CN114063456B - Fault prediction and early warning method using autoregressive model and Kalman filtering algorithm - Google Patents

Fault prediction and early warning method using autoregressive model and Kalman filtering algorithm Download PDF

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CN114063456B
CN114063456B CN202111350431.7A CN202111350431A CN114063456B CN 114063456 B CN114063456 B CN 114063456B CN 202111350431 A CN202111350431 A CN 202111350431A CN 114063456 B CN114063456 B CN 114063456B
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CN114063456A (en
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王振华
张文瀚
沈毅
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Harbin Institute of Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
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Abstract

The invention discloses a fault prediction and early warning method by utilizing an autoregressive model and a Kalman filtering algorithm, which comprises the following steps: 1. establishing a state equation and a measurement equation of a fault autoregressive degradation model; 2. computing stateVector predictor
Figure DDA0003355663040000011
3. Calculating a predicted value P (k|k-1) of the system state covariance matrix; 4. calculating a filter gain coefficient K (K); 5. calculating an estimated value P (k) of the system state covariance matrix; 6. calculating state vector estimates
Figure DDA0003355663040000012
7. Calculating a predicted value of a measurement vector
Figure DDA0003355663040000013
To the point of
Figure DDA0003355663040000014
8. Calculating the upper boundary of the measured vector predictor
Figure DDA0003355663040000015
To the point of
Figure DDA0003355663040000016
And lower boundaryy(k) To the point ofy(k+m); 9. determining a fault early warning result; 10. and repeating two to nine times at each k moment, and iteratively realizing fault prediction and early warning. The method can give out a reliable early warning signal before the control system fails, effectively ensures the safety and the robustness of the system, has clear principle and simple algorithm, and is easy for practical engineering realization.

Description

Fault prediction and early warning method using autoregressive model and Kalman filtering algorithm
Technical Field
The invention relates to a fault early warning method of a control system, in particular to a method for predicting and early warning the fault of the control system by utilizing an autoregressive model and a Kalman filtering algorithm.
Background
Modern control systems are becoming more and more complex, and accordingly, the safety and reliability of the control system are particularly important. However, various failures occurring when an actual system is operated may reduce reliability thereof and even destroy stability of the system, thereby causing serious safety accidents. Therefore, in order to improve the safety and reliability of the control system, it is necessary to diagnose the fault and take countermeasures in time to minimize damage to the system caused by the fault. In general, it is more likely that there will be various ramp faults within the control system that are caused by performance degradation of system components. The slow-change faults are small in amplitude at the initial stage, the performance of the control system is not seriously affected, but the fault amplitude is gradually increased along with the continuous accumulation of the time dimension, and the threat to the safety and the reliability of the system is higher. In particular, when a ramp fault evolves to a system component performance failure threshold, the safety and stability of the overall control system can be severely compromised. Therefore, the degradation process modeling is performed aiming at the control system gradual change failure, the future failure development trend is predicted, and the failure early warning is timely performed before the performance failure, so that the method has important significance for improving the reliability and the safety of the control system.
According to published related documents at home and abroad, the current fault prediction and early warning algorithm is mainly used for carrying out fault degradation modeling and prediction based on a fuzzy network or a deep learning technology, has higher requirements on the data quantity of faults and the calculated quantity of model training, and has a limited application range. Meanwhile, most of the existing methods need to train a fault model offline in advance, cannot update and adjust model parameters online in real time, and are difficult to provide accurate fault prediction results, so that the reliability of fault early warning can be affected to a certain extent.
Since the use of fuzzy network or deep learning techniques for fault degradation modeling and prediction requires a high computational effort, such algorithms are only suitable for use in forming a control system of simple construction and low part count. In the face of the development trend of increasingly complex structure and increasing number of component parts of the modern control system, the failure prediction and early warning algorithms can not meet the requirements of high-efficiency and accurate failure early warning performance, and the modern control system has urgent need for a failure early warning algorithm with good applicability, small calculated amount and reliable performance.
Disclosure of Invention
The invention aims to provide a fault prediction and early warning method by utilizing an autoregressive model and a Kalman filtering algorithm, which can give a reliable early warning signal before a control system breaks down, effectively ensure the safety and the robustness of the system, has clear principle and simple algorithm, and is easy to realize in actual engineering.
The invention aims at realizing the following technical scheme:
a fault prediction and early warning method using an autoregressive model and a Kalman filtering algorithm comprises the following steps:
step one, establishing a state equation and a measurement equation of a fault autoregressive degradation model:
Figure BDA0003355663020000021
wherein x (k) and x (k-1) are respectively state vectors of the system at the moment of k and the moment of k-1; a (k-1) is a one-step transfer matrix of a degradation model state; w (k-1) is the disturbance vector of the system; c is a measurement matrix; v (k) is the measurement noise vector of the system; y (k) is a measurement vector;
step two, calculating a state vector predicted value
Figure BDA0003355663020000031
Figure BDA0003355663020000032
In the method, in the process of the invention,
Figure BDA0003355663020000033
an estimated value of the state vector at the moment k-1;
calculating a predicted value P (k|k-1) of the system state covariance matrix:
P(k|k-1)＝A(k-1)P(k-1)A(k-1) T +Q；
wherein P (k-1) is an estimated value of a systematic error covariance matrix at the moment k-1; q is a system noise covariance matrix;
step four, calculating a filter gain coefficient K (K):
K(k)＝P(k|k-1)C T /(CP(k|k-1)C T +R)；
wherein R is a covariance matrix of a measurement noise vector;
step five, calculating an estimated value P (k) of the system state covariance matrix:
P(k)＝P(k|k-1)-K(k)CP(k|k-1)；
wherein P (k) is an estimated value of a state covariance matrix at the moment k;
step six, calculating a state vector estimated value
Figure BDA0003355663020000034
Figure BDA0003355663020000035
In the method, in the process of the invention,
Figure BDA0003355663020000036
an estimated value of the state vector at the moment k;
step seven, calculating the predicted value of the measurement vector
Figure BDA0003355663020000037
To->
Figure BDA0003355663020000038
Figure BDA0003355663020000039
In the method, in the process of the invention,
Figure BDA00033556630200000310
to->
Figure BDA00033556630200000311
Measuring the predicted value of the vector for the time k to k+m; />
Figure BDA00033556630200000312
State vector estimate for time k>
Figure BDA00033556630200000313
Is the i-th element of (a); n is the order used by the fault autoregressive degradation model; m is the step length of predicting the fault from the current moment to the back;
step eight, calculating the upper boundary of the predicted value of the measurement vector
Figure BDA0003355663020000041
To->
Figure BDA0003355663020000042
And lower boundaryy(k) To the point ofy(k+m)：
Figure BDA0003355663020000043
In the method, in the process of the invention,
Figure BDA0003355663020000044
to->
Figure BDA0003355663020000045
Measuring the upper boundary of the vector predicted value for the time k to k+m;y(k) To the point ofy(k+m) measuring the lower boundary of the vector predictor for the time k to k+m;
step nine, determining a fault early warning result:
Figure BDA0003355663020000046
and is also provided withy(k+j)≤y th And->
Figure BDA0003355663020000047
Wherein y is th A set fault detection threshold;
if m is taken from 0 for j, all j enable the above formula to be established, indicating that no fault is predicted at the moment k;
if j is taken from 0 to m, and one j exists so that the above formula is not established, the fact that the fault is predicted at the moment k is indicated, and the predicted future fault occurs at the moment k+j * ；
And step ten, repeating the step two to the step nine at each k moment, and iteratively realizing fault prediction and early warning.
The invention provides a fault early warning method by utilizing an autoregressive model and a Kalman filtering algorithm, which can predict the future change value of a fault according to the degradation characteristic of the fault and perform effective early warning of the fault on the basis, and has the following advantages and beneficial effects:
(1) An autoregressive model is adopted to describe the degradation process of the fault, the model is simple in form and wide in description range, and the fault early warning method is good in applicability;
(2) The model parameters are updated on line in real time by using a Kalman filtering algorithm, the parameter estimation accuracy is high, the calculation amount of on-line operation is small, and the realization of an actual hardware platform is easy;
(3) The fault early warning signal is given through the fault prediction value and the upper and lower bounds thereof, so that the reliability of an early warning result is ensured, and the safety of a control system is further improved.
Drawings
FIG. 1 is a flow chart of a fault prediction and early warning method using an autoregressive model and a Kalman filtering algorithm according to the present invention.
Fig. 2 shows the result of the dc motor ramp fault data.
Fig. 3 shows motor creep fault prediction and early warning results when k=180.
Fig. 4 shows motor creep fault prediction and early warning results when k=185.
Fig. 5 shows motor creep fault prediction and early warning results when k=190.
Fig. 6 shows motor creep fault prediction and early warning results when k=195.
Detailed Description
The following description of the present invention is provided with reference to the accompanying drawings, but is not limited to the following description, and any modifications or equivalent substitutions of the present invention should be included in the scope of the present invention without departing from the spirit and scope of the present invention.
The invention provides a fault prediction and early warning method by utilizing an autoregressive model and a Kalman filtering algorithm, which comprises the following steps:
step one, a state equation and a measurement equation of a fault autoregressive degradation model are established, and the formula is as follows:
Figure BDA0003355663020000051
wherein x (k) and x (k-1) are respectively state vectors of the system at the moment of k and the moment of k-1; a (k-1) is a one-step transfer matrix of a degradation model state; w (k-1) is the disturbance vector of the system; c is a measurement matrix; v (k) is the measurement noise vector of the system; y (k) is the measurement vector.
Step two, calculating a state vector predicted value
Figure BDA0003355663020000061
The formula is as follows:
Figure BDA0003355663020000062
/>
in the method, in the process of the invention,
Figure BDA0003355663020000063
is the estimate of the state vector at time k-1.
Step three, calculating a predicted value P (k|k-1) of the system state covariance matrix, wherein the formula is as follows:
P(k|k-1)＝A(k-1)P(k-1)A(k-1) T +Q (3)；
wherein P (k-1) is an estimated value of a systematic error covariance matrix at the moment k-1; q is the system noise covariance matrix.
Step four, calculating a filter gain coefficient K (K), wherein the formula is as follows:
K(k)＝P(k|k-1)C T /(CP(k|k-1)C T +R) (4)；
where R is the covariance matrix of the measured noise vector.
Step five, calculating an estimated value P (k) of the system state covariance matrix, wherein the formula is as follows:
P(k)＝P(k|k-1)-K(k)CP(k|k-1) (5)；
where P (k) is an estimate of the state covariance matrix at time k.
Step six, calculating a state vector estimated value
Figure BDA0003355663020000064
The formula is as follows:
Figure BDA0003355663020000065
in the method, in the process of the invention,
Figure BDA0003355663020000066
is an estimate of the state vector at time k.
Step seven, calculating the predicted value of the measurement vector
Figure BDA0003355663020000067
To->
Figure BDA0003355663020000068
The formula is as follows:
Figure BDA0003355663020000069
in the method, in the process of the invention,
Figure BDA0003355663020000071
to->
Figure BDA0003355663020000072
Measuring the predicted value of the vector for the time k to k+m; />
Figure BDA0003355663020000073
State vector estimate for time k>
Figure BDA0003355663020000074
Is the i-th element of (a); n is the order used by the fault autoregressive degradation model; m is the fault slaveStep size of prediction is carried out from time to time.
Step eight, calculating the upper boundary of the predicted value of the measurement vector
Figure BDA0003355663020000075
To->
Figure BDA0003355663020000076
And lower boundaryy(k) To the point ofy(k+m) having the formula:
Figure BDA0003355663020000077
in the method, in the process of the invention,
Figure BDA0003355663020000078
to->
Figure BDA0003355663020000079
Measuring the upper boundary of the vector predicted value for the time k to k+m;y(k) To the point ofy(k+m) the lower boundary of the vector predictor is measured for the times k to k+m.
Step nine, determining a fault early warning result, wherein the fault early warning strategy is as follows:
if m is taken from 0 for j, all j will be such that:
Figure BDA00033556630200000710
and is also provided withy(k+j)≤y th And->
Figure BDA00033556630200000711
It is stated that no fault is predicted at time k.
If j is taken from 0 to m, and if one j is present such that equation (9) is not satisfied, it is indicated that a fault is predicted at time k, and the predicted future fault occurs at time k+j * Wherein y is th Is the set fault detection threshold.
And step ten, repeating the step two to the step nine at each k moment, and iteratively realizing fault prediction and early warning.
Preferably, in step one, the system state vector is given by equation (10):
x＝[f c 1 c 2 …c n ] T (10)；
wherein f is the fault value of the engineering system, and c is obtained by measurement 1 、c 1 、…、c n Coefficients that are autoregressive models; measurement y=f; the matrices A (k-1) and C are given by equation (11) and equation (12), respectively:
Figure BDA0003355663020000081
C＝[1 0 0 … 0] (12)。
preferably, in step two, the state vector estimation values at the first n+1 times are given by equation (13) and equation (14), respectively, starting from k=n+1:
Figure BDA0003355663020000082
Figure BDA0003355663020000083
preferably, in step seven, the predicted value of the vector is measured
Figure BDA0003355663020000084
To->
Figure BDA0003355663020000085
Iterative calculation is carried out by an autoregressive model method, and the autoregressive model parameter is a state vector estimated value +.>
Figure BDA0003355663020000086
The 2 nd to n+1th elements of (c).
Preferably, in step eight, the upper boundary of the vector predictor is measured
Figure BDA0003355663020000087
To->
Figure BDA0003355663020000088
And lower boundaryy(k) To the point ofy(k+m) 3 sigma interval by Kalman filtering,/g>
Figure BDA0003355663020000089
Is->
Figure BDA00033556630200000810
Standard deviation of (2).
Preferably, in step nine, the fault pre-warning result is determined by measuring the magnitude relation between the vector predicted value and the upper and lower boundaries thereof and the fault detection threshold.
In summary, the present invention first describes the dynamic degradation process of the fault using an autoregressive model; then, the Kalman filtering algorithm is utilized to update the model parameters on line in real time, so that the future time fault value and the upper and lower bounds of the future time fault value are predicted; and finally, combining a fault prediction result and a set fault threshold value to realize effective early warning of faults. The invention utilizes the autoregressive model to describe the fault degradation dynamics, has simple model form and good applicability, and widens the application range of the algorithm; the method utilizes Kalman filtering to update model parameters on line in real time, has high parameter estimation accuracy, simple algorithm and clear principle, and is convenient for actual hardware realization; the algorithm gives out fault early warning signals by using the fault prediction value and the upper and lower boundaries thereof, and ensures the reliability of early warning results.
Application instance
The following describes the application of the fault prediction and early warning method using an autoregressive model and a kalman filter algorithm with an example of a dc motor. Considering that the performance of the direct current motor gradually deteriorates along with the increase of the running time, so that the motor temperature under the normal running condition is continuously increased, therefore, the difference value between the motor temperature under the normal running condition and the nominal temperature can be used as an indication signal of the gradual failure of the direct current motor, and the gradual failure data result shown in figure 2 can be drawn through the collected motor temperature data under the normal running condition.
In the simulation of the fault prediction algorithm, the step length of predicting the fault from the current moment to the back is set to be m=20, the order of the fault autoregressive degradation model is set to be n=20, and the output matrix C= [ 10 0 0 … 0 of the degradation model]. The fault detection threshold is selected as y th Initial value of kalman filter algorithm =3.8
Figure BDA0003355663020000091
Selected as->
Figure BDA0003355663020000092
Status initial value->
Figure BDA0003355663020000093
The covariance matrix of (1) is selected as P (k) =10 -4 I n+1 ，I n+1 An identity matrix of n+1 dimensions; the covariance matrix of the disturbance w (k) and the noise v (k) is q=10 -4 I n+1 And r=10 -4 . Based on given simulation parameters and measured slow-varying fault data, a fault prediction simulation result as shown in fig. 3-6 can be obtained.
As can be seen from the simulation results in fig. 3 to 6, when k=180 and k=185, the actual ramp fault data of the dc motor does not exceed the given fault detection threshold, and the designed method does not give a fault early warning signal. When k=190 and k=195, the actual slow-change fault data of the direct current motor exceeds a given fault detection threshold, and at the moment, the proposed method can rapidly and accurately predict faults and give out early warning signals, so that the effectiveness of the proposed method is verified.

Claims (4)

1. The fault prediction and early warning method by utilizing an autoregressive model and a Kalman filtering algorithm is characterized by comprising the following steps of:
step one, establishing a state equation and a measurement equation of a fault autoregressive degradation model:
Figure FDA0004184352900000011
wherein x (k) and x (k-1) are respectively state vectors of the system at the moment of k and the moment of k-1; a (k-1) is a one-step transfer matrix of a degradation model state; w (k-1) is the disturbance vector of the system; c is a measurement matrix; v (k) is the measurement noise vector of the system; y (k) is a measurement vector;
the state vector of the system is given by the following formula:
x＝[f c 1 c 2 …c n ] T ；
wherein f is the fault value of the engineering system, c 1 、c 1 、…、c n Coefficients that are autoregressive models;
the matrix A (k-1) is given by the following formula:
Figure FDA0004184352900000012
wherein f is a fault value of the engineering system;
step two, calculating a state vector predicted value
Figure FDA0004184352900000013
Figure FDA0004184352900000014
In the method, in the process of the invention,
Figure FDA0004184352900000015
an estimated value of the state vector at the moment k-1;
calculating a predicted value P (k|k-1) of the system state covariance matrix:
P(k|k-1)＝A(k-1)P(k-1)A(k-1) T +Q；
wherein P (k-1) is an estimated value of a systematic error covariance matrix at the moment k-1; q is a system noise covariance matrix;
step four, calculating a filter gain coefficient K (K):
K(k)＝P(k|k-1)C T /(CP(k|k-1)C T +R)；
wherein R is a covariance matrix of a measurement noise vector;
step five, calculating an estimated value P (k) of the system state covariance matrix:
P(k)＝P(k|k-1)-K(k)CP(k|k-1)；
wherein P (k) is an estimated value of a state covariance matrix at the moment k;
step six, calculating a state vector estimated value
Figure FDA0004184352900000021
Figure FDA0004184352900000022
In the method, in the process of the invention,
Figure FDA0004184352900000023
an estimated value of the state vector at the moment k;
step seven, calculating the predicted value of the measurement vector
Figure FDA0004184352900000024
To->
Figure FDA0004184352900000025
/>
Figure FDA0004184352900000026
In the method, in the process of the invention,
Figure FDA0004184352900000027
to->
Figure FDA0004184352900000028
Measuring the predicted value of the vector for the time k to k+m; />
Figure FDA0004184352900000029
State vector estimate for time k>
Figure FDA00041843529000000210
Is the i-th element of (a); n is the order used by the fault autoregressive degradation model; m is the step length of predicting the fault from the current moment to the back;
step eight, calculating the upper boundary of the predicted value of the measurement vector
Figure FDA00041843529000000211
To->
Figure FDA00041843529000000212
And lower boundaryy(k) To the point ofy(k+m)：
Figure FDA00041843529000000213
In the method, in the process of the invention,
Figure FDA0004184352900000031
to->
Figure FDA0004184352900000032
Measuring the upper boundary of the vector predicted value for the time k to k+m;y(k) To the point ofy(k+m) measuring the lower boundary of the vector predictor for the time k to k+m; measuring the upper boundary of vector predictors +.>
Figure FDA0004184352900000033
To->
Figure FDA0004184352900000034
And lower boundaryy(k) To the point ofy(k+m) 3 sigma by Kalman filteringInterval determination, tight>
Figure FDA0004184352900000035
Is->
Figure FDA0004184352900000036
Standard deviation of (2);
step nine, determining a fault early warning result:
Figure FDA0004184352900000037
and is also provided withy(k+j)≤y th And->
Figure FDA0004184352900000038
Wherein y is th A set fault detection threshold;
if m is taken from 0 for j, all j enable the above formula to be established, indicating that no fault is predicted at the moment k;
if j is taken from 0 to m, and one j exists so that the above formula is not established, the fact that the fault is predicted at the moment k is indicated, and the predicted future fault occurs at the moment k+j * ；
And step ten, repeating the step two to the step nine at each k moment, and iteratively realizing fault prediction and early warning.
2. The method for predicting and warning a fault using an autoregressive model and a kalman filter algorithm according to claim 1, wherein in the first step, C is given by the following formulas:
C＝[1 0 0…0]。
3. the method for predicting and warning a fault using an autoregressive model and a kalman filter algorithm according to claim 1, wherein in the second step, the state vector estimated values at the first n+1 times are given by the following formula:
Figure FDA0004184352900000039
Figure FDA00041843529000000310
4. the method for predicting and warning a fault using an autoregressive model and a Kalman filtering algorithm as defined in claim 1, wherein in said step seven, the predicted value of the vector is measured
Figure FDA0004184352900000041
To->
Figure FDA0004184352900000042
Iterative calculation is carried out by an autoregressive model method, and the autoregressive model parameter is a state vector estimated value +.>
Figure FDA0004184352900000043
The 2 nd to n+1th elements of (c). />
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