Academic Open Internet Journal

Volume 15, 2005

RELIABILITY ESTIMATE OF RISK TECHNICAL

SYSTEMS USING SPLAIN FUNCTIONS

Nikolay Ivanov Petrov

Tracian University – St. Zagora, Yambol - Bulgaria

The using of splain functions in the diagnostic and prediction tasks of the technical systems state is examined in references [1;2]. The proposed methods are elaborated with supposition for a closed and limited examined region in the multitude of conditions of the phase spase. In situations with multidimensional and many times repeated failures, intrinsic to the technical systems (aircrafts, automobiles, railway, sea and river transport, chemical installations, munitions and other), the using of these methods demands additional examinations, made in the present article.

Introduction: The reliability estimate of risk technical systems (RTS) by using of splain functions is connected with an analysis of the problem for existing and construction of these functions. So that, lets scrutinize in advance the question for existing and unity of a multidimensional splain in the frontier areas of using and the methods for its design, too. This question is examined in the present article on the base of the determination of splains in accordance with references [1;2].

Presentation: We initiate the following indications: X, Y, Z -Hilbert space; and , where and are the kernels of the operators. On the base both of such a formulation and the initiated indicates we define the relevant interpolation and polishing splains. If the condition is fulfilled, then interpolation splain will be every submultitude of , for which is executed:

(1)

Let’s consider the multitude and to construct a functional in as follows:

(2)

The element is called polishing splain, if the equality (3) is executed:

(3)

The shown variables are interpret physically by the following way: - an operator, defining the structure of the made measurements (periodicity, dependence of the parameters, precision of the measurements); - a vector of the measured parameters; - a concrete functional space, including functions of such degree of fluent, which is defined by a - priory information about the character of the processes, carrying out in the RTS; - an operator defining the criterion of correspondence with the modeling function of the examined process.

For investigation of the existing and uniqueness of - measured splain for prediction of the RTS state, it is formulated the following theorem, which is proved in the article.

Theorem: If the multitude is closed in and , than interpolation splain , appears as a decision of the tasks and , exists and it is defined unique for every , satisfying the condition .

Evidence: Let the condition is done, which follows from the physical considerations for RTS functionality. Let’s examine the element . From the definition for nucleus and from the attribute for linearity for the operator A it can be seen that:

(4) .

Consequently, from the geometry considerations we have a replacing of the linear space of the vector , i.e. is an affinity variety in space . Let’s examine the expression:

(5) .

Because it is a linear operator, than is also an affinity variety, but now by and by condition from theorem is closed in . It is follows that the expression is equivalent to minimum of the interval from to the affinity variety . More over, in this closeness exists an element assuring such a minimum. Due to condition (1) it follows, that , where is a linear operator, than follows that exists.

Now it follows a proof for the uniquness of the element . Let two points and , exist so that . Hence, follows:

(6) , .

Because the operators and are linear, than follows:

(7) .

From equations (6) and (7), using the quality linearity of the standard the equations are written:

(8) ,

(9) .

We summarize member by member inequalities (7) and (8), where we get:

(10) .

From inequality (10) we get:

(11) .

We rise the left and the right part of (11) to second degree and we get:

(12) .

From equation (12) follows:

(13) ,

Therefore: .

Now it can be proved, that for a given , exists only one splain , for which it is done and . We suppose that exist splains and . It follows that , and , . Let couseder . While change to the operators and , due to their linearity we get . Similarly we get . Hence, . From the conditions of the theorem it follows that , therefors, or , which should be proved.

Now let we prove the existing and the uniqueness of the polishing splain. We put (2) on the right part of (3), and we get

(14)

Because and are constant quantities, it follows, that the prove for existing and uniqueness of the polishing splain reduces to the evidence of the existing and uniqueness of the expression , which was already proved.

Now let examine the behaviour of RTS near to the limit of the area, for which are defined the peculiarity of design of the splains.

1. Self – removing the design failures

At border values of the parameters of RTS , equal to and 0, must be done near to the avalanche rise out of the permit. However, the area of all uninterrupted functions with values in the interval does not describe fully the examined process (fig. 1).

Fig. 1. Jump of the values of the parameter of RTS

out of the admissible borders of the Producer

In the whole multitude of splains , located in the interval , by the theorem for existing and uniqueness, proven above, it could be found splains so that the following to be fulfiled:

(15)

For the design of the modeling function of the process of examination, by splains in case, when it is necessary an examination of the possibility for their design in the field of functions. Then, the conditions (15) define the region of used splains .

2. Alternate failures

In work couditions, which define the limit values of the parameters of RTS, parameter changes beyond their admissible value (alternate failures) come, for small time intervals (fig. 2).

Fig. 2. Small deflections of relevant parameter of RTS beyond its

admissible value

For the modeling of the process of alternate failures, using splain functions it is necessary to be proven, that in the multitude of splains for every group of three functions are done the operations: multiplication, collecting, commutativeness and associativeness, so it is possible a work in circle or semicircle of splains describing the multitude of real number .

That is why, without being violated the community of the ratiocinations, a task for design of polynomial functions in arbitrary areas is soluted. For such a purpose, must be soluted the task for design of splains in circle or in field. The using of operations with splains over field of circle is necessary, due to the fact, that design of functions, modeling the state of RTS can be done, by algebraic operations over the numbers (estimates of parameters) with different dimensions.

Here is the moment for reminding, that the multitude R of a few elements is called circle, if in them are defined (possible) two operations-collecting and multiplication, together with the operations commutativeness and associativeness, connected with the laws of the distributivity (because the collecting is connected with the opposite operation- deduction) [1].

The multitude of all splains, defined for , will be a circle, if in them can be done the operations collecting and multiplication and the qualities commutativeness, associativeness and distributiveness. If the quality distributiveness is not applicable , than the multitude will be semicircle of splains, includins the multitude of real numbers .A circle of splains forming an area , and including will be field, if some zeros are containing (of course in them must be applicable the operation of partition) [2]. For all elements of the area , for which , exists a splain function and in such a time the element will be uniqudy. Obvious from these definitions follow the following conclusions:

-for arbitrarily basic field could be used the theory of the linear dependent vectors, the theory for solving of systems with linear equations (but only in case of an endless field ), and also a matrix algebra;

- for an arbitrary field , could be used the theory of the linear areas and their linear transformations.

3. Mathematical models of failures

3.1. Model of self-removing failures

Self-removing failure from the point of view of the analysis of the parameters of RTS, is called every violation of the admissible values of the parameters during casual interval of time (much less than the technical resource to main repair of RTS put by the producer ) [3, 4]. The missing of information for the real value of the parameters of RTS in the frames of the admissible bords, must be examined as a question, connected with the using of splains for the prediction of the technical state. For that purpose it is examined the design of splain-function from the point of view for minimization of an integral function chosen for a criterion (in the particular case for and third degree of the splain the above property is the minimum rate of Holliday) [5].

The minimum of the multiple integral must be found:

(16)

where: of , and

is a function describing the behaviour of the process around the limit of the area.

If is the exact solution of that task, then with could be noted the value of the minimum. If it is possible in the field D to be build a function, reaching the minimum value , which satisfies (16), and is near to , then will be a passable decision of the task. If it is possible to be minimized the sequence , i.e. the sequence from functions satisfying (16) for which , then it is possible a search for a solution in the field of splains. It is suggested an examination of family splain functions:

(17) ,

where: are arguments, characterizing numbers of the approximation function and depending on some parameters.

Then, for , is done. After substitution of formula (17) in (16), we get :

(18) .

Equation (18) is minimized from the condition:

(19) .

After solution of the system of equations (19) we get values , for which it is done:

(20) .

It is chosen from the family functions (20), such for which replays to the value , where we get the searching model of a self-removing failure:

(21) .

We must mention, that the process of searching of that function is comparatively simple, except in the cases of work over a field (area), because all possible actions over the elements which are in them must be done.

From what has been said above it follows, that the examined field of splains in the section of breaking of the function from fig.1 (coming out of the informative parameter out of the permit), allows the same to be used for prediction of the technical state in the condition of self- removing failures.

3.2. Model of alternate failures

For design of that model it is necessary to be examined the application of splains for prediction of the technical state in RTS. For that purpose must be examined the conditions, where the developed in this article method could be used for design of splains in the circle . These families of splain functions are examined:

(22) ,

each of them is wider let the one before in the circle as a result of adding an addition parameters. Let be a function, defining the least value of defined by (16), in comparison with all other functions of the family. Because of every adding of parameters, the class of admissible functions is widening, then .

It could be defined the condition, for which the succession of functions is minimum, i.e. the sequence is striving for the minimum, defined by:

(23) .

Sufficient condition for fulfillment of (23) is the relative completness of the family splain functions from (22). Then the equality will be fulfhboot:

(24) ,

and the inequalities .

From the realization of (22) and (23), follows, that every function could be very precisely described together with its private derivatives, by splain functions from the corresponding family in the circle . That condition is sufficient for the requirement the order to strive for minimum. Actually if the condition for fulfelement is done, then using the solution of the task from (16), is possible the selection of an appropriate splain function from the - family for a model of alternate failures:

(25) ,

so that to be done the inequality:

(26) .

From the execution of (26) and from the uninterruption of F, follows that the difference:

(27) ,

would be possibly least in the circle D. And the difference defined also from the next equation:

(28) ,

would also be possibly least. Then .

From (28), follows, that is one of the functions of such family. Then the condition must always be executed :

(29) или .

Because is as least as possible, than follows the equation, which should be proved:

(30) .

CONCLUSION

As a conclusion it could be done the following inferences:

1. From the suggested and proved theorem for interpolation by splain functions and from the created models of self-removing and alternating failures follows the possibility for a reliable realization of the prediction of the technical state of risk technical systems.

2. The approach of splain-functions is applicable both in the modeling of the behaviour of risk technical systems, and for estimation of hidden faults.

REFERENCES

[1]. Сендов Бл., Попов В. Числени методи. Първа част. София, Университетско издателство “Св. Климент Охридски”, 1996.

[2]. Стойлик Б. Стохастические модели и информационные системы. Доклад ВЦ СО АН Русия, 1997.

[3]. Petrov N. Use Reliability of Risk Technical Systems. Prof. “Asen Zlatarov” University, Burgas, Bulgaria, ISBN 954-9978-26-5, 2002.

[4]. Петров Н. Оптимизация и управление на техническата експлоатация на военни авиационни системи. Дисертация за присъждане на научна степен “Доктор на науките”, ВА “Г. Раковски”, 2001.

[5]. Калчев Ив. Измервания базирани на модели.Докторат, София, 2002.

Technical College - Bourgas,