CN113743670B - GRU model-based circuit fault real-time prediction method and verification circuit - Google Patents

GRU model-based circuit fault real-time prediction method and verification circuit Download PDF

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CN113743670B
CN113743670B CN202111047941.7A CN202111047941A CN113743670B CN 113743670 B CN113743670 B CN 113743670B CN 202111047941 A CN202111047941 A CN 202111047941A CN 113743670 B CN113743670 B CN 113743670B
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闫理跃
王厚军
刘震
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Abstract

The invention provides a GRU model-based circuit fault real-time prediction method and a GRU model-based circuit fault verification circuit, which combine deep learning and interval estimation together, and simultaneously utilize the advantages of the two methods to obtain more accurate probability distribution of reliability prediction. The cyclic neural network based GRU model has the property of retaining historical information of time series data, which is used herein to estimate initial predictions and to assist in calculating parameters of the initial reliability probability distribution. And then updating time-varying parameters through on-site operation data by using a zone estimation algorithm based on Bayesian estimation, and continuously updating the reliability probability distribution of the potential tested object in real time. In the experimental stage, the experimental result of the output frequency domain signal deviation of the elliptical filter circuit is adopted to prove that the method can effectively utilize real-time data, continuously correct the prediction precision, update and optimize the time-varying parameters of the reliability performance, and further predict the reliability probability distribution condition of the circuit in real time under the condition of considering uncertainty.

Description

GRU model-based circuit fault real-time prediction method and verification circuit
Technical Field
The invention relates to the technical field of engineering reliability analysis, in particular to a GRU model-based circuit fault real-time prediction method and a verification circuit.
Background
Unlike popular deep learning application scenarios such as autopilot, computer vision, etc., engineering reliability analysis typically encounters many external constraints: small sample size, less experiment times, etc. Therefore, many conventional machine learning based state prediction methods have limited utility in field applications for engineering reliability analysis. These conventional models typically not only require a large amount of historical data to support their computation, but their prediction accuracy can also be affected by the inherent uncertainties of data acquisition and model design. For example, in previous studies, both simplified failure modes and indirect measurement measures have resulted in the propagation of uncertainties throughout the prediction process, which may further affect the accuracy of the prediction.
Disclosure of Invention
Aiming at the problems in the prior art, the application provides a GRU model-based circuit fault real-time prediction method and verification circuit considering uncertainty, and the reliability probability distribution condition of the circuit can be predicted in real time.
The invention discloses a GRU model-based circuit fault real-time prediction method, which comprises the following steps:
s1: selecting a performance variable having a time-varying parameter;
s2: acquiring and storing prior density functions of corresponding time-varying parameters;
s3: combining the field data and Bayesian estimation to obtain a corresponding posterior density function;
s4: and evaluating the real-time performance of the tested object.
Further, step S3 further includes updating the field data in real time.
Further, the performance variable with the time-varying parameter in step S1 is a specific reliability failure index.
Further, step S2 further includes: and estimating probability distribution of the preliminary performance failure variable by using a gating circulating unit neural network.
Further, step S2 further includes step S21: it is assumed that the failure distribution of the electronic product belongs to a normal distribution.
Further, step S2 further includes step S22: state value of failure index FI at any momentExpressed as f (x) to N (mu, sigma) 2 ) The relevant probability density function for the state at time t can be expressed as:
Figure BDA0003251617660000021
where x represents the time scale of product degradation (typically equally spaced test time nodes), μ and σ 2 The mean value and the variance of the variable distribution respectively representing the failure indexes of the product;
taking the average value of a history sample group at a certain moment as a statistical object, taking the average value and the variance of the average value as the prior average value and the prior variance of the average value mu, and obtaining normal distribution N (mu) 0 ,σ 0 2 ) A priori distribution of μ:
Figure BDA0003251617660000022
/>
μ 0 sum sigma 2 Representing the mean and variance of the set of historical samples at a time instant, respectively.
Further, in step S22, mu is calculated using the mean value of each group of samples as a statistical object 0 Sum sigma 0 2 When the first data is used as a first sample, the average value of the first two data is used as a second sample, the average value of the first three data is used as a third sample, and the average value of the first N data is used as an N-th sample, so that N samples are generated, and the number of the data of the N-th sample is N.
Further, in step S3, the distribution density function of the FI parameter variable under the given μ condition is:
Figure BDA0003251617660000023
Figure BDA0003251617660000024
site sample y t Posterior probability density function under update:
Figure BDA0003251617660000025
the posterior probability density function of the FI parameter variable at any moment is:
f(x|y t )＝∫f(x|μ)f(μ|y t )dμ
further, in step S4, the real-time reliability model is calculated as:
Figure BDA0003251617660000026
μ c and
Figure BDA0003251617660000027
and respectively recording the current distribution parameters containing posterior information obtained by combining the historical data samples with the field data.
On the other hand, the application also relates to a verification circuit of a circuit fault real-time prediction method based on the GRU model, which comprises 15 resistors, 7 capacitors and 3 operational amplifiers, wherein:
the first end of the resistor R1 is connected with the input voltage, and the second end of the resistor R1 is connected with the inverting input end of the first operational amplifier;
the first end of the resistor R2 is connected with the second end of the resistor R1, and the second end of the resistor R2 is connected with the output end of the first operational amplifier;
the first end of the resistor R3 is connected with the output end of the first operational amplifier, and the second end of the resistor R3 is connected with the first end of the resistor R7;
the first end of the resistor R4 is connected with the output end of the first operational amplifier, and the second end of the resistor R4 is connected with the first end of the resistor R5;
the second end of the resistor R5 is grounded;
the first end of the resistor R6 is connected with the second end of the capacitor C3, and the second end of the resistor R6 is connected with the second end of the capacitor C2;
the second end of the resistor R7 is connected with the inverting input end of the second operational amplifier;
the first end of the resistor R8 is connected with the non-inverting input end of the second operational amplifier and the output end of the second operational amplifier, and the second end of the resistor R8 is connected with the first end of the resistor R9;
the second end of the resistor R9 is connected with the first end of the resistor R10;
the second end of the resistor R10 is grounded;
the first end of the resistor R11 is connected with the output end of the second operational amplifier, and the second end of the resistor R11 is connected with the first end of the resistor R12;
the second end of the resistor R12 is connected with the inverting input end of the third operational amplifier;
the first end of the resistor R13 is connected with the second end of the capacitor C5, and the second end of the resistor R13 is connected with the second end of the capacitor C6;
the first end of the resistor R14 is connected with the non-inverting input end of the third operational amplifier and the output end of the third operational amplifier, and the second end of the resistor R14 is connected with the first end of the resistor R13 and the first end of the resistor R15;
the second end of the resistor R15 is grounded;
the first end of the capacitor C1 is connected with the second end of the resistor R1, and the second end of the capacitor C1 is connected with the output end of the first operational amplifier;
the first end of the capacitor C2 is connected with the second end of the resistor R4, and the second end of the capacitor C2 is connected with the first end of the capacitor C4;
the first end of the capacitor C3 is connected with the second end of the resistor R3, and the second end of the capacitor C3 is connected with the first end of the resistor R6;
the first end of the capacitor C4 is connected with the second end of the capacitor C2, and the second end of the capacitor C4 is connected with the inverting input end of the second operational amplifier;
the first end of the capacitor C5 is connected with the second end of the resistor R11, and the second end of the capacitor C5 is connected with the first end of the resistor R13;
the first end of the capacitor C6 is connected with the second end of the resistor R9, and the second end of the capacitor C6 is connected with the first end of the capacitor C7;
the second end of the capacitor C7 is connected with the inverting input end of the third operational amplifier;
the non-inverting input end of the first operational amplifier is grounded;
the output end of the third operational amplifier is connected with the output voltage.
The above-described features may be combined in various suitable ways or replaced by equivalent features as long as the object of the present invention can be achieved.
Compared with the prior art, the method for predicting the circuit faults based on the GRU model in real time and the verification circuit provided by the invention have the following beneficial effects: it can simultaneously utilize the advantages of the two methods to obtain more accurate reliability probability prediction distribution. In the method, the GRU model utilizes the deviation of the frequency domain waveform of the circuit output signal to estimate the initial (pseudo) prediction result and the parameters of the initial reliability probability distribution, wherein the time-varying parameters can be continuously updated through Bayesian estimation and on-site operation data, so that the probability distribution condition of the circuit reliability can be continuously updated in real time. Simulation results show that the method can effectively utilize real-time data, continuously update and optimize the reliability performance time-varying parameters, and continuously correct the prediction precision.
This integration method provides a series of values with known probabilities of capturing the overall parameters to achieve uncertainty quantization. This probabilistic representation of the predicted outcome will provide statistical basis for the relevant studies of PHM. Unlike other applications of deep learning, such as intelligent driving, computer vision, etc., in actual equipment reliability maintenance, it is often the case that the sample size is small and the number of experiments is small, the methods presented herein can provide as much as possible a range of predictive results under such severe conditions, providing more intuitive understanding and assistance for the following reliability maintenance decisions.
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The invention will be described in more detail hereinafter on the basis of embodiments and with reference to the accompanying drawings. Wherein:
FIG. 1 shows a conceptual diagram of probability of failure;
FIG. 2 is a schematic diagram of a real-time performance evaluation method framework of the present invention;
FIG. 3 shows a schematic diagram of a GRU algorithm model;
FIG. 4 shows a schematic view of an update gate in a GRU model;
FIG. 5 shows a schematic diagram of a reset gate in a GRU model;
FIG. 6 shows a schematic diagram of the current memory content selection calculation process in the GRU model;
FIG. 7 shows a schematic diagram of the final memory content calculation process in the GRU model;
FIG. 8 shows an experimental verification circuit diagram;
FIG. 9 shows an initial and fault waveform display;
FIG. 10 shows a circuit performance degradation trend;
FIG. 11 is a graph showing the results of state evaluation of different algorithms at different times;
fig. 12 shows a real-time reliability evaluation result graph.
Detailed Description
The invention will be further described with reference to the accompanying drawings.
The invention provides a circuit fault real-time prediction method and a verification circuit based on a GRU model, which take uncertainty into consideration.
In one embodiment, the reliability R (t) of the measured object may be defined as a probability form: p (T > T) =P (T > T) (1)
If the failure probability of the product is subjected to normal distribution, the calculation formula of the reliability R (t) is as follows:
Figure BDA0003251617660000051
wherein F (t) is a normal distribution function. Φ (t) is a standard positive-ethernet distribution function. In practice, the real-time reliability of the product can be obtained by the field data, and then the mean mu and the variance sigma of the corresponding distribution function can be obtained by the variable distribution of the initial failure index 2 。
Therefore, probability of failure P f Can be defined as:
Figure BDA0003251617660000052
in this context, as shown in fig. 1, we still assume that the system has only 1 failure mode, defined as y t 。y Th Representing the most likely threshold of failure, f t (y t ) Is the potential probability distribution of the reliability performance of the measured object at the moment t.
In fig. 1, the object to be measured may fail or fail at any time t according to the probability distribution of the failure index. In field applications, however, individual products typically do not follow a common model due to dynamic environmental conditions and work operations. Thus, the reliability of an individual should be defined as the conditional probability based on continuously updating data over time: r (t) f |t c )，t f Is the time of failure, t c Is the current time index.
The integration method proposed herein can utilize both deep learning and parameter estimation, which relates reliability probability distributions to historical data and updates in real-time using online data. Thus, the integrated model may reflect the actual behavior of the individual units in terms of their environment and application and provide a more accurate prediction.
Bayesian methods provide a way to calculate hypothesis probabilities. It applies the a priori probabilities of the hypotheses, the probabilities of the observed data, and the observed data itself. Bayesian methods use field data and combine them. In the method presented herein, the specific steps are as follows: first, selecting a performance variable (generally referred to as a specific reliability failure index) with a time-varying parameter; then, acquiring and storing a priori density function of the corresponding time-varying parameter; and then, combining the field data and Bayesian estimation to obtain a corresponding posterior density function. A specific algorithm framework is shown in fig. 2.
In general, assuming that failure distribution of electronic products is normal, the state values of the failure index FI at any time are expressed as f (x) to N (μ, σ) according to fig. 1 2 ) The associated probability density function for the state at time t can be expressed as:
Figure BDA0003251617660000061
mu and sigma 2 The mean and variance of the variable distribution representing the product failure index, respectively. This stage requires two key preconditions: first, a more accurate preliminary prediction model will yield a more reasonable FI initial state probability distribution. Second, for a stable product and system, to facilitate the subsequent parameter estimation and posterior probability density calculation by the Z-test method, it is necessary to assume that μ is unknown, σ 2 Are known.
Taking the average value of a historical sample group at a certain moment as a statistical object, taking the average value and the variance as the prior average value and the prior variance of the average value mu, namely taking the normal distribution N to mu 0 ,σ 0 2 ) A priori distribution of μ:
Figure BDA0003251617660000062
in reality, μ tends to stabilize as the number of field data increases, i.e., μ 0 Sum sigma 0 2 Will get closer and closer to the true value. Two approaches to generating samples in general: one is small grouping sample capacity and large grouping quantity, namely historical data are uniformly divided into more groups as much as possible, and the data in each sample are less; the other is that the grouping sample capacity is large, the grouping number is small, i.e. the historical data is divided into less groups, and more data is in each sample. Mu is calculated by taking the mean value of each group of samples as a statistical object 0 Sum sigma 0 2 。
Both of these methods have certain problems. For the first method, since the number of data in the sample is small, there are μ and σ which are scattered even after a large amount of field data is obtained 0 2 Approaching 0, a risk of a significant erroneous result. For the second method, although the amount of data in the sample is large, there is no risk as aboveBut because of the small number of samples, a statistically significant parameter μ is calculated 01 Sum sigma 0 2 In some cases, the accuracy is not high, and the real situation cannot be reflected.
In order to solve the problems of the two methods, a sliding window sample segmentation method is provided. Taking the first data as a first sample, taking the average value of the first two data as a second sample, taking the average value of the first three data as a third sample, …, taking the average value of the first N data as an N-th sample, and generating N samples, wherein the data quantity of each sample is 1,2,3, … and N respectively. The method for generating the samples can ensure the number of the samples and the number of data in the samples to a certain extent, so that the problems and risks of the two methods are avoided, and a more optimized sample generation method is realized.
The GRU (gate loop unit) can be considered a variant of LSTM, as both are designed similarly and in some cases produce equally superior results. GRU is suitable, two door structure, reset gate r t Adjusting the combination of the new input and the previous memory, updating gate z t Controlling the preservation of the previous memories. These two vectors will determine what information should be passed to the output. The GRU can be trained to retain long-before information without the need to time wash or delete information that is not relevant to the prediction. Therefore, in recent years, GRUs are very popular in predicting sequence data.
As shown in the diagram of FIG. 3, σ represents the sigmoid function, and by Hadamard (element-wise) product, i.e., the corresponding element product between matrices, tan h represents the hyperbolic tangent function.
The transfer function in the GRU hidden unit can be divided into the following steps:
at time t, the update gate z first needs to be calculated using the following formula t ：
z t ＝σ(W z x t +V z h t-1 +b z ) (6)
Wherein x is t Is the input vector at time t, i.e. the t-th component of the input sequence, which passes through aLinear transformations (and weight matrix moments W z Multiplication). h is a t-1 The information at the last time t-1 is stored, and the information is subjected to linear transformation. The update gate adds the two pieces of information and drops them into the Sigmoid activation function, thus compressing the activation result to between 0 and 1. Fig. 4 is a schematic diagram and position of the update door throughout the unit.
The update gate helps the model decide how much past information to pass to the future, or how much information to pass on to the previous and current times. The model can thus decide on the information needed to replicate from the past to reduce the risk of gradient extinction.
The reset gate mainly determines how much past information needs to be forgotten, and can be calculated using the following expression:
r t ＝σ(W r x t +V r h t-1 +b r ) (7)
this expression is consistent with the expression of the update gate, but the parameters and usage of the linear transformation are not the same. FIG. 5 shows a schematic diagram of this expression in the GRU model.
H as described in the update door above t-1 And x t Firstly, linear transformation is carried out, and then, a Sigmoid activation function is added to output an activation value.
In the use of a reset gate, new memory contents will use the reset gate to hold information related to the past, and its calculation expression is:
Figure BDA0003251617660000081
input information x t Information h from the last time t-1 First, the current weight matrix W is multiplied right by a linear transformation c And V c 。
Calculating reset gate r t And h t-1 Hadamard product of (a), i.e. r t And h t-1 Corresponding element products between the two matrices of (a). Because the reset gate calculated previously is a reset gate from 0 to1, which automatically measures the magnitude of the gate opening. For example, if the gating value corresponding to an element is 0, then the information representing the element is completely forgotten. Thus, the Hadamard product will determine the past information to be retained and forgotten.
And adding the calculation results of the two parts, and putting the added calculation results into the hyperbolic tangent activation function. A schematic of this calculation process can be represented as shown in fig. 6.
In the last step, the GRU network needs to calculate h t This vector will retain the information of the current cell and pass it on to the next cell. In this process we need to use the update gate z t It decides the current memory content
Figure BDA0003251617660000082
And the previous time h t-1 What information needs to be collected. This process can be expressed as:
Figure BDA0003251617660000083
wherein z is t To update the activation result of the gate, it also controls the inflow of information in a gated fashion. z t And h t-1 The Hadamard product of (c) represents the information that was retained to the final memory for the previous time step, and the information that was retained to the final memory for the current memory plus the information that was retained to the final memory is equal to the content output by the final gating loop unit.
The above process may be illustrated in fig. 7.
To sum up, in the GRU network model, at time t, when x t When placed in a network element, it will be multiplied by its own weight W. Likewise, h t-1 The state information of the previous time t is saved and is multiplied by its own weight V as well. The two results are added together and a sigmoid activation function is applied to compress the results. Then a nonlinear activation function tanh is applied to obtain the current memory content
Figure BDA0003251617660000084
FinallyUpdating the current hidden output h t . In this way, information is stored and filtered by the update gate and reset gate. The gated loop unit does not clear previous information over time, it retains relevant information and passes it on to the next unit, so it uses all the information to avoid the gradient vanishing problem.
Therefore, no matter how long the time series data is, the GRU model can keep the information of the initial moment, and can well complete the task of acquiring the initial variable probability distribution parameters. Then, the GRU model can continuously update the field data to obtain the probability distribution of real-time update.
For the real-time reliability R (t) of the field product at each moment, the field sample state y at the moment t can be utilized t Updating the probability density function of the FI variable at the moment to obtain a posterior probability density function f (x|y) of the FI parameter variable at any moment t ) The individual characteristics of the field product are reflected by fusing the field information.
The distribution density function of the FI parameter variables at a given μ is:
Figure BDA0003251617660000091
applying Bayes formula, on-site sample y can be obtained t The posterior probability density function under update is:
Figure BDA0003251617660000092
wherein, according to the conjugation characteristic of normal distribution, can obtain
Figure BDA0003251617660000093
μ α And->
Figure BDA0003251617660000094
Mean and variance in the posterior density function of μ, respectively.
Then it is possible to obtain:
Figure BDA0003251617660000095
at this time, f (x|y) t ) The following steps are:
f(x|y t )＝∫f(x|μ)f(μ|y t )dμ (13)
from this, f (x|y) t ) Is a mean value of mu α Variance is
Figure BDA0003251617660000096
Is a normal distribution of (c). Simplified expression as
Figure BDA0003251617660000097
Wherein mu c ＝μ α ,/>
Figure BDA0003251617660000098
μ c And->
Figure BDA0003251617660000099
And respectively recording the current distribution parameters containing posterior information obtained by combining the historical data samples with the field data.
Parameter mu c And
Figure BDA00032516176600000910
by updating the field data, instead of equation->
Figure BDA00032516176600000911
Mu and sigma of (a) in (b) so that +.>
Figure BDA00032516176600000912
This model avoids the additional uncertainty problem that arises in the fitting formula stage.
As shown in fig. 8 and 9, the blue curve represents the frequency domain output waveform when the circuit is free from faults, and the red curve represents the frequency domain output waveform at a certain time during the gradually induced circuit faults. Wherein, the capacitor C2 is set as a fault capacitance.
In fig. 10, the horizontal axis represents the degradation process over time, and the vertical axis represents the Failure Index (FI) value. In this experiment, the FI value here is a gradually increasing waveform difference, which can be calculated by the cumulative euclidean distance between the fault and the initial frequency domain output signal. The prediction results based on the GRU and FI data from the first 50 moments are shown in fig. 11.
As shown in fig. 11, the blue line represents the actual data curve, the red curve represents the prognosis curve fitted by the GRU model, and the green curve represents the ARMA model update. As shown, the GRU has better prediction accuracy.
Figure BDA0003251617660000101
Sum mu j The probability distribution of the estimated initial failure indicator can be extrapolated from the historical data sets from different times, respectively. Mu is obtained first 0 And->
Figure BDA0003251617660000102
The updated parameter mu can be obtained according to the formulas (12) and (13) c And->
Figure BDA0003251617660000103
Finally, the updated parameter mu c And->
Figure BDA0003251617660000104
Substitution equation: />
Figure BDA0003251617660000105
Thus, the real-time reliability of the circuit under test can be evaluated (see fig. 12).
As can be seen from fig. 12, the blue curve is the prediction result of the first 30 data and the field data, the red curve is the prediction result of the first 40 data and the field data, and the green curve is the prediction result of the first 30 data and the field data. The prediction results integrate the first 50 data with the field data. Furthermore, different predictions point to different probability distributions of circuit reliability.
From time 0, the circuit has been set to the ideal full reliability. The reliability of the test is gradually changed to 0 along with the time of the test. According to the Bayesian estimation theory, the larger the sample size is, the narrower the confidence interval is. As field data increases, the prediction curve gradually approaches the actual reliability decline trend of the circuit. The simulation results show that the more the field information is, the higher the prediction accuracy is.
Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the different dependent claims and the features described herein may be combined in ways other than as described in the original claims. It is also to be understood that features described in connection with separate embodiments may be used in other described embodiments.

Claims (5)

1. The circuit fault real-time prediction method based on the GRU model is characterized by comprising the following steps of:
s1: selecting a reliability failure index with a time-varying parameter;
s2: acquiring and storing prior density functions of corresponding time-varying parameters, and estimating probability distribution of preliminary performance failure variables by using a neural network of a gating circulation unit; the method specifically comprises the following steps:
step S21: assuming that failure distribution of the electronic product belongs to normal distribution;
step S22: the state value of the failure index FI at any time is expressed as f (x) to N (mu, sigma) 2 ) The relevant probability density function for the state at time t can be expressed as:
Figure FDA0004156571250000011
wherein x represents the time scale of product degradation, μ and σ 2 The mean value and the variance of the variable distribution respectively representing the failure indexes of the product;
taking the average value of a history sample group at a certain moment as a statistical object, taking the average value and the variance of the average value as the prior average value and the prior variance of the average value mu, and obtaining normal distribution N (mu) 0 ,σ 0 2 ) A priori distribution of μ:
Figure FDA0004156571250000012
μ 0 sum sigma 2 Respectively representing the mean value and the variance of a history sample group at a certain moment;
s3: the corresponding posterior density function is obtained by combining the field data and Bayesian estimation, and the distribution density function of the FI parameter variable under the given mu condition is as follows:
Figure FDA0004156571250000013
Figure FDA0004156571250000014
site sample y t Posterior probability density function under update:
Figure FDA0004156571250000015
the posterior probability density function of the FI parameter variable at any moment is:
f(x|y t )＝∫f(x|μ)f(μ|y t )dμ；
s4: and evaluating the real-time performance of the tested object.
2. The method for predicting circuit failure in real time based on the GRU model of claim 1, wherein said step S3 further comprises updating the field data in real time.
3. The method according to claim 1, wherein in step S22, μ is calculated using the mean value of each group of samples as a statistical object 0 Sum sigma 2 When the first data is used as a first sample, the average value of the first two data is used as a second sample, the average value of the first three data is used as a third sample, and the average value of the first N data is used as an N-th sample, so that N samples are generated, and the number of the data of the N-th sample is N.
4. The method for predicting circuit failure in real time based on the GRU model according to claim 1, wherein the calculating the real-time reliability model in step S4 is:
Figure FDA0004156571250000021
/>
μ c and
Figure FDA0004156571250000022
and respectively recording the current distribution parameters containing posterior information obtained by combining the historical data samples with the field data.
5. A verification circuit for a GRU model based circuit fault real time prediction method according to claim 1, comprising 15 resistors, 7 capacitors and 3 operational amplifiers, wherein:
the first end of the resistor R1 is connected with the input voltage, and the second end of the resistor R1 is connected with the inverting input end of the first operational amplifier;
the first end of the resistor R2 is connected with the second end of the resistor R1, and the second end of the resistor R2 is connected with the output end of the first operational amplifier;
the first end of the resistor R3 is connected with the output end of the first operational amplifier, and the second end of the resistor R3 is connected with the first end of the resistor R7;
the first end of the resistor R4 is connected with the output end of the first operational amplifier, and the second end of the resistor R4 is connected with the first end of the resistor R5;
the second end of the resistor R5 is grounded;
the first end of the resistor R6 is connected with the second end of the capacitor C3, and the second end of the resistor R6 is connected with the second end of the capacitor C2;
the second end of the resistor R7 is connected with the inverting input end of the second operational amplifier;
the first end of the resistor R8 is connected with the non-inverting input end of the second operational amplifier and the output end of the second operational amplifier, and the second end of the resistor R8 is connected with the first end of the resistor R9;
the second end of the resistor R9 is connected with the first end of the resistor R10;
the second end of the resistor R10 is grounded;
the first end of the resistor R11 is connected with the output end of the second operational amplifier, and the second end of the resistor R11 is connected with the first end of the resistor R12;
the second end of the resistor R12 is connected with the inverting input end of the third operational amplifier;
the first end of the resistor R13 is connected with the second end of the capacitor C5, and the second end of the resistor R13 is connected with the second end of the capacitor C6;
the first end of the resistor R14 is connected with the non-inverting input end of the third operational amplifier and the output end of the third operational amplifier, and the second end of the resistor R14 is connected with the first end of the resistor R13 and the first end of the resistor R15;
the second end of the resistor R15 is grounded;
the first end of the capacitor C1 is connected with the second end of the resistor R1, and the second end of the capacitor C1 is connected with the output end of the first operational amplifier;
the first end of the capacitor C2 is connected with the second end of the resistor R4, and the second end of the capacitor C2 is connected with the first end of the capacitor C4;
the first end of the capacitor C3 is connected with the second end of the resistor R3, and the second end of the capacitor C3 is connected with the first end of the resistor R6;
the first end of the capacitor C4 is connected with the second end of the capacitor C2, and the second end of the capacitor C4 is connected with the inverting input end of the second operational amplifier;
the first end of the capacitor C5 is connected with the second end of the resistor R11, and the second end of the capacitor C5 is connected with the first end of the resistor R13;
the first end of the capacitor C6 is connected with the second end of the resistor R9, and the second end of the capacitor C6 is connected with the first end of the capacitor C7;
the second end of the capacitor C7 is connected with the inverting input end of the third operational amplifier;
the non-inverting input end of the first operational amplifier is grounded;
the output end of the third operational amplifier is connected with the output voltage.
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