Fault diagnosis of rolling bearing based on fuzzy neural network and chaos theory

2.1. Phase space reconstruction

Through the study of chaos by the time series, Packard et al. proposed phase space reconstruction in 1980 [9]. Phase space reconstruction is the base of chaotic time series analysis, while the key of reconstruction is its time delay selection and embedding dimension determination. Phase space reconstruction is to return to the chaotic attractor in high-dimensional space. According to Takens theory [10], we can use one-dimensional chaotic time series $x\left(t\right)$, $t=\mathrm{}$1, 2, 3,… to define the points of $d$ dimensional $Y\left(t\right)$, $t=\mathrm{}$1, 2, 3,… so:

where $d$ is called system’s embedding dimension while $\tau$means time delay.

By assuming $D$ is the real dimension of chaotic attractor, Takens proved that we can find $d$ as a suitable embedding dimension. When $d$ meets the condition:

So we can recover the regular trajectory attractor in the dimension. That is to say, the original dynamic system keeps differential homeomorphism in the trajectory of reconstructed $R^d$ space.

2.2. Grassberger-Procaccia (G-P) algorithm

Grassberger-Procaccia or G-P algorithm [11] proposed by Grassberger and Procaccia in 1983 is a relatively common method of solving system correlation dimension, which refers to the true dimension estimated value of the attractor. Where $D_m$ is the estimated value of correlation dimension.

Correlation dimension can be approximated with the following formula:

where the associated vector is any distance less than a given positive number $r$. If the phase space reconstruction has $N$ points or vectors. $H$ is the Heaviside unit function.

In fact, it is not until the double logarithm $\mathrm{l}\mathrm{n}{C}_d\left(r\right)~\mathrm{l}\mathrm{n}\left(r\right)$ slope of the linear portion substantially keeps constant that the small embedding dimension $d$ usually becomes bigger one. That is to say, the embedding dimension curve reaches saturation. The slope of the best fit straight line except for zero or infinite is the correlation dimension $D_m$. After solving it, system phase space reconstruction obtains the minimum embedding dimension $m$ as follows i.e. the $d$ meets the minimum conditions:

2.3. The structure of fuzzy neural network

The fuzzy neural network (FNN) model designed in this study use the time series phase space reconstruction to obtain the minimum embedding dimension $m$ a as the network’s input nodes. Since the minimum embedding dimension $m$ may reflect intrinsic properties of one-dimensional time-series data reconstruction. Hence the designed input nodes have certain significance.

The fuzzy network model used to predict is the Takagi-Sugeno model [12]. It is a MISO–based network, $x\left(t\right)\text{,}x(t-1)\text{,...,}x(t-m+1)$ are the input nodes and the predicted value $\widehat{x}(t+1)$ of time series are the output nodes. The specific structure is shown in Fig. 1.

Fig. 1The structure of FNN

4.1. The performance simulation of virtual signals

Assuming a chaotic time series is expressed as Eq. (10):

Fig. 3 shows 1200 data points of time series of the $X$ axis. We select the first 1000 data from it for simulation to validate the diagnosis method, and the rest of the data are usefulness.

Fig. 3Time series data in the X axis

Where the first 500 training samples were trained for the fuzzy neural network, and the remaining 500 test samples were used to test the prediction performance of the fuzzy neural network.

According to the G-P algorithm, with the embedding dimension is increasing, the corresponding diagrams of $\mathrm{l}\mathrm{n}{C}_m\left(r\right)-\mathrm{l}\mathrm{n}\left(r\right)$ of time series data as showed in Fig. 4. We can obtain its time delay and the $m$ minimum embedding dimensions values are ten and four respectively. Therefore, the input node of the fuzzy neural network from the minimum embedding dimensions is 4.

Fig. 4The corresponding diagrams of lnCm(r)-ln(r) of time series data

Fig. 5a) The results of single step prediction of timing data based on fuzzy neural network, b) The absolute prediction error

a)

b)

Through an experiment on the first 500 training samples, we can obtain the single prediction step of the fuzzy neural network, as showed in Fig. 5(a), Fig. 5(b) shows in the absolute error that the prediction the Mean Square Error ($MSE=$ 6.3817×10^-6) using the fuzzy neural network to predict.

4.2. Fault diagnosis of rolling bearing based on chaotic fuzzy neural networks

We put the normal data normalization processing by phase space reconstruction and use G-P algorithm to obtain the corresponding diagrams of $\mathrm{l}\mathrm{n}{C}_m\left(r\right)-\mathrm{l}\mathrm{n}\left(r\right)$ of time series data with the $m$ embedding dimensions increasing in the normal state of the rolling bearing, as shown in Fig. 6.

Fig. 6Training and test data

Accordingly, the corresponding time delay $\tau$ and the $m$ minimum embedding dimensions are respectively ($\tau =$3, $m=$ 4), that is to say, the number of input nodes of the fuzzy neural network is 4. And in the predication data, the first 600 data are used as training sets, and the rest of 400 data are used as testing sets of the diagnosis model.

Fig. 7(a) is the 100 (800-900) data points’ prediction results, Fig. 7(b) is its absolute errors which are relatively clear display. We can see that the experimental prediction results are satisfactory, and moreover its predicted trajectory tracking performs well. The MSE value is only 2.8851×10^-5 which gives a perfect proof.

Fig. 7a) The results of single step prediction of rolling bearing based on chaos fuzzy neural network, b) The absolute prediction error of rolling bearing the 100 (800-900) test data

a)

b)