Journal of Control Science and Engineering

Volume 2017 (2017), Article ID 9517385, 10 pages

https://doi.org/10.1155/2017/9517385

## A Bayesian Approach to Control Loop Performance Diagnosis Incorporating Background Knowledge of Response Information

Department of Automation, Xiamen University, Xiamen 361005, China

Correspondence should be addressed to Sun Zhou

Received 20 June 2017; Accepted 3 August 2017; Published 28 September 2017

Academic Editor: Chunhui Zhao

Copyright © 2017 Sun Zhou and Yiming Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To isolate the problem source degrading the control loop performance, this work focuses on how to incorporate background knowledge into Bayesian inference. In an effort to reduce dependence on the amount of historical data available, we consider a general kind of background knowledge which appears in many applications. The knowledge, known as response information, is about what faults can possibly affect each of the monitors. We show how this knowledge can be translated to constraints on the underlying probability distributions and introduced in the Bayesian diagnosis. In this way, the dimensionality of the observation space is reduced and thus the diagnosis can be more reliable. Furthermore, for the judgments to be consistent, the set of posterior probabilities of each possible abnormality that are computed from different observation subspaces is synthesized to obtain the partially ordered posteriors. The eigenvalue formulation is used on the pairwise comparison matrix. The proposed approach is applied to a diagnosis problem on an oil sand solids handling system, where it is shown how the combination of background knowledge and data enhances the control performance diagnosis even when the abnormality data are sparse in the historical database.

#### 1. Introduction

Fault diagnosis is a topic of practical significance in process industries. In complex control loop systems, the control performance could be degraded due to various reasons [1]. Sensors and instrumentation problems are usually revealed as the systematic errors in mass or energy balance equations [2]. Poor control loop performance might also be due to changes in the model or modeling error [3].

One challenge of control loop diagnosis is that some similar evidences could be shared among different faults [4]. In a complex industrial control loop system, there may be lots of observations. For example, a large-scale industrial process includes thousands of process measurements [5]. Many diagnostic algorithms are often designed to identify specific components, while the faults may propagate and influence other components which are not to be detected [6]; thus, the methods could be influenced by possible faults in other components [7]. Moreover, all processes may run subject to uncertainty due to missing information, noise, and so on. Therefore, the occurrence of one fault may lead to flood of abnormal measurements and alarms and make it difficult to distinguish true underlying source.

Fault diagnosis methods for control loop systems may be classified into three categories, qualitative model-based methods, quantitative model-based methods, and data-driven methods [8–13]. However, these methods result in problems when multiple abnormalities have the same influence on the measurements. To deal with these problems, the methods were extended based on qualitative information about signs, magnitudes, and so on, to consider the direction and the magnitude of change [14, 15]. Also, Bayesian method has been proposed; for example, a Bayesian network was constructed from expert knowledge [16]. However, in these methods, the models are assumed to be known. Besides, fuzzy logic methods were proposed [17]. All these previous works rely on prior knowledge only.

To overcome the drawbacks of both quantitative and qualitative model-based diagnosis approaches, data-driven methods are developed [18]. For example, the support vector machine (SVM) methods were proposed that take the diagnosis problem as a classification one [19]. Besides, multivariate statistical process monitoring methods were suggested [20]. Nevertheless, the problem sources of abnormality may not be explicitly identified by means of the variable contribution methods [21].

Then, a systematic probabilistic approach based on Bayesian inference was proposed that considers all possible abnormal observations. A Bayesian framework for control loop performance diagnosis was developed in [4]. The measurements from plenty of monitors are synthesized to generate a probabilistic result to diagnose the fault. Pernestal [22] proposed using Bayesian approaches to isolate faults in diesel engines. In a similar way, a data-based Bayesian approach is suggested in [23], to diagnose underlying sources of control performance degradation.

The main disadvantage with the data-driven or statistical approaches in diagnosis is that their performance relies heavily on the amount of available historical data. However, the requirement of sufficient training data is hardly met in diagnosis applications since faults are rare in normal processes. On the one hand, in their general form, they require sufficient historical samples from all faulty cases. However, in practical applications, there is only a limited amount of data available. On the other hand, the large number of monitors is a principle challenge for Bayesian diagnosis to be applied in industries. It is required in Bayesian inference to estimate the joint likelihood probability density of the observations from all monitors. In previous works, it is shown that the computational effort in estimating the probabilities grows exponentially with respect to the number of monitors [24, 25] That phenomenon is also regarded as the* curse of dimensionality*. It makes it difficult to correctly estimate the likelihood probability in more than five dimensions with practical sample sizes. These works using Bayesian methods are based on training data only, and no explicit background knowledge is integrated about the process under diagnosis.

There is also a general type of background knowledge available. In this paper, we consider incorporating the background knowledge together with the training data under the Bayesian framework in order to improve the diagnosis even if the historical data are insufficient with respect to the monitors number. Regarding the process knowledge, it is possibly known that one measurement in observation vector is equally distributed given different abnormalities. This type of knowledge is very general and can be formulated as constraints on the underlying likelihood probability distributions [22, 27]. It can express several types of process knowledge and appear in many diagnosis applications naturally.

In this paper, the background information is expressed in terms of response signature matrix (RSM). With a translation from RSM to the constraints of the marginal probability of the likelihoods, the background knowledge is explicitly taken into account in Bayesian control loop diagnosis. Moreover, we also suggest using a moving window method to consider a sequence of observation rather than a single observation in the diagnosis. In order to evaluate the proposed approach, we applied it to an oil sand solid handling system in such a case where only a few samples from abnormalities are available.

The rest of this paper is organized as follows. A description of the Bayesian control performance diagnosis problem is introduced first, and in Section 2 some terminologies are reviewed. In Section 3, the problem studied in this paper is stated formally, and in Section 4 Bayesian diagnosis evaluating multiple consecutive observations is presented. The computations of the posterior probabilities for different modes considering historical data only are presented first, and, the approach is extended to incorporate process knowledge in Section 5. In Section 6, the proposed approach is evaluated on the diagnosis problem on the oil sand solids handling system, using training and background knowledge. The conclusion is given in Section 7.

#### 2. Preliminaries: Bayesian Diagnosis for Control Loop Systems

Before going into the details of the Bayesian diagnosis, some terminologies are introduced based on the definition proposed in [23].

*Component*. Assume that the process under diagnosis consists of some components which is possible to fail or not fail. In a typical control loop, the components can be sensors, actuators, controllers, process models, and so on [28]. Assume there are components of interest. Each component may have several different states. For example, a sensor may consist of three states: unbiased, moderate biased, and severe biased.

*Mode*. A mode of the process is defined as an assignment of the states of all the components. It indicates the state of the system. For example, a mode can be as follows: is normal, activator has severe stiction, process model has no error, and sensor has moderate . Each mode represents the status under which the process is operating. It can be normal state or abnormal state. Assume the total number of the modes is . If each component has states, . The mode can be considered as a random scalar variable described by with values , .

*Observation*. There are some monitors, sensors, or add-on indirect measurements such as “ad hoc tests” conducted by engineers, model-based diagnostic tests, and monitoring algorithms that are designed to measure certain parameters. They can be represented by a general designation,* monitors*. the th monitor is denoted by . Assume there are monitors; then . Each monitor output or measurement signal can be represented by a discrete value; for example, low, medium, and high are 3 possible values. Define that* observation *is a -dimension vector composed of the discrete values of all monitor outputs. These outputs may be preprocessed, for example, in diagnostic tests. The observation vector with the domain . Denote an assignment of the observation vector by , , where is the number of different observations. If the th monitor output has discrete values, . Each value () is an -dimensional vector, and we write to explicitly denote the elements. Consider the observation as a random variable.

*Training Data*. A training sample at time consists of simultaneous values of the mode variable and the observation vector at that time. The value is denoted by . All training samples collected from different modes of the system form the training dataset. A realization of training data is denoted by . And denotes the subset of training data entries where the underlying mode is .

##### 2.1. Bayesian Diagnosis

The Bayesian control loop diagnosis proposed in [23] is briefly reviewed in this section. Each component is possible to suffer from some abnormal operating conditions that may degrade the control performance. Also, any fault in one component may influence the monitors for the other components [4]. Consider there are certain probabilistic interconnections between problem causes and monitor outputs [4]. Bayesian inference is applied to compute the probability of mode variable given a current observation and the training observation data set . The posterior probability of every operating mode can be computed based on Bayes’ rule.where is the likelihood probability and is the probability of mode which is typically specified by a priori knowledge. The mode with the highest posterior probability can be determined as the underlying mode based on the* maximum a posteriori *(MAP) principle, and the related abnormality is generally regarded as the fault source.

Thus, the main issue to construct a Bayesian diagnostic system is estimating the likelihood in line with the training observation data . Following the results of [22, 23], a Bayesian algorithm is presented for the likelihood estimation for control loop diagnosis.

#### 3. Formal Problem Formulation

Consider that there is a general type of background information and multiple consecutive observations available. The task is to determine which fault(s) which has caused the measurements, given consecutive observations , training data , and background knowledge that is described as follows.

*Background Knowledge in Terms of Probability Constraints*. Background information usually comes from expert or process knowledge. It can be described as *.* It can be considered to consist of two parts of information. One specifies the prior probability for the modes, and the other defines that there are elements, representing the monitor outputs, in the observation vector which are equally distributed under different modes.

In addition, rather than considering a single observation as [23], assume that consecutive observations are recorded and that the same fault is present during collection of these observations. Now, the fault diagnosis problem studied in this work can be stated formally as to computethat is, to compute the probabilities that each mode is present at a time instant , given the training dataset , the background knowledge , and the observation values from the control loop process under diagnosis. The subscripts on are used to denote observation vectors from consecutive instants and those on to enumerate the observation values.

In the following, the posterior probabilities of each mode given consecutive observations are calculated with the training data only.

#### 4. Bayesian Inference Using Training Data Only

To solve the stated problem (2), a new method is proposed for learning the likelihood probability distribution. Before going into the details, first let us present a previous result on inference based on training data only.

According to Bayes’ rule, to compute (2), the likelihood probability needs to be calculated:where denotes the training data under mode .

Assume that the likelihood of all possible values of observation under mode is parameterized by ,

Let denote the space of all the likelihood parameters when the mode . Also, the prior probability of these parameters is Dirichlet distributed

It can be shown that the Dirichlet distribution is the only possible choice for under certain, not very restrictive assumptions [29]. One attractive property of Dirichlet distribution is that it is conjugate to the multinomial distribution [30], and the distribution for the training samples is proportional to the multinomial distribution. This makes the computations particularly simple. Further, the parameters of the distribution are required. is the gamma function. For real number .

By marginalizing over all the likelihood parameters, we haveFor the first factor of the integral (6), given the likelihood parameters and assuming that these observations from mode are independent,And for the second factor, following the derivation of [23], we can writeFurther, the likelihood of training data subset related to the operating mode can be calculated aswhere is the number of training data samples with the observation from the mode .

Then combining (7)–(9) and substituting in (6), likelihood (3) can be obtained.

To introduce the consecutive observations, first some notations are needed. Let denote the set of distinct values present in consecutive observations , and let be the total number of observations in with the value . Following [31], the likelihood probability is given by the expressionwhere is the count of the hypothetical samples and is the count of training samples. Theorem A.1 in the appendix can be referred to for the derivation of (10).

#### 5. Bayesian Diagnosis Incorporating RSM and Data

To combine the background knowledge with training data, first, the problem dimensionality needs to be reduced utilizing the probability constraints implied in the available background information, and in the dimension-reduced subspaces, estimate the likelihoods with Bayesian inference. In this way, the estimation accuracy can be improved in the case of small amount of available historical samples that is often encountered in real applications since abnormalities are rare in normal process operations. Then, from the set of posterior probabilities that might be inconsistent as they are computed from different subspaces, derive the partially ordered posteriors that are consistent in the original probability space.

##### 5.1. Background Knowledge Expressed as RSM

In many applications, there are only a few historical samples available. Therefore, the process knowledge must be explicitly handled. We consider a general type of process knowledge about what abnormalities can possibly affect each of the monitors. It can be expressed in terms of the following: “observation has the same but maybe unknown probability distribution under and .”

Table 1 gives an example of such knowledge. “” at the th row and the th column represents a response signature meaning that the th element of the observation that is from the th monitor is affected under abnormal mode , compared with that under the normal mode. The likelihood distribution of the th observation element given is different from that under the normal mode . In other words, the th monitor output would respond when the operating mode turns into the th abnormal operating mode. And “0” in the table indicates that the likelihood probability distribution is the same as that under the fault free mode. Or to say, the th monitor measurement shows zero response to the th abnormal mode.