Ph Dr. Eng. Nikolai Ivanov Petrov; eng. Nikolai Veskov Nikolov
University “Prof. As. Zlatarov”-Bourgas
MAAAD “P. Volov” - Shumen
The methods for prognosing the technical status (TS) in this
number the optimal filters, are build on the base of classical procedures for
analyzing and data processing, demanding the knowledge of the fully expectable
characteristics of the measuring mistakes 
and the prognosible process 
. Usually
these data are rarely given. They could not be received and by imitation modeling
due to the TS complication of from the statistical data exerpt in our disposition
because of it’s insufficiency. The salving precision of the problem is defined
by the mistake of the modeling 
. For
the complicate TS the prior adequate model defining of the prognosed accidental
process is extremely difficult. The influence of external factors may result
in receiving of mistakes in the mode making. These mistakes should not be accepted
as accidental mistakes, but they must be taken like accidental quantities. Salving
the TS prognosing problem in such conditions with statistical methods assistance
(optimal filters) may lead to unjustifiably optimistically valuations 
.
Practically, this is completely inadmissible [4], because the aim of the TS
is the case of receiving pessimistically (quaranted ) valuation
for the technical state.
A base for prognosing the TS parameters alterations are the
data of their control. The completeness of these data depend a many factors
mainly from such control characteristics, like reliability and realization form.
The influence of the surround, the imperfection of the control
devices, the insufficient qualification of technical staff and etc. lead to
this, that the control results 
and
its measured value 
differs with same
accidental quantity 
- control mistake.
Usually it’s given up, according to [4], that:
(2.2.1)
The reliability valuation of the control in the accidental
mistake presence practically may be received by known from the mathematical
statistic procedures with independence supposition homeogeneonsy and normality
of 
. The examination for normality
may be made according to Shapiro-Willkock’s agreement criteria [1]. This criterion
is the most suitable for TS in Republic of Bulgaria Air Forces (RBAF) conditions,
because of its little sentient of statistical observations. By this the mistakes
characteristics from the controls ,
subordinating to the Gaus distibution are identical to the quality of the accidental
process of the type “white noise”. Descriptions of
like a “white noise” have received wide distribution [4]. Practically however
experimental confirmation of the description truth by prognosing the parameter
dreif of TS is seldom possible (because of the initial statistic limit).
By data missing for the accepted Gaus model advantage for
distribution, the valuation of the control precision may be made over the expectable
inequality
(2.2.2)
where : 
- given
function
- given probability
for realization of (2.2.2).
In the most of the practical situations, the not relative,
stohastical nature of the valuations ,
gives the opportunity to define prior 
and the probability value. By this
is assotiating with the quantity
almost mistake of the control clevires and may be used for the control precision
index.
By realization form we distinguish two control types – a discrete
and an intermittent control. Practically the opportunity for accomplishing of
intermittent control (receiving the
values 
) is limited because of the
large accomplishing difficulties and the too high resources consumption. More
often used is the discreet control of TS of AT because it is single and cheaper.
According by every algorithm for TS parameters dreif prognosing, prefending
for practical usage, must read this factor. Consequently it must allow filtration
of the intermittent and of the discrete measuring information giving priority
to the last, this means giving the information (control data) in recruitment
values form 
The usage of guarantied prognosing results for optimal regime
defining for technical exploitation and maintenance of TS, demands taking measures
for damages and break-down warning in the limited initial data conditions. For
the model of the accidental process 
from (2.1.1), the salving of the problem situation, when the prior knows is
only the dependence structure (2.1.1) [it’s determinate basis 
],
and the qualities of the control mistake 
are described from the expectable inequality (2.2.2).
This method for TS prognosing, suitable for use in limited
initial data conditions of RBAF way be build on the basis of the idea for external
(guarantied) or minimaximal appraisement [5]. The minimax principle is the best
case calculation in comparison with the accepted in the classical statistic
principal for average risk minimization allows.
- receiving of guarantied reliability and precision for the prognosing results;
- solving the problem without drawing any hypothesizes and supposes for
the prognosed process stohastical nature;
- fully initial information usage.
Minimaximal valuation consists of guarantied (about initial
data) limits for valuated value alternation [5].
The offered method for TS prognosing is directed to the making
of guarantied limits for 
alternation
by 
, i.e. for interval valuation receiving
. Interval prognosing of
may be execution based by known statistical information such as type maximum
alike, the best little square etc. with However our authentically must not be
guarantied without additional data for stohastics natural of
and 
[5]. This is consist and the
principal difference of the minimax approach for salving the TS prognosing problem.
By presenting the suggested method for guarantied prognosing,
firstly we examine the situation the situation in which the TS status is characterized
by a summary parameter (SP) - 
. The
SP alteration in the time is introduced by the realization of accidental function
from the type:
(2.2.3)
where: m – fixed number accidental quantities;
- accidental
quantities;
- intermittent
determinated time function
In time interval 
TE of TS, where 
is the technical
resource, given by the AT producer, or by the factory making it basically
or current repair. Let it be possible a SP intermittent control in time
interval
The control mistake
(i.e. the process identification,
the mistake provoked from the reversible fluctuations insisting etc.) will
we examine like same kind of disturbance (noise) - 
,
accumulating in the respective process realization (2.2.3). For the central
mistake is typical that it must not overflow the given limit 
.
Consequently, the control precision depends on execution degree of inequality.
(2.2.4)
Let it in result from the SP control, made in the interval
is received a piece of realization
. Taking presence the mistake
existing in the measuring (noises) can we downwrite the next equation.
(2.2.5)
On the basis of (2.2.4) and (2.2.5) we downwrite the following
inequality, which describes she prognosing process realization:
(2.2.6)
From (2.2.6) follows, that in the interval 
the real process realization 
in blocked in “prognosing area”, limited from the function 
above and 
underneath. These function
are defined by (2.2.6), accordingly:
(2.2.7)
,
which we call acceptable. They are shown on fig 2.2.1: 
we share from the number of function “The worse” i.e. these functions, which get above or underneath the “prognosing area” after the prognosing moment 
(fig. 2.2.1.).
to the system extremal polynoms. For propriety of the used mathematical apparatus we consider that the function system 
. Typical for the previous function system is the Tshebishev’s nature, meaning, that every summary polynom in the functions number 
, where 
- arbitrary gather of material numbers by the condition 
, has no more than m – number different zeros in the measuring interval 
and 
may be disposed so:
(2.2.8)
- positive constants
- functional system polynom 
are calculated by separated k-values of the approximating function 
which describes the TS corresponding parameter behavior according (2.2.5). The quantities 
and 
, may be desposed like TS work disturbing parameters.
and 
limited in the field 
, shown in fig. 2.2.1, by the functions 
(2.2.9)
is a polynom minimally diverting by module from zero in the field 
among all polynoms of the function system 
with single coefficients by 
. The coefficient 
is so selected that 
in the field 
is, that in the field 
points 
in which 
, and receive from (2.2.9):
(2.2.10)
(2.2.11)
and 
have the qualities formulated in the Karlin’s theorem [2].
, so in this case 
. In particular case, when 
the polynom 
analytical aspect is already knows (first kind Tsebishev’s polynom, normalized in the field in the field 
(2.2.12) 
and 
presenting in appearance (2.2.8), corresponds to a case, when the exanimate field 
from fig. 2.2.1 has the same section 
on the whole length, i.e. the relative mistake by the TS control is permanent