kalman filter. 

The solution to dynamic measurement problems in specific 
applications is characterised by the great variety thereof and 
particular methods of resolving them. The correction of a 
dynamic error is undertaken on the basis of setting acceptable 
dynamic properties for a specific problem.  

[69] considers the correction of a dynamic error of an 
auxiliary wale heat flux sensor. This sensor is described by the 
differential equation of the first or second order. The correction 
of a dynamic error is performed by the inverse transfer 
function, the parameters of which are selected according to an 
acceptable dynamic error. 

There are studies wherein analysis of the dynamic properties 
of measuring equipment helps to develop the optimal structure 
of the measuring equipment.  

There are studies that found that the dynamic measuring 
equipment has nonlinearities. 

In [70], an approach to develop magnetic-field sensors using 
a nonlinear dynamic equation describing the operation of the 
sensors is considered. The basic dynamic equation is the 
following:  

𝜏
𝑑𝑥

𝑑𝑡
= −𝑥 + tanh [

𝑥 + ℎ(𝑡)

𝑇
] , 

where 𝑥 is the strength of a magnetic field; ℎ(𝑡) is an external 
magnetic field; 𝜏 is the system time constant; and 𝑇 is the 
temperature. 

On the basis of an analysis of this equation (2), states of the 
sensor having minimum potential energy are identified. This 

essentially nonlinear differential equation is used to create an 
optimal sensor circuit. 

In [71], dynamic measurements of the composition of gases 
in chemical industry are considered. Chemical sensors are 
characterised by inertia and have nonlinear statistical 
characteristics. The following inverse model of a dynamic 
measuring system is given. 

𝑢[𝑛 − 𝑘] = ∑ 𝐵𝑗 (𝑞)𝑦[𝑛]

𝑛

𝑗=1

+ 𝑐[𝑛] , 

where 𝐵𝑗 (𝑞) is a polynomial in the delay operator 𝑞 (𝑞 𝑢[𝑛] =

𝑢[𝑛 − 1]), 𝑢[𝑛] and 𝑦[𝑛] are output and input signals of the 
inverse system, respectively, and 𝑐[𝑛] is a residual term. The 
parameters of the model are determined by minimising the 
measuring error in the test experiments. 

The studies on this topic also give information about 
completely unexpected effects in dynamic measurements. In 
[72], four algorithms for the compensation for static 
measurement error in the operation of the measuring device in 
dynamic mode are considered. These algorithms are based on 
the linear statistical characteristics of the measuring device, with 
the following methods: the piecewise linear interpolation 
compensation algorithm, polynomial progressive compensation 
algorithm, improved polynomial progressive compensation 
algorithm, and artificial neural network compensation 
algorithm. There are no parameters of dynamic characteristics 
in the algorithms. Testing them in dynamic mode has shown 
that they have quite certain frequency characteristics. In this 
case, the bandwidth of the first three algorithms is 
approximately the same, but the bandwidth of a neural 
algorithm is slightly larger. 

[73] considers the system of pipelines in the form of a set of 
concentrated masses described by the system of differential 
equations. The problem of determining the frequencies of the 
oscillations of the system is solved by the method of the theory 
of inverse problems solution, with boundary conditions at the 
ends of the pipeline.  

[74] is devoted to the use of the Bayesian estimation 
problem to measure spatial coordinates of the objects. Here, in 
the general case, a discrete variant of a nonlinear dynamic 
system is considered, in which a coordinate vector of the state 
to be measured is connected with a vector of the observed 
coordinates by the known model. The technique used for 
computing spatial coordinates is traditional and requires the 
knowledge of their probability density function and noise 
probability densities. 

[75] studies the rheological properties of drops by means of 
experimental installation with a capillary, piezo translator, and 
pressure sensor. A particular characteristic of the experimental 
installation is obtained. The results of the research are obtained 
on the basis of the Fourier transform of pressure sensor signals 
accounting for the particular characteristic of the experimental 
installation. 

In [76], the properties of thermal electric transducers are 
studied by measuring their electrical resistance at various 
frequencies. The dynamic model of an integrated electrical 
resistance is given. The natural frequencies of this model’s poles 
provide information on the thermal electrical transducer. 
Although the problem of restoring a dynamically distorted 
signal or correcting a dynamic progression is not solved here, 
the studies conducted are based on an analysis of the dynamic 
properties of the object. 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 74 

The researchers in [77] propose a method of estimation of 
square error criterion for measuring the signals by an 
accelerometer with a dynamic model of the second order. It is 
shown therein that the maximum error occurs when measuring 
triangular and trapezoidal signals. For measuring signals of 
different forms, the methods of their approximation are given 
on the basis of a genetic algorithm to obtain the maximum 
quadratic measuring error of these signals. The authors suggest 
using this method to calibrate accelerometers in dynamic 
measurement. 

[78] considers the design of a dynamic compensator for a 
second-order temperature sensor. The compensator parameters 
of the second order are determined by the Kalman filter 
adapted to evaluate the parameters of a dynamic compensator. 

The problem of reducing a dynamic error in the 
measurement of forces by the tri-component dynamometer of a 
micro-machine is considered in [79]. At the initial stage, the 
matrix of the frequency characteristics of the tri-component 
dynamometer is identified accounting for the cross links on the 
measured coordinates. Compensation for a dynamic error is 
performed on the basis of multiplying the observed force 
vector by the inverse matrix of the frequency response. In the 
article, the problem of the regularity of the inverse matrix and 
suppression of the influence of high-frequency noise when it is 
reversed is not discussed. 

[80] considers a dynamic model of a sensor to measure the 
convective heat transfer coefficient of buildings. This dynamic 
model is presented by a differential equation of the third order. 
On the basis of the developed model, the design parameters of 
the sensor, ensuring its higher dynamic characteristics, are 
selected. The dynamic measuring error is determined by a 
simple operation of multiplication time interval by the rate of 
change in the transfer of convective energy. 

CONCLUSION 

In conclusion, it is important to note that this report does 
not fully cover the topic of dynamic measurements. The author 
has simply attempted to discuss the basic results of the theory 
of dynamic measurements and to give examples of several of 
their specific applications. The number of the examples 
presented in the articles is large. The author has been able to 
analyse only a small part of them. At present, the author feels 
the growing interest in such measurements. One of the 
objective reasons for this growing interest is the necessity to 
create new technology.  

REFERENCES 

[1] A. Cornu, Sur la condition de stabilite du movement d’un 
systeme oscilant soumis a une liaison synchrinique pendulaire, 
C.r. Acad. sci. 104, 22 (1887) p. 1463 (in French). 

[2] A. Blondel, Oscillographes, nouveaux apparelis pour l’etude des 
oscillations electriques lentes, C.r. Acad. Sci. 116, 10 (1893) p. 
502 (in French). 

[3] D. I. Mendeleev, About methods of precise or metrological 
weighing, in: Papers on Metrology, VNIIM, Standart gid, 
Moscow, 1936, pp. 67-164. 

[4] A. N. Krylov, Some notes about crushers and indicators, in:  
Izvestia Saint Petersburg A. N., 1909, pp. 623-655. 

[5] V. A. Granovsky, Dynamical measurements: fundamentals of 
metrology software, Energoizdat, Leningrad, 1984 (in Russian). 

[6] A. L. Shestakov, Analysis of dynamic error and selection of a 
measuring transducer’s parameters at stepwise, linear and 

hierarchic segments, Izmeritelnaya tekhnika [Measuring 
Engineering] 6, 13 (1992). 

[7] E. Layer, W. Gawedzki, Theoretical principles for dynamic errors 
measurement, Measurement 8, 1 (1990), pp. 45-48. 

[8] E. Layer, Mapping Error of linear Dynamic systems caused by 
reduced-order model, IEEE Transactions on Instrumentation 
and Measurement 50, 3 (2001) p. 792. 

[9] E. Layer, Non-standard input signals for the calibration and 
optimization of the measuring systems, Measurement 34 (2003) 
p. 179. 

[10] K. Tomezyk, E. Layer, Energy density for signals maximizing the 
integral-square error, Measurement 90 (2016) p. 224. 

[11] E. Layer, Theoretical principles for establishing a hierarchy of 
dynamic accuracy with the integral-square-error as an example, 
IEEE Transactions on Instrumentation and Measurement 46, 5 
(1997) p. 1178. 

[12] V. V. Solodovnikov, Statistical dynamics of linear systems of 
automated control, Fizmatgiz, Moscow, 1960. 

[13] H. James, N. Nichols, R. Phillips, Theory of Servomechanisms 
(I.L., 1951). 

[14] A. F. Verlan, V. S. Sizikov, Integral Equations: Methods, 
Algorithms, and Programmes (Resource Book) Naukova dumka, 
Kiev, 1986.  

[15] V. I. Krylov, Approximate calculation of integrals (M.: Nauka, 
1967). 

[16] V. I. Krylov, L. T. Shulgina, Reference book on numerical 
integration (M.: Nauka, 1966). 

[17] G. I. Vasilenko, Signals renewal theory: from reduction to the 
ideal device in physics and technique (M.: Sov.radio, 1979). 

[18] S. G. Rautian, Real spectral apparatus, Sov. Phys. Usp 66, 3 
(1958) p. 27. 

[19] E. I. Buzz, Sharpening of observational data in two dimensions, 
Australian J. of Phys. 7, 1 (1955) p. 67. 

[20] G. N. Solopchenko, Ill-posed problems of measuring 
engineering, Izmeritelnaya tekhnika [Measuring Engineering] 1, 
51 (1974). 

[21] K. Hein, A. Lesniewski, Digital correction of the dynamic error 
arising in the input circuits of measuring instruments, 
Measurement 2, 4 (1984) p. 170.  

[22] A. N. Tikhonov, V. Ya. Arsenin, Solutions of ill-posed problems 
(M.: Nauka, 1979). 

[23] V. A. Morozov, On regularization methods for ill-posed 
problems and regularization parameter selection, DANSA 6, 1 
(1967) p. 43. 

[24] V. A. Morozov, On the solution of functional equations by the 
method of regularization, DANSA 6 (1967) p. 175. 

[25] N. Wiener, Extrapolation, interpolation and smoothing of 
stationary time series with engineering applications, John Wiley, 
New York, 1950.  

[26] S. Twomey, The application of numerical filtering of the solution 
of integral equations encountered in indirect sensing 
measurements (J. Franklin Inst., 1965). 

[27] S. Engelberg, Control theory, part 1, IEEE Instrumentation and 
Measurement Magazine 11, 3 (2008) p. 34. 

[28] K. H. Ruhm, Dynamic and stability: A proposal for related terms 
in metrology from a mathematical point of view, Measurement 
79 (2016) p. 276. 

[29] A. L. Shestakov, Measuring transducer of the dynamic 
parameters with the iterative principle for signal restoration, 
Devices and systems. Management, control, diagnostics 10 
(1992) p. 23. 

[30] B. N. Petrov, V. Yu. Rutkovskii, I. N. Krutova, Construction 
principles of self-adjusting control system, Mashinostroenie, 
Moscow, 1972. 

[31] A. L. Shestakov, Dynamic error correction method, IEEE 
Transactions on Instrumentation and Measurement 45, 1 (1996) 
p. 250. 

[32] A. L. Shestakov, The automatic control theory methods in 
dynamic measurements (Chelyabinsk, South Ural State 
University, 2013) (in Russian). 



 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 75 

[33] A. L. Shestakov, D. Yu. Iosifov, Zeros and transfer function 
poles control of measuring converter with measurable sensor 
status vector, SUSU Bulletin, Computer Science, Management, 
Electronic Engineering Series, 4, 2 (2003) p. 42. 

[34] A. L. Shestakov, Detector element of dynamic parameters with 
signal restriction iteration principles, Apparatus and Control 
Systems, 10 (1992) p. 23. 

[35] A. L. Shestakov, M. N. Bizyaev, Dynamically distorted signals 
recovery of testing measurement system by sliding mode 
approach, Proc. of RAS. Energetics, 6, 119 (2004), pp. 119-130. 

[36] A. S. Volosnikov, A. L. Shestakov, Neural network dynamic 
model of measurement system with regenerate signal filtration 
SUSU Bulletin, Computer Science, Management, Radio 
Engineering Series, 14, 69 (2006), p. 16-20. 

[37] A. L. Shestakov, E. V. Soldatkina, Measuring Transducer With 
Parameters Self-Adjusted by Dynamic Error Criterion in 
Information-Measuring and Operated Systems and Devices: 
Subject Collection of Research Papers, SUSU Publishing, 
Chelyabinsk, 2000, pp. 126-134. 

[38] K. H. Ruhm, Measurement plus observation: A new structure in 
metrology, Measurement 126 (2018) 421-432,  
DOI: https://doi.org/10.1016/j.measurement.2017.03.040 

[39] A. T. Ruban, Adaptive Control with Identification, Tomsk 
University Publishing, Tomsk, 1983. 

[40] A. Monti, F. Ponci, Uncertainty evaluation under dynamic 
conditions using polynominal chaos theory, IEEE Transactions 
on Instrumentation and Measurement 59, 11 (2010) p. 2825. 

[41] P. Eikhof, System Identification, John Wiley, London, 1974. 
[42] G. N. Solopchenko, Method of statistical regularization, Proc. of 

VELNII, 5, 43 (1970). 
[43] N. Kh. Ibragimova, G. N. Solopchenko, Assessment of the 

weighing and transfer function using statistical regularization, 
Proc. of VELNII, 16, 86 (1973). 

[44] L. Ying-jun, G.-C. Wang, Y. Zhang, et al., Dynamic 
characteristics of piezoelectric six-dimensional heavy 
force/moment sensor for large-load robotic manipulator, 
Measurement 45, 5 (2012) p. 1114. 

[45] R. Kumme, Investigation of the comparison method for the 
dynamic calibration of force transducers, Measurement 23, 4 
(1998) p. 239. 

[46] R. S. Michelini, G. B. Rossi, Computerized test scheme for the 
motion recognition and evaluation of a dynamic system, 
Measurement 5, 2 (1987) p. 61. 

[47] R. Engel, H.-J. Baade, Qualifying impacts on the measurement 
uncertainty in flow calibration arising from dynamic flow effects, 
Flow Measurement and Instrumentation 44 (2015) p. 51. 

[48] R. S. Oliveira, S. Winter, H. A. Lepikson, et al., A new approach 
to test torque transducers under dynamic reference regime, 
Measurement 58 (2014) p. 3554. 

[49] M. Schetzen, The Volterra and Wiener Theories of Nonlinear 
Systems, John Wiley, New York, 1980. 

[50] M. Alizadeh, S. Amin, D. Ronnow, Measurement and analysis of 
frequency-domain Volterra kernels of nonlinear dynamic 3x3 
MIMO systems, IEEE Transactions on Instrumentation and 
Measurement 66, 7 (2017) p. 1893. 

[51] Y. Zhang, J. Zu, H.-Y. Zhang, Dynamic calibration method of 
high-pressure transducer based on quasi-δ function excitation 
source, Measurement 45, 8 (2012) p. 1981. 

[52] J. Schoukens, G. Vandersteen, R. Pinelon, et al., Bounding the 
polynomial approximation errors of frequency response 
functions, IEEE Transactions on Instrumentation and 
Measurement 62, 5 (2013) p. 1346. 

[53] T. J. Esward, Investigation of dynamic measurement applications 
through modeling and simulation, Technisches Messen 83, 10 
(2016) p. 557. 

[54] R. Sanchis, I. M. Tkachenko, G. Verdu, et al., Dynamic analysis 
of the NPP pressure sensor system, Progress in Nuclear Energy 
29, 3-4 (1995) p. 321. 

[55] H. Chang, P. K. Tzenog, Analysis of the dynamic characteristics 
of pressure sensors using ARX system identification, Sensors and 
Actuators A: Physical 141, 2 (2008) p. 367. 

[56] T. Kobata, A. Ooiwa, Method of evaluating frequency 
characteristics of pressure transducers using newly developed 
dynamic pressure generator, Sensors and Actuators A: Physical 
79, 2 (2000) p. 97. 

[57] H. Zhuang, S. C. Fan, Z. S. Guo, D. Z. Zheng, Dynamic 
characteristics analysis of vibrating cylinder pressure transducers 
(VCPT), Sensors and Actuators A: Physical 157, 2 (2010) p. 219. 

[58] Y. Durgut, E. Aksahin, E. Batci, et al., Preliminary dynamic 
pressure measurement system at UME, in Proc. of the 21st 
IMEKO World Congress 30 Aug.- 4 Sept. 2015, Prague, Czech 
Republic, pp. 181-184. 

[59] Y. Zhang, J. Zu, H.-Y. Zhang, Dynamic calibration method of 
high-pressure transducer based on quasi-δ function excitation 
source, Measurement 45, 8 (2012) p. 1981. 

[60] A. Svete, M. Stefe, A. Macek, et al., Development of a 
measurement system for dynamic calibration of pressure sensors, 
in Proc. of the XXI IMEKO World Congress, 30 Aug. – 4 Sept. 
2015, Prague, Czech Republic, pp. 1666-1671. 

[61] A. Svete, M. Stefe, A. Macek, et al., Dynamic pressure generator 
for dynamic calibrations at different average pressures based on a 
double-acting pneumatic actuator, Sensors and Actuators A: 
Physical 247 (2016) p. 136. 

[62] S. Razzak, J.-P. Damion, N. B. Hsouna, Dynamic pressure 
calibration, in Proc. of the XXI IMEKO World Congress, 30 
Aug. – 4 Sept. 2015, Prague, Czech Republic, p. 1660. 

[63] Z. Zhang, W. Wang, Research on frequency response of pressure 
transducer, in Proc. of the XXI IMEKO World Congress, 30 
Aug. – 4 Sept. 2015, Prague, Czech Republic, p. 1683. 

[64] F. R. F. Theodoro, M. L. C. C. Reis, C. d’Andrade Souto, et al., 
Measurement uncertainty of a pressure sensor submitted to a 
step input, Measurement 88 (2016) p. 238. 

[65] M. Kunimatsu, T. Mizuide, K. Yamane, Measurement of 
dynamic pressure using piezoelectric sensors at extremely low 
temperatures, JSAE Review 22, 4 (2001) p. 553. 

[66] U. Rhyner, R. Mai, H. Leibold, et al., Dynamic pressure 
measurements of a hot gas filter as a diagnostic tool to assess the 
time dependent performance, Biomass and Bioenergy 53 (2013) 
p. 72. 

[67] A. M. Hurst, J. VanDeWeert, A study of bulk modulus, entrained 
air, and dynamic pressure measurements in liquids, Journal of 
Engineering for Gas Turbines and Power 138, 10 (2016) p. 34. 

[68] W. Shu, Dynamic weighing under nonzero initial condition, IEEE 
Transactions on Instrumentation and Measurement 42, 4 (1993) p. 
806. 

[69] E. Kaiser, Dynamic measuring error correction of encapsulated 
auxiliary wall heat flux sensor made of film resistance 
thermometer, Measurement 8, 1 (1990) p. 12. 

[70] S. Baglio, A. Bulsara, B. Audo, et al., Exploiting nonlinear 
dynamics in Novell measurement strategies and devices: from 
theory to experiments and applications, IEEE Transactions on 
Instrumentation and Measurement 60, 3 (2011) p. 667. 

[71] A. Pardo, S. Marco, Nonlinear inverse dynamic models of gas 
sensing systems based on chemical sensor arrays for quantitative 
measurements, IEEE Transactions on Instrumentation and 
Measurement 47, 3 (1998) p. 644. 

[72] J. R. Mejia, J. V. Arrovo, J. V. Pineda, et al., Comparison of 
compensation algorithms for smart sensors with approach to 
real-time or dynamic applications, IEEE Sensors Journal 15, 12 
(2015) p. 7071. 

[73] S. Saha, A new method using inverse theory for estimation of 
dynamic response in frequency domain for operation piping, 
Procedia Engineering 86 (2014) p. 843. 

[74] A. Garcia, T. Hausotte, A. Amthor, Bayes filter for dynamic 
coordinate measurements: accuracy improvement, data fusion, 
and measurement uncertainty evaluation, Measurement 46, 9 
(2013) p. 3734. 

https://doi.org/10.1016/j.measurement.2017.03.040


 

ACTA IMEKO | www.imeko.org March 2019 | Volume 8 | Number 1 | 76 

[75] A. Javadi, J. Kragel, A. V. Makievski, et al., Fast interfacial 
tension measurements and dilational rheology of interfacial layers 
using the capillary pressure technique, Collaids and Surfaces A: 
Physiochemical and Engineering Aspects 407, 5 (2012) p. 159. 

[76] W. Zhu, Y. Deng, Y. Wang, Dynamic thermal response 
characteristics and feasibility analysis of thermoelectric module in 
impedance measurement, Applied Thermal Engineering 111, 25 
(2017) p. 1417. 

[77] K. Tomczyk, E. Layer, Accelerometer errors in the measurement 
if dynamic signals, Measurement 60 (2015) p. 2306.  

[78] S. Yao, R. Gao, Nonlinear dynamic compensation research of 
temperature measurement system in coal mine movable refuge 
chamber, Procedia Engineering 2 (2011) p. 2306. 

[79] R. Kozkmaz, B. A. Gozen, B. Bediz et al., High-frequency 
compensation of dynamic distortions in micromachine force 
measurements, Procedia Manufacturing 1 (2010) p. 534. 

[80] K. E. Anders Ohlsson., R. Ostin, S. Gzundberg, et al., Dynamic 
model for measurement of convective heat transfer coefficient at 
external building surfaces, Journal of Building Engineering 7 
(2016) p. 239. 

[81] S. Eichstadt, C. Elster, I. M. Smith, T. J. Esward, Evaluation of 
dynamic measurement uncertainty: an open-source software 
package to bridge theory and practice, Journal of Sensors and 
Sensor Systems 6 (2017) p. 97. 

[82] L. Klaus, T. Bruns, H. Volkers, Calibration of bridge-, charge- 
and voltage amplifiers for dynamic measurement applications, 
Metrologia 52, 1 (2015), 72– 81. WOS:000348119700011 

 

 

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