Performance assessment of hydraulic servo system based on bi-step neural network and autoregressive model

2.1. Servo valve

The physical parameters of the servo valve are shown in Table 1.

2.2. Cylinder

Considering the injection of internal leakage fault, the cylinder consists of three hydraulic component design modules: 2 piston modules, 1 leakage and viscous friction module.

The physical parameters of the piston modules are shown in Table 2.

The physical parameters of the leakage and viscous friction module are shown in Table 3.

2.3. Mass and displacement limit

The physical parameters of the mass and displacement limit module are shown in Table 4.

2.4. Displacement sensor

The physical parameters of the displacement sensor are shown in Table 5.

2.5. Force sensor

The physical parameters of the force sensor are shown in Table 6.

3.1. Observer model based on 1st RBF neural network

Suppose that the hydraulic servo system can be described as:

where $X\left(t\right)$, $Y\left(t\right)$, $U\left(t\right)$ and $f\left(t\right)$ denote the state vector, output vector, input vector, and failure vector, respectively, and $g$ and $h$ are the nonlinear vector functions.

The state observer is defined as:

The state error is defined as:

If $\underset{t\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}e\left(t\right)=0$ when $f\left(t\right)=0$ or $f\left(t\right)\ne 0$, then Eq. (2) is called the fault observer of Eq. (1).

To effectively describe the nonlinearity in the hydraulic servo system, the RBF neural network shown in Fig. 4 is adopted as a fault observer, where $y_r$ and $r$ denote the output and input of the system, respectively, and ${\widehat{y}}_r$ is the estimated output of the fault observer.

The adopted RBF neural network consists of three layers: the input layer, the hidden layer and output layer. The input layer of an RBF neural network is constituted by input nodes; the second layer is hidden layer, and the number of nodes can be determined as required; the third layer is output layer. The output of an RBF neural network is:

Normally, a Gaussian function is chosen as the radial basis function:

where the input vector $x$ is the independent variable vector of Gaussian function; $C$ is the constant vector, namely, the center of radial basis function; and $\Phi (x-C)$ is the radial basis function.

For the training of the RBF neural network observer, the mean squared error goal is set to 0.08; the spread of radial basis function, 2; and the maximum number of neurons, 50, respectively.

The residual error is generated based on a comparison between the actual and the estimated system response. The latter is generated by the RBF neural network observer. The residual error is defined as the difference between the actual and the estimated output:

Under normal conditions, the residual error is only due to the influence of noise and modeling errors, and it is close to zero. The residual error in this case is defined as the baseline residual and is denoted as ${\epsilon }_0\left(k\right)$. However, in the presence of some faults, the residual error deviates from zero in characteristic ways.

3.2. Adaptive threshold generation based on 2nd RBF neural network

The main factors influencing the threshold of the hydraulic servo system are the system input, system output, and some random factors such as the modeling error, random disturbance, random noise, and parameter drift. Considering that the observer is very robustness, the authors assume that the threshold only depends on the system input and system output. Then, the mapping relationship among the system input, system output, and threshold can be established. Another RBF neural network is selected to fit this relationship because RBFs have good nonlinear mapping capability and training efficiency. Like the RBF neural network mentioned above, the RBF neural network that is used to generate threshold also contain three layers, and the radial basis function is Gaussian function.

The second RBF neural network is trained by using the system input and estimated output as the input training datasets and the expected threshold of the system as the output training datasets. The expected threshold of the system depends on the baseline threshold $\left({\epsilon }_0\left(k\right)\right)$ and influencing factors (see Eq. (7)):

In Eq. (10), ${\epsilon }_0\left(k\right)$ is the baseline threshold generated by the first neural network observer and $\beta$ is the correction coefficient.

Regarding the RBF network training process for the adaptive threshold generation, the mean squared error goal, the spread of radial basis functions, and the maximum of neurons are defined as 0.001, 2, and 30, respectively.

3.3. Coefficient extraction of AR model

An AR model is simply a linear regression of the current value against one or more prior values of the series. An AR model of order $n$ is defined as:

where ${\phi }_i$ are the AR coefficients and $a_t$ is a Gaussian white-noise series with zero mean and variance ${\sigma }_a^2$. An AR model is a time sequence analysis method whose parameters contain important information about the system condition, and it has been successfully applied to the fault diagnosis in recent years. In fact, in view of the stationarity in the input signal of hydraulic servo systems, the spectrum of the AR moving average process can be represented purely in terms of AR coefficients, without the need of computing the moving average model coefficients. The advantage of the AR model is: differing from a moving average model (the moving average part of ARMA), its parameters can be determined by solving a linear set of equations. Generally, an AR model requires far fewer coefficients than the corresponding moving average model, and thus, it is more effective for those signals with sharp peaks (e.g. residual error and adaptive threshold).

The selection of the order of the AR model is a crucial step because spurious spectral peaks and general statistical instability will be caused if the order is too large, whereas smoothed spectral peaks will be cause if the order is too small. In this study, the order is selected based on the AIC. The AIC, which reflects the effect of the spectral variance due to the increase in model order $n$ and prediction errors computed in estimating the AR coefficients, is given by:

In this study, an order of 3 is selected for the AR model.

When the signal is stationary, the coefficients of the AR model are most commonly determined either by using the autocorrelation (second-order statistical characteristic) of the signal or by solving the Yule-Walker equations, and the latter can be done by several methods. The two most commonly used methods are the Levinson-Durbin recursion (LDR) algorithm and the Burg (maximum entropy) method (BM). Both ARMA and ARIMA models can be applied to both stationary and non-stationary signals, but the increasing computation time is needed. Considering the input command of hydraulic servo system is stationary signal, an AR model is sufficient for the purpose of performance assessment. In this study, the Burg method is adopted.

3.4. MD calculation and normalization

MD, introduced by P.C. Mahalanobis in 1936, is a multivariate generalized measure used to determine the distance of a data point to the mean of a cluster. It is measured in terms of the standard deviations from the mean of the samples and provides a statistical measure of how well the unknown data set matches with the ideal one. The advantage of the MD is that it is sensitive to the inter-variable changes in the reference data. In this study, the MD is used to assess the performance of hydraulic servo systems.

The MD is calculated as follows:

a) Calculate the mean for each characteristic in a normal dataset as:

b) Calculate the standard deviation for each characteristic:

c) Normalize each characteristic, form the normalized data matrix $\left(Z_{ij}\right)$, and take its transpose $\left(Z_{ij}^T\right)$:

d) Form the correlation matrix $\left(C\right)$ for the normalized data. Calculate the matrix elements $\left(C_{ij}\right)$ as follows:

e) Finally, calculate the MD as:

As mentioned above, the order of the AR model is 3 in this study. The AR(3) process is given by Eq. (15), where ${\phi }_1$, ${\phi }_2$ and ${\phi }_3$ are the coefficients of the process and $a_t$ is an independent and identically distributed random noise process with zero mean and variance ${\sigma }_a^2$:

The AR(3) model divides $x_t$ into two parts: deterministic part and stochastic part. The former is determined by the expectation of $x_t$, that is:

Therefore, the coefficients ${\phi }_1$, ${\phi }_2$ and ${\phi }_3$ reflect the system performance.

Suppose that the residual error contains $N$ epochs, and one spot $A_i({\phi }_{i1},{\phi }_{i2},{\phi }_{i3})$ in three dimensions is extracted from $M$ epochs. Then, $n$ spots $(n=[N/M\left]\right)$ are extracted from the residual error. Similarly, $n$ spots $B_1\left({\psi }_{11},{\psi }_{12},{\psi }_{13}\right),B_2\left({\psi }_{21},{\psi }_{22},{\psi }_{23}\right),\dots ,B_n\left({\psi }_{n1},{\psi }_{n2},{\psi }_{n3}\right)$ are extracted from the adaptive threshold.

Under normal conditions, the value of the difference between the coefficients of the residual error and the coefficients of the adaptive threshold contains $n$ spots, and these spots are denoted as follows:

where $\left({\phi }_{0_{11}},{\phi }_{0_{12}},{\phi }_{0_{13}}\right),\left({\phi }_{0_{21}},{\phi }_{0_{22}},{\phi }_{0_{23}}\right),\dots ,({\phi }_{0\_n1},{\phi }_{0\_n2},{\phi }_{0\_n3})$ and

$\left({\psi }_{0_{11}},{\psi }_{0_{12}},{\psi }_{0_{13}}\right),\left({\psi }_{0_{21}},{\psi }_{0_{22}},{\psi }_{0_{23}}\right),\dots ,({\psi }_{0\_n1},{\psi }_{0\_n2},{\psi }_{0\_n3})$ are the coefficients of the AR(3) model extracted from the residual error and the adaptive threshold, respectively, under normal conditions. Below, $\{C_1,C_2,\cdots ,C_n\}$ is used to construct the Mahalanobis space as the benchmark reference.

Similarly, $n$ spots can be extracted from the residual error and adaptive threshold under unknown conditions or the most recent measurement. These spots are denoted as follows:

where $({\phi }_{11},{\phi }_{12},{\phi }_{13}),({\phi }_{21},{\phi }_{22},{\phi }_{23}),\cdots ,({\phi }_{n1},{\phi }_{n2},{\phi }_{n3})$ and

$({\psi }_{11},{\psi }_{12},{\psi }_{13}),({\psi }_{21},{\psi }_{22},{\psi }_{23}),\cdots ,({\psi }_{n1},{\psi }_{n2},{\psi }_{n3})$ are the coefficients of the AR(3) model extracted from the residual error and the adaptive threshold under unknown conditions.

By using the method above, the MD can be obtained. As aforementioned, the Mahalanobis space is constructed by using the differences between the coefficients of the AR model for the generated residual error and the adaptive threshold under normal conditions. Then, the distance between $X_1,X_2,\cdots ,X_n$ and the constructed Mahalanobis space, which is defined as Mahalanobis distance, actually indicates how far the most recent feature vector deviates from normal condition. Hence, the degradation trend can be visualized by the curve of MD over time. As the MD increases, the performance degradation becomes severer accordingly.

Then, the MD is normalized into [0, 1] as a confidence value (CV). The CV, representing the performance of a hydraulic servo system, can be formulated as below:

where $d$ is the MD and $a$ is a shape parameter. The MDs obtained from normal states are closer to the constructed Mahalanobis space, compared with those obtained from performance degradation or fault states. Meanwhile, the normalization function is a monotone decreasing function, thus, the CV will be higher as the system shows better performance.

4.1. Experimental design

A simulation model with fault injection modules was used to evaluate the proposed method. The hydraulic servo system consists of various components, as shown in Figs. 2 and 3. Statistical data of hydraulic servo system faults indicates that the main fault modes of these systems are electronic amplifier faults, actuator cylinder faults, leakage faults, and sensor faults. In this study, electronic amplifier fault, sensor fault and internal leakage fault were injected to the simulation model. The detail of fault injection is described below.

(1) Electronic amplifier fault injection

Electronic amplifier fault injection module is shown in Fig. 2. The parameters of normal amplifier and fault amplifier are shown in Table 7.

(2) Sensor fault injection

The physical parameters of fault sensor are shown in Table 8.

(3) Leakage fault injection

In order to compare the CVs of hydraulic servo system under different degradation level, slight leakage fault, moderate leakage fault and severe leakage fault were injected. The physical parameters of leakage fault are shown in Table 9, Table 10 and Table 11.

The input signal for the simulation model of the hydraulic servo system is:

Faults were introduced into the actuator cylinder, electronic amplifier, and sensor. For data acquisition, a sampling rate of 10 is used; the simulation time is 600 s. Four tests were conducted, and the details of each test are shown in Table 12. The correction coefficient in this case is 0.2. Note that one spot was extracted from 50 epochs of residual error/adaptive threshold, and therefore, the confidence value contains 120 epochs.

4.2. Procedure and details of the test

The approach adopted in this study is shown in Fig. 4, and more details and the test procedure are shown in Fig. 5. In this flowchart, the authors take test 4 as an example. The others tests were conducted similarly.

4.3. Comparison of test results

The performance confidence values of the hydraulic servo system as calculated by using the proposed method are shown in Fig. 6. This value indicates the performance of the system. The curve of this value over time is representative of the degradation of the hydraulic servo system. In test 1, no fault was injected into the system; therefore, the system operated normally, with a performance confidence value of ~0.9. In test 2, an electronic amplifier fault was injected into the model at $t=$ 200 s; then, it was observed that between 1 and 200 s (1st to 40th epochs in figure), the performance confidence value fluctuated around 0.9, before it decreased abruptly at $t=$ 200 s (40th epoch in figure). In test 3, a sensor fault was injected, then, at $t=$ 200 s (40th epoch in figure), the performance confidence value decreased to 0.7. In test 4, three internal leakage faults were injected into the cylinder, with different fault severity; at $t=$ 150 s (30th epoch in figure), an internal leakage fault with clearance diameter of 1mm was injected, and the performance CV decreased to 0.6; at $t=$ 300 s (60th epoch in figure), a severer leakage fault with clearance diameter of 1.1 mm was injected to the cylinder, then, the performance CV decreased to ~0.55; at last, the performance CV decreased to ~0.4 as a more severer leakage fault was injected. These test results indicate that the performance confidence value can not only detect the occurrence of faults but also reflect the degradation in performance. Thus, the performance confidence value obtained by using the proposed method can serve as an effective index for determining whether maintenance needs to be carried out.

Fig. 5Step-by-step diagram of the proposed performance assessment method

Fig. 6Comparison of results for each fault in the hydraulic servo system