Identification based fault detection: Residual selection and optimal filter☆
Introduction
Process safety and product quality are two important issues of modern industrial processes [1]. Process monitoring and fault diagnosis for the industrial processes is an important area of the scientific research and technological development. Fault refers to an unpermitted deviation of at least one characteristic or variable in a system [2]. From a control engineering viewpoint, industrial fault types can be divided into input (actuator) faults, output (sensor) faults and process faults [3]. The fault detection, isolation and diagnosis technique have been extensively investigated over several decades.
Three representative branches of fault diagnosis techniques are data-driven, model-based and signal-based. Along with the popularity of big data, data-driven methods have now become a hotspot in fault diagnosis [4], [5]. The advantages of data-driven method lie in the fact that they are easy to implement, requiring very little modeling effort and prior knowledge [6]. In general, they utilize multivariate analysis approaches to extract useful features and information from process historic data; fault diagnosis is realized by analyzing the changes between new measurements and process history in the feature space. These methods are represented by principal component analysis (PCA), partial least squares (PLS), independent component analysis (ICA) and canonical correlation analysis (CCA) [7]. For deeper insights and comparisons of these methods, we recommend the review papers [4] and [8].
Signal-based methods directly utilize measured signals, rather than models, for fault diagnosis [9]. The measured signals, also known as condition monitoring data or alarm data, can either be time-domain or frequency-domain. More specifically, the signals can be vibration, temperature, strain, acoustic etc. as long as they can indicate the status of the equipment. Signal-based methods first extract features and symptoms from the signals, then diagnose faults based on symptom analysis and the corresponding prior knowledge. The methods can be classified into time-domain, frequency-domain and time–frequency [9], [10]. Signal-based fault diagnosis have attracted much attention and been widely used in mechanical systems, e.g. induction motors, gearbox, rolling mills, wind turbines [11], [12], [13], [14]. Recently, with the aid of data fusion as well as feature extraction and reduction technique, signal-based methods can take more relevant information into account such that their accuracy and reliability can be further improved [10], [15], [16], [17].
Unlike the aforementioned two branches, model-based methods require causal input–output models. Such models, after proper simplification and treatment, are often in the class of differential equations in view of mathematical modeling, and expressed in the forms of state–space equations or transfer-function matrices from control engineering perspective. Based on the models, the consistency between the measured outputs and the model outputs can be checked for fault diagnosis [18]. Model-based methods have developed and matured through years and several textbooks are available for thorough review and discussion [18], [19], [20], [21]. Industrial processes are essentially dynamic and there are casual relationships between various variables. These two features are captured by state–space models or by transfer-function models. However, dynamics and causality cannot be properly described by data-driven models and signal-based methods, although for data-driven methods, many dynamic variations of the original ones have been proposed, for instance dynamic inner principal component analysis (DiPCA), dynamic inner canonical correlation analysis (DiCCA), and dynamic inner partial least squares (DiPLS) [22].
The main obstacle that hinders the widespread use of model-based methods is the high cost in the process modeling because most model-based fault diagnosis methods are based on first-principle state–space models, also called rigorous models. As reported, the application fields has so far been concentrated in the aerospace, mechanical and electrical engineering [23], [24], [25], successful applications in process industries are rare [26] because accurate rigorous models are very hard to develop in process industries. Although data-driven and signal-based methods are easier to implement, in process industries, data are noisy, lack of excitation. Moreover, many process variables are operated in closed loops which means that there will be little useful information about the real process dynamics to extract and analyze from the normal operation data. These factors pose challenges for applications of both data-driven and signal-based methods in process industries.
System identification is a powerful tool for building industrial process models, and has become the standard tool in industrial advanced process control (APC) in the last three decades [27], [28]. System identification uses test signals to excite the dynamic characteristics of the systems, generating informative data, then estimates dynamic models. Although identified models provide less information about processes than first-principle models (if available), it gives much deeper insight and is more interpretable than data-driven (multivariate analysis) models. Concerning identification related fault diagnosis, early works focus on parameter estimation of some first-principle models [29], [30], [31], [32], which is different from black-box system identification [33] that will be used in this work.
Recently, some scholars applied subspace identification method in fault detection [26], [34], [35], [36]. These works use subspace identification method to obtain process models and use the models to generate residuals, or directly generate residuals from test data using the subspace identification framework, for fault detection. However, many fault detection specific questions remain to be answered. Among many, this work will answer the following fault detection relevant questions: Which error (prediction error or output error) should be used for generating residual signal? Can the detection problem be studied from a frequency domain perspective? How model error should be considered? How to design an optimal detection filter?
In this work and what follows, inspired by the tremendous success of system identification in industrial APC projects, we aim at developing an identification based fault diagnosis scheme for process industries. As a first step in the direction, this work focuses on fault detection, in which residual generation and filtering will be studied. The problem will be approached from a frequency domain perspective. In model-based fault diagnosis literature, contributions exist in this topic [18], [19], [37], [38], which have close relations to and robust control theory. In this work, a fault detection performance index is proposed from a statistical viewpoint which has a natural relation with system identification. Using the performance index, the frequency domain information of faults can be considered and handled. A comparison study on fault detection performance in Tennessee Eastman process (TEP) will be carried out where the proposed method is compared with many existing data-driven methods and a subspace method. The paper unfolds as follows: In Section 2, some basic assumptions and concepts are introduced; Section 3 briefly discusses some system identification methods; Section 4 proposes a detection performance index, studies residual selection and optimal detection filter; Section 5 discusses practical implementation of the method; a numerical example study is given in Section 6 to illustrate the method; in Section 7 the proposed method is applied to Tennessee Eastman process benchmark and the results are analyzed and compared to other existing data-driven methods; Section 8 is the conclusion.
Vectors and matrices are in bold-face. is a time shifter satisfying . denotes probability, , , denote mathematical expectation, variance and covariance operators. is the modulus of a complex number . The norm of a transfer function matrix is defined as: . Gaussian distribution and distribution are denoted as and . With probability is abbreviated as .
Section snippets
Process description
First consider a nominal (fault-free) linear time-invariant (LTI) system with scalar input and output: where and are scalar rational functions in , with , proper and stable, is minimum-phase. Here is the process input signal, is the observed (measured) process output signal, denotes process disturbance and is zero mean white noise signal with variance . The input signal is assumed to be quasi-stationary (see
Prediction error method and asymptotic method
It is necessary to choose identification methods that can deliver sufficient accurate estimates, and applicable under both open and closed loop operations. Prediction error method is such a standard tool in system identification, in which a quadratic cost function of the prediction error is minimized: is the number of data samples, is the estimate. Prediction error method can offer consistent estimates with minimum variance (reaching the Cramér–Rao lower
Residuals selection and optimal filter for fault detection
Once the process model is identified, then utilizing on-line measurements of and , a certain residual can be built. By investigating (9), one could see that when there is no fault, the residual dynamics will be governed by model error and disturbance; when fault occurs the residual will present abnormal deviations. Inspired by this intuition, fault detection is usually carried out according to the following logic: where and are some fault
Filter design
In practice, an approximation to the optimal filter ought to be made, for instance a Butterworth band-pass or low-pass filter which is minimum-phase. The passband width or the cutoff frequency of the filter is a designable parameter. For brevity we only use the word “passband” below. In general, there is an inverse relationship between the width of the passband of a filter and its response to inputs, i.e. the narrower the passband is, the longer the response time, this can be formulated as an
A numerical example
This section studies a relatively simple numerical example, aiming at: (1) show how the assumptions in Section 4 can be realized and verify the theoretical results; (2) illustrate the implementation considerations in Section 5 and how to use the proposed method to handle various kinds of faults. The nominal system is given by: and are both zero mean white noise sequence. The output
Application to Tennessee Eastman process
In this section, the proposed method is applied to Tennessee Eastman process (TEP) benchmark. Tennessee Eastman process was developed to provide a realistic simulation of an industrial process for the evaluation of monitoring methods [60]. Fig. 6 shows its flow diagram with 5 major units: reactor, condenser, compressor, separator and stripper. In [1], data sets corresponding to Tennessee Eastman process were given, which are widely used for fault diagnosis studies. The data sets are generated
Conclusion and discussion
In this work, an identification based fault detection method is developed. Using system identification, process models and model error bounds can be obtained which form the basis of model based fault detection. A fault detection performance index is introduced in a statistical framework that has a natural link to system identification. Based on the performance index, it is shown that the output error residual is better than the prediction error residual for use in fault detection. Then optimal
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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2022, Journal of Process ControlCitation Excerpt :However, the reliance on fault data becomes a restriction according to issue (3). Following the same line of [19], this paper aims at overcoming the fault data reliance and improving the applicability of the optimal filter framework. Specifically, the paper has following contributions: (1) it proposes a scheme that automatically determines an optimal detection filter online without using fault data and that can handle faults with varying spectra; (2) a method of threshold setting based on kernel density estimation (KDE) and an online decision rule are developed to further refine the detection performance.
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2022, Computers and Chemical EngineeringCitation Excerpt :Bottleneck of the model-based methods is the difficulty in model building, especially in process industries. In Zhou and Zhu (2021), we have studied the identification based residual generation and FD problem. Following the same line, this work aims at developing an efficient FI algorithm based on the identified model.