CN106295975A - System reliability fuzzy assessment method under a kind of change factor - Google Patents

Abstract

The invention discloses system reliability fuzzy assessment method under a kind of change factor, it is characterized in that, for solving the multifactor change impact on component failure, and then impact comprises the system fault conditions of these elements, propose the improved method of a kind of cloud model fuzzy overall evaluation, result cloud model is the function with factor independent variable, and then can determine that the system reliability under different conditions, have an advantage in that and can represent multiple cloud model uniformly, and obtain the cloud model of transitive state between these states, and it is continuous print；It comprises the steps: that in multiple synthetic evaluation matrixs that the method will obtain under different conditions, the characteristic parameter of cloud model correspondence position is as dependent variable, influence factor's value of different conditions is fitted as independent variable, replaces cloud model characteristic parameter to be multiplied with weight cloud function after matching and obtains final evaluation result；The present invention can be used for subway work Scheme Choice.

Description

System reliability fuzzy assessment method under a kind of change factor

Technical field

The present invention relates to safety system engineering, particularly relate to subway work Scheme Choice.

Background technology

The focus of systems reliability analysis Dou Shi educational circles all the time research.Which concrete skill is systematology do not belong to Art subject, but the general science of one on these subjects.Systematology is through among the research of each subject, it is ensured that quilt The function of object of study is it is necessary to study the reliability of this object.Actual productive life is also such, various industry production Be unableing to do without sorts of systems, especially for fields such as military project, space flight, medical treatment, its system reliability is particularly important.But how to determine System reliability is a key issue.On the one hand can be by analyzing system structure and building block, and known elements is reliable System reliability is determined under implementations；On the other hand can be by system practical work process obtains system to system outage statistics Reliability.

More research existing for system reliability at present.Polymorphic manufacture system is carried out by Zhou Fengxu etc. based on state entropy Fail-safe analysis；Gao Jun etc. have studied complication system operation reliability evaluation method under considering multi-invalidation mode；Zhang Yong enters Based on virtual out-of-service sequence sample, system reliability in time is studied；Hou Yushen etc. study based on quasi Monte Carlo method Model in Reliability Evaluation of Power Systems method；The section east mission reliability that is based on proposes multiphase system equipment investment policy optimization Model.If but system bulky structure is complicated, determine that reliability is the most inadvisable by analyzing system structure.Another kind of universal It is unalterable that situation is that system reliability is not, its impact being likely to be due to various factors changes.Propose so One problem: such as the diode in electric system, its probability of malfunction just with the length of working time, the size of operating temperature, Direct relation is had by electric current and voltage etc..If be analyzed this system, the working time of each element and work are suitable The temperature answered etc. may be the most different, the working time overall along with system and the change of ambient temperature, the probability of malfunction of system Also it is different.

For this problem, method mentioned above is insurmountable.Propose space fault tree (Space Fault Tree, SFT) theoretical, can be divided into further continuous space fault tree (Continuous SpaceFault Tree, CSFT) and Discrete type space fault tree (Discrete Space Fault Tree, DSFT).As this theory being supplemented and further Explore the probability using cloud model as the fault rate characteristic function in this theory.Utilize under the influence of different factors Cloud model represents that the feasibility of system failure rate is analyzed.With reference to existing cloud model fuzzy synthetic appraisement method, carry out structure Make the cloud model fuzzy synthetic appraisement method changed with factor change.

Cloud model is that the firm academician of Li De can describe with quantitative value with qualitative language in the one that the nineties in 20th century proposes The model of uncertain conversion, its application actual effect gets the nod and promotes.The uncertainty that cloud model is changed as qualitative, quantitative Model, it is possible to fully demonstrate randomness and the ambiguity of language concept, is the effective tool realizing qualitative, quantitative conversion.Use cloud Under model representation different condition, the relation between component failure and the system failure is appropriate.If but having obtained two kinds of environment Under cloud model, then the cloud model before how determining both of these case is a problem.Such as temperature is for component failure Impact is relatively big, adds up component failure when having obtained 10 DEG C and 20 DEG C and system failure relation according to an expert view with fault data Cloud model, then need a kind of method to determine cloud model between the two temperature (such as 7 DEG C).

Propose in multiple synthetic evaluation matrixs of will obtaining under different conditions the characteristic parameter of cloud model correspondence position as Dependent variable, influence factor's value of different conditions is fitted as independent variable, and function after matching is replaced cloud model characteristic parameter It is multiplied with weight cloud and obtains final evaluation result.The cloud model obtained is the function with factor independent variable, and then can determine that System reliability under different affecting factors.

Summary of the invention

1 Field Using Fuzzy Comprehensive Assessment based on cloud model

1.1 cloud models and numerical characteristic thereof

If U is a quantitative domain represented with exact value, C is the qualitativing concept on U, if quantitative value x ∈ U, and x is qualitative Stochastic implementation of concept C, x degree of membership μ (x) ∈ [0,1] to C, is the random number μ with steady tendency, i.e. μ: U → [0,1],x→μ(x).Then x distribution on domain U referred to as cloud is designated as C (x), each x and is referred to as water dust (x, a μ (x))。

The numerical characteristic of cloud reflects the quantitative characteristic of qualitativing concept, characterizes with expectation Ex, entropy En and super entropy He, is designated as C (Ex, En, He).Expect Ex representation theory domain space the most representational qualitativing concept value, reflect the central value in domain space.Entropy On the one hand En is the comprehensive measurement of qualitativing concept ambiguity and randomness, reflects and can be accepted by qualitativing concept in domain space The span of water dust, on the other hand can reflect again the dispersion degree of water dust.Super entropy He describes the uncertainty measure of entropy, reflection The cohesion degree of water dust in domain space, He is the biggest, the thickness of water dust is the biggest.

1.2 cloud generator

The algorithm or the hardware that generate water dust are referred to as cloud generator, occur including forward cloud, reverse cloud, X condition cloud and Y condition cloud Device.Normal Cloud Generator achieves scope and the regularity of distribution obtaining quantitative data in the qualitative information that prophesy value is expressed, and has Forward direction, direct feature.Backward cloud generator is that a number of exact numerical is effectively converted to appropriate qualitative Linguistic Value, Have inversely, indirectly feature.Here using Normal Cloud Generator, its water dust process generating requirement is as follows:

1) generating with En for expectation, He is the normal random number En ' of standard deviation；

2) generation one is with En for expectation, and the absolute value of En ' is the normal random number x of standard deviation_i, x_iIt is referred to as domain space U's One water dust；

3) μ is calculated_i=exp [-(x_i-E_x)²/2(E_n')²], then μ_iFor x_iDegree of membership about C.

4) circulation 1)～3), generate n water dust, then stop.

1.3 evaluation index cloud methods

Forward cloud model carries out cloud to index quantification numerical value, and the numerical characteristic computing formula of cloud model is:

In formula: i is constant, specifically can adjust according to the fuzzy threshold degree of index variable itself.

1.4 Field Using Fuzzy Comprehensive Assessment based on cloud model

Determine evaluation object, i.e. determine the several elements maximum on system reliability impact, C={x₁,x₂,…,x_q}.These yuan Part inefficacy and the importance degree weight causing the system failure are A={a₁,a₂,…,a_q}.Synthesis of System Reliability Evaluations matrix is R= {r₁,r₂,…,r_q}^T.This algorithm is use cloud model to replace membership function, and then calculates weight matrix and overall merit square Battle array.

Weight coefficient can collect data by forms such as Expert questionnaires, then adds up, uses reverse cloud to send out Raw device obtains the characteristic parameter (Ex of cloud model_a,En_a,He_a).For synthetic evaluation matrix, according to risk class and with cloud mould The relation that type is corresponding, causes component failure the risk of the system failure to give a mark, utilizes backward cloud generator to be converted to parameter (Ex,En,He)。

Shown in importance degree weight coefficient matrix such as formula (2).Component failure causes the Risk Comprehensive Evaluation matrix of the system failure As shown in formula (3).

Field Using Fuzzy Comprehensive Assessment based on cloud model calculates shown in process such as formula (4).

Wherein: Ex=Ex_a1×Ex₁+Ex_a2×Ex₂+…Ex_aq×Ex_q,

By Ex compared with each expected value of comment cloud model, immediate comment is evaluation result.

The change factor impact of 3 comprehensive evaluations improves

If component failure is affected by multiple factors, then be equivalent to the system failure relevant with multiple factors, then different because of The difference that the different values of element will cause component failure to cause system failure probability.

If d₁,…d_iFor affecting the extraneous factor of component failure degree, synthetic evaluation matrix R is represented by formula (5).

Work as d₁,…d_iDuring value difference, represent the Evaluations matrix obtained under different conditions, so the cloud model in R is special Levy parameter and be probably different.If S represents a kind of state, this state is by d₁,…d_iI factor determines altogether, different factor values Difference result in different states, i.e. S=S (d₁,…d_i)。

The problem that needs solve is, if the synthetic evaluation matrix R known under multiple state, then would represent the most uniformly These evaluation informations, and and then understand the R of transitive state between these states.The most how according to the expression (5) that formula (6) is comprehensive Form, and this expression is continuous print.

The problem that needs solve is, if the synthetic evaluation matrix R known under multiple state, then would represent the most uniformly These evaluation informations, and and then understand the R of transitive state between these states.The most how according to the expression (5) that formula (6) is comprehensive Form, and this expression is continuous print.

For solving the problems referred to above, propose different conditions S₁Under cloud model pair in multiple synthetic evaluation matrixs (formula (6)) of obtaining The characteristic parameter answering position is worth d as dependent variable, the influence factor of different conditions₁,…d_iIt is fitted as independent variable, will intend After conjunction, function replacement cloud model characteristic parameter is multiplied with weight cloud A and obtains final evaluation result.

The cloud model finally given is with factor independent variable d₁,…d_iFunction, and then can determine that under different affecting factors System reliability.So can unified representation formula (6), R can be made again to become continuous function.If shown in fitting function such as formula (7). If influence factor is one, then carry out one-dimensional matching；If factor is two carries out two dimension matching, by that analogy.

Wherein: k represents kth element, k ∈ [1, q]；Ex/En/He represents the object asking matching；S₁,…,S_NRepresent affect because of Element d₁,…d_iThe state that different values obtain；Represent S₁,…,S_NThe i dimension space vector of different factor value compositions under state；Represent S₁,…,S_NThe value vector of the Ex/En/He of lower correspondence；Q represents matching number of times.

According to formula (7), it is considered to different factors affect descending manner (5) and are rewritten as formula (8).Corresponding Field Using Fuzzy Comprehensive Assessment meter Calculate as shown in formula (9).

Wherein:

4 algorithm application

The failure condition of one system is added up for a long time.Find that the system failure is mainly derived from the mistake of wherein three elements Effect.Statistical data is analyzed the inefficacy finding these three element and has relatively Important Relations with the temperature factor in working environment.Separately On the one hand three component failures are the most different to the influence degree of the system failure.Engage associated specialist system to be carried out point for this Analysis, and analyze on the basis of to system in condition of different temperatures under reliability be evaluated.

For the problems referred to above, first fault data is analyzed.If system is T, the main element affecting T reliability is X1, x2 and x3, affect factor d of these part failure rates change₁For temperature, take three kinds of temperature that is 10 DEG C, 20 DEG C and 30 DEG C and enter Row statistical analysis, i.e. S={S₁=10 DEG C, S₂=20 DEG C, S₃=30 DEG C }, N=3.

Use expert survey combine with field data survey method Expression element lost efficacy to the system failure generation can Can property and seriousness.According to the basic thought of security risk classification, risk assessment can be divided into 4 grades, can accept, having ready conditions can Accept, be not intended to unacceptable.Grade and the relation corresponding with cloud model of risk are as shown in table 1.The calculating of cloud index is such as Shown in formula (1).

The grade of table 1 risk and the relation corresponding with cloud model

Table 1Level ofrisk and its relationship with cloud model

According to table 1 and formula (3), comprehensive expert survey and the result of accumulative fault data survey method, use backward cloud generator It is calculated the comprehensive evaluation matrix at each temperature (state S), as shown in formula (10).

According to the result that expert investigation and Field Research are fed back, the relative importance of the system failure is carried out by each component failure Marking.According to formula (2), backward cloud generator is used to be calculated the weight coefficient matrix of corresponding cloud model, such as formula (11).

According to formula (7) to R (S₁),R(S₂),R(S₃In), same position cloud characteristic parameter is fitted, and uses multinomial after deliberation Fitting function.When matching number of times is 4, the highest term coefficient is 0, and is that error when 3 is essentially identical with matching number of times, so Q =3.Under the influence of so obtaining temperature, element x1 failure effect system failure degree cloud model characteristic parameter Ex₁Change matching Shown in function such as formula (12).

Wherein: X represents temperature variable ,/DEG C.

In like manner obtain remaining cloud model parameter.

So the function of above-mentioned cloud characteristic parameter is substituted into formula (9), can get three elements under the influence of considering temperature and lose The effect Risk Calculation process to the system failure, as shown in formula (13).

Wherein:

Ex=0.00098X³-0.0499X²+0.7447X

It can be seen that each characteristic parameter of the fuzzy overall evaluation cloud model ultimately formed is with representing temperature from formula (13) The function of independent variable X.As long as bringing actual temp into X, system reliability cloud model at such a temperature just can be tried to achieve.So I.e. illustrate the fuzzy overall evaluation cloud model that synthetic evaluation matrix that original three determine according to real data is corresponding, also make Can continuous representation at the cloud model considered under variations in temperature, and then the system reliability feelings in this temperature range can be evaluated Condition.

Detailed description of the invention

The failure condition of one system is added up for a long time.Find that the system failure is mainly derived from wherein three elements Inefficacy.Statistical data is analyzed the inefficacy finding these three element and has relatively high point with the temperature factor in working environment System.On the other hand three component failures are the most different to the influence degree of the system failure.Engage associated specialist that system is entered for this Row analyze, and analyze on the basis of to system in condition of different temperatures under reliability be evaluated.

For the problems referred to above, first fault data is analyzed.If system is T, the main element affecting T reliability is X1, x2 and x3, affect factor d of these part failure rates change₁For temperature, take three kinds of temperature that is 10 DEG C, 20 DEG C and 30 DEG C and enter Row statistical analysis, i.e. S={S₁=10 DEG C, S₂=20 DEG C, S₃=30 DEG C }, N=3.

Use expert survey combine with field data survey method Expression element lost efficacy to the system failure generation can Can property and seriousness.According to the basic thought of security risk classification, risk assessment can be divided into 4 grades, can accept, having ready conditions can Accept, be not intended to unacceptable.Grade and the relation corresponding with cloud model of risk are as shown in table 1.The calculating of cloud index is such as Shown in formula (1).

The grade of table 1 risk and the relation corresponding with cloud model

Table 1Level ofrisk and its relationship with cloud model

According to table 1 and formula (3), comprehensive expert survey and the result of accumulative fault data survey method, use backward cloud generator It is calculated the comprehensive evaluation matrix at each temperature (state S), as shown in formula (10).

According to the result that expert investigation and Field Research are fed back, the relative importance of the system failure is carried out by each component failure Marking.According to formula (2), backward cloud generator is used to be calculated the weight coefficient matrix of corresponding cloud model, such as formula (11).

According to formula (7) to R (S₁),R(S₂),R(S₃In), same position cloud characteristic parameter is fitted, and uses multinomial after deliberation Fitting function.When matching number of times is 4, the highest term coefficient is 0, and is that error when 3 is essentially identical with matching number of times, so Q =3.Under the influence of so obtaining temperature, element x₁Failure effect system failure degree cloud model characteristic parameter Ex₁Change matching letter Number is as shown in formula (12).

Wherein: X represents temperature variable ,/DEG C.

In like manner obtain remaining cloud model parameter.

So the function of above-mentioned cloud characteristic parameter is substituted into formula (9), can get three elements under the influence of considering temperature and lose The effect Risk Calculation process to the system failure, as shown in formula (13).

Wherein:

Ex=0.00098X³-0.0499X²+0.7447X

It can be seen that each characteristic parameter of the fuzzy overall evaluation cloud model ultimately formed is with representing temperature from formula (13) The function of independent variable X.As long as bringing actual temp into X, system reliability cloud model at such a temperature just can be tried to achieve.So I.e. illustrate the fuzzy overall evaluation cloud model that synthetic evaluation matrix that original three determine according to real data is corresponding, also make Can continuous representation at the cloud model considered under variations in temperature, and then the system reliability feelings in this temperature range can be evaluated Condition.

Claims (6)

1. system reliability fuzzy assessment method under a change factor, it is characterised in that element is lost for solving multifactor change The impact of effect, and then affect the system fault conditions comprising these elements, it is proposed that changing of a kind of cloud model fuzzy overall evaluation Entering method, result cloud model is the function with factor independent variable, and then can determine that the system reliability under different conditions, and it is excellent Point is to represent uniformly multiple cloud model, and obtains the cloud model of transitive state between these states, and is continuous print； It comprises the steps: the feature of cloud model correspondence position in multiple synthetic evaluation matrixs that the method will obtain under different conditions Parameter is fitted as independent variable as dependent variable, influence factor's value of different conditions, and function after matching is replaced cloud model Characteristic parameter is multiplied with weight cloud and obtains final evaluation result；The present invention can be used for subway work Scheme Choice.

Fuzzy assessment method the most according to claim 1, it is characterised in that to changing of the Field Using Fuzzy Comprehensive Assessment of cloud model Entering, component failure is affected by multiple factors, then be equivalent to the system failure relevant with multiple factors, then different factors are not The difference that component failure will be caused to cause system failure probability with value, if d₁,…d_iFor affecting the external world of component failure degree Factor, synthetic evaluation matrix R is represented by formula (5),

$R(d_1,...d_i)={\left(\begin{array}{ccc}{\mathrm{Ex}}_1& {\mathrm{En}}_1& {\mathrm{He}}_1\\ {\mathrm{Ex}}_2& {\mathrm{En}}_2& {\mathrm{He}}_2\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ {\mathrm{Ex}}_q& {\mathrm{En}}_q& {\mathrm{He}}_q\end{array}\right)}_{d_1,...d_i}---\left(5\right),$

Work as d₁,…d_iDuring value difference, represent the Evaluations matrix obtained under different conditions, so the cloud model feature in R Parameter is probably different, if S represents a kind of state, this state is by d₁,…d_iI factor determines altogether, and different factor values are not With result in different states, i.e. S=S (d₁,…d_i)。

The improvement of the Field Using Fuzzy Comprehensive Assessment to cloud model the most according to claim 2, it is characterised in that need solution Problem is, if the synthetic evaluation matrix R known under multiple state, then would represent these evaluation informations the most uniformly, go forward side by side And understand the R of transitive state between these states, the most how according to the form of the comprehensive expression (5) of formula (6), and this table Show it is continuous print,

$R\left(S_1\right)={\left(\begin{array}{ccc}{\mathrm{Ex}}_1& {\mathrm{En}}_1& {\mathrm{He}}_1\\ {\mathrm{Ex}}_2& {\mathrm{En}}_2& {\mathrm{He}}_2\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ {\mathrm{Ex}}_q& {\mathrm{En}}_q& {\mathrm{He}}_q\end{array}\right)}_{S_1},R\left(S_2\right)={\left(\begin{array}{ccc}{\mathrm{Ex}}_1& {\mathrm{En}}_1& {\mathrm{He}}_1\\ {\mathrm{Ex}}_2& {\mathrm{En}}_2& {\mathrm{He}}_2\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ {\mathrm{Ex}}_q& {\mathrm{En}}_q& {\mathrm{He}}_q\end{array}\right)}_{S_2},...,R\left(S_N\right)={\left(\begin{array}{ccc}{\mathrm{Ex}}_1& {\mathrm{En}}_1& {\mathrm{He}}_1\\ {\mathrm{Ex}}_2& {\mathrm{En}}_2& {\mathrm{He}}_2\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ {\mathrm{Ex}}_q& {\mathrm{En}}_q& {\mathrm{He}}_q\end{array}\right)}_{S_N}---\left(6\right),$

For solving the problems referred to above, propose different conditions S₁Under in multiple synthetic evaluation matrixs (formula (6)) of obtaining cloud model corresponding The characteristic parameter of position is worth d as dependent variable, the influence factor of different conditions₁,…d_iIt is fitted as independent variable, by matching Rear function replacement cloud model characteristic parameter is multiplied with weight cloud A and obtains final evaluation result.

The improvement of the Field Using Fuzzy Comprehensive Assessment to cloud model the most according to claim 2, it is characterised in that finally give Cloud model is with factor independent variable d₁,…d_iFunction, and then can determine that the system reliability under different affecting factors, so Can unified representation formula (6), R can be made again to become continuous function, if fitting function such as formula (7) shown in, if influence factor is one Individual, then to carry out one-dimensional matching；If factor be two carry out two dimension matching, by that analogy,

$F_{Ex/En/He}^k(S_1,...,S_N)=polyfit(\stackrel{\→}{X},\stackrel{\→}{Y},Q)---\left(7\right),$

K represents kth element, k ∈ [1, q]；Ex/En/He represents the object asking matching；S₁,…,S_NRepresent influence factor d₁,… d_iThe state that different values obtain；Represent S₁,…,S_NThe i dimension space vector of different factor value compositions under state；Represent S₁,…,S_NThe value vector of the Ex/En/He of lower correspondence；Q represents matching number of times.

The improvement of the Field Using Fuzzy Comprehensive Assessment to cloud model the most according to claim 2, it is characterised in that according to formula (7), Considering that different factor affects descending manner (5) and is rewritten as formula (8), corresponding Field Using Fuzzy Comprehensive Assessment calculates as shown in formula (9),

$R\left(d_1,...d_i\right)={\left(\begin{array}{ccc}{\mathrm{Ex}}_1& {\mathrm{En}}_1& {\mathrm{He}}_1\\ {\mathrm{Ex}}_2& {\mathrm{En}}_2& {\mathrm{He}}_2\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ {\mathrm{Ex}}_q& {\mathrm{En}}_q& {\mathrm{He}}_q\end{array}\right)}_{d_1,...d_i}=\left(\begin{array}{ccc}{F}_{Ex}^1\left(S_1,...,S_N\right)& F_{En}^1\left(S_1,...,S_N\right)& F_{He}^1\left(S_1,...,S_N\right)\\ F_{Ex}^2\left(S_1,...,S_N\right)& F_{En}^2\left(S_1,...,S_N\right)& F_{He}^2\left(S_1,...,S_N\right)\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ F_{Ex}^q\left(S_1,...,S_N\right)& F_{En}^q\left(S_1,...,S_N\right)& F_{He}^q\left(S_1,...,S_N\right)\end{array}\right)---\left(8\right)$

(Ex, En, He) the most according to claim 5, it is characterised in that

$Ex={\mathrm{Ex}}_{a1}\×F_{Ex}^1\left(S_1,...,S_N\right)+{\mathrm{Ex}}_{a2}\×F_{Ex}^2\left(S_1,...,S_N\right)+...{\mathrm{Ex}}_{aq}\×F_{Ex}^q\left(S_1,...,S_N\right);$

$En=\sqrt{\begin{array}{c}{\left(|{\mathrm{Ex}}_{a1}\×F_{Ex}^1\left(S_1,...,S_N\right)\sqrt{{\left(\frac{{\mathrm{En}}_{a1}}{{\mathrm{Ex}}_{a1}}\right)}^2+{\left(\frac{F_{En}^1\left(S_1,...,S_N\right)}{F_{Ex}^1\left(S_1,...,S_N\right)}\right)}^2}|\right)}^2+\\ ...+{\left(|{\mathrm{Ex}}_{aq}\×F_{Ex}^q\left(S_1,...,S_N\right)\sqrt{{\left(\frac{{\mathrm{En}}_{aq}}{{\mathrm{Ex}}_{aq}}\right)}^2+{\left(\frac{F_{En}^q\left(S_1,...,S_N\right)}{F_{Ex}^q\left(S_1,...,S_N\right)}\right)}^2}|\right)}^2\end{array}};$

$He=\sqrt{\begin{array}{c}{\left(|{\mathrm{Ex}}_{a1}\×F_{Ex}^1\left(S_1,...,S_N\right)\sqrt{{\left(\frac{{\mathrm{He}}_{a1}}{{\mathrm{Ex}}_{a1}}\right)}^2+{\left(\frac{F_{He}^1\left(S_1,...,S_N\right)}{F_{Ex}^1\left(S_1,...,S_N\right)}\right)}^2}|\right)}^2+\\ ...+{\left(|{\mathrm{Ex}}_{aq}\×F_{Ex}^q\left(S_1,...,S_N\right)\sqrt{{\left(\frac{{\mathrm{He}}_{aq}}{{\mathrm{Ex}}_{aq}}\right)}^2+{\left(\frac{F_{He}^q\left(S_1,...,S_N\right)}{F_{Ex}^q\left(S_1,...,S_N\right)}\right)}^2}|\right)}^2\end{array}}.$