105

1 INTRODUCTION

To obtain current information about the state of the

high-tech complex (HTC), as an object of control of a

complex technical system, the information-measuring

system (IMS) must perceive the measured input

values and convert them into signals necessary for:

formation and implementation of norms in analog

and digital types; comparison of values of input

signals or functions from them with norms (settings);

formation of quantitative judgment and its issuance in

the information model and/or in the circuit of the

automated control system (ACS) [25]. And some

studies prove that the use of non-classical theory of

measurement errors for processing time series allows

to detect the presence of weak, not removed from the

processing of sources of systematic errors [15].

On the other hand, the aerohydrodynamic and

climatic conditions of HTC operation and the

aggressive impact of the marine environment

significantly reduce the resource of HTC. This feature

requires the use of methods for forecasting the

technical condition of HTC in order to assess the

possible time of its safe operation [7]. The specificity

of the problem is mainly in the assessment of causal

relationships between the controlled parameters of the

equipment and defects that may cause their change,

and the possibility of further operation of HTC [16].

The presence of errors in measuring and control

devices leads to specific errors that should be taken

when assessing the quality of control and solving

management problems of HTC [4]. That is why the

task of creating diagnostic and forecasting tools that

operate in complex operating conditions and adapted

for continuous, long-term and reliable monitoring of

the state of HTC elements under the action of

concentrated destabilizing factors is relevant and in

demand.

Diagnosis of the Technical Condition of High-tech

Complexes by Probabilistic Methods

V. Budashko, A. Sandler & V. Shevchenko

National University “Odessa Maritime Academy”, Odessa, Ukraine

ABSTRACT: When designing multilevel systems for monitoring the parameters and characteristics of high-tech

complexes, it is necessary to effectively control the state of the elements of generating units. Existing control

systems for their specification and technical characteristics do not fully meet the monitoring objectives. The

capabilities of existing known systems have limitations on the depth of use and compensation for the impact of

operational factors. The article proposes and substantiates the feasibility of using the principle of automated

measurement and control of load in high-tech complexes based on the probabilistic approach. It is established

that the presence of errors in the means of measurement and control leads to specific errors that should be taken

when assessing the quality of control, solving management and control tasks. To register the transition of

parameters beyond the limit values, a new sensor circuitry based on fiber-optic elements is proposed. The main

difference of the proposed diagnostic tool is the invariance to operational destabilizing factors.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 16

Number 1

March 2022

DOI: 10.12716/1001.16.01.11

106

2 PURPOSE OF WORK

Ensuring high efficiency and reliability of HTC, which

is achieved by introducing new tools for diagnosing

and forecasting the technical condition, increasing the

adequacy of estimating the parameters of HTC

elements through the use of the principles of partial

invariance to external uncontrolled influences on

measurements.

Achieving this goal involves solving the following

tasks:

− determination of noise-resistant to the impact of

operational destabilizing factors (DF) means of

diagnosis and forecasting of the technical

condition of the elements of HTC;

− synthesis of the model of automated diagnostics of

the technical condition of HTC taking the

metrological characteristics of noise-tolerant

sensors.

3 CONTENTS AND RESULTS OF THE RESEARCH

The analysis of the known solutions proves that for

modern technical operation of HTC the newest means

of diagnosing and forecasting of a technical condition

are demanded, namely: devices of fiber optics

insensitive to the majority of operational DF [22].

Most scientific research does not define the methods

of construction, principles and features of synthesis of

diagnostic and technical condition of equipment

operated under the influence of concentrated DF, does

not consider structural and technological features of

construction and synthesis of such tools, does not

assess the stability of their characteristics in difficult

conditions operation [34].

On the other hand, using large data from IMSs and

current forecast data corresponding to time-

dependent operational situations and

hydrometeorological conditions, respectively, models

for detecting DF based on avoidance behavior are

presented to identify potential DF scenarios [28].

Research shows that DF risk assessments can be

extremely diverse depending on the operational

regime, and in 97.5% of potential scenarios, warning

actions are triggered only when the risk is 45% or

more of its maximum value [29]. At present, these

scenarios are not taken in databases of such cases,

determining outside the agreed risk criteria for the

operation of protection and assessing the nature of the

risk during the operational mode [11].

During the operation of HTC it should be borne in

mind that the change in the geometry of the

controlled elements due to wear occurs in the case of

frictional contact of parts [18]. Therefore, determining

the amount of wear is a necessary prerequisite for the

safe operation of HTC. As a rule, the sensitive

elements of measuring devices are built into the

elements whose technical condition is controlled [10].

Comparative analysis of fiber-optic wear sensors

showed that in most cases, there are practical

implementations of sensors based on Bragg fiber

lattice [1]. Also known is the circuit design of the wear

sensor, which consists of a fiber with a Bragg fiber

lattice, which is built perpendicular to the wear

surface (Fig. 1).

Figure 1. Block diagram of the measuring system of the

wear sensor with Bragg fiber lattice: 1 – radiation source; 2 –

splitter; 3 – primary fiber; 4 – fiber with a Bragg lattice that

is in contact with the tribo-surface; 5 – secondary fiber; 6 –

photodetector; 7 – controller for determining the amount of

wear

In addition to the wear of parts of the elements of

the HTC, it is necessary to take the peak loads

(twisting) on the shafts, which can lead to their

physical destruction. Assuming that the measured

load x and the measurement error in the probabilistic

sense are independent, the control result can be

obtained by operating with the composition of the

distribution density F(x) and θ(ε), Fig. 2, a, b.

For practical purposes, it is of interest to use

approximate estimates of these probabilities, for

example, using nomograms linking the standard

deviation of the error of the measuring devices σε and

the controlled value σx, as well as the tolerance zone r.

Error reduction can be achieved by repeating control

operations repeatedly or by double-checking, using,

taking the required accuracy, different control systems

[30].

At the same time, it is known that friction in the

tribo-combination depends on the microrelief of the

surface and is a source of vibration of the high

frequency range, the presence of which requires

appropriate measures for their filtration at the level of

power sources IMS. Wear sensors focused on

controlling the vibration of the surfaces to be worn

consist of a sealed housing with the base and the light

guide with a cantilever-mounted mirror (Fig. 3) [21].

In the process of measurement and control there is

a problem of sampling of one or another controlled

quantity, i.e. the task of determining the allowable

value of the control interval. Assume that at time ti

the operation of load control HTC is performed and it

is established that it is in the zone {PH, PD} (Fig. 2). In

this case, the estimation of the control result can be the

probability p (x ϵ r) that at ti+1 = ti + ∆t the load will not

go beyond {PH, PD}, i.e. the conditions will be met

[26]:

PHiPD

xttxx + )(

(1)

were x(ti+∆t) = ẋ(ti)∆t аnd ẋ(ti) – derivative.

107

Figure 2. Distribution densities: a) probability distribution

density of the controlled variable F(x); b) the density of the

error probability distribution θ(ε); c) the composition F(x)

and θ(ε); d) the composition f1(x) and f2(x); e) the graph

explaining the process of determining the probabilities p0(δ)

Figure 3. Vibration wear sensor: 1 – optical fiber; 2 – shell; 3

– console support; 4 – mirror; 5 – basis

The problem of increasing the stability and

reliability of measurements of precision sensors [20]

by minimizing their temperature drift is solved using

methods based on design and technological

improvements that minimize destabilizing factors

initiated by a variable intensity temperature field [31].

The most expedient and economically justified is the

way to improve fiber-optic wear sensors, based on the

use of passive methods to minimize temperature drift

in the most thermostable known circuit solutions [2].

In [9] it is proposed to use as a material that

provides the necessary mechanical characteristics of

the oscillating system for the control of the elements

of rotary machines of fibers based on artificial

sapphire. It is investigated that the use of single-mode

optical fibers made [33] of artificial sapphire with a

depressed core and a wavelength of 1.55 μm will

increase not only the resistance of the sensor elements

to DF, but also increase the relative amount of

radiation power (Fig. 4).

Changing the linear dimensions of the sensor base

at elevated temperatures up to 250 can reach

1,25·10

-5

m. Thermo-deformation of the biscle plate,

when connected with its base and secondary fiber

(Fig. 5), will create an axial shift of the latter in the

direction of the primary fiber [24].

Figure 4. The relative intensity K depending on the gap xs

between the primary and secondary fibers of artificial

sapphire for different wavelengths of optical radiation λ,

μm: 1 – 0.85; 2 – 1.33; 3 – 1.55

Figure 5. Fiber-optic vibration sensor wear: 1 – base; 2 –

primary fiber; 3 – secondary fiber; 4 – reflective layer; 5 –

braided console plate

The following initial data were used to estimate

the value of possible compensation of thermal

propagation of sensor elements due to the use of the

plate: material of primary and secondary fibers –

artificial sapphire with refractive indices (RI) n = 1.75;

base material quartz glass with software n = 1.48; the

composition of the glass plate – glass made of

artificial sapphire with a coefficient of thermal linear

expansion 1 = 5,6·10

-6

and glass brand KRS5 with 1 =

5,8·10

-5

; the thickness of the glass plate



h = 5·10

-4

temperature range



Т = 50…250 С; due to the

bending of the glass plate and the thermal

propagation of the base [3].

Based on these limitations, it is possible to

determine the relative intensity depending on the gap:

( ) ( )

2 2 2 2 2 2 2

1 4 coth ,

K z z z



  

−





= − + + +















(2)

where

( )

0,5

sin 1









= −





for radiation polarized perpendicular to the plane of

incidence

( )

0,5

, sin 1 ;

cos



−

= = −  −



for radiation polarized in the plane of incidence

108

( )

0,5

0 2 1

3 3 0

cos

, sin 1

l T h

x x x x l T









= = −  −



−  

= +  = + − + 







(3)

If the calculations take the average value for both

possible polarizations, then when using a glass plate,

the addition of the value of K in the temperature

range



Т = 50…250С will be within 6...9,5·10

-3

This value of the application of the coefficient of

relative intensity can be neglected.

If we limit the measurement error xPH – xPD > εmax

and use the conclusions [23], we can find the

probability of error of the first MIS (1) and the second

MIS (2) kind:

()

MIS(1)= ( ) ( ) ( ) ;

PH l

PD l x x

F x d d dx

     

−



− −







  

)

MIS(2)= ( ) ( ) ( ) ( ) ;

PH PH

PD PD

x x x x

x x PH x x

F x d dx f x d dx

     

−−



− − −















   

The probability p(x(ti + ∆t)) depends on the width

of the interval r, the dynamic properties of the load

and the value of x(ti). It can be found by integrating

the conditional density f(x, ẋ/xPD) < x < xPH in regions I

and II (Fig. 2, с). Assuming that the distribution ẋ (3) is

also normal, then

( ) ( ) ( )

,f x x f x f x=

(4)

where

( )













−=













−

−=

exp

;

exp

)(











Probability (1 – p) can be found as

.),/()(

),/()(

),/()(1

)(

/)(



−



−

+=−

PHPD

PDPDPH

PDPH

PHPD

txx

PDPH

txx

PHPD

txx

dxxxxfxdxf

dxxxxfxdxfp





At small ∆t (see Fig. 2, d), the probability p will be

equal

( )

1 / 0p t dp d t t= −   =

From here

( )

dp d t t

−

=



. Differentiating the

expression (1 – p) by ∆t and substituting it in the

expression for ∆t, you can get the allowable value of

the control interval:

( )

(

)

2 2 2

(1 )

exp ( ) / 2 ) exp( ( / 2 )

PH x x PD x x

x M x M









−

=

− − + − −

where р and р0* – apriority and given probabilities of

finding the controlled size of loading.

To simplify the determination of the control

interval with a known maximum error εmax and the

maximum modulus of the first derivative of the

controlled load, you can use the expression:

( )

max

//t t dx t dt



 = 

and thus take εmax = 3σε, and (dx(t)/dt)max determine by

conducting an appropriate analysis of the function

f(ẋ).

Further consideration will be given only for

emissions for the upper level of the load HTC [12].

The most important issues are to determine the law of

distribution of the time of the random function above

a given level and the law of distribution of the

number of emissions [17].

The load output beyond the upper limit is

described by the inequalities

xdttxtx + )()(



PHPH

xtxdttxx − )()(



, and the probability of ЕPH

emission will be determined by the expression

( ( ) ( ) ) ( ( ), ( ))

PH PH PH

x xdt

E x x t dt x t x f x t x t dxdx



−

−   =



(5)

Since ẋdt << xPH, i.e. the limits of integration differ

little, and also assuming that the emission probability

is proportional to the value of the time interval [19],

simplify expression (5) and introduce the concept of

time density for the emission probability

( )

PH PH

l x t

( / ) ( , / ) ;

( / ) ( , / ) ,

PH PH PH

PD PD PD

l x t f x x t dxdx





(6)

where f(ẋ/t) – conditional density distribution of the

value of ẋ over time.

In this case, the average residence time of the load

above the specified limit for the period of time T will

be found as

( / ) ,

PH PD

T f x t dxdt







(7)

and the average duration of the single emission [14]

(from the practical point of view, this is the most

important characteristic):

109

( / ) )

( , / )

f x t dxdt

f x dx t dxdt











(8)

If we now assume that the conditional density of

the ordinate distribution of the random function f(x/t)

and the function f(dx/t) does not depend on time, then

the search task is greatly simplified [13]. In this case,

similarly to (4), the distribution laws f(x) and f(x, ẋ) are

uniquely expressed through the mathematical

expectation Mx and the variance σх and σ

ẋ

, since the

mathematical expectation of the derivative M

ẋ

due to

the stationaryness of the random process is zero.

The variance σ

ẋ

is determined by the correlation

function of the velocity at zero (τ = 0):

()

d K r





= − =

As a result of conversions we will receive:









dxxfТТ ;)(







;),( xdxxdxfTN



;

),(

)(





dxxxdxf

dxxf





(9)

( )

exp

;

exp

























−

−













−

−=















−

−=



xPH

MxMx















(10)

where Ф(z) = Ф(Мх, σх, хPH) – Laplace function:

( ) , .

PH x

z e dx z





−

 = =



It should be noted that, using the tables of Laplace

functions Ф(z), for example [27], it is possible to

calculate the probability of feedback of the predicted

load x ϵ N(Mx, σx) in the interval xPH, xPD. To do this,

you must first calculate

PD x



−

and

PH x



−

and then, using Laplace [5] tables to find Ф(z1) and

Ф(z2). In this case, the probability is defined as

)()()(

zzxxxp

PHPD

−=

. (11)

When determining the control processes of HTC

with the use of electricity storage, you need to forecast

the value of energy required to ensure the emission of

the load above a given level. It is clear that in this case

the average area

limited by the implementation of

the normal and stationary random function above a

given level of xPH at the time of release, and therefore

determined:

( )

1 exp .

PH x







−

= + 



−



−



 −  −





(12)

Let us also determine the request for estimating the

global mathematical expectation Mx load on the

considered interval, which is stored in the computer

memory and should be periodically adjusted (if

required by the evaluation results). For this purpose,

we use the Monte Carlo method [8]. In this case, the

computer memory must have the average of the

sample value x load for the last N > 10 observations.

Assuming that the distribution of random values

is asymptotically normal and taking the well-known

rule of "three sigmas" [32]:

( 3 ) 2 (3) 0,997,

P M x



−    

we formulate the conditions for the need to correct the

global mathematical expectation in the form of a

predicate Пмх:

( )

,,,3

NxM

xMNxM

xxMX

xxx



→













−

(13)

where

)(













−





Then we find the static estimate of the constant

mathematical expectation [6] by averaging:

()

m x t dt





(14)

and to determine the estimate of the correlation

function we use known formulas:

( )

( ) ( ) ( ) ( ( ) ) ;

( ) ( ( ) )( ( ) ),

x i i

K t x t m x x t m dx

K t x t x x t x



−

  

−



= − + −

−

= − + −

−−





(15)

where m – is the number of quantization intervals on

the entire interval T, and n is the number of

quantization intervals on the interval τ.

4 CONCLUSIONS AND RECOMMENDATIONS

The proposed tool for diagnosing the operation of

HTC is invariant to external uncontrolled influences

on diagnostic processes, which allows for continuous

monitoring, preventive diagnosis of the technical

condition of the elements of VTK, improve the quality

of their technical operation and repair. We emphasize

that formulas (14) and (15) are valid in assuming the

ergodicity of the process, which can be judged by the

behavior of its correlation function. Thus, if Kx(τ)

110

converges to zero on the finite interval τ, then this fact

serves as a sufficient condition for the ergodicity of

the process itself x(t) (see Fig. 7, b), because

( ) 0,05К





In Figs. 7, 8 and 9 show the results of statistical

processing of experimental studies of HTC for

different types of vessels.

Figure 7. The results of experimental evaluation load

characteristics (a) and the normalized correlation function,

designed for many implementations (b)

It is determined that the introduction of a new

means of preventive diagnosis of the technical

condition of tribodes will increase the efficiency of use

and reliability of HTC by reducing the accident rate

10 %, increasing the maintenance period and

reducing operating costs with an average load.

σx

τ, ч

2 4 6 8 10 12 14 16 18 20

22 24

σср x

Figure 8. Daily changes in mathematical expectations

relative to the average values

τ, ч

2 4 6 8 10 12 14 16 18 20

22 24

Mср x

Figure 9. Daily changes in standard deviation relative to

average values

Errors of the first and second kind are

distinguished, the probability of false indication and

omission of registration of transition of parameters X

of limit value, accordingly, is estimated. The

measured parameters X are connected with the

generated power in the running mode of l-parallel

generating sets, the change of which should be

controlled between the upper ХPH and lower ХPD

loading thresholds.

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