Performance degradation forecast technology for quantitatively assessing degradation states of aeroengine using exhaust gas temperature is an important technology in the aeroengine health management. In this paper, a GM (1, 1) Markov chain-based approach is introduced to forecast exhaust gas temperature by taking the advantages of GM (1, 1) model in time series and the advantages of Markov chain model in dealing with highly nonlinear and stochastic data caused by uncertain factors. In this approach, firstly, the GM (1, 1) model is used to forecast the trend by using limited data samples. Then, Markov chain model is integrated into GM (1, 1) model in order to enhance the forecast performance, which can solve the influence of random fluctuation data on forecasting accuracy and achieving an accurate estimate of the nonlinear forecast. As an example, the historical monitoring data of exhaust gas temperature from CFM56 aeroengine of China Southern is used to verify the forecast performance of the GM (1, 1) Markov chain model. The results show that the GM (1, 1) Markov chain model is able to forecast exhaust gas temperature accurately, which can effectively reflect the random fluctuation characteristics of exhaust gas temperature changes over time.

Prognostic and health management for aeroengine are the main concerns for many researchers and users in order to provide more useful information for the safe operation [

In the past half-century, different methods have been developed to analyze aeroengine gas path performance parameters for performance degradation [

Several approaches have been introduced to forecast the gas path performance parameters of aeroengine. Based on the research of aeroengine performance parameters relativity, with the condition of small samples and variables with multiple correlations, Shi et al. proposed a partial least-squares regression method to build short time forecasting model of aeroengine performance parameter under the condition of small samples [

From the literature described above, statistical and artificial intelligence based approaches are the two main techniques. Auto Regressive (AR), Moving Average (MA), Auto Regressive Moving Average (ARMA), and Auto Regressive Integrated Moving Average (ARIMA) can be mentioned as statistical models, while Artificial Neural Network (ANN) and Support Vector Machines (SVM) have been most widely used as artificial intelligence approaches. The essences of the above approaches are establishing the appropriate time series model by analyzing historical data. The modeling processes of the statistical based approaches are relatively simple [

In practical applications, it is difficult to obtain the complete information because of many reasons. Besides, the aeroengine gas path performance parameters are often highly nonlinear, stochastic, and nonstationary. Therefore, not only the conventional statistical models are not as accurate as the artificial neural network-based approaches for aeroengine gas path performance parameters trend forecast problems, but the traditional methods may also be too complex to be used in forecasting future values of time series.

Grey system theory, proposed by Deng in 1982 [

The goal of the paper is to introduce the time series forecast based on grey system theory to the forecast modeling of aeroengine gas path performance parameters. Firstly, a type of time series forecast method based on GM (1, 1) model is introduced for aeroengine exhaust gas temperature (EGT). This method can effectively solve the trend forecast problems of EGT under incomplete information and discrete small sample data. Then, Markov chain model is integrated into GM (1, 1) model in order to enhance the forecast performance, which can solve the influence of random fluctuation data on forecast accuracy and achieve an accurate estimate of the nonlinear EGT. A real case of aeroengine EGT from CFM56 aeroengine of China Southern is used to test the capability of the proposed improved model.

The rest of this paper is organized as follows. In Section

The performance of an aeroengine will deteriorate over the time due to different gas path component degradations such as fouling, erosion, corrosion, and foreign object damage [

Outlet temperature of combustor chamber is the most important performance parameter for aeroengine. Not only does it affect the overall performance of the engine, but also it directly determines the ultimate strength of turbine blade. For example, the creep life of hot channel components can reduce the order of magnitude when outlet temperature of combustor chamber increases 50°C [

Considering the gas path performance monitoring parameters, the multiple linear regression models for the relationship between EGT and other parameters were established by Song et al. [

Aeroengine EGT can be divided into take-off EGT and cruise EGT in accordance with different data acquisition stages during flight [

Figure

EGTM sequence and trend analysis for CFM56 aeroengine [

From Figure

Based on the above analysis, aeroengine performance degradation forecast can be solved as EGTM time series forecasting problem. However, it is difficult to establish a precise mathematical model to describe EGTM that can be affected by many uncertain factors. Therefore, the key problem lies in how to establish precise forecast model under incomplete information and discrete small sample data in order to achieve an accurate estimate of the nonlinear EGTM parameters.

Based on the temporal variation characteristics of aeroengine EGTM mentioned above, the system of aeroengine performance parameter EGTM can be regarded as a grey dynamic system. This section briefly describes the modeling methodologies about GM (1, 1) model and provides an EGTM trend forecasting framework based on GM (1, 1) model.

In grey systems theory, the most commonly used grey forecast model is GM (1, 1) model, which is successfully employed in time series forecast applications with the uncertain problems under discrete data and incomplete information [

Generally speaking, forecast based on GM (1, 1) model can be regarded as curve fitting analysis in time series [

The forecast system based on GM (1, 1) model [

Consider the following nonnegative EGTM time sequence

To increase the forecast of the GM (1, 1) model, the first order accumulating generation operation (1-AGO) is derived from the original sequence

Then the accumulated sequence

The generated mean sequence

The grey model GM (1, 1) can be expressed by one variable, and the grey difference equation is defined as

And its whitening equation is

The solutions

Hence, the time response sequences of (

To obtain the forecast value of the primitive data at time

And the predicted value of the primitive data at time

Compared with the statistical models, GM (1, 1) model need not find the statistics features of original series. So GM (1, 1) model gets rid of the shadow of large-sample statistics in terms of information availability degree [

According to the above method, a GM (1, 1) model based approach for aeroengine performance degradation forecast using EGTM signatures is illustrated in Figure

Flowchart of EGTM forecast based on GM (1, 1) model approach.

Generated sufficient EGTM samples from the historical database and the essential preprocessing upon EGTM data are carried out before data analysis, such as supplementary data, eliminating noise and outliers. After that, the samples can be divided into training samples and testing samples.

The EGTM forecast model based on GM (1, 1) method is established by using the training samples.

The testing samples are used to verify the forecast performance of the GM (1, 1) model. Step

Apply the GM (1, 1) model that meets the accurate requirement to EGTM measured signatures obtained from real aeroengine to forecast.

As a first order single variable grey model, GM (1, 1) model provides an excellent approach to forecast uncertain systems [

A GM (1, 1) Markov chain based approach for aeroengine performance degradation forecast using EGTM signatures is illustrated in Figure

Flowchart of EGTM forecast based on GM (1, 1) Markov chain approach.

According to the forecast values

The real values of

Let the state space of a Markov chain

The matrix

The transition probability of state is written as

The transition probability matrix of states

Generally speaking, consider

When the possibility of a certain state of the next step is determined by the probabilities in

The main assumption in a Markov chain model is that knowledge of the current state occupied by the process can be sufficient to describe the future probabilistic behavior of the process. Another unique property of this Markov chain model is the existence of a steady state matrix.

In this section, the forecast approach based on GM (1, 1) Markov chain introduced in this paper will be applied to EGTM forecast of CFM56 aeroengine to demonstrate the potential capability of the new approach. The comparisons between the EGTM forecast capabilities using the GM (1, 1) model, GM (1, 1) Markov chain model, and other traditional methods are adopted. Four performance measures are used to examine the forecast accuracy of forecast models in this paper. The relative percentage error (RPE), mean square error (MSE), absolute mean error (AME), and absolute mean percentage error (AMPE) are calculated using the following functions, respectively:

For this study, the investigators gathered data samples from CFM56 aeroengine of China Southern, which have been described in Figure

Sample sets partition for EGTM forecast.

After the generation of samples, GM (1, 1) model is established by (

The forecast results of EGTM based on GM (1, 1) model and the original series are plotted in Figure

Forecast values of EGTM based on GM (1, 1) model.

The accumulated data of EGTM using the 1-AGO formation.

In order to compare other models, the linear regression model

The comparison results of GM (1, 1) model and GM (1, 1) Markov chain model.

Cycle | Forecast model | |||||
---|---|---|---|---|---|---|

Original values | Linear model | Nonlinear model | ARIMA model | RBF model | GM (1, 1) model | |

4200 | 86 | 73.088 | 74.66925168 | 76.24820454 | 83.893778 | 74.850179 |

4400 | 86 | 71.808 | 73.63116571 | 85.04659158 | 75.345291 | 73.899849 |

4600 | 76 | 70.528 | 72.60751169 | 85.09212012 | 66.440678 | 72.961583 |

4800 | 72 | 69.248 | 71.59808897 | 79.58775994 | 77.213138 | 72.035231 |

5000 | 64 | 67.968 | 70.60269971 | 69.85924741 | 75.907728 | 71.12064 |

5200 | 66 | 66.688 | 69.6211488 | 69.85924741 | 51.433499 | 70.217661 |

5400 | 72 | 65.408 | 68.65324386 | 70.37022346 | 77.51105 | 69.326147 |

5600 | 70 | 64.128 | 67.69879518 | 74.57638112 | 68.512595 | 68.445951 |

5800 | 58 | 62.848 | 66.75761567 | 72.84613016 | 52.139635 | 67.576931 |

6000 | 62 | 61.568 | 65.82952088 | 62.6255194 | 75.584126 | 66.718945 |

6200 | 66 | 60.288 | 64.91432888 | 65.48119394 | 63.496734 | 65.871852 |

6400 | 58 | 59.008 | 64.01186029 | 68.51302315 | 64.808649 | 65.035514 |

6600 | 60 | 57.728 | 63.12193824 | 61.40259349 | 54.990815 | 64.209794 |

6800 | 60 | 56.448 | 62.24438829 | 62.78391435 | 60.518705 | 63.394558 |

7000 | 60 | 55.168 | 61.37903843 | 62.49368451 | 62.428861 | 62.589673 |

7200 | 64 | 53.888 | 60.52571907 | 62.25825584 | 61.24122 | 61.795007 |

7400 | 60 | 52.608 | 59.68426294 | 65.60171099 | 57.527047 | 61.01043 |

7600 | 68 | 51.328 | 58.85450511 | 62.02015608 | 58.92226 | 60.235815 |

7800 | 62 | 50.048 | 58.03628296 | 68.8277359 | 53.706033 | 59.471034 |

8000 | 68 | 48.768 | 57.22943611 | 63.72946594 | 60.035962 | 58.715964 |

8200 | 66.5 | 47.488 | 56.4338064 | 68.75093548 | 60.237787 | 57.97048 |

8400 | 65 | 46.208 | 55.64923791 | 67.47264457 | 67.031388 | 57.234461 |

8600 | 55 | 44.928 | 54.87557684 | 66.17459205 | 53.436919 | 56.507787 |

8800 | 55 | 43.648 | 54.11267156 | 57.5174732 | 55.602918 | 55.79034 |

9000 | 57 | 42.368 | 53.36037254 | 57.31506291 | 58.04245 | 55.082001 |

In the previous literature, there is no unified standard to determine number of state and the boundary of the

According to the absolute error series

State division for the time series of EGTM.

From Figure

When the possibility of a certain state of the next step is determined according to 1-step probability transition matrices

: The forecast values by GM (1, 1) model and GM (1, 1) Markov chain model for EGTM.

The forecast RPE by linear regression model and nonlinear regression model for EGTM.

The forecast values by ARMIA (1, 0, 0) model and RBF model for EGTM.

The forecast values by GM (1, 1) model and GM (1, 1) Markov chain model for EGTM.

The comparison of forecast accuracy of the different models is listed in Table

The comparative analysis results of different models.

Models | Evaluating standards | ||
---|---|---|---|

MSE | AME | AMPE/% | |

Linear model | 108.1934 | 8.572960000 | 13.07414 |

Nonlinear model | 37.67832 | 4.861358116 | 7.209246 |

ARIMA model | 37.36459 | 4.785827391 | 7.447413 |

RBF model | 48.05668 | 5.59155023 | 8.443975 |

GM (1, 1) model | 34.2345 | 4.673887228 | 6.982612 |

GM (1, 1) Markov chain model | 11.75861 | 1.981937814 | 2.727393 |

According to the characteristics of the aeroengine gas path performance parameters, EGTM is used to realize aeroengine performance degradation forecast. Based on the change law of aeroengine EGTM, EGTM forecast is solved as a grey system forecast problem. However, it is shown that GM (1, 1) model is only able to accurately forecast the trend of EGTM and the forecast accuracy of GM (1, 1) model is not satisfactory when the EGTM data show great randomness. In order to enhance the forecast performance of GM (1, 1) model, Markov chain model is integrated into GM (1, 1) model. The comparison results show that the forecast accuracy of the improved model named GM (1, 1) Markov chain model is better than other models, especially for the small samples. GM (1, 1) Markov chain model can solve the influence of random fluctuation data on forecast accuracy and achieve an accurate estimate of the nonlinear EGTM.

The authors declared that there is no conflict of interests regarding the publication of this paper.

The present work is supported by the Fundamental Research Funds for the Central Universities of China (no. HEUCFZ1005).