CN111966966A - Method and system for analyzing feasible domain of sensor measurement error model parameters - Google Patents
Method and system for analyzing feasible domain of sensor measurement error model parameters Download PDFInfo
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Abstract
The invention discloses a method and a system for analyzing feasible domains of sensor measurement error model parameters, wherein the method comprises the following steps: the method comprises the steps of firstly determining the real life of the equipment at the first arrival time, then determining the pseudo life of the equipment, and finally determining the feasible regions of the mean value and the standard deviation of a measurement error model according to the real life of the equipment at the first arrival time and the pseudo life, so that the problem of measurement errors existing in degradation data of the high-reliability equipment residual service life estimation is solved, and the maintenance decision analysis is carried out subsequently.
Description
Technical Field
The invention relates to the technical field of feasible analysis of measurement errors, in particular to a feasible domain analysis method and a feasible domain analysis system for a sensor measurement error model parameter.
Background
In recent years, a statistical data driving method is widely developed and applied in the fields of degradation modeling and life estimation as an effective method capable of depicting the randomness of a degradation process and reflecting the uncertainty of life estimation. The measurement and the acquisition of the degradation data are the premise and the basis of degradation modeling and life estimation, and measurement errors often exist in the obtained degradation data due to the influence of factors such as the performance of test equipment, the level of operators, the test method and the like in actual engineering. If the degradation data with the measurement error is directly adopted for modeling and life estimation, the prediction result is inevitably deviated.
Compared with a service life estimation result with deviation, the equipment system is maintained and managed by using an accurate prediction result, so that the safety risk can be reduced, and the economic loss can be reduced. If the measurement errors can be detected and calibrated on-line, the bias in lifetime estimation can be avoided by culling the measurement errors in the degraded data, so that this inverse problem is not meaningful. Unfortunately, for many degraded devices, the measurement error is difficult to identify and estimate online. Therefore, it is necessary to limit the sensor measurement error to a reasonable range to ensure the accuracy of the lifetime estimation result. Currently, only a few of the scholars have conducted research on it.
For example, sequisite and tang holy et al studied the feasible domain problem of measurement error given the performance accuracy requirements of the life estimate based on a linear Wiener process degradation model and further analyzed the impact of measurement error on maintenance decisions. The research still has certain defects and shortcomings, firstly, the measurement error is defined as a random variable and is subjected to a parameter fixed time-independent random distribution, such as normal distribution, Gamma distribution and the like. In practice, however, firstly, the measurement error may vary in a trend as the sensor performance degrades. For example, a metal thermocouple, which is a temperature sensor commonly used to reflect the degradation of the wall of a blast furnace, has a measurement performance that deteriorates with the increase in the use time. If only random variables are used to describe the variation of the measurement error, the time dependence of the error variation cannot be reflected. Secondly, in order to describe the deviation between the lifetime estimation result without the influence of the measurement error and the lifetime estimation result with the influence of the measurement error, the above documents have proposed a plurality of measurement indexes for measurement. However, there is randomness in the lifetime of the device obtained under the statistical data driven approach, i.e., a random variable rather than a fixed constant. The measurement index for measuring the deviation between two random variables proposed by the above documents can only reflect partial statistical characteristics of distribution, and has certain limitations. Besides, the above document mainly studies the deviation between the true lifetime in the first-arrival time sense and the false lifetime (the time at which the degradation process first-arrives at the failure threshold in the presence of measurement errors). In practical engineering applications, the pseudo-lifetime is often the result of applying the degradation data with measurement errors directly to a degradation model that does not take into account the measurement errors.
Disclosure of Invention
Based on this, the invention aims to provide a method and a system for analyzing the feasible domain of the sensor measurement error model parameters.
In order to achieve the above object, the present invention provides a method for analyzing a sensor measurement error model parameter in a feasible domain, the method comprising:
step S1: determining the real life of the equipment at the first arrival time;
step S2: determining a pseudo-lifetime of the device;
step S3: determining a feasible region of a measurement error model parameter according to the real life and the pseudo life of the equipment under the first arrival time; the measurement error model parameters include a mean of the measurement error model and a standard deviation of the measurement error model.
Optionally, the determining a feasible region of the measurement error model parameter according to the real life and the pseudo life of the device at the first arrival time specifically includes:
step S31: determining probability density distribution corresponding to the real life estimation value and probability density distribution corresponding to the pseudo life estimation value;
step S32: determining a KL distance formula;
step S33: and substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine the measurement error model parameters of the sensor.
Optionally, determining the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value specifically includes:
step S311: determining a probability density distribution function of the equipment under the condition of no measurement error; the probability density distribution function of the equipment under the condition of no measurement error is the probability density distribution corresponding to the estimated value of the real life;
step S312: determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time;
step S313: determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time; the probability density distribution corresponding to the pseudo life estimation value comprises a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is irrelevant to time and a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is relevant to time.
Optionally, the substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine the sensor measurement error model parameter specifically includes:
step S331: substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameter under the condition that the measurement error model is not related to time;
step S332: and substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time into the KL distance formula to determine the feasible region of the measurement error model parameter of the sensor under the condition that the measurement error model is related to time.
Optionally, the feasible region of the sensor measurement error model parameter under the condition that the measurement error model is not related to time is determined, and a specific formula is as follows:
wherein σMean value, D, representing a model of measurement errormaxRepresenting the maximum acceptable lifetime estimation deviation, σBReferred to as the diffusion coefficient, at represents the sampling interval time.
Optionally, the feasible region of the sensor measurement error model parameter under the condition that the measurement error model is time-dependent is determined, and a specific formula is as follows:
wherein σMean, μ, representing the measurement error modelMean value, D, representing a model of measurement errormaxRepresenting the maximum acceptable lifetime estimation deviation, σBReferred to as the diffusion coefficient, at represents the sampling interval time, μ represents the drift coefficient, and ξ represents the failure threshold of the device.
The invention also provides a sensor measurement error model parameter feasible region analysis system, which comprises:
the real life determining module is used for determining the real life of the equipment at the first arrival time;
a pseudo-lifetime determination module for determining a pseudo-lifetime of the device;
the feasible region determining module is used for determining a feasible region of the measurement error model parameters according to the real life and the pseudo life of the equipment under the first arrival time; the measurement error model parameters include a mean of the measurement error model and a standard deviation of the measurement error model.
Optionally, the feasible region determining module specifically includes:
a probability density distribution determining unit for determining a probability density distribution corresponding to the real life estimation value and a probability density distribution corresponding to the pseudo life estimation value;
the KL distance formula determining unit is used for determining a KL distance formula;
and the sensor measurement error model parameter determining unit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine sensor measurement error model parameters.
Optionally, the probability density distribution determining unit specifically includes:
a first probability density distribution function determining subunit, configured to determine a probability density distribution function of the device without measurement error; the probability density distribution function of the equipment under the condition of no measurement error is the probability density distribution corresponding to the estimated value of the real life;
the second probability density distribution function determining subunit is used for determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time;
a third probability density distribution function determining subunit, configured to determine a probability density distribution function corresponding to the pseudo life estimation value under a condition that the measurement error model is time-dependent; the probability density distribution corresponding to the pseudo life estimation value comprises a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is irrelevant to time and a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is relevant to time.
Optionally, the sensor measurement error model parameter determining unit specifically includes:
the first feasible region determining subunit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameter under the condition that the measurement error model is not related to time;
and the second feasible region determining subunit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is related to time.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
the invention discloses a method and a system for analyzing feasible domains of sensor measurement error model parameters, wherein the method comprises the following steps: the method comprises the steps of firstly determining the real life of the equipment at the first arrival time, then determining the pseudo life of the equipment, and finally determining the feasible regions of the mean value and the standard deviation of a measurement error model according to the real life of the equipment at the first arrival time and the pseudo life, so that the problem of measurement errors existing in degradation data of the high-reliability equipment residual service life estimation is solved, and the maintenance decision analysis is carried out subsequently.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a flow chart of a feasible domain analysis method for sensor measurement error model parameters according to an embodiment of the present invention;
FIG. 2 is a drawing showing an example of the present inventionWhen the value is 0, the life estimation probability density PDF and the cumulative distribution function CDF under the influence of measurement errors are shown in a schematic diagram;
FIG. 3 shows a graph of σ according to an embodiment of the present inventionWhen the value is 0, the life estimation probability density PDF and the cumulative distribution function CDF under the influence of measurement errors are shown in a schematic diagram;
FIG. 4 shows an embodiment D of the present inventionmaxDetermining a schematic diagram of feasible domains of the measurement error model parameters under the three conditions of being equal to 0.05, 0.1 and 0.2;
FIG. 5 shows an embodiment D of the present inventionmaxComparing the pseudo life probability density PDF with a real value under the three conditions of being equal to 0.05, 0.1 and 0.2;
FIG. 6 is a graph showing actual blast furnace wall degradation data according to an embodiment of the present invention;
FIG. 7 shows a lifetime estimation bias D according to an embodiment of the present inventionKLAssociated with muA variation graph;
FIG. 8 is a PDF comparison graph of probability densities under different measurement error model parameters according to an embodiment of the present invention;
FIG. 9 is a comparison of two maintenance strategies according to an embodiment of the present invention;
FIG. 10 is a diagram of expected cost rate bias according to an embodiment of the present invention
Deviation from life estimation DKLA relationship graph;
FIG. 11 shows an embodiment D of the present inventionmaxWhen the value is 0.26, the measurement error model parameters can be used for determining a schematic diagram.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a feasible domain analysis method and a feasible domain analysis system for a sensor measurement error model parameter.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Step S1: determining the real life of the equipment under the first arrival time, wherein the specific formula is as follows:
T＝inf{t:X(t,θ)≥ξX(0；θ)＜ξ}(1)；
wherein T represents the real life of the equipment at the first arrival time and is a random variable, X (T; theta) represents a time T-related degradation process, theta represents a parameter vector, and xi represents a failure threshold of the equipment.
Step S2: determining the pseudo life of the equipment specifically comprises:
step S21: constructing a degradation model containing measurement errors, wherein the specific formula is as follows:
Y(t)＝X(t；θ)+(t)(2)；
wherein, Y (t) represents a degradation model containing measurement errors, X (t; theta) represents a degradation model which is related by time t and does not contain the measurement errors, theta represents a parameter vector, and t represents the measurement errors.
Step S22: determining a first observation and a second observation; the first observation value is an observation value Y corresponding to a degradation model Y (t) containing measurement errors0:k，Y0:k＝[y0,y1…,yk]，yk＝xk+k，ykDegradation data, x, representing measurement errors in the k-th observationkDegradation data indicating that the k-th observation does not contain measurement error,kindicating the error of the k-th measurement. The second observation value is the observation value X corresponding to the degradation model X (t; theta) without measurement error0:k，X0:k＝[x0,x1…,xk]。
Step S23: performing parameter estimation on the first observation value and the second observation value to respectively obtain estimation values containing measurement errors
And estimated values without measurement errors
Step S24: according to the estimated value containing the measurement error
Substituting into a degradation model without considering measurement errors to obtain the pseudo life, wherein the specific formula is as follows:
wherein, TThe representation is based onThe estimated value containing the measurement error
Is the failure threshold of the device,
indicating that the estimated value contains a measurement error
Substituting the value obtained by the degradation model not considering the measurement error, X (t; theta) represents the degradation model not containing the measurement error in relation to time t.
Step S3: according to the real life T and the false life T of the equipment under the first arrival timeDetermining a feasible region of the measurement error model parameters, specifically comprising:
step S31: determining probability density distribution and pseudo life T corresponding to real life T estimated valueThe probability density distribution corresponding to the estimated value specifically includes:
step S311: a probability density distribution function of the device without measurement error is determined.
1) Determining a degradation process X (t) without considering observation errors based on a diffusion process, wherein the specific formula is as follows:
X(t)＝μt+σBB(t),t≥0(4)；
wherein X (t) represents the degradation process of the device at time t without considering observation errors, mu is a drift coefficient, and sigma isBCalled diffusion coefficient, B (t) is the standard Brownian motion.
2) Determining a probability density distribution function of the equipment under the condition of no measurement error, wherein the specific formula is as follows:
wherein f isT(t) represents the probability density distribution function of the device without measurement error, ξ represents the failure threshold, μ is the drift coefficient, σBReferred to as diffusion coefficient, t denotes the sampling time.
3) Determining a drift coefficient mu and a diffusion coefficient sigma according to the maximum likelihood estimation and the properties of the Wiener degradation process modelBThe specific formula of the estimated value of (1) is as follows:
wherein the content of the first and second substances,
and
respectively representing the drift coefficient mu and the diffusion coefficient sigmaBEstimated value of, Δ ti＝ti-ti-1Representing the sampling interval time, tiDenotes the ith sample time, k denotes the total number of samples, Δ xiRepresenting delta data of degradation, Δ xi＝xi-xi-1，xiRepresents the i-th observation without considering the observation error degradation process x (t).
If the data size samples are large enough, estimate
And
will progressively converge to mu and sigmaBThe true value of (d). In addition, since in actual engineering, the data is generally acquired by adopting an equal time interval sampling method, there is Δ tiΔ t. In practice, the actual degradation data x takes into account the presence of measurement errorskIt is often difficult to obtain, by measuring with a sensor, monitoring data with measurement errors, i.e. yk＝xk+k. From the previous analysis, the present invention considers the case of time-independent measurement error models and time-dependent measurement errors, respectively.
Step S312: and determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time.
1) Assuming that a measurement error model (t) of each measurement is a normal random variable-N (mu) which is independent and identically distributed,σ2)，μ,σRespectively, the mean and standard deviation of the measurement error model, then Δ y can be obtainedi＝yi-yi-1＝xi-xi-1+i-i-1，ΔyiIncrement representing monitoring degradation data, yiRepresenting degradation data with measurement error, x, obtained from the i-th testiRepresenting the degraded data without measurement error obtained by the i-th test,iindicating the measurement error of the i-th test. In practical engineering, calibration is often carried out to eliminate the system constant value error, namely mu0. In this way, the delta Δ y of the monitored degradation data can be obtainediCompliance parameter of
Is normally distributed. The estimated values of the drift coefficient and the diffusion coefficient under the condition that the measurement error model is independent of time are as follows:
wherein the content of the first and second substances,
and
respectively representing the estimated values of the drift coefficient and the diffusion coefficient under the condition that the measurement error model is independent of time, delta t represents the sampling interval time, k represents the total times of sampling, and delta yiRepresenting the increments of monitored degradation data.
2) according to the estimated values of the drift coefficient and the diffusion coefficient under the condition that the measurement error model is not related to time, and normal random variable-N (mu),σ2) And monitoring the delta deltay of the degradation datai＝yi-yi-1＝xi-xi-1+i-i-1The method comprises the following steps of determining a probability density distribution function corresponding to a pseudo life estimation value under the condition that a measurement error model is not related to time by utilizing normal distribution properties, wherein the specific formula is as follows:
wherein f isT′(t) represents a probability density distribution function corresponding to the pseudo-life estimation value under the condition that the measurement error model is not related to time, ξ represents a failure threshold value of the equipment, mu is a drift coefficient, and σ representsBReferred to as diffusion coefficient, Δ t represents the sampling interval time, and the measurement error (t) follows a normal distribution, μMeans, σ, representing the measurement error model (t)The standard deviation of the measurement error model (t) is shown, and t represents the sampling time.
Step S313: and determining a probability density distribution function corresponding to the pseudo-life estimation value under the condition that the measurement error model is related to time.
1) Considering that the measurement error model (t) has a linear variation trend, namely the sensor error model changes into a linear Gaussian process:
wherein, muti+0A systematic error is indicated that is time-dependent,0which is indicative of the error in the initial measurement,
representing random errors and obeying a normal distribution N (0, σ)2). Parameter estimation
And
progressive convergence to μ + μAnd
2) determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time, wherein the specific formula is as follows:
wherein, f ″)T(t) represents a probability density distribution function corresponding to the pseudo-life estimation value under the condition that the measurement error model is related to time, ξ represents a failure threshold value of the equipment, mu represents a drift coefficient, and sigma represents a drift coefficientBRepresents the diffusion coefficient, Δ t represents the sampling interval time, and the measurement error model (t) follows a normal distribution, μMeans, σ, representing the measurement error (t)The standard deviation of the measurement error model (t) is shown.
Step S32: determining KL distance formula.
From previous analysis, the life estimation result is a random variable rather than a constant value. Then the traditional mahalanobis distance cannot measure the real life T and the pseudo life TThe deviation therebetween. Relative entropy, also called divergence or distance (KL for short), is an effective method for measuring the distance between two random variables, and is basically defined as that if P and Q represent two random variables respectively, the probability density distribution function is fP(z) and fQ(z), then the relative entropy of P and Q, D (P | | Q), should be expressed as:
d (P | | Q) has nonnegativity and does not meet symmetry, i.e., D (P | | Q) ≠ D (Q | | P). To satisfy symmetry, therefore, the KL distance formula is defined specifically as:
that is, the true lifetime T and the pseudo lifetime T are both true and only trueWhen the probability distribution density functions of (2) are completely consistent, DKL(TI | T) | 0. Given an acceptable maximum deviation of DmaxThen there is DKL(T||T)≤Dmax. In other words, if the real life T and the pseudo life TA distance D betweenKL(TI T) meets the requirement, on the one hand, T is statedApproaching the true life T, on the other hand, illustrates that the effect of sensor measurement errors on the life estimation is acceptable.
Next, T and T will be quantified using the KL distanceIn combination with DKL(T||T)≤DmaxTo analyze the feasible domain of sensor measurement error model parameters.
Step S33: substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine the measurement error model parameters of the sensor, wherein the method specifically comprises the following steps:
step S331: substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time into the KL distance formula to determine the feasible region of the measurement error model parameter of the sensor under the condition that the measurement error model is not related to time, wherein the specific formula is as follows:
wherein σIndicating measurement errorMean value of the model, DmaxRepresenting the maximum acceptable lifetime estimation deviation, σBReferred to as the diffusion coefficient, at represents the sampling interval time.
Specifically, the method comprises the following steps: if the initial amount of degradation x0Is 0 and the maximum acceptable life estimate bias is DmaxMean value σ of the measurement error modelThe following conditions must be satisfied:
further deducing the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is not related to time according to the formula (13) as follows:
equations (13) and (14) require attention to the following three points: first, if the maximum estimated deviation D of the life is acceptablemaxMean value σ of given, measured error modelThe requirements deduced 2.1 need to be met; secondly, it is noted that the formula (13) and the formula (14) also include xi and sigmaBAnd Δ t, these parameters need to be known in advance or can be identified from historical monitoring data; third, the lifetime estimation deviation DKL(T| T) is only given by the parameter σDecided upon, and with it there is a monotonically increasing relationship.
Step S34: substituting the probability density distribution corresponding to the real life T estimated value and the probability density distribution function corresponding to the pseudo life estimated value under the condition that the measurement error model is related to time into the KL distance formula to determine the feasible region under the condition that the measurement error model is related to time, wherein the specific formula is as follows:
wherein σMean, μ, representing the measurement error modelMean value, D, representing a model of measurement errormaxRepresenting the maximum acceptable lifetime estimation deviation, σBReferred to as the diffusion coefficient, at represents the sampling interval time, μ represents the drift coefficient, and ξ represents the failure threshold of the device.
First consider a special case, if-muMu is more than or equal to mu, namely the data measured by the sensor can not reflect the real degradation condition at all. In this case, a calculated pseudo-life estimate T resultsInfinity, and the resulting safety risk can be severely underestimated. In view of this, the present invention gives the following assumptions.
Assuming that the monitored degradation data can reflect the degradation trend of the device, i.e., -muMu, i.e. if the device is degraded, it can be reflected in the monitoring data measured by the sensor.
If the initial amount of degradation x0Is 0 and the maximum acceptable measurement deviation is DmaxMeasuring mean value μ of error modelAnd standard deviation σThe following conditions must be satisfied:
wherein-muMu should also be satisfied.
Thus, once DmaxGiven, muAnd σThe allowable range is obtained. Like equation (13), the degradation process parameters μ and σBIt needs to be known in advance or can be estimated from historical monitoring data. In addition, it should be noted that in the expression of equation (15), the sensor error range is a two-dimensional interval.
Further, consider several special cases:
first, if the initial amount of degradation x0Is 0 and mu0 and the maximum acceptable measurement deviation is DmaxThen equation (15) can be converted to equation (13).
Second, if the initial amount of degradation x0Is 0 and σ0 and the maximum acceptable measurement deviation is DmaxThen measure the error model mean μThe following conditions are satisfied:
μ∈([D2,D3]∪(-∞,D1])∩(-μ,+∞) (16)；
wherein D is1＜D2＜D3Expression equation
The three real roots of (2) can be obtained by using a root-solving formula according to the characteristics of the unitary cubic equation.
Third, σIt is known to find μThe range of (1). Then muThe following conditions should be satisfied:
wherein the content of the first and second substances,
three real roots of the following equation.
Measurement error model parameters for maintenance decision analysis:
replacement and repair, as a widely adopted maintenance method, is often used as a benchmark problem for comparison and discussion of models and methods. The invention discusses the influence of a measurement error model on maintenance decision based on a theoretical framework of replacement and maintenance. Generally, the replacement and repair method is mainly based on the expected cost rate, and if the influence of the measurement error is not considered, the expected cost rate is expressed as follows:
where CR (τ) represents the expected rate of charges without taking into account the effect of measurement errors, τ represents the maintenance time, C represents all the charges,e (C) stands for expectation, E (T)mξ) represents the average lifetime of the device, cpRepresenting the cost of preventive replacement maintenance, cfIndicates a maintenance cost of failure (and satisfies c)f＞cp)，FT(t) denotes the life CDF, and reflects the probability of failure occurring until time t. For better representation, let
Indicating the condition reliability. According to the nature of the Wiener process, FT(t) has the form:
where Φ (·) represents a standard normally distributed CDF.
Similarly, if the effect of measurement error is taken into account, the CDF of lifetime will be affected, defined
Representing a pseudo-life estimate TThe CDF of (1). The expected cost function in this case is then:
wherein, CR(τ) represents the expected cost rate taking into account the effect of measurement errors,
representing a conditional reliability function taking into account the influence of measurement errors. In this case, the maintenance time obtained by optimizing the formula (21) necessarily has a deviation, and according to the formulas (19), (20) and (21), it can be concluded that,
theorem 2.3: if the initial device degenerates to Wiener degeneration and its measurement error is in the form of equation (7), then there is:
if σ0 and μIs > 0, then the optimum periodThe expected charge rate satisfies that CR (tau) is less than or equal to CR(τ) and the deviation CR (τ) -CR thereof(tau) is followed by muAnd increased by an increase.
If σ0 and- μ < μIf < 0, then the optimal expected cost rate satisfies CR (tau) ≦ CR(τ) and the deviation CR (τ) -CR thereof(tau) is followed by muDecrease and increase.
The invention also provides a sensor measurement error model parameter feasible region analysis system, which comprises:
and the real life determining module is used for determining the real life of the equipment at the first arrival time.
And the pseudo life determining module is used for determining the pseudo life of the equipment.
The feasible region determining module is used for determining a feasible region of the measurement error model parameters according to the real life and the pseudo life of the equipment under the first arrival time; the measurement error model parameters include a mean of the measurement error model and a standard deviation of the measurement error model.
As an optional implementation manner, the feasible region determining module of the present invention specifically includes:
and the probability density distribution determining unit is used for determining the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value.
And the KL distance formula determining unit is used for determining the KL distance formula.
And the sensor measurement error model parameter determining unit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine sensor measurement error model parameters.
As an optional implementation manner, the probability density distribution determining unit of the present invention specifically includes:
a first probability density distribution function determining subunit, configured to determine a probability density distribution function of the device without measurement error; the probability density distribution function of the equipment under the condition of no measurement error is the probability density distribution corresponding to the estimated value of the real service life.
And the second probability density distribution function determining subunit is used for determining the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time.
A third probability density distribution function determining subunit, configured to determine a probability density distribution function corresponding to the pseudo life estimation value under a condition that the measurement error model is time-dependent; the probability density distribution corresponding to the pseudo life estimation value comprises a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is irrelevant to time and a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is relevant to time.
As an optional implementation manner, the sensor measurement error model parameter determining unit specifically includes:
and the first feasible region determining subunit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is not related to time.
And the second feasible region determining subunit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is related to time.
The present invention uses the following numerical examples to illustrate the effect of measurement errors on life estimation. Where the parameters are given below, μ ═ 0.1, σ B1 and ξ 10. For better illustration, consider the following three special cases: 1) mu.s0, and σFrom 0 to 10; 2) sigma0, and μIncreases from- μ to 0.1; 3) mu.sAnd σIs increased.
As shown in fig. 2, if μ0 and σIncrease, expectation of lifetime estimation is constant, variance thereofWill follow sigmaAnd increases gradually, which also indicates that the uncertainty of the pseudo-life estimation result becomes large. As can be seen from fig. 2(b), the deviation between the true lifetime cumulative distribution function CDF and the pseudo lifetime cumulative distribution function CDF becomes positive first and then negative as the abscissa increases. As shown in fig. 3, if σ0 and muAn increase results in an increase in the pseudo-lifetime expectation and the variance. In particular, if> 0, then have
Otherwise, then
Given maximum acceptable life estimation deviations of 0.05, 0.1, and 0.2, respectively, then according to inference 2.2 and inference 2.3, the acceptable ranges for the corresponding measurement error model parameters are shown in table 1.
TABLE 1 given DmaxRequired measurement error model parameter feasible region
According to theorem 2.2, the measurement error model parameter mu can be obtainedAnd σThe allowable variation range is shown in fig. 4. (a) Drawing Dmax0.05 measurement error model parameter muAnd σAllowable variation range, (b) is shown as Dmax0.1 measurement error model parameter muAnd σAllowable variation range, (c) is shown as Dmax0.2 measurement error model parameter muAnd σAllowable variation range in which the measurement error model parameter variation range is satisfied, that is, the measurement error model parameter μ falling within the shadow rangeAnd σMeet a given life estimation bias requirement, i.e. DKL(T||T)≤DmaxOtherwise, it is not satisfied.
FIG. 5 (a) is DmaxWhen the value is 0.05, the pseudo life probability density PDF is compared with the true value, and (b) is DmaxWhen the value is 0.1, the probability density PDF of the pseudo life is compared with the true value, and the graph (c) is DmaxWhen the pseudo lifetime probability density PDF is 0.2, the real value is compared with the pseudo lifetime probability density PDF, and it can be seen that fig. 5 compares all the maximum acceptable error parameters, that is, the values of the parameters at the shaded edges of fig. 4. By comparison, it can be found that if DmaxThe smaller the pseudo lifetime probability density PDF is, the closer the true lifetime probability density PDF. This also illustrates a measure D based on the KL distanceKLThe deviation between the two random variables can be well reflected.
The effectiveness of the method provided by the invention is verified by taking an actual blast furnace as an example, and the method for analyzing the sensor error model parameters based on the KL distance can be used for a domain analysis. As shown in fig. 6, the degradation data was derived from blast furnace wall degradation data for up to 300 days or more.
The present invention performs the estimation to obtain the following result, mu + mu＝1.9879、σB6.4430 and σ4.0020. First, the present invention considers the simplest case, namely, μWhen the value is 0, then μ is 1.9879, and the estimated deviation of the lifetime is DKL0.1680. In this case, the lifetime estimation bias is only affected by σThe influence of (c). In practical engineering, the failure threshold of the blast furnace is generally set to 800 ℃, so that the method can obtain the equivalent muWhen different values are taken, the change situation of the life estimation deviation is shown in fig. 7. It can be seen that if and only if μThe lifetime estimation bias takes the minimum value of 0.
Deviation D if maximum acceptable life estimationmaxGiven, muThe allowable range of variation will be determined. For example, if DmaxWhen the average molecular weight is 0.2, then-0.0645. mu. or lessLess than or equal to 0.0659. FIG. 8 shows the pseudo-lifetime probability density PDF as a function of μIn which the solid bold line indicates the time when μ0 and σResults of 0, the black line with bold dashed line indicates when μ0 and σResults of 4.0020, the black thin dashed line indicates when μNot equal to 0 and σ4.0020 time pseudo lifetime probability density PDF. It can be seen that the peak of the pseudo-lifetime probability density PDF is dependent on μIs increased and is shifted to the right.
The invention only considers the case of alternative maintenance. According to the actual situation, the cost of replacement repair, i.e. blast furnace overhaul cost, usually exceeds 2 billion (yuan). And because the failure of the blast furnace often brings about safety accidents and causes huge personnel and property losses, the failure cost of the blast furnace is difficult to accurately quantify. In view of this, the invention uses the maintenance cost after the burning-through of the historic blast furnace as the failure cost, then there is cp＝0.2×109(element) and cp＝1×109(Yuan). If a simple case is considered, i.e. muThen, the expected rate of charge changes with the maintenance time as shown in fig. 9, and it can be seen from the figure that the measurement error affects the formulation of the optimal maintenance time.
In fact, muGenerally not equal to 0 and difficult to identify by monitoring only the resulting degradation data. In view of this, in Table 2, the present invention compares the different μMaintenance decisions in case. As can be seen from Table 2, with | μThe deviation between the real optimal maintenance time and the pseudo optimal maintenance time is larger and larger as | is increased. In particular, for D at presentmaxHas not yet formed a consistent conclusion. Thus, the present invention determines D by measuring the economic impact of errormax. As shown in FIG. 10, if a deviation from the optimal expected cost rate is desired
Not more than 0.3 × 105Yuan/day, then DmaxShould be less than 0.26. If given DmaxThe feasible field of measurement error is shown in fig. 11, 0.26.
It has been found that both the lifetime estimation and the maintenance strategy are affected by measurement errors, so it makes sense to select the appropriate sensors for measuring the degradation data. In addition, since the measurement error time-related parameter is often difficult to be identified only by the acquired degradation data, it is more necessary to select before the degradation device is used for operation to ensure the accuracy of the lifetime estimation.
TABLE 2 Effect of measurement errors of different parameters on maintenance strategy
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.
Claims (10)
1. A method for performing domain analysis on a sensor measurement error model parameter, the method comprising:
step S1: determining the real life of the equipment at the first arrival time;
step S2: determining a pseudo-lifetime of the device;
step S3: determining a feasible region of a measurement error model parameter according to the real life and the pseudo life of the equipment under the first arrival time; the measurement error model parameters include a mean of the measurement error model and a standard deviation of the measurement error model.
2. The method for analyzing the feasible region of the sensor measurement error model parameters according to claim 1, wherein the determining the feasible region of the measurement error model parameters according to the real life and the pseudo life of the device at the first arrival time specifically comprises:
step S31: determining probability density distribution corresponding to the real life estimation value and probability density distribution corresponding to the pseudo life estimation value;
step S32: determining a KL distance formula;
step S33: and substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine the measurement error model parameters of the sensor.
3. The method for performing domain analysis on the sensor measurement error model parameters according to claim 2, wherein determining the probability density distribution corresponding to the true lifetime estimation value and the probability density distribution corresponding to the pseudo lifetime estimation value specifically comprises:
step S311: determining a probability density distribution function of the equipment under the condition of no measurement error; the probability density distribution function of the equipment under the condition of no measurement error is the probability density distribution corresponding to the estimated value of the real life;
step S312: determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time;
step S313: determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time; the probability density distribution corresponding to the pseudo life estimation value comprises a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is irrelevant to time and a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is relevant to time.
4. The method according to claim 3, wherein the step of determining the sensor measurement error model parameters by substituting the probability density distribution corresponding to the true lifetime estimation value and the probability density distribution corresponding to the pseudo lifetime estimation value into the KL distance formula includes:
step S331: substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameter under the condition that the measurement error model is not related to time;
step S332: and substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time into the KL distance formula to determine the feasible region of the measurement error model parameter of the sensor under the condition that the measurement error model is related to time.
5. The method for analyzing the feasible region of the sensor measurement error model parameters according to claim 4, wherein the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is not related to time is determined by the following specific formula:
wherein σMean value, D, representing a model of measurement errormaxRepresenting the maximum acceptable lifetime estimation deviation, σBReferred to as the diffusion coefficient, at represents the sampling interval time.
6. The method for analyzing the feasible region of the sensor measurement error model parameters according to claim 4, wherein the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is related to time is determined by the following specific formula:
wherein σMean, μ, representing the measurement error modelMean value, D, representing a model of measurement errormaxIndicating a maximum acceptable life estimateDeviation, σBReferred to as the diffusion coefficient, at represents the sampling interval time, μ represents the drift coefficient, and ξ represents the failure threshold of the device.
7. A sensor measurement error model parameter feasible region analysis system, the system comprising:
the real life determining module is used for determining the real life of the equipment at the first arrival time;
a pseudo-lifetime determination module for determining a pseudo-lifetime of the device;
the feasible region determining module is used for determining a feasible region of the measurement error model parameters according to the real life and the pseudo life of the equipment under the first arrival time; the measurement error model parameters include a mean of the measurement error model and a standard deviation of the measurement error model.
8. The system of claim 7, wherein the feasible region determining module specifically comprises:
a probability density distribution determining unit for determining a probability density distribution corresponding to the real life estimation value and a probability density distribution corresponding to the pseudo life estimation value;
the KL distance formula determining unit is used for determining a KL distance formula;
and the sensor measurement error model parameter determining unit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution corresponding to the pseudo life estimation value into the KL distance formula to determine sensor measurement error model parameters.
9. The system of claim 7, wherein the probability density distribution determining unit specifically comprises:
a first probability density distribution function determining subunit, configured to determine a probability density distribution function of the device without measurement error; the probability density distribution function of the equipment under the condition of no measurement error is the probability density distribution corresponding to the estimated value of the real life;
the second probability density distribution function determining subunit is used for determining a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time;
a third probability density distribution function determining subunit, configured to determine a probability density distribution function corresponding to the pseudo life estimation value under a condition that the measurement error model is time-dependent; the probability density distribution corresponding to the pseudo life estimation value comprises a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is irrelevant to time and a probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is relevant to time.
10. The system of claim 8, wherein the sensor measurement error model parameter determining unit specifically comprises:
the first feasible region determining subunit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is not related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameter under the condition that the measurement error model is not related to time;
and the second feasible region determining subunit is used for substituting the probability density distribution corresponding to the real life estimation value and the probability density distribution function corresponding to the pseudo life estimation value under the condition that the measurement error model is related to time into the KL distance formula to determine the feasible region of the sensor measurement error model parameters under the condition that the measurement error model is related to time.
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