CN111966966B - Method and system for analyzing feasible domain of sensor measurement error model parameters - Google Patents

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

Method and system for analyzing feasible domain of sensor measurement error model parameters

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 feasible region of the sensor measurement error model parameter.

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 feasible region of 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 measurement error model parameter of the sensor 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 σ_εMean value, D, representing a model of measurement error_maxRepresenting the maximum acceptable lifetime estimation deviation, σ_BReferred 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 by a specific formula:

wherein σ_εMean, μ, representing the measurement error model_εMean value, D, representing a model of measurement error_maxRepresenting the maximum acceptable lifetime estimation deviation, σ_BKnown as the diffusion systemThe number, Δ t, 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 feasible region 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 the feasible region of the 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 feasible region 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 parameter 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.

Drawings

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 invention_εWhen 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 invention_εWhen 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 invention_maxDetermining 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 invention_maxComparing 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 invention_KLAssociated with mu_εA 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 D_KLA relationship graph;

FIG. 11 shows an embodiment D of the present invention_maxWhen 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)≥ξ|X(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)+ε(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 epsilon (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 errors_0:k，Y_0:k＝[y₀,y₁…,y_k]，y_k＝x_k+ε_k，y_kDegradation data, x, representing measurement errors in the k-th observation_kDegradation data, ε, indicating that the kth observation contains no 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 error_0:k，X_0:k＝[x₀,x₁…,x_k]。

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, T_εRepresenting estimated values based on said contained measurement errors

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 time_εDetermining 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 value_εThe 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+σ_BB(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 is_BCalled 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 is_T(t) represents the probability density distribution function of the device without measurement error, ξ represents the failure threshold, μ is the drift coefficient, σ_BReferred 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 model_BThe 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 sigma_BEstimated value of, Δ t_i＝t_i-t_i-1Representing the sampling interval time, t_iDenotes the ith sample time, k denotes the total number of samples, Δ x_iRepresenting delta data of degradation, Δ x_i＝x_i-x_i-1，x_iRepresenting a degradation process X (t) without taking into account the observed error) The ith observation of (1).

If the data size samples are large enough, estimate

And

will progressively converge to mu and sigma_BThe 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 t_iΔ t. In practice, the actual degradation data x takes into account the presence of measurement errors_kIt is often difficult to obtain, by measuring with a sensor, monitoring data with measurement errors, i.e. y_k＝x_k+ε_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 epsilon (t) of each measurement is a normal random variable epsilon-N (mu) which is independent and identically distributed_ε,σ_ε ²)，μ_ε,σ_εRespectively, the mean and standard deviation of the measurement error model, then Δ y can be obtained_i＝y_i-y_i-1＝x_i-x_i-1+ε_i-ε_i-1，Δy_iIncrement representing monitoring degradation data, y_iRepresenting degradation data with measurement error, x, obtained from the i-th test_iRepresenting the degradation data without measurement error, obtained at 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 mu _ε0. In this way, the delta Δ y of the monitored degradation data can be obtained_iCompliance parameter of

Is normally distributed. Drift coefficient and expansion under the condition of measuring error model and time independenceThe estimated values of the dispersion coefficients 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 y_iRepresenting the increments of monitored degradation data.

And

progressive convergence on mu and

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 the normal random variable epsilon-N (mu)_ε,σ_ε ²) And monitoring the delta deltay of the degradation data_i＝y_i-y_i-1＝x_i-x_i-1+ε_i-ε_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'_Tε(t) probability density distribution corresponding to pseudo-lifetime estimation value under the condition that measurement error model is independent of timeFunction, ξ denotes the failure threshold of the plant, μ is the drift coefficient, σ_BCalled diffusion coefficient, Δ t denotes the sampling interval time, the measurement error ε (t) follows a normal distribution, μ_εMeans, σ, representing the measurement error model ε (t)_εThe standard deviation of the measurement error model epsilon (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 epsilon (t) has a linear change trend, namely the sensor error model changes into a linear Gaussian process:

wherein, mu_εt_i+ε₀Representing a systematic error, epsilon, related to time₀Which is indicative of the error in the initial measurement,

representing random errors and obeying a normal distribution N (0, σ)_ε ²). Parameter estimation

And

progressive convergence to μ + μ_εAnd

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 ″)_Tε(t) watchShowing a probability density distribution function corresponding to a pseudo life estimation value under the condition that a measurement error model is related to time, showing a failure threshold value of equipment by xi, showing a drift coefficient by mu and showing a sigma_BRepresents the diffusion coefficient, Δ t represents the sampling interval time, and the measurement error model ε (t) follows a normal distribution, μ_εMeans, σ, representing the mean of the measurement error ε (t)_εThe standard deviation of the measurement error model epsilon (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 T_εThe 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 f_P(z) and f_Q(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 true_εWhen the probability distribution density functions of (2) are completely consistent, D_KL(T_εI | T) | 0. Given an acceptable maximum deviation of D_maxThen there is D_KL(T_ε||T)≤D_max. In other words, if the real life T and the pseudo life T_εA distance D between_KL(T_εI T) meets the requirement, on the one hand, T is stated_εApproaching the true lifetime T, on the other hand, illustrating the effect of sensor measurement errors on the lifetime estimateThe noise is acceptable.

Next, T and T will be quantified using the KL distance_εIn combination with D_KL(T_ε||T)≤D_maxTo 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 feasible region of the sensor measurement error model parameters, and specifically comprising 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 σ_εMean value, D, representing a model of measurement error_maxRepresenting the maximum acceptable lifetime estimation deviation, σ_BReferred 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 D_maxMean value σ of the measurement error model_εThe following conditions must be satisfied:

further deducing the feasible range 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:

the equations (13) and (14) requireThe following three points are noted: first, if the maximum estimated deviation D of the life is acceptable_maxMean value σ of given, measured error model_εThe requirements deduced 2.1 need to be met; secondly, it is noted that the formula (13) and the formula (14) also include xi and sigma_BAnd Δ t, these parameters need to be known in advance or can be identified from historical monitoring data; third, the lifetime estimation deviation D_KL(T_ε| T) is only given by the parameter σ_εDecided upon, and with it there is a monotonically increasing relationship.

Step S332: 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 σ_εMean, μ, representing the measurement error model_εMean value, D, representing a model of measurement error_maxRepresenting the maximum acceptable lifetime estimation deviation, σ_BReferred 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-mu_εMu 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 results_εInfinity, 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., -mu_εMu, 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 D_maxMeasuring mean value μ of error model_εAnd standard deviation σ_εMust satisfyThe following conditions were used:

wherein-mu_εMu should also be satisfied.

Thus, once D_maxGiven, mu_εAnd σ_εThe allowable range is obtained. Like equation (13), the degradation process parameters μ and σ_BIt 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 mu _ε0 and the maximum acceptable measurement deviation is D_maxThen equation (15) can be converted to equation (13).

Second, if the initial amount of degradation x₀Is 0 and σ _ε0 and the maximum acceptable measurement deviation is D_maxThen measure the error model mean μ_εThe following conditions are satisfied:

μ_ε∈([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, σ_εIt is known to find μ_εThe range of (1). Then mu_εThe 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 charge without taking into account the effect of measurement errors, τ represents maintenance time, C represents all costs, E (C) represents expectations, E (T)_mξ) represents the average lifetime of the device, c_pRepresenting the cost of preventive replacement maintenance, c_fIndicates a maintenance cost of failure (and satisfies c)_f＞c_p)，F_T(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, F_T(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 affectedTo influence, define F_Tε(T) pseudo life estimation value T_εThe CDF of (1). The expected cost function in this case is then:

wherein, CR_ε(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 σ _ε0 and μ_εIs more than or equal to 0, then the optimal expected cost rate satisfies CR (tau) less than or equal to CR_ε(τ) and the deviation CR (τ) -CR thereof_ε(tau) is followed by mu_εAnd increased by an increase.

If σ _ε0 and- μ < μ_εIf < 0, then the optimal expected cost rate satisfies CR (tau) ≦ CR_ε(τ) and the deviation CR (τ) -CR thereof_ε(tau) is followed by mu_εDecrease 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 feasible region 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 the feasible region of the 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 feasible region determining unit of the present invention 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 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 parameter 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.s _ε0, and σ_εFrom 0 to 10; 2) sigma _ε0, and μ_εIncreases from- μ to 0.1; 3) mu.s_εAnd σ_εIs increased.

As shown in (a) of FIG. 2, if μ _ε0 and σ_εIncreasing, the life estimate expectation is constant, with variance as a function of σ_εAnd increases gradually, which also indicates that the uncertainty of the pseudo-life estimation result becomes large. As can be seen from (b) in fig. 2, 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 σ _ε0 and mu_εAn increase results in an increase in the pseudo-lifetime expectation and the variance. In particular, if_ε> 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 D_maxRequired measurement error model parameter feasible region

According to theorem 2.2, the measurement error model parameter mu can be obtained_εAnd σ_εThe allowable variation range is shown in fig. 4. (a) Drawing D_max0.05 measurement error model parameter mu_εAnd σ_εAllowable variation range, (b) is shown as D_max0.1 measurement error model parameter mu_εAnd σ_εAllowable variation range, (c) is shown as D_max0.2 measurement error model parameter mu_εAnd σ_εAllowable 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 range_εAnd σ_εMeet a given life estimation bias requirement, i.e. D_KL(T_ε||T)≤D_maxOtherwise, it is not satisfied.

FIG. 5 (a) is D_maxWhen the value is 0.05, the pseudo life probability density PDF is compared with the true value, and (b) is D_maxWhen the value is 0.1, the probability density PDF of the pseudo life is compared with the true value, and the graph (c) is D_maxWhen 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 D_maxThe 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 distance_KLThe 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_ε＝1.9879、σ_B6.4430 and σ_ε4.0020. First, the present invention considers the simplest case, namely, μ_εWhen the value is 0, then μ is 1.9879, and the estimated deviation of the lifetime is D_KL0.1680. In this case, the lifetime estimation bias is only affected by σ_εThe 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 mu_εWhen 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 μ_εThe lifetime estimation bias takes the minimum value of 0.

Deviation D if maximum acceptable life estimation_maxGiven, mu_εThe allowable range of variation will be determined. For example, if D_maxWhen the average molecular weight is 0.2, then-0.0645. mu. or less_εLess than or equal to 0.0659. FIG. 8 shows the pseudo-lifetime probability density PDF as a function of μ_εIn which the solid bold line indicates the time when μ _ε0 and σ_εResults of 0, the black line with bold dashed line indicates when μ _ε0 and σ_εResults of 4.0020, the black thin dashed line indicates when μ_εNot equal to 0 and σ_ε4.0020 time pseudo lifetime probability density PDF. It can be seen that the peak of the pseudo-lifetime probability density PDF is dependent on μ_εIs 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 c_p＝0.2×10⁹(element) and c_p＝1×10⁹(Yuan). If a simple case is considered, i.e. mu_εThen, 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, mu_εGenerally 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 μ_εMaintenance decisions in case. Can send out through the table 2Now, with | μ_εThe 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 present_maxHas not yet formed a consistent conclusion. Thus, the present invention determines D by measuring the economic impact of error_max. As shown in FIG. 10, if a deviation from the optimal expected cost rate is desired

Not more than 0.3 × 10⁵Yuan/day, then D_maxShould be less than 0.26. If given D_maxThe 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 (8)

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 value of the measurement error model and a standard deviation of the measurement error model, and specifically include:

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 feasible region of the sensor measurement error model parameter.

2. The method for performing domain analysis on the sensor measurement error model parameters according to claim 1, 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.

3. The method according to claim 2, wherein the step of determining the feasible region of the sensor measurement error model parameter 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 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 measurement error model parameter of the sensor 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.

4. The method for analyzing the feasible region of the sensor measurement error model parameters according to claim 3, 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 σ_εMean value, D, representing a model of measurement error_maxRepresenting the maximum acceptable lifetime estimation deviation, σ_BReferred to as the diffusion coefficient, at represents the sampling interval time.

5. The method for analyzing the feasible region of the sensor measurement error model parameters according to claim 3, 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 σ_εMean, μ, representing the measurement error model_εMean value, D, representing a model of measurement error_maxRepresenting the maximum acceptable lifetime estimation deviation, σ_BReferred 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.

6. 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 comprise a mean value of the measurement error model and a standard deviation of the measurement error model;

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 feasible region 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 the feasible region of the sensor measurement error model parameters.

7. The system of claim 6, 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.

8. The system according to claim 6, wherein the feasible region 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 parameter under the condition that the measurement error model is related to time.

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