Acta IMEKO, Title


ACTA IMEKO 
ISSN: 2221-870X 
June 2018, Volume 7, Number 2, 65-72 

 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 65 

Numerical Investigation of Optimal Dynamic Measurements  
Aleksandr L. Shestakov, Georgy A. Sviridyuk, Alevtina V. Keller, Alyona A. Zamyshlyaeva, Yurii V. 
Khudyakov  

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

 

 

Section: RESEARCH PAPER  

Keywords: mathematical model of measuring transducer; optimal dynamic measurement; set of admissible measurements 

Citation: Aleksandr L. Shestakov, Georgy A. Sviridyuk, Alevtina V. Keller, Alyona A. Zamyshlyaeva, Yurii V. Khudyakov, Numerical Investigation of Optimal 
Dynamic Measurements, Acta IMEKO, vol. 7, no. 2, article 12, June 2018, identifier: IMEKO-ACTA-07 (2018)-02-12 

Section Editor: Franco Pavese, Italy 

Received December 16, 2017; In final form May 11, 2018; Published June 2018 

Copyright: © 2018 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: Alevtina Keller, e-mail: kellerav@susu.ru 

 

1. INTRODUCTION 
Currently, the theory and practice of dynamic measurements 

are developing in various directions. For a long time the theory 
of inverse problems [1], methods of direct and inverse Fourier 
transforms were the mathematical basis of dynamic 
measurements. In addition, for more than 20 years methods of 
the remote control theory were successfully applied for solving 
various problems of dynamic measurements. In the recent 
years, a new approach was developed [2] to restore the input 
signal distorted by the measuring transducer and external 
interference, called by the authors “the theory of optimal 
dynamic measurements”. We emphasize that the theory of 
optimal dynamic measurements is based on three fundamental 
components: 1) the solution of dynamic measurements 
problems using the ideas and methods of the remote control 
theory; 2) the development of the descriptor systems theory 
and its applications [3]; 3) the development of methods for the 
solution of Sobolev type equations and corresponding optimal 
control problems, including numerical methods for solving 
optimal control problems for Leontief type systems. Leontief 
type systems are finite-dimensional case of Sobolev-type 
equations   and  a   special  case   of  descriptor  systems   (with  

 
 
 

constant matrices). This report presents recent results in the 
theory of optimal dynamic measurements. 

2. MATHEMATICAL MODELING OF OPTIMAL DYNAMIC 
MEASUREMENTS  

We first introduce mathematical model of optimal dynamic 
measurements (Figure 1). 

Mathematical model of measuring transducer (MT) is 
represented by the Leontief type system of equations 

,
,

Lx Ax Bu G
y Cx D

ς
η

= + +


= +


 (1) 

where L and A are matrices that characterize the structure of 
the MT. In some cases it is possible that det 0L =  (this will be 
discussed in the next section); ( )x t  and ( )x t are vector 
functions of the state of the MT and the velocity of the state 
change, respectively; y(t) is a vector-function of observation; C 
and D are rectangular matrices characterizing the interrelation 
between the system state and observation; ( )u t  is a vector-
function of measurements; B   is the matrix characterizing the 

ABSTRACT 
The basic ideas of mathematical modeling of optimal dynamic measurements are presented. The key thing is to construct a mathematical model of the 
measuring transducer, which allows simulating also a complex measuring transducer. In this manuscript we propose such a mathematical model and 
we discuss its results obtained through numerical studies in a variety of cases: from the simplest, without considering disturbances, to the case when a 
deterministic disturbance, for example resonances, occur at the input and at the output of the measuring transducer. Lastly, the importance of the 
description of the set of admissible measurements in the modeling of dynamic measurements is discussed. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 66 

interrelation between the system state and measurement; ( )tς  
is a vector-function of resonances in the circuits of the MT (see 
Figure 1); G  is the matrix characterizing influence of 
resonances in the circuits of the MT; ( )tη  is a vector-function 
of disturbance at the output of the the MT (see Figure 1). If L 
is not degenerate then system (1) can be reduced to 
 

,
,

x Mx Fu H
y Cx D

ς
η

= + +


= +


 (2) 

where 1 ,M L A−=  1 ,F L B−=  1 .H L G−=   
This system appeared in the dynamic measurements theory 

from the remote control theory, where (2) was obtained in the 
study of the transfer function of the MT. Finally, in accordance 
to new approaches in the field of measurements, it is possible 
to replace the terms "input signal" with "measurement" and 
"output signal" with "observation" [4], respectively. We 
emphasize that all observations and measurements in systems 
(1) and (2) are simulated or "virtual". 

The initial Showalter – Sidorov condition  

( ) ( )
11

0(0) 0
p

L A L x xα
+− − − = 

                       (3) 

for some 
0 ,

nx R∈  ( )L Mα ρ∈ , reflects the initial state of the 
MT. The initial Showalter – Sidorov condition is equivalent to 
the initial Cauchy condition ( ) 00x x=  when 0detL ≠ .  

For the investigation of problem (1), (3) introduce the space 
of states 

2 2{ ((0, ), ) : ((0, ), )}
n nX x L R x L Rτ τ= ∈ ∈  and the 

space of observations [ ]Y C X= . The key concept is the 
optimal measurement simulated by solving the optimal control 
problem for (1), (3). To find it, we introduce the space of 
measurements ( 1)

2 2{ ((0, ), ) : ((0, ), )}
n p nU u L R u L Rτ τ+= ∈ ∈  

and allocate it in a closed convex set of admissible 
measurements U U∂ ⊂   

2( )

0 0

( )q
q

U u U u t dt d
τq

∂
=

 
= ∈ ≤ 
 

∑∫: . (4) 

The latter contains a priori information about measurements 
(this will be discussed in the sixth section of the paper). In the 
optimal measurements theory, as in the control theory, several 
types of cost functionals can be used depending on the goals of 

restoration of a dynamically distorted signal.  
Consider a general cost functional  

( ) ( )1 0 2( ) ( , , , ) ( ) , ,J u J y u t y t J uς η ς= − +  (5) 

where 0 ( )y t , [0, ]t τ∈  is a continuously differentiable function 
constructed on the basis of the observed values 0iY  at the 
output of the real MT, which is called “real observation” in the 
mathematcal model.  

The first term of the cost functional reflects the closeness 
of the real observation 0 ( )y t  and the virtual observation ( )y t , 
obtained on the basis of mathematical model of the MT. The 
second term acts as a filter in the presence of resonances ( )tς  
in the chains of the MT. In case of lack of resonances ( )tς  in 
the chains of the MT, functional (5) consists only of the first 
term. 

The minimum point ( )v t  of the functional (5) on the set 
U∂ , being a solution of optimal control problem 

( ) min ( )
u U

J v J u
∂∈

=  (6) 

is called an optimal dynamic measurement. In practice, there is 
only indirect information about ( )v t . 

Note that quadratic functionals containing the 
observational error and its derivative have also been considered 
in the remote control theory, moreover issues of stability arising 
in connection with this were actively investigated. One of the 
first works that used the criterion of quadratic error is the 
article by A. A. Kharkevich [5]. It describes methods for 
estimating distortions introduced by linear systems. In the 50s 
the integral criteria of quadratic error began to be used: 1) to 
search the optimal parameters of the system modeling the 
dynamics [6], [7] and 2) to solve the problem of synthesis of the 
regulator [8]. It should be noted that numerical methods for the 
calculation of the real automatic systems were developed by K. 
Merriam [9]. 

Solutions were also made by working in the frequency 
domain and by using the spectra of disturbances and formulas 
for squared deviations, which allowed to reduce the problem of 
finding the best operator to the search of a function, delivering 
an extremum to the mean-square functional [10]. 

3. MATHEMATICAL MODEL OF A COMPLEX MEASURING 
TRANSDUCER AS A LEONTIEF TYPE SYSTEM  

The mathematical model of a complex measuring transducer 
involves several subsystems, each of which represents a model 
of one measuring transducer (or simple MT). Differential 
equations in the system reflect a complex of dynamic system 
elements (subsystems), and algebraic equations represent the 
connection between dynamic elements. 

Theoretical aspects of the construction of mathematical 
models of the complex MT are based on the ideas of the 
descriptor systems theory [11]. As an example, we present the 
mathematical model of the complex MT, using the iterative 
principle for the restoration of a dynamically distorted signal, 
which allows to achieve a high dynamic accuracy during 
dynamic measurements [12].  

A structural scheme of the iterative MT is shown in Figure 
2 for the case N=3.  

The mathematical model of each simple transducer is 
represented by system: 

 
Figure 1. Structure elements of mathematical model of optimal dynamic 
measurements.  



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 67 

,
.

i Mi i Mi Mi

Mi Mi i

z A z B u
y C z

= +


=



 
The error identified by the previous iteration 

1 , 1Mi M iu u y −= −  becomes the input of the i-th simple 

transducer Miu , = 2, 3....i n . Matrices , , , 1, 2,...Mi Mi MiA B C i N=  
have the same meaning as the matrices , ,A B C  in (1).  

Therefore a mathematical model of the iterative MT can be 
represented by system 

1 1

1 1 1 1 1

1 1 1 1 1

2 1 1

2 2 2 2 2

2 2 2 2 2

3 1 2

1 1 2 1 1 2

,
,
,

,
,
,

,
...

,
,

... ... .

M

M M M

M M M M

M M

M M M

M M M M

M M

N MN N MN MN

MN MN N MN MN

M M MN M M MN

u u
z A z B u

y C z D
u u y

z A z B u
y C z D
u u y

z A z B u
y C z D

y u u u u y y y

η

η

η

=
 = +
 = +


= −
 = +


= +
 = −


 = +
 = +


= + + + + − − − −







  
(7) 

which reflects the idea of correction of dynamic error in the 
iterative principle, which consists in the sequential use of a 
number of simple MTs. The output of the system is presented 
by the sum of the observed signal and errors simulated by 
iterative chains. 

We rewrite the mathematical model of the MT, which is 
represented by the scheme of the iterative MT, in the form of 
matrix equations for the case = 2N . By introducing the 
notation 

( )1( ) ( ), ( ), ( ),
T

M Mx t z t u t y t y= , 1 1 2( ) ( , 0, , )
T

M Mu t u η η= , 

1( ) ( )y t y t= .  
 (7) can be represented in the form of a Leontief type system 

,
,

Lx Ax Bu
y Cx
= +


=


 

where 

0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0

I
I

L

 
 
 
 
 

=  
 
 
 
 
 

,                  

1

2

1 2

0 0 0 0
0 0 0 0
0 0 0
0 0 0

0 0 0
0 0 0
0

M

M

D
B D

I
I
I D D

 
 
 
 
 

=  
 
 
 
 
 

,         (0, 0, 0, 0, 0, 0, )C I= .    

The use of Leontief type systems in modeling of the MT 
allows: 1) to model complex measuring devices with different 
types of connections between the transducers, considering the 
various types of feedback; 2) to solve not only the problem of 
restoration of dynamically distorted signals, but also problems 
connected with restoration of the system states. 

4. MATHEMATICAL MODEL OF OPTIMAL DYNAMIC 
MEASUREMENTS WITH INERTIA AND METHODS FOR ITS 
NUMERICAL STUDY SUBSECTIONS 

Considering the mathematical model of optimal dynamic 
measurements taking into account only inertia [14] of the MT, 
i.e. under the assumption of absence of disturbance or their 
insignificance, system (1), which models the MT, can be written 
in more simple way 

,
.

Lx Ax Bu
y Cx
= +


=

  (8) 

The initial condition (3) remains unchanged. By putting 
( ) ( , )y t y u t=  in the cost functional (5), we obtain  

1 2( ) ( )
1 0

1 0

( ) ( ) ( , ) ( )q q
q

J u J u Sy u t Sy t dt
τ

=

= = −∑∫  (9) 

where ( )0 01 02( ) ( ), ( ),..., ( )
T

ory t y t y t y t=  are real observations, 
0 ( )Sy t are significant real observations, i.e. the ones that are 

used for searching the optimal measurement (though one can 
watch a greater number of characteristics of the process); ( )y t
are virtual observations, obtained during mathematical 
modeling of the restoration of dynamic measurement; ( )Sy t  are 
significant virtual observations, i.e. the ones that are simulated 
by "the virtual MT", which are used for the comparison with 
significant real observations. We emphasize that the cost 
functional (9) minimizes the difference between the real and 
virtual observations, and the difference between the derivatives 
of these processes  

( ) min ( )
u U

J v J u
∂∈

= . 

An approximate solution to the optimal measurement 
problem is searched in the form of vector-polynomials  

1 2
0 0 0

( ) , ,..., .
T

j j j
j j nj

j j j
u t a t a t a t

= = =

 
=  
 
∑ ∑ ∑
  

  (10) 

 
Figure 2. Structural scheme of the iterative measuring transducer. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 68 

The polynomials power is determined based on the 
requirements of numerical methods and calculation accuracy. In 
the numerical procedure [13] we search the coefficients of the 
vector-polynomials for which the cost functional is minimal. 
Note that the actual observation 0 ( )y t  was constructed by an 
interpolation procedure based on observed values 0iY  at the 
output of the real MT. The initial values of the coefficients 
vector of the polynomial (10) were taken equal to zero. Because 
of this, the search for optimal dynamic measurement with a 
given accuracy used to spend considerable time. We suggested 
the modification of the numerical method by introducing an 
additional procedure into the algorithm that uses linear splines 
[14]. As a result, the algorithm contains the following steps. 

Step 1. Procedures of input and calculation of auxiliary 
parameters. 

Step 2. Splitting [0, ]τ  into n intervals. Consistent solving of 
the optimal measurement problem at each i -th interval 

1[ , ]i iτ τ−  (where 1,...,i n= ) using linear functions for real 
observation 0 ( )iy t , measurement ( )iu t  and optimal 
measurement ( )iv t .  

As a result, on [0, ]τ  we get a piecewise linear function with 
the values 0 0 ( )i i iV V τ=  (Figure 3). 

Step 3. Determination of the vector-function of real 
observations 0 ( )y t  using the values 0iY  and the interpolation 
procedure (Figure 4). 

Step 4. Determination of the vector-function (10), whose 

coefficients will be the starting values for the numerical 
algorithm for the solution of the optimal measurement problem 
in the whole interval [0, ]τ  (Figure 4), using the values 0iV  and 
the interpolation procedure.  

Step 5. Search for virtual optimal measurement on the 
segment [0, ]τ . It uses the idea of the multistage alternating-
variable descent method with memory (Figure 5). 

5. MATHEMATICAL MODEL OF OPTIMAL DYNAMIC 
MEASUREMENTS WITH INERTIA AND RESONANCES ON 
THE OUTPUT OF THE MEASURING TRANSDUCER WITH 
DIFFERENT A PRIORI INFORMATION  

One of the most important problems in the theory of 
dynamic measurements is the restoration of the input signal 
distorted by deterministic disturbance, i.e. interference that does 
not change from experiment to experiment. 

By deterministic we refer to disturbance caused by 
• resonances in the MT, which lead, for example, to the blur 

of the lance-like input signals. Note that in this case it is 
assumed that frequencies of resonance disturbance are known;  

• impacts of different nature, with a known form of useful 
signal, for example, a variable of voltage power sources.  

We consider the modeling of optimal dynamic measurement 
under the assumption that the impact of deterministic 
disturbance is only on the output of the MT. In this case, the 
system modeling the MT has the form  

,
.

Lx Ax Bu
y Cx Dη
= +


= +

  (11) 

The initial condition (3) remains unchanged. By putting 
( ) ( , , )y t y u tη=  in the cost functional (5), we obtain  

1 2( ) ( )
1 0

1 0

( ) ( ) ( , , ) ( )q q
q

J u J u Sy u t Sy t dt
τ

η
=

= = −∑∫  (12) 

Note that the functionals (9) and (12) differ only by the 
functions of virtual observations, which are defined in systems 
(8) and (11) respectively. The virtual optimal measurement is 
determined by finding the minimum of the cost functional (12)  

( ) min ( )
u U

J v J u
∂∈

= . 

If the frequency of resonances in the output of the MT is 
known as a priori information then the function of real 

 
Figure 3. Step 2 of the proposed algorithm. 

 

 
Figure 4. Steps 3 and 4 of the proposed algorithm. 

 
Figure 5. Step 5 of the proposed algorithm. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 69 

observations 0 ( )y t  is constructed by an interpolation of the 
observed values 0iY  at the output of the real MT. 

If the form of the measurement function is known as a 
priori information, then the function of real observations 0 ( )y t  
is constructed by an approximation procedure of the observed 
values 0iY  at the output of the real MT. 

 The method of numerical solution of the optimal 
measurement problem taking into account inertia and 
disturbance at the output of the MT in detail was described 
earlier. An approximate solution is sought in the form 

( )1 2( ) , ,...,
T

nu t u u u= =
   

1 2
0 0 0

sin , sin ,..., sin
T

j j j j nj j
j j j

a t a t a tϖ ϖ ϖ
= = =

 
=  
 
∑ ∑ ∑
  

.  (13) 

The deterministic disturbance is considered as a vector 
function, each component of which is the sum of the 
resonances at known frequencies 

1 2

1 21 2
0 0 0

( ) sin , sin ,..., sin
m

m

TMM M

j j j j j mj
j j j

t a t a t a tω ω ωη ω ω ω
= = =

 
=  
 
∑ ∑ ∑ .          (14) 

For the conduction of computational experiments, we 
construct a mathematical model of the MT, specified by a 
transfer function  

( )gW p =  

( )( )( )( )2 2 2 21 1 1 2 2 2 3 4
1

2 1 2 1 1 1T p T p T p T p T p T pξ ξ
=

+ + + + + +
 

with known parameters 1 0, 01T s= , 1 0, 6ξ = , 2 0, 002sT = , 

2 0, 2ξ = , 3 0, 0005sT = ,  4 0, 0001sT = . 
The mathematical model of the MT taking into account 

resonances has the form: 

1 1 2

2 1 2

3 2 3 4

4 3 4

5 4 5

6 5 6

6

0, 9881 ,
10000 119 ,

0, 999204 ,
250000 119 ,
2000 2000 ,
10000 10000 ,

.

x x x u
x x x
x x x x
x x x
x x x
x x x
y x η

= − − +
 = −
 = − −


= −
 = −


= −
 = +













 (15) 

For simplicity, B, C and D will be identity matrices. To 
evaluate the accuracy of the method, the found approximate 
optimal dynamic measurement is compared to the known input 
signal (precise dynamic measurement). Note that in real 
problems the precise dynamic measurement is not known. 

Example 1. We consider the case when the resonance 
appears only at the output of the MT. Its frequency 5000ω =  
is known, so sin 5000a tωη = . 

Figure 7 presents the real observation 0 ( )y t  found as a result 
of computational experiment, the optimal virtual measurement 

( )kv t
  and a precise dynamic measurement ( )v t . In (13) 

frequencies 6, 25j jϖ = , 0,1, 2,... ,j =   32,=  were taken, 
due to the amplitude-frequency properties of the transfer 
function of the MT (Figure 6). 

As a result of computational experiment (Figure 7) the 

amplitude of the resonance disturbance 0, 30151aω =  was 
found, so 0, 30151sin 5000tη = .  

A virtual optimal measurement as an approximate solution 
of the optimal dynamic measurement problem was obtained 

0,15638671sin(2 6,25 ) 0,24740745 sin(2 12,5 )v t tk π π= ⋅ + ⋅ +


0, 25821506 sin(2 18, 75 ) 0, 21717253sin(2 25 )t tπ π+ ⋅ + ⋅ +
0,15941696 sin(2 31, 25 ) 0,10642029 sin(2 37, 5 )t tπ π+ ⋅ + ⋅ +
0, 06576689 sin(2 43, 75 ) 0, 03746629 sin(2 50 )t tπ π+ ⋅ + ⋅ +
0, 018902669 sin(2 56, 25 ) 0, 00722448 sin(2 62, 5 )t tπ π+ ⋅ + ⋅
0, 00014199 sin(2 68, 75 ) 0, 00397622 sin(2 75 )t tπ π+ ⋅ − ⋅ −
0, 00622392 sin(2 81, 25 ) 0, 007312764 sin(2 87, 5 )t tπ π− ⋅ − ⋅ −
0, 00769625 sin(2 93, 75 ) 0, 00766148 sin(2 100 )t tπ π− ⋅ − ⋅ −
0, 00738759 sin(2 106, 25 ) 0, 0069865873 in(2 112, 5 )t s tπ π− ⋅ − ⋅ −
0, 00652671sin(2 118, 75 ) 0, 00604986 sin(2 125 )t tπ π− ⋅ − ⋅ −
0, 00558016 sin(2 131, 25 ) 0, 00513176 sin(2 137, 5 )t tπ π− ⋅ − ⋅ −
0, 00471147 sin(2 143, 75 ) 0, 00432264 sin(2 150 )t tπ π− ⋅ − ⋅ −
0, 00396555 sin(2 156, 25 ) 0, 00363959 sin(2 162, 5 )t tπ π− ⋅ − ⋅ −
0, 00334281sin(2 168, 75 ) 0, 00307338 sin(2 175 )t tπ π− ⋅ − ⋅ −
0, 00282881sin(2 181, 25 ) 0, 002607121 in(2 187, 5 )t s tπ π− ⋅ − ⋅ −

0, 00698658726010777sin(2π 112,5t)⋅ −
0, 0073875884086711sin(2π 106,25t)− ⋅ −
0, 00240593sin(2 193, 75 ) 0, 00222346 sin(2 200 ).t tπ π− ⋅ − ⋅   

 
Figure 6. The amplitude-frequency properties of the transfer function of the 
MT. 

 
Figure 7. Results of computational experiment for the case of resonances at 
the output of the MT with known resonance frequency. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 70 

Example 2. We consider the case when the resonances 
appear at the output of the MT, their frequencies are not 
known, and the form of the signal 22

1 3( )
a tu t a t e a−= +  is 

known. Figure 8 presents the observed values 0iY  at the output 
of the the real MT, real observation 0 ( )y t  approximating these 
values, virtual optimal measurement ( )kv t

  found as a result of 
the computational experiment and precise dynamic 
measurement ( )v t .  

As a result of the computational experiment an approximate 
solution of the optimal dynamic measurement problem 

2 250,24( ) 115002, 332 tkv t t e
−=  was obtained.  

6. MATHEMATICAL MODEL OF OPTIMAL DYNAMIC 
MEASUREMENTS WITH INERTIA AND RESONANCES AT 
THE OUTPUT AND INPUT OF THE MEASURING 
TRANSDUCER 

We consider the model of an optimal dynamic measurement 
under the assumption on the impact of deterministic 
disturbances (resonances) both in the circuits of the MT and at 
the output of the MT. Note that it is assumed that the 
frequency of resonance disturbance is known. In this case, the 
system that models the MT has the form (1) 

,
.

Lx Ax Bu G
y Cx D

ς
η

= + +


= +


 

The initial condition (3) remains unchanged. The cost 
functional (5) has the form 

( )J u =

( )
1 2

( ) ( ) ( )
0 0

1 0

( , , , ) ( , , , ) ( , , )q q q
q

Sy u t Sy u t Sy t dt
τ

ς η ς η ς η
=

= − − +∑∫

 
1

( ) ( )

1 0

( ) ( ), ( ) ( )q qq
q

N u t u t dt
τ

ς ς
=

+ + +∑∫      (16) 

where 0 ( , , )ty ς η  are real observations with zero input signal, 

0 ( , , )y tS ς η  are those real observations with zero input signal 
that are  used for restoration, 0 ( , , , )u ty ς η  are real observations 
with measurement, 0 ( , , , )y u tS ς η  are those real observations 
that are used for restoration of a dynamically distorted signal 
with nonzero input signal; ( , , , )y u tς η  are virtual observatins 
obtained in mathematical modeling of dynamical measurement, 

( , , , )Sy u tς η  are those virtual observations that are used for 
restoration of a dynamically distorted signal in mathematical 
model. Virtual optimal measurement is determined by finding 
the minimum of the cost functional (16): 

( ) min ( ).
u U

J v J u
∂∈

=  

Example 3. We consider the case when the frequencies of 
resonance  at the output of the MT 5000ω =  and resonance in 
the circuits of the MT 500ϕ =  are known, so 

sin 5000a tωη =  and sin 500a tϕς = . Figure 9 represents 
the real observations 0 0( ) ( , , , )t uy y tς η=  with zero input 
signal (this function is constructed by an interpolation 
procedure on the basis of observed values 0iY  at the output of 
the real MT with zero values of measured signal).  

The real observations are 0 0( ) ( , , , )t uy y tς η=  with 
nonzero input signal (this function is constructed by an 
interpolation procedure on the basis of observed values 0iY  at 
the output of the real MT with nonzero values of measured 
signal); the virtual optimal measurement ( )kv t

  is found in the 
computational experiment; the precise dynamical measurement 
is ( )v t . 

As a result of computational experiment the approximate 
amplitudes of the resonances were found:  0, 0300098aω =  and 

0, 2999991aϕ = , so 0, 0300098 sin 5000tη =  and 
0, 2999991sin 500tς = . 

7. THE SET OF ADMISSIBLE MEASUREMENTS IN 
MATHEMATICAL MODELING OF DYNAMICAL 
MEASUREMENTS 

An essential part of the optimal measurement model is a set 

of admissible measurements U ∂ , which in fact is a compact 
convex subset in the space of observations U and plays the 
same role as the set of admissible controls in optimal control 
theory: 

2( )

0 0

( )q
q

U u U u t dt d
τq

∂
=

 
= ∈ ≤ 
 

∑∫: ,                              (17) 

where q  can take integer values from 0 to 1p + . The integral 
form of the set of admissible measurements allows to use a 
priori information about the nature of the measured process in 
time. Note that the time integral of the square of the measured 

 
Figure 8. Results of computational experiment for the case of disturbances 
at the output of the MT with the known form of measured signal. 

 
Figure 9. Results of computational experiment for the case of disturbances 
at the input and output of the MT with the known resonance frequences.  



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 71 

process is the energy released during this process in the time 
interval [ ]0,τ . This information can be obtained on the basis of 
laboratory tests,  physical model, values of its parameters, etc. 
Moreover, this information may allow to allocate time intervals 

1[0, ]τ , 2[0, ]τ  [0, τ2], ..., [0, ]τ   (or 1[0, ]τ , 1 2[ , ]τ τ , ..., 

1[ , ]i iτ τ− ,…), reflecting several stages of the process, and to 
assess the set of admissible measurements at each of these 
intervals with constants 1d , 2d ,..., d , respectively. Note that 
the more accurate estimates are the more significant is the use 
of the set of admissible measurements at various stages of the 
measuring process. As a measure of accuracy of the 
mathematical model one can use the difference  

2( )

0 0

( )qk
q

d d d v t dt
τq

=

− = −∑∫ 
 

where  ( )kv t
  is an approximate virtual optimal measurement. 

This approach was proposed to be used in a numerical 
algorithm to determine the stopping criteria of the algorithm, 
and selection of the sets of admissible measurements at 
different stages for more effective finding of an approximate 
solution [15]. 

We provide here an example about a combustion chamber: 
when comparing the speed and volume of delivery of the fuel 
mixture into the chamber, the calculated expected energy and 
burning rate, we can allocate time periods defining the stages of 
the combustion process, which will allow to estimate the value 
of energy released during these intervals. Data from laboratory 
tests, physical models of process, etc, can be used. In the 
combustion process  two phases can be selected – transition or 
start of combustion and the operating mode of the mixture 
combustion in the combustion chamber. The transition process 
begins at the moment of ignition, when the contained working 
mixture is ignited. The operating mode of combustion starts 
with the moment when the volume of flammable mixture 
becomes constant in time (i.e. the pressure in the chamber 
becomes constant in time). Information about these phases 
allows to specify the set of admissible measurements, which will 
further improve the accuracy of virtual dynamic measurements. 

We consider two cases of definition of the set of admissible 
measurements. 

S1. The set of admissible measurements U∂  is given in the 
form (4) with 211, 6097d = , 0, 08τ =  and 1q = , i.e. 

0,081 2( )

0 0

( ) 211, 6097q
q

U u U u t dt∂
=

  
= ∈ ≤ 
  

∑ ∫: . 

S2. The set of admissible measurements *U∂  is given as 
follows: 

3
*

1

,i
i

U u U u U∂ ∂
=

= ∈ ∈
 
 
 

1:  

0,0041 2( )
1

0 0

( ) 123, 006q
q

U u U u t dt∂
=

  
= ∈ ≤ 
  

∑ ∫: , 

0,0141
2( )

2
0 0,004

( ) 124, 6636q
q

U u U u t dt∂
=

= ∈ ≤
 
 
 

∑ ∫: , 

0,081
2( )

3
0 0,014

( ) 38, 9579q
q

U u U u t dt∂
=

= ∈ ≤
 
 
 

∑ ∫: . 

Figure 10 represents the solution of the optimal 
measurement problem in both cases S1 and S2: ( )kv t

 and 
* ( )kv t
 are the virtual optimal measurement for the case S1 and 

S2 respectively; 0 ( )y t  is the real observation. 

8. CONCLUSION 
This paper presents a number of mathematical models of 

optimal dynamic measurements with the inclusion of 
deterministic disturbances in various cases. The results of 
computational experiments show the adequacy of the 
mathematical models and the efficiency of computational 
methods.  

In addition, we note that the theory of optimal dynamic 
measurements is developing in other directions, the results of 
which were not included in the paper:  

- development of numerical methods using parallel 
computing; 

- study of mathematical models of optimal dynamic 
measurements with a deterministic multiplicative effect on the 
state of the MT; 

- study of mathematical models of optimal dynamic 
measurements with "noise" [16].  

REFERENCES 

[1] V.A. Granovsky, Dynamical Measurements: Fundamentals of 
Metrology Software  (Energoizdat, Leningrad, 1984). (in Russian) 

[2] A.L. Shestakov and G.A. Sviridyuk, A new approach to 
measurement of dynamically perturbed signals, Bulletin of the 
South Ural State University. Series: Mathematical Modelling, 
Programming and Computer Software 16 (192), 116 (2010). (in 
Russian)  

[3] A.A. Belov and A.P. Kurdyukov, Descriptor Systems and 
Control Problems (FIZMATLIT, Moscow, 2015). (in Russian) 

[4] K.H. Ruhm, Measurement Plus Observation – a Modern 
Structure of Metrology, in Measurement in Research and 
Industry World Congress International Measurement 
Confederation (XXI IMEKO), (Prague, Czechia, 2015).  

[5] A. A. Kharkevich, Application of quadratic error criterion to 
estimate a linear distortion, Technical Physics  7 (5), 515 (1937). 
(in Russian) 

[6] V.V. Solodovnikov, Introduction to the Statistical Dynamics of 

 
Figure 10. Results of computational experiment for different sets of 
admissible measurements.  



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 72 

Remote Control Systems (Gostekhteorizdat, Moscow, Leningrad, 
1952). (in Russian) 

[7] Sh. Chang, Synthesis of Optimal Systems of Remote Control 
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[8] D. Newton, L. Gould and L.D. Kaiser, Theory of Linear 
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[9] K. Merriam, Optimization. Theory and calculation of systems of 
feedback control (Mir, Moscow, 1967).  

[10] A. Bryson and Ho Yu-Chi, Applied Optimal Control theory 
(translated from the edition of 1969, Mir, Moscow, 1972). 

[11] Guang-Ren Duan. Analysis and Design of Descriptor Linear 
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[12] A. L. Shestakov, Measuring transducer of the dynamic 
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Devices and systems. Management, control, diagnostics 10, 23 
(1992). 

[13] A.L. Shestakov, A.V. Keller and E.I. Nazarova, Numerical 
solution of the optimal measurement problem, Automation and 
Remote Control  1, 107 (2012). 

[14] A.A. Ebel, On algorithm for numerical solution of optimal 
measurement problem using linear splines, Journal of 
Computational and Engineering Mathematics 3 (1), 37 (2016). 

[15] Yu.V. Khudyakov, Оn adequacy of the mathematical model of 
the optimal dynamic measurement, Journal of Computational 
and Engineering Mathematics. 4 (2), 14 (2017). 

[16] A.L. Shestakov, G.A. Sviridyuk, A.V. Keller and Yu.V. 
Khudyakov, The numerical algorithms for the measurement of 
the deterministic and stochastic signals, in Springer Proceedings 
in Mathematics & Statistics, vol. 113 (2015), pp. 183-195. 

 

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	Numerical Investigation of Optimal Dynamic Measurements