2.2. Problem in the UKF
In order to ensure the good accuracy of the output parameters of unscented Kalman filter, a model that can reflect the real working state of the monitored target must be established. In the condition monitoring of two-spool turbojet based on the unscented Kalman filter, the Component-level Gas Path Model (CGPM) is usually used to predict the values of health parameters [
26,
27,
28,
29,
30,
31,
32]. The health parameters are the indicators of the health status of turbojet and can be used to illustrate the flowing ability and working efficiency. The CGPM is essentially a series of physical equations based on the principle of aerothermodynamics. By the operation of CGPM, the health parameters (flow coefficients and efficiency coefficient of gas path components) and the measured parameters (total temperature and total pressure of gas path components) can be calculated. The structure of CGPM is shown in
Figure 3.
The Component-level Gas Path Model of two-spool turbojet can be described as follows [
1]:
In Formulas (1) and (2),
represents the
kth sampling time.
is the state parameters vector. Furthermore, all of these parameters are the estimated objects.
is the control variable. There are:
and
are the efficiencies and flow coefficients of different components, respectively, include the low-pressure compressor, high pressure compressor, low pressure turbine, and high pressure turbine.
is the flow of fuel.
and
are the state transmission noise and measurement transmission noise.
represents the process of predicting the health parameters based on the Component-level Gas Path Model.
is the vector of health parameter. The value of
is determined by
and
. Thus, the calculation of
can be realized by the health parameters calculation module, as shown in
Figure 3. The content of
is similar with that of
.
represents the process of predicting the measured parameters based on the Component-level Gas Path Model.
is the vector of measured parameters at the
k + 1th sampling time. By the operation of measured parameters calculation module, function
can be.
The working state transformation of turbojet is a continuous process, the Component-level Gas Path Model can not accurately reflect all the working state. If the working state of turbojet changes suddenly, the output measured parameters of CGPM deviate greatly from those of the turbojet due to the defect of CGPM. Consequently, the estimations of UKF may be distorted. This circumstance can be illustrated by
Figure 4.
In
Figure 4, the values of
are estimated by the unscented Kalman filter based on the Component-level Gas Path Model. The working state of the turbojet changed abruptly after working for an hour, and the measured parameters include
changed widely in short time. However, due to the defect of Component-level Gas Path Model, the estimations of unscented Kalman filter are not consistent with the true values of measurements, as shown in
Figure 4.
2.3. Resolution
To overcome above drawback, a strong tracking filter based on the UKF is proposed. The strong tracking filter (STF) satisfies following condition [
33,
34,
35]:
and are the residuals of measurements and outputs of model at the and sampling times. and are the measurements and outputs of model, respectively. Equation (3) means that the residuals of measurements and outputs of model is orthogonal if the UKF is working normally. When the working state of the engine changes abruptly, the residuals are not orthogonal anymore. Paper design STF to adjust the variance ratio of measurements at different times to force the residuals keep orthogonal so that the accuracy of estimated parameters remains high. The steps of STF are as follows:
Samples collection and weight calculation.
are the estimated objects (state variable) which consist of , , , , , ,, and . denotes the mean vector of the estimated objects. is the weight of estimated object. is the parameter to reduce prediction error. is the number of estimated objects and there is . is the covariance matrix of the estimated objects.
State variable calculation based on model.
State variable is calculated by Equation (4). is the value of fuel flow. and are the estimated variables (state variable) at the and sampling time. is the mean vector of estimated variables and is the covariance matrix. and are the estimated values and the mean value of measurements respectively. is the covariance matrix of and . and are the variances of and respectively. is the measurement vector obtained by sensors. is the Kalman gain. and are the state transmission noise and measurement transmission noise. .
The need to pay attention is that
is fading factor vector. There is
.
denotes the fading factor. By regulating the proportion of fading factors, the residual of measurements at the current and last sampling time can be kept orthogonal. The emphasis of STF is to calculate the value of
. For
, set each fading factor as:
is the common parameter and
is the ratio parameter. The ratio value of fading factors can be determined by experience, there is:
Obviously,
can be calculated if
is obtained. Equation (3) can be transformed as:
is the residual covariance matrix of measurements. The condition to satisfies Equation (16) is that:
is the noise statistical matrix of measurements. Compute the trace of Equation (18), and the expression of
can be obtained.
named scale factor is used to adjust the ratio of residual covariance matrix at the sampling time. The greater the value of , the greater the proportion of . Otherwise, the greater the proportion of . Usually, the value of is determined by experience, and there is a drawback that unreasonable value of may lead to the distortion of . Paper proposes a method to obtain . The steps are as follows:
- (1)
Construct a variance vector which consist of the diagonal elements of . Furthermore, obtain the residual vector which consist of diagonal elements of .
- (2)
Similarity calculation between
and
.
is the cosine value between and , and . Considering the Equation (20), coefficients of and are and respectively. Obviously, the sum of and is 1.
Set the angle between
and
as
, there is:
Replace the original coefficients of
and
with
and
. Equation (20) can be transformed as:
According to the working principle of gas turbine and taking into account that the proportion of current (the sampling time) information should be greater than that of previous sampling time. Equation (22) ensures that the coefficient of is greater than that of . Estimate by above method.
Compare with
Figure 4 and
Figure 5 accurately reflects the sudden change of measurements. It shows the validity of STF proposed by paper, which compensates the model error and enhances the estimation accuracy.
2.4. Health Parameters Estimation
Paper adopt particle filter to estimate the health parameters. The measurements filtered by the STF are used to determine the posterior probability. Weight degradation that may lead to the accuracy decrease of estimations is a commonly problem exists in the process of particle filter Particle resampling is a traditional way to solve this problem. By increasing the number of larger-weight particles and make all particles have the same weight, the weight degradation has been effectively solved. But the above-mentioned method will lead to another problem, that is, the loss of particle diversity. In order to coordinate these two issues, paper proposes a weight optimization method in the PF. The core idea of this method is to adjust the posterior probability density function of health parameters. By properly increasing the weight of small weight particles and reducing the weight of large weight particles, the diversity of particles can be kept, and the high accuracy of probability density function can be ensured. The steps of health parameters estimation algorithm based on weight optimization PF are as follows [
33,
34]:
(1) k = 0, particles initialization.
k denotes the sampling time. Set the number of particles is 100. Each particle represents the value of health parameter. Generate particles according to the importance probability density function . consists of health parameters which include the efficiency coefficients of LPC, HPC, HPT, LPT, and the flow coefficients of LPC, HPC, HPT, and LPT. is the uniform distribution function.
(2) k = 1,2, 3, …. weight update.
Predict the health parameters based on the prior probability distribution function:
Above calculation can be realized based on the component-level model of engine. Weight update:
y consists of different measurements filtered by STF. There are total temperatures at the outlet of LPC, HPC, HPT, LPT, and total pressures at the outlet of LPC, HPC, HPT, and LPT.
Weight optimization. Calculate the mean of weights, there is:
Adjust the weight of each particle:
R is regulator and
. The function of
R is to regulate the weight of particles. Normalize the weights:
- (3)
- (4)
Optimize the health parameters:
In order to verify the validity of proposed method, a simulation to detect the failure occurrence of two-spool turbojet is conducted. By suddenly changing the value of health parameters, failure occurrence can be simulated [
1,
13]. According to the research of previous chapters, by estimate the values of health parameters, failure detect can be realized [
13]. The steps to conduct the simulation are as follows:
- 1
Generate the measured parameters from a software named Gasturb13 (Gasturb 13 is a simulation software for gas turbine performance calculation with high accuracy). Add noise w to these measured parameters. , N is the normal probability density function.
- 2
Establish the Component-level Gas Path Model of turbojet. This model is the detailed expression of the Equations (1) and (2).
- 3
Build the module of strong tracking filter according the method introduced in
Section 2.3. The measured parameters including noise are input into the module and output to the particle filter after being processed by the strong tracking filter.
- 4
Build the module of particle filer with weight optimization according to the method introduced in
Section 2.4. This module is used to estimate the health parameters.
- 5
Input the measured parameters to the particle filter and estimate the health parameters. The way to simulate the failure are listed as follows:
FLPC and are the latest values of low-pressure compressor’s flow coefficient and efficiency coefficient after the failure is simulated. Fini and are the initial values of low-pressure compressor’s flow coefficients and efficiency coefficient before failure are simulated. . named the failure factor is variation volume of Fini. denotes the degradation value of flow coefficients during every sampling time if failure happen. The meaning of and are similar to that of and . T and Tfailure represent current sampling time and failure occurrence time. The design working parameters of engine are as follows:
Efficiency of LPC: ELPC = 0.868 | Pressure ratio of LPC: |
Efficiency of HPC: EHPC = 0.878 | Pressure ratio of HPC: |
Efficiency of high-pressure rotator: EHPR = 0.98 | Efficiency of low-pressure rotator: ELPR = 0.98 |
Efficiency of burning room: EBR = 0.98 | Air intake coefficient of cabin: EAI = 0.01 |
Cooling coefficient of HPT: CHPT = 0.03 | Efficiency of HPT: EHPT = 0.89 |
Cooling coefficient of LPT: CLPT = 0.01 | Efficiency of LPT: ELPT = 0.91 |
Design rotating speed of Low Pressure Rotator: SLPR = 104r/m |
Design rotating speed of High Pressure Rotator: SHPR = 1.6 × 104r/m |
Total temperature at the outlet of burning room: Tt4 = 1600 K |
Heat value of fuel: FHV = 4.29 × 104 |
Due to limitation of space, the estimation of low-pressure compressor’s flow coefficient and efficiency coefficient are listed only. The estimation processes of other health parameters of high-pressure compressor, high-pressure turbine, low-pressure turbine are similar with that of low-pressure pressure. Assure that 100 measured parameters are collected. When the engine performance degrades slowly, the efficiency coefficient decreases by 0.6% compared with the initial value, and the flow coefficient decrease by 0.7%. To simulate the failure, at the 11th sampling time, set the flow coefficient and efficiency coefficient decreased by 0.3%.
Figure 6 shows the Estimated health parameters of low-pressure compressor based on the traditional unscented Kalman filter and particle filter. Set there are 100 sampling times. The initial theoretical values of the efficiency coefficient and the flow coefficient are 0.868 and 0.92, respectively. During each sampling time, the variation of efficiency coefficient and flow coefficient are 5.2 × 10
−6 and 6.4 × 10
−6, as shown in
Table 1. There are:
and are the values of efficiency coefficient and flow coefficient after the performance degrades slowly. It can be seen that the estimations are close to the theoretical values of health parameters basically. However, the estimations curve fluctuates greatly, and the accuracy degree of estimations is not high.
Figure 7 shows the estimated values of LPC’s flow coefficient and efficiency coefficients when the working state of two-spool turbojet is steady based on the proposed hybrid filter. Under this working condition, the variations range of health parameters (flow coefficient and efficiency coefficient of LPC) are small and slow degradation of performance is happened due to the poor working circumstance of turbojet. The purple curve consists of the estimated values and black curve consists of the theoretical values. Obviously, the method proposed in this paper can accurately characterize the change trend of health parameters. Furthermore, the accuracy of the estimations is also consistent with the theoretical values of health parameters. The estimations variance of efficiency coefficient and flow coefficient are 2.59 × 10
−6 and 4.05 × 10
−6 respectively, as shown in
Table 2.
Figure 8 shows the estimated values of low-pressure compressor’s flow coefficient and efficiency coefficients based on the method proposed by this paper when the working state of two-spool turbojet breaks down. At the 11th sampling time, set the efficiency coefficient and flow coefficient have a sudden change of 0.3%. There are:
Paper simulate the failure by changing the health parameters at the tenth sampling time. Due to the occurrence of failure, the measured parameters have a sudden change. By the application of strong tracking filter, the measured parameter can be estimated with high accuracy. According to the introduction of
Section 2.1, the output of strong tracking filter is input to the particle filter. Due to the high accuracy tracking ability of the STF to the state mutation and the weight optimization of particle filter, the health parameters are estimated with high accuracy by the particle filter, as shown in
Figure 8. The value of efficiency coefficient reduced from 0.8675 to 0.8649, and the value of flow coefficient reduced from 0.9196 to 0.9168. From
Figure 8, the mutations in health parameters are accurately reflected on the curve and the occurrence of failure can be detected.