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Equation
of the casual relative error during measuring of basic parameters of technical
systems
Assoc. Prof. Vesselin Atanasov Ph. D,Nikolay Petrov Ph. D,Karmen Aleksandrova Ph. D
Perspective
Researches of Defence,
Aviation Research Base of The Bulgarian
Military Airtforces.
nikipetrov@descom.com
Abstract: The
dependency between the output characteristic of the technical system and its
basic parameters is exmined in the paper. A
relative decorrelation is done and for this reason each of their
possibilities for parametric functional
correspondation must be chosen big enough. This allows statistical analysis of
every basic parameter and its stohastic prognostication.
A
non-linear equation of the relative indefinition of the output characteristic
and its wide interpretation.
Key
words: output characterictic of technical system; relative decorrelation of basic parameters; relative
indefinition.
Introduction: The parametric valuation and prognostication of technical
systems(TS) is determined from their output characteristic(OC). OC on the other
hand is determined by the observation of specific basic parameters(BP),
determining the correspondence of TS to their functional purpose. For example
for the board aviation radiostation, OC is the maximal distance between the
aircraft and the land control tower and a stabile radioconnection is realized.
BP for Ð-862 are the power of the transmitter and the sensitivity of the
receiver. For more complicated TS, ñèëîâè óñòàíîâêè, êîëåñíèê, hydraulic
systems, àãðåãàòè and so on, the basis includes consideraly larger amount of BP
- 
in a more geeneral case they are intercorrelater.
TS, which is characterized by the
value of the i-th BP for observed interval of time 
in the stationary and final period of exploitation, is
determined by [1;5]:
, (1)
where: 
- range of the fluctuation of the i-th BP for observed
interval of time 
;
- nominal value of the i-th BP - 
;
- quadratic mean diversion of 
calculated for 
;
s
– accepted number of quadratic mean diversion, which guarantee the functional
value of 
in the confidence range
[1;2;5;8].
The teorethical and experimental
analysis show that the distribution of BP for AS, is satisfactory approximated from the normal law of distribution
[1;3;4;5;6].
The possibility for fulfillment of
condition (1) in the interval 
for the i-th BP is
determined by:
, (2)
where: 
- possibility for fulfillment of condition (1) for the parameter
in separated moments
of 
.
In formula (2) the conditional possibility from the second multiplier, threats
the fulfillment of condition (1) for the rest 
- BP, if it is
fulfilled for the 
- th BP.
The stohastic observation and
determination of the functional dependencies (2) is complicated and in
applicative plan is definitely excessive. Considering this for relative
decorrelation of 
a good method is
each one of the possibilities 
to be chosen big
enough (practically probable event 
). For example, in (1), at 
the possibility
for 
to be in the
range, determined by the standart technical documentation of the manufacturer
is 
, which means that for all parameters 
correlation (2)
can be decorrelated relatively through the expression for the ñóìàðíàòà
possibility for the parametric functional correspondence of TS, considering
the following:
,
(3)
where: 
- summary possibility for parametric functional
correspondence of TS.
Formula (3) gives the opportunity for separated
statistical analysis of each BP - 
and its
stohastical prognostication. For TS, the decorrelation of (3) is
determined from the high probability for fulfillment of condition (1) for each
i-th BP, 
[1;2] .
If condition (3) is fulfilled we can represent the
functional dependency for the OC of the TS - 
from 
- basic
parameters of the corresponding TS,
through the decomposition of Taylor:
, (4)
where: 
- average value of OC, determined for the nominal
values(mathematical expectancies) of ÎÏ - 
;
- consecutive partial derivative of Z, for the
corresponding 
th BP - 
, determined for corresponding nominals 
;
- the diversion of 
from 
, due to variations of the 
th BP - 
.
When 
, the fixed equation (4) changes like this:
(5)
where: 
- coefficients of influence (weight coefficients),
determining the level of influence of the relative fluctuations 
of the 
th BP over the
relative fluctuations 
of OC.
Equation (5) is determined as a non-linear
equation of the casual relative error and is valid when 
and 
. When the non-linear members in equation (5) are
considerable (determined from TS with expressed non-linear behaviour of BP), it
can be represented as a fixed:
(6)
In equation (6), the non-linear members with
subsequent derivatives after the first
derivative also have to be considered.
Considering the separate BP the order
(6) consists from partial rows(for the i-th BP):
,
(7)
where
the fixed function 
has the value:
(8)
In (8) the dimentionless coefficients 
are
determined from:
,… .
The multiplier 
it the right side of (8) is exponential level
of the variable 
. Considering the axiom
for regularity [9] in commensurable a
priori and
a posteriori ranges of fluctuation of the variable, it can be represented in
the following way with a considerable accuracy:
(9)
where 
and 
are constant coefficients, derrived from the
immediate a priori data.
Considering correlation (9),
formulas (6) and (7) have the following view:
, (10)
(11)
Equation (11) is a
widened intepretation of the equation of the relative error from (5). It is characterized with the availability of
a “partial” coefficient of influence, derrived from the multiplier 
, which shows the
influence of the non-linear members in row (4) from second and higher order.
A priori determination of the
parameters of the partial rows from (11).
The basis of the a priori usage of the influence of
the artificial formed fluctuation of the 
-th BP - 
, over OC
when the rest of the BP
are constan).
Figure 1
It is done using the
following input data and procedures:
1. For BP - 
two a priori
points with coordinates as follow – ò.1
, ò.2 
, shown on figure 1 where 
are known
nominals of the 
-th BP and OC; 
are
given fluctuations of the 
-th BP and measured fluctuation of 
.
2. For Point 1 we determine the
derivative 
and the coefficient 
.
3. We valuate the coefficient 
according to the coordinates of
Point 1 
(end of the a
priori range, beginning of the a posteriori range of prediction):
(12)
4.
From proportion (11), represented in àïðèîðåí view:
,
we
obtain: 
. (13)
Prognostication of the relative
error of the output characteristoic of TS when it is complexly dependent on
basic parameters.
When supplying a stable work modes of TS, the folowing
is valid for formulas (12)¸(14):
.
We can change (12) according to these conditions:
(14)
Equation (14) is “Extended
equation of the relative error 
”, giving the
opportunuty for its prognostication during complex fluctuation of the defining
BP ÎÏi(
)
out of the range 
, i.e. for 
according to:
. (15)
Generally, equation (15) is a sum of relatively casual errors, depending on
a large amoung of nondominating factors distributed normally [7].
It is important to say that the distribution of the
process of variation of 
, due to its non-;inear
character is multiplication of distributions of Veibl[4].
When 
and execution in
general case of condition 
[4], for the mathematical expectancy 
and the
dispersion 
of the different ÎÏi , the mathematical expectancy and the dispersion of OC of
TS 
can be
valuated according to:
(16)
. (17)
Expression (17) means that the calculated through
“Extended interpretation of the equation of the relative error” dispersion of
OC 
exceeds the defined through “Equation of the
small relative error” with the values:
(18)
Equations (17) and (18) mean
that when 
practically the distribution of 
of OC of TS is close to the normal when 
.
Results and conclusions:
1. An extended interpretation of the equation
of casual realtive error considering the non-linear characteristic of the
fluctuation of output characteristics of the technical systems.
2. The parameters(moments)
of distribution of the OC of TS are analytically determined on the base of the
moments of distribution of the different BP.
3. A condition for
normality of the distribution of the OC is derrived..
REFERENCE
[1]. BDS 27.002-86. Reliability in the technique.Basic
terms and definitions, Quality comitee to the Ministry Council, Sofia, 1987.
[2]. Anceliovich L. L., Reliability
and safety, Moscow, “Mashinostroenie”,
1985.
[3]. Statistical analysis of the
exploitation reliability of Mig-29 airplane, Sofia, 15.11.1999.
[4]. Kocev A. I., Asenov Sv