Machine health condition (MHC) prediction is useful for preventing unexpected failures and minimizing overall maintenance costs since it provides decision-making information for condition-based maintenance (CBM) (Vachtsevanos et al. 2006). Typically, MHC prediction methods can be divided into two categories, namely, model-based data-driven methods (Jardine et al. 2006). Due to the difficulty of deriving an accurate fault propagation model (Yu et al. 2012;, Yu et al. 2011), researches have focused more on the data-driven method in recent years (Si et al. 2011). The neural network (NN)-based approach, which falls under the category of the data-driven method, has been considered to be very promising for MHC prediction due to the adaptability, nonlinearity, and universal function approximation capability of NNs (Tian and Zuo 2010). Batch learning and sequential learning are two major training schemes of NNs. MHC prediction is essentially an online time-series forecasting problem which should perform real-time prediction while updating the NN. Thus, to save updating time and to maintain consistency of the NN, the sequential learning should be employed in such a problem.
The most popular NNs applied to MHC prediction are recurrent NNs (RNNs) and fuzzy NNs (FNNs). In Gu and Li (2012), an extended RNN which contains both Elman and Jordan context layers was developed for gearbox health condition prediction. In Zhao et al. (2009), a FNN in Brown and Harris (1994) was applied to predict bearing health condition. In Wang et al. (2004), an enhanced FNN was developed to forecast MHC. Next, in Wang (2007) and Liu et al. (2009), a recurrent counterpart of the approach in Wang et al. (2004) and a multistep counterpart of the approach in Wang (2007) were presented to predict MHC, respectively. An interval type-2 FNN was also proposed to predict bearing health condition under noisy uncertainties in Chen and Vachtsevanos (2012). Note that the batch learning was employed in Tian and Zuo (2010), Zhao et al. (2009), and Chen and Vachtsevanos (2012). Common conclusions from Tian and Zuo (2010), Zhao et al. (2009), and Wang et al. (2004), Wang (2007), Liu et al. (2009), and Chen and Vachtsevanos (2012) are that the RNN usually outperforms the feedforward NN and the FNN usually outperforms the feedforward perceptron NN, feedforward radial basis function (RBF) NN, and RNN. Recently, to improve prediction performance under measurement noise, an integrated FNN and Bayesian estimation approach was proposed for predicting MHC in Chen et al. (2012), where a FNN is employed to model fault propagation dynamics offline and a first-order particle filter is utilized to update the confidence values of the MHC estimations online. In Chen et al. (2011), a high-order particle filter was applied to the same framework of Chen et al. (2012). A question in the approaches of Chen et al. (2011, 2012) is that the FNNs should be trained by the system state data (rather than the output data) which are assumed to be immeasurable.
Extreme learning machine (ELM) is an emergent technique for training feedforward NNs with almost any type of nonlinear piecewise continuous hidden nodes (Huang et al. 2006). The salient features of ELM are as follows (Huang et al. 2006): (i) All hidden node parameters of NNs are randomly generated without the knowledge of the training data; (ii) it can be learned without iterative tuning, which implies that the hidden node parameters are fixed after generation and only output weight parameters need to be turned; (iii) both training errors and weight parameters need to be minimized so that the generalization ability of NNs can be improved; and (iv) its learning speed is extremely fast for all types of learning schemes. ELM demonstrates great potential for MHC prediction due to these salient features. Nonetheless, the original ELM proposed in Huang et al. (2006) is not appropriate for predicting MHC since it belongs to the batch learning scheme. To enhance the efficiency of ELM, online sequential ELM (OS-ELM) was developed in Liang et al. (2006) and was further applied to train the FNN in Rong et al. (2009). Due to its extremely high learning speed, the OS-ELM-based FNN in Rong et al. (2009) seems to be suitable for MHC prediction. Yet, there are two drawbacks in Rong et al. (2009) as follows: (i) It is not good to yield generalization models since only tracking errors are minimized and (ii) it may encounter singular and ill-posed problems while the number of training data is smaller than the number of hidden notes.
To further improve the efficiency of MHC prediction, a novel FNN with an enhanced sequential learning strategy is proposed in this paper. The design procedure of the proposed approach is as follows: Firstly, a ellipsoidal basic functions (EBFs) FNN is proposed; secondly, the FNN approximation problem is transformed into the bi-objective optimization problem; thirdly, an enhanced online sequential learning strategy based on the ELM is developed to train the FNN; and finally, a multistep direct prediction scheme based on the proposed learning strategy is presented for MHC prediction. The developed enhanced online sequential learning-FNN (EOSL-FNN) is applied to predict bearing health condition by the use of real-world data from accelerated bearing life. Comparisons with other NN-based methods are carried out to show the effectiveness and superiority of the proposed approach.

Architecture of Fuzzy Neural Network

For MHC prediction, the n-input single-output system is considered. Yet, the following results can be directly extended to the multi-input multi-output (MIMO) system. The FNN is built based on an EBF NN. It is functionally equivalent to a Takagi-Sugeno-Kang (TSK) fuzzy model that is described by the following fuzzy rules (Wu et al. 2001):

where xi ∈ ℝ and ŷ ∈ ℝ are the input variable and output variable, respectively; Aij is the antecedent (linguistic variable) of the ith input variable in the jth fuzzy rule; wj is the consequent (numerical variable) of the jth fuzzy rule, i = 1, 2, . . ., n, j = 1, 2, . . ., L; and L is the number of fuzzy rules.

As illustrated in Fig. 6, there are in total four layers in the FNN. In Layer 1, each node is an input variable xi and directly transmits its value to the next layer. In Layer 2, each node represents a Gaussian membership function (MF) of the corresponding Aij as follows:

where cij ∈ ℝ and σij ∈ ℝ+ are the center and width of the ith MF in the jth fuzzy rule, respectively. Note that the MF in Eq. 8 is an EBF since all its widths σij are different (Wu et al. 2001). In Layer 3, each node is an EBF unit that denotes a possible IF-part of the fuzzy rule. The output of the jth node is as follows:

Online Sequential Learning Strategy