# Fisher Discriminative Sparse Representation Based on DBN for Fault Diagnosis of Complex System
## 1 Introduction  
With the increase in the complexity of operation and multiloop control processes, a robust fault detection and diagnosis (FDD) method has become increasingly necessary for industrial processes. Fault detection and diagnosis involves developing a method that detects faults quickly, diagnosing the fault type, and determining the root causes of the fault with the least number of false alarms [1; 2]. A good fault detection and diagnosis system not only guarantees the quality of the product but also improves the safety of the system. Fault diagnosis methods are categorized into data-based, model-based, knowledge-based, and hybrid methods [3]. Knowledge-based methods are mature and we do not discuss them here. Model-based methods have been proposed by [PERSON] (1971) to replace hardware redundancy with analytical redundancy, and [PERSON] (1976) proposed the key concepts of analytical redundancy for model-based fault detection and diagnosis. Next, parameter estimation, observer-based algorithms, and parity space methods were introduced and later improved. They were used to solve the control problem in heating, ventilating, air conditioning, motor, and engine systems [3; 4]. Based on the model constructed using the physical principles and system identification techniques, fault diagnosis methodswere developed to monitor the error between the outputs of practical systems and model-predicted outputs [5; 6; 7; 8; 9]. Generally, obtaining an explicit model to accurately describe the dynamic behavior of a complex system is difficult, and model-based algorithms [3; 5; 6; 7] have considerable limitations. Due to the extensive applications of modern measurements and the quick development of data analysis methods, data-driven process monitoring techniques have attracted the attention of experts [9; 10; 11; 12; 13]. Many fault detection and diagnosis methods have been proposed based on some widely used dimensionality reduction techniques, such as partial least squares (PLS) [14], independent component analysis (ICA) [15], principle component analysis (PCA) [16], and locally preserving projections (LPP) [17]. These multivariate statistical process monitoring (MSPM) methods are commonly used to extract features based on the established distribution, determine the latent variables, and then construct a monitoring model using the extracted features for fault detection and diagnosis. One common drawback of the above dimensionality reduction methods is that the extracted features are the linear combination of all input variables. Furthermore, the lack of sparsity in the features hides the physical connection of complex systems. To reveal the deeper connection between the variables and create a proper representation of physical meaning, this paper proposes a novel feature extraction scheme for complex system fault diagnosis using deep learning and sparse representation. Deep learning allows computational models composed of multiple processing layers to learn the representation of data [18; 19; 20; 21]. The backpropagation algorithm (BP algorithm) is applied after the weights training step in deep learning to discover intricate structures in large data sets, and the BP algorithm makes up the disadvantage of gradient diffusion in pretraining weights of the network. The convolutional neural network proposed by [PERSON] was the first deep learning algorithm that trained multilayer networks successfully, but it was not sufficiently effective during application. A breakthrough in deep learning occurred in 2006 when [PERSON] introduced the deep belief network (DBN), a novel deep learning architecture [18]. Since then, ever-increasing research has been performed in the fields of pattern recognition, computer vision, and data mining. [PERSON] et al. applied DBN to fault feature extraction and diagnosis of bearings and introduced PCA to extract two principle components for visualization [22]. The authors determined the DBN can both extract the features of fault types in single work conditions and the features of blended conditions.