Effective fault detection and isolation using bond graph-based domain decomposition

Computers and Chemical Engineering 35 (2011) 132–148

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Effective fault detection and isolation using bond graph-based domain decomposition Xi Zhang 1 , Karlene A. Hoo ∗ Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409-3121, United States

a r t i c l e

i n f o

Article history: Received 7 November 2008 Received in revised form 25 July 2010 Accepted 31 July 2010 Available online 18 August 2010 Keywords: Wastewater treatment plant Principal component analysis Wavelet transform Bayesian network

a b s t r a c t The problem of fault detection and isolation in complex plants can be effectively addressed by a hierarchical strategy involving successive narrowing of the search space of potential faults. A bond graph network is one means of achieving a hierarchical strategy based on the physical domains present in the plant. First, the multivariate statistical method of principal component analysis is used to reduce the data dimension. Second, a discrete wavelet transform is applied to abstract the dynamics at different scales. Thirdly, the Mahalanobis distance is applied to calculate the confidence level. Following a conclusion of the existence of a fault, isolation is achieved by comparing the time scale at which the violation occurred to the time scale associated with a physical domain. In the final step, a Bayesian network is employed to describe the conditional dependence between faulty domains and fault signatures. Two examples are presented to demonstrate these concepts. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The complete reliance on human operators to cope with process faults is increasingly difficult due to several factors. Primary among these are the complexity and the size of process. Furthermore, the unreliability of human operators adds to inefficient, unreliable fault management. Without an effective response to faults, minor local faults may evolve to plant-wide failures, emergency shut-downs, and sometimes an environmental disaster with large economical losses and occupational injuries. Effective and rapid fault detection and isolation (FDI) is the first step in developing a successful fault management system. The majority of the past efforts on developing a fault management system focused on solving the problem as a two-stage procedure of detection followed by isolation. The various methods proposed can be divided into two categories according to the extent of prior process knowledge – model and data-based approaches. The former utilizes a mathematical model (often based on conservation laws) to formulate the diagnostic conclusion. While the latter relies on historical operating data and evidential observations. Most approaches on FDI are proposed at a single level of abstraction. These non-hierarchical methods require sufficient details to resolve faults at the unit level. For today’s complex plants, the

∗ Corresponding author. Tel.: +1 806 742 4079; fax: +1 806 742 3552. E-mail address: [email protected] (K.A. Hoo). 1 Supported by TTU Process Control & Optimization Consortium. 0098-1354/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2010.07.033

cost to develop high resolution models and collect high quality noise-free data can be extremely expensive. Also non-robust fault management systems may become unacceptably inefficient because of the large plant scale and simultaneous large search space. Finch and Karmer (1988) suggest that a multi-tiered, hierarchical approach to FDI may be more suitable for large or complex systems. Thus, a modeling formalism is needed to describe the system at an appropriate level of abstraction for rapid FDI. One example of a model formalism is one that abstracts and represents the complex and integrated system as subsystems that correspond to the physical groups or functional behavior. Bond graph, a graphical modeling language, provides a model formalism that decomposes the system into subsystems that map to the physical connections (Karnopp & Rosenberg, 1968). The resulting subsystems are essentially physical fields including mechanics, electronics, hydraulics, and chemistry. The time granularity for these domains are usually distinct. For instance, the dynamic response of a mechanical pump is usually on the scale of seconds when compared to the consumption or generation of a chemical species (usually on a scale of minutes or greater). Thus, in this work, we propose a multi-scale analysis based on the time features of the system phenomena corresponding to a system decomposition based on bond graph theory for the purposes of FDI. A four-stage procedure is developed to fulfill the task of a multiscale analysis. First, the multivariate statistical method of principal component analysis (PCA) is used to remove outliers and reduce the data dimension whenever the data are colinear. Second, a discrete wavelet transform (DWT) is applied to the resulting scores of the PCA to abstract the dynamics at different scales. In the third step,

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Nomenclature Symbol e f n˙ i n˙ in i n˙ out i Ci Ji Jdecay

effort (power variable in bond graph) flow (power variable in bond graph) molar rate of species i (mol h−1 ) inflow molar rate of species i (mol h−1 ) outflow molar rate of species i (mol h−1 ) molar concentration of species i (mol L−1 ) rate of consumption/accumulation for species i (mol h−1 ) biomass decay rate (mol h−1 )

the Mahalanobis distance is determined and used to calculate the confidence levels. Based on the degree of violation from the nominal confidence level, the presence of a potential fault is assessed to be either true or false. The idea of jointly using PCA and wavelets is not a new concept (Bakshi, 1998; Kosanovich & Piovoso, 1997; Lu, Wang, & Gao, 2003; Misra, Yue, Qin, & Ling, 2002). Kosanovich and Piovoso (1997) applied this combination to analyze industrial data to develop an online monitoring system; Bakshi (1998) developed an MSPCA (multi-scale PCA) formulation and demonstrated its effectiveness for monitoring multivariate processes; and Misra et al. (2002) applied MSPCA to two industrial datasets for real-time sensor fault detection and process fault diagnosis. In Misra et al. (2002) work, the approach simultaneously extracts both cross-correlation across the sensors (PCA) and auto-correlation within a sensor (wavelet). Using wavelets, the individual sensor signals are decomposed at different scales. Contributions from each scale are used to provide data to construct a PCA model at each scale. The approach of applying PCA followed by DWT proposed for FDI is very similar to that of Lu et al. (2003). In their work, the high dimensional and correlated process data are preprocessed by PCA. The PCA scores are transformed into time and frequency by an application of an appropriate wavelet transform. There are, however, some major differences between (Lu et al., 2003) work and this one. Specifically: • This approach combines data-based detection and causalitybased isolation in a single framework. Once a fault is detected via comparing the current signal features to the calibration model, the causality inherited from the bond graph is employed to locate the domain of the root cause. Lu et al. (2003) approach is based solely on historical knowledge of the process. • This work proposes to use a domain decomposition inherited from the process bond graph to identify the distinct time-scale phenomena. Once a bond graph model for the process is designed, various physical domains with different dominant time constants are naturally emphasized by the network structure. It is the sampling interval and the slowest time scale that determine the decomposition levels needed for the wavelet analysis. • A Bayesian network is used in this work to describe the conditional dependence between faulty domains and fault signatures. As a consequence, a probabilistic fault detection and isolation approach is presented. Following a conclusion of true, fault isolation is achieved by comparing the abstracted fault signature at the time scale at which the violation of the nominal probability level occurred, to the time scale associated with each physical domain. Then, in the final step, to address the challenging task of reliable fault isolation which suffers from noise and model mismatch, a Bayesian approach is

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employed to compute the probabilities, given all the information including the abstracted fault signatures, that certain faults are present. The idea is to build a dynamic Bayesian network based on the probabilistic information on how the fault propagates among domains as well as how the faulty domains impact the abstracted fault signatures. From the Bayesian network, we can infer the most probable faulty domains; this completes the proposed FDI procedure. It is expected that by narrowing the search space to locate the fault, the response actions and the response time to correct the faults are improved. The paper is organized as follows. In Section 2, a brief overview of existing FDI approaches is presented. In Section 3 the relevant theory on bond graphs is reviewed. In Section 4, the concept of a hierarchical FDI is introduced as a process decomposition that is based on the process bond graph. The concepts in the proposed FDI procedures outlined above are introduced in Section 5. The detailed approach for on-line detection and isolation is presented in Section 6 along with two examples to demonstrate the approach. Lastly, in Section 7 a summary of the findings and some recommendations are presented.

2. Fault detection and isolation Fault detection is a well studied area in many disciplines by the very nature that FD is essential to safe operations. The companion problem of fault isolation (FI) also has been studied and is a much more difficult problem. Simply stated, a fault is a malfunction of the system because of some unexpected change. The malfunction disturbs normal operation and if unchecked may further deteriorate the system’s performance (Chen & Patton, 1999). While a fault is considered nonnormal behavior, in contrast, a failure is when a process is unable to perform its required functions. Generally, a fault is minor when compared to a failure, but most failures tend to stem from ignored or undetected faults (Nimmo, 1995). There are different ways to classify faults according to various standards. Faults can be characterized by their temporal features (Isermann, 2005): (i) drifting faults occur slowly over time (minutes to hours). Such faults usually are linked to component usage and drift in control parameters. (ii) Intermittent faults are present only for very short periods of time (seconds to minutes), but sometimes they can have disastrous consequences. (iii) Abrupt faults are dramatic and persistent, and are usually accompanied by significant deviations from steady state operations. A classification most often used in the process industry is to cite the origin of the fault: (i) Equipment malfunction. In many cases, errors occur with actuators or sensors. A faulty actuator is not able to function accurately and promptly to provide proper input to the process. With the evolvement of system dynamics, this local malfunction may lead to deterioration of the entire system. If the failed sensor happens to be a part of a feedback control loop the malfunction is rapidly propagated through the causal chain of the loop. Examples of a faulty actuator and a sensor are a stuck valve and drifting thermocouple, respectively. (ii) Structural process change. Despite its infrequency, this type of fault tends to result in catastrophic consequences if no effective response action is promptly taken. The challenge of a structural fault is the lack of an accurate mathematical description. Examples of structural faults include: broken tubing or a malfunctioning controller. (iii) Parameter changes. Such changes arise when a disturbance enters the process through one or more exogenous inputs. An example is a change in the feed composition. It should be noted that the more common faults in the process industry are parameter changes and equipment faults.

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The task of responding to a fault involves timely detection, identifying the causal origin of the fault, estimating the degree of the fault, and finally taking the necessary steps to bring the system to within the normal operating limits. This work will concern itself only with detection and isolation. Fault detection usually involves making a binary decision – either that something has gone wrong or that everything is within normal operating limits. The outcome is simple, but never the procedure to arrive at the outcome. For a single variable not within its normal range, limit checking may be enough to make a decision. But most industrial systems consist of monitoring and collecting hundreds of process variables. Moreover, process and measurement noise are unavoidable and may contribute to false alarms. Thus, robust detection is a desirable feature in the design of an effective fault detection algorithm. Fault isolation, also called fault localization, locates the possible root causes for the detected fault. Of primary importance is the resolution of the isolation, which indicates the depth of isolation nd the effort required for fault recovery. With a finer resolution (component level) the fault origin is clearly targeted but locating the component may be too time consuming. In contrast with a coarser resolution (unit level), location is more rapid (smaller search space) but it may not reveal the true fault origin. The detection and isolation functions are often implemented as a component of the process supervisory system. Comparable to process control design, FDI requires computational efficiency to obtain real-time performance. The logic of fault detection is to provide either an affirmative or a negative as to the existence of a fault(s). While the aim is straightforward, the approach is non-trivial. There are two types of errors that are challenging when designing the fault detection module, false alarms (type I) and the inability to detect faults (type II) (Luo, Misra, & Himmelblau, 1999). If the detection design is too sensitive to the deviations from nominal operations a type I error is triggered. On the other hand, if the threshold of the detection design is set too high it is possible that a fault is masked which leads to a type II error. Also for closed-loop processes, compensating effects such as dynamic feedback may begin to mask the effects of faults, thus capturing the faulty behavior in a timely manner is important to avoid masking or compounding effects which may result in catastrophic failures. If the assumption of a single fault is valid either brute force or smart analysis can be applied to isolate the root cause. The former may rely on a scheme that checks exhaustively for all potential faulty components, however such an approach is time consuming. The latter may include approaches such as knowledge-based reasoning, cluster analysis, pattern recognition and signature analysis, qualitative reasoning, statistical analysis, or any number of parametric and nonparametric models. During the past 20 years, research activities in several sub-areas related to fault detection and isolation have included: determining what kind of prior-knowledge is better to represent the process for FDI applications to achieve accuracy in real-time (e.g. first-principle model and historical data, etc.); and deciding which approaches (quantitative or qualitative methods) will result in effective FDI. Also the more focused areas of sensor faults (Luo, Misra, Qin, Barton, & Himmelblau, 1998; Luo et al., 1999); actuator faults, and process faults (Samantaray, Medjaher, & Bouamama, 2004) may differentiate among the approaches. With the ever increasing complexity of process systems, more and hybrid FDI approaches are being investigated (Bakshi, 1998). Maurya and coworkers developed qualitative trend analysis (QTA) based on principal components to reduce computational complexity (Maurya, Rengaswamy, & Venkatasubramanian, 2005). Lee, Tosukhowong, and Lee (2006) studied the combination of a signed digraph and partial least-squares (PLSs) to develop a hybrid fault diagnosis method which they demonstrated performed well in the

Table 1 Power variables for different physical domains. Variables

Electrical

Hydraulic

Mechanical (translation)

Mechanical (rotation)

Effort (e) Flow (f)

Voltage Current

Pressure Flow rate

Force Velocity

Torque Frequency

presence of multiple faults. And Cheng, Nikus, and Jamsa-Jounela (2008) investigated the application of a causal digraph for fault diagnosis. Although chemical processes are inherently nonlinear, the application of FDI to nonlinear system continues to apply linear methods based on the assumption that the disturbance caused by the fault is within a small region of the designed operating condition. Using an approximate linear model derived from a nonlinear model of the process, various algorithms have been proposed to obtain analytical redundancies, such as parity relations (Gertler, 1991; Gertler & Singer, 1990), Kalman filters (Basseville, 1988; Willsky, 1976) and parameter estimation (Isermann, 1984; Young, 1981). Excellent reviews on FDI can be found in Basseville & Nikiforov (1993), Frank (1990), Gertler (1993), Isermann (1984), Venkatasubramanian, Rengaswamy, and Kavuri (2003a), Venkatasubramanian, Rengaswamy, Kavuri, and Yin (2003b) and Venkatasubramanian, Rengaswamy, Yin, and Kavuri (2003c). 3. Bond graph Bond graph is an explicit graphical tool for capturing the structures among the physical systems and representing them as an energy network based on the exchange of power (Paynter, 1961). Others (Bouamama, Busson, Dauphin-Tanguy, & Staroswiecki, 2000; Dauphin-Tanguy, Rahmani, & Sueur, 1999; Gawthrop & Bevan, 2007) have extended the bond graph concept to represent phenomena such as chemical kinetics and to extract causal models and control structures from the bond graph networks. Power is the common variable among the different physical domains. Indeed, the product of variables such as voltage (V) and current (A) in the electrical domain, and pressure (N/m2 ) and flowrate (m3 /s) in the hydraulic domain is power. In bond graph theory, these variables are referred to as effort (e) and flow (f). Table 1 gives some examples of effort and flow in different physical domains. The elements of a system are represented as nodes in a bond graph. Bonds connect nodes. The bond denotes the energy transferred between nodes. The direction of the transfer of energy and the associated flow between nodes is based on the orientation of a half arrow () added to the bond. The bond graph of a system reflects the physical structure in which the effort and flow variables are used to construct a path between nodes to track power exchanges and the dynamics associated with power conversions. Bond graph theory provides a series of standard elements to model the flow of power among heterogeneous domains. Each element can be mapped to a specific physical device within a given domain (Karnopp, Margolis, & Rosenberg, 2000). A graphical representation of a process using bond graph elements defines a bond graph network. 3.1. Basic elements The basic bond graph elements provide mappings for fundamental physical phenomena in various fields. Resistance (R) is introduced to represent power dissipation by imposing a constitutive relationship between effort and flow: e = Rf

f =

e R

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The R-element is analogous to electrical resistors, mechanical dampers or dashpots, porous plugs in fluid lines and other passive power elements. The concepts of a capacitor (C) and inductance (I) also are basic bond graph elements. Both are storage elements that are related to flow and effort as follows (left for C and right for I, respectively):

t e=C

fdt 0

A capacitor can be the idealization of springs in a mechanical system, a tank in a hydraulic system, or reactants in a chemical system. An inductance is used to represent physical elements with inertial effects such as the mass in the mechanical domain. 3.2. Converting between domains Power variables from different domains cannot be connected without some conversion. The bond graph elements called the transformer (TF) and the gyrator (GY) provide conversion capability. Both TF and GY assure that power is conserved. The TF element has the property that the ratio of the efforts is the inverse of the ratio of the flows. For example, the boundary between a mechanic domain and a hydraulic domain is denoted by a TF element, which represents a pump that converts mechanical motion into hydraulic motion (Karnopp et al., 2000). In the case of the GY element, the flow is dependent on the effort. In some cases, information rather than power is the link between domains, for instance, the hydraulic flow carrying reactants imposes a modulating effect on the chemistry. In the bond graph network, a dashed line is used to represent this nonenergetic interaction (see Fig. 3 – dashed line from the hydraulic to the reaction domain). By identifying TF and GY elements and information bonds from the physical system, the boundaries among the physical domains are naturally inherited by the resulting bond graph network.

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tion focus is rapidly narrowed from an initially broad scope to a restricted, feasible space. In the first pass, the faulty subsystems are isolated. The resolution may be coarse but the tradeoff is efficiency. The coarse isolation always can be followed by more detailed isolation about the domain in question. 4.2. Domain decomposition There are at least two distinct dimensions of decomposition, a structural dimension corresponding to the physical groupings of components and a functional dimension related to the purpose of the equipment. In many instances, it is not easy to discern the functional from structural features of a fault. In contrast, the domain decomposition natural to a bond graph network is a hybrid between structure and function of the system. One disadvantage of functional decompositions is the nonunique correspondence between the inspected fault and candidate units. Frequently, a single unit may perform two or more functions, or several units are collectively responsible for the performance of a single function. For example, the tubing in a continuous process may be used for fluid conductance, heat transfer and/or material holdup. But also, the process of heat transfer requires tubing and a heat exchanger. This makes narrowing the search space for the root cause inefficient due to overlap among multiple subsystems. Similar disadvantages exist for a structural decomposition. The domain decomposition provided by bond graph theory resolves conflicts by imposing clear boundaries among the physical domains with a unique mapping from the bond graph elements to the physical units of the system. Consider the hydraulic tubing mentioned above, the functions that the hydraulic tubing performs are represented by one element in the hydraulic domain for fluid conductance and another in the thermal domain for heat transfer. Thus, by isolating the origin of the fault to within distinct domains, the search space for the following procedures is definitely restricted to within a scope that has no overlap with other subsystems. 5. Procedures

3.3. Representing measurement devices In a bond graph network, either effort or flow within their respective domains is measurable. To capture this concept of sensing in a bond graph network, the bond graph elements De and Df are used for effort and flow detection, respectively. Both the location and type of measurement are made explicit by the bond graph network. The real sensor and its corresponding representation by effort or flow are necessary to represent a measurement in a bond graph. When there is a direct correspondence between the physical sensor and the effort or flow variable, then the sensor is either a De or Df in the domain it belongs to. Otherwise, a conversion is necessary to arrive at the corresponding effort or flow variable. With a complete representation of the system by a bond graph network, we will show that the tasks of detection and isolation can be done more efficiently. For example, if the fault is due to a single equipment malfunction, it will not be difficult to detect and isolate the faulty equipment due to the domain separation achieved by the bond graph network. 4. Hierarchical system decomposition 4.1. Hierarchical FDI Since the tasks of fault detection and isolation are complicated by the high dimensionality search space of the system it is not unreasonable to consider a hierarchical approach to narrowing the search space. Here a procedure is proposed in which the isola-

In this section fault detection and isolation is addressed based on a domain decomposition provided by an application of bond graph theory. PCA is applied first to remove multivariate outliers and to reduce the data size due to colinearity. Next, an application of a discrete wavelet transform to the score representation of the data signals will provide a multiple time-scale decomposition. The third step is to calculate the Mahalanobis Distance (MD). Lastly, based on the degree of violation from the nominal confidence level set at each time scale, the decision that a potential fault exists is made. The steps are illustrated in Fig. 1. Fault isolation is accomplished by comparing the time scales at which the violation of the nominal confidence level occurred (fault signature) to the time scales associated with each physical domain in the bond graph network. A Bayesian network which incorporates the conditional dependence between faulty domains and fault signatures is employed to reason about the possible fault origin at different levels. 5.1. Multivariate process monitoring Multivariate statistical analysis methods such as principal component analysis (PCA) assist in the identification of process correlations, thereby improving the existing process knowledge (Kosanovich & Piovoso, 1997). Theoretically, PCA is based on an orthogonal decomposition of the covariance matrix of the measured variables along directions ˆ represent a matrix of m that explain the variability in the data. Let X ˆ can be normalized sample data points of n variables. Assume that X

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Fig. 1. Schematic of the FDI procedure.

The new scores can be compared against expected scores using different statistical measures.

frequencies (short time resolution) and long windows at low frequencies (long time resolution). Basically, the wavelet transform is a signal decomposition onto a set of basis functions, called wavelets that are generated from a single basic mother wavelet. The family of wavelets are generated by stretching (dilation), compressing, and shifting (translation) of the mother wavelet (Daubechies, 1990). Analogous to the Fourier concept there are continuous, discrete and series wavelet transforms. The discrete wavelet transform (DWT) is signal processing with highpass (g = [gm , gm−1 . . . g2 , g1 ]) and lowpass (h = [hm , hm−1 . . . h2 , h1 ]) filters. The results of the filters are two sets of coefficients, one describing the details of the signal, and another describing a smooth approximation of the signal. The application of the band pass filter can be carried out to a desired number of scales, by recursively applying the filter to the smooth approximation at the previous scale. For each filter there are corresponding reconstruction filters, g* and h*. By applying g* and h*, the original signal can be reconstructed without loss of information. Alternatively, it is possible to reconstruct a variant of the signal using different combinations of the highpass and lowpass filtered signals.

5.2. Multi-resolution analysis using wavelet transform

5.3. Mahalanobis distance

The wavelet transform has been used to identify events that are localized in time and space (Rioul & Vetterli, 1991). The wavelet transform is an alternative to the classical windowed Fourier transform (WFT). However, in contrast to WFT, which uses a single analysis window, wavelet analysis uses short windows at high

The Mahalanobis distance of the ith data sample is calculated as the distance between the ith sample, xi , and the centroid, , of the calibration model:

to X such that each variable has a zero mean and unit variance. The application of PCA: X = TP + Ex

(1)

produces a set of projection vectors or eigenvectors P of size n × r with r > n and a set of scores T (projection of X onto P) of size m × r. The residual Ex is a measure of the error in the fit. The columns of the T matrix are orthogonal and represent linear combinations of the data variables. The first column of P is the first eigenvector and corresponds to the direction with the largest variability. The second column describes the next dominant direction of variability and so on. Determination of the number of eigenvectors can be established by several techniques, such as cross validation. Once a calibration model of normal variations is established, the model can be used to classify new data points. That is, given xk , the corresponding tk can be determined as follows: tk = P xk .

MDi2 = (xi − ) ˙ −1 (xi − )

(2)

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Fig. 2. Left: dynamic Bayesian network as a first-order Markov chain. Right: dynamic Bayesian network for fault isolation.



where is the reference set’s variance–covariance matrix that explains the dispersion of data around the centroid. Classification based upon the MD is done by calculating the confidence level obtained from the Chi-squared distribution of MD. Classification also can be established based on the Q-statistic or Square Prediction Error (SPE) (Hoo, Tvarlapati, Piovoso, & Hajare, 2002). Whereas the MD provides a measure of “inside the model space” the Q-statistic gives a measure of “outside the model space”. This work is concerned primarily with the MD since we are looking at the distance within the model plane from an observation to the origin. 5.4. Dynamic Bayesian network It is easier to confirm the existence of a fault than to locate its origin. Hence, probabilities are needed to reason about the root cause (Pernestal, 2007). There are two main ways to interpret probability either as a frequency or as a degree of belief. The former view is more related to classical statistics, whereas the second view is adopted in this work as a Bayesian approach for reasoning given the known knowledge including the current observation. The core of the Bayesian approach is Bayes’ rule (Jensen & Nielsen, 2007): ℘(A|B, C) =

℘(B|A, C)℘(A|C) ℘(B|C)

(3)

to update our belief about an event A given that we have information about another event B and other knowledge C (Jensen & Nielsen, 2007). In Eq. (3), ℘(A|C) represents the prior probability of A, whereas, ℘(A|B, C) represents the posterior probability of A, and ℘(B|A, C) indicates what is the likelihood of B given A. The procedure to determine the posterior probability from the prior probability is called inferencing. A Bayesian network as a system-

atic graphic model that describes the conditional dependence of multiple events provides a suitable tool for diagnostic inference of fault isolation of complex systems (Jansson, 2004; Pernestal, 2007; Schwall & Gerdes, 2002). Considering the temporal characteristics of a fault, if there is a specific fault at time t, there will be a higher probability of that fault persisting at time t + 1 as compared to other faults (Schwall & Gerdes, 2002). Hence, we can assume the fault propagation process is a first-order Markov chain, meaning that the future of a faulty event is dependent on only the current information. Thus, the conditional dependence in the Bayesian network only exist for t + 1 and t, excluding any event prior to t. We can employ a Dynamic Bayesian Network with two consecutive time slices (Lerner, Parr, Koller, & Biswas, 2000; Schwall & Gerdes, 2002) (left panel of Fig. 2) where the Bayesian network within each time slice is identical and the edge between the two slices denotes fault propagation. Two nodes are assigned to each slice of the dynamic Bayesian network, as shown in the right panel of Fig. 2 where Dt represents a faulty domain at time t and St represents the abstracted fault signature. All the physical domains in the bond graph are in Dt and each is a candidate for the origin of the fault. All the time scales are in St after an application of the discrete wavelet transform on the PCA scores plus a 0 state that represents a fault-free signature. These time scales are those that are in violation of the preset nominal confidence levels. The principle of indifference which states that when there is no reason to believe that any specific outcome is more probable than any other, then all outcomes should be assigned equal probability (Pernestal, 2007) will be applied in assigning prior probabilities and conditional probabilities. The principle of indifference leads to uniform prior probabilities meaning that any domain can be faulty at any time. It should be pointed out that as the dynamic Bayesian network evolves the prior probabilities are updated.

Table 2 Operating condition and parameters of CSTR.

Operating condition

Parameter

Variable

Value

Definition

CA0 Se0 Se9 TF C12 , C16 , C20 K Ks I2 R3 C6 R8

0.001 mol/mL 66 N cm 1 atm 0.25 mL/min rpm 2000 mL 0.0015/min 0.000 mol/mL 0.1 N cm min/rpm 1 N cm/rpm 150 mL/atm 0.03 atm min/mL

Feed concentration Rated torque of the pump Atmosphere pressure Pump coefficient Reactor working volume Maximum utilization rate Saturation concentration

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Fig. 3. CSTR. Top: schematic. Bottom: bond graph network.

Table 3 Conditional probabilities for St on Dt in Example A. Wavelet scales

Nominal

Mechanics T = 0.25 min,  = 1

Hydraulics T = 2.5 min,  = 1

Reaction T = 100 min,  = 1

1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0

0.68 0.28 0.04 0 0 0 0 0 0

0.01 0.08 0.28 0.38 0.20 0.04 0 0 0

0 0 0 0 0 0 0.08 0.33 0.59

X. Zhang, K.A. Hoo / Computers and Chemical Engineering 35 (2011) 132–148 Table 4 Conditional probabilities for fault propagation based on a first-order Markov process for s CSTR. Dt+1

Dt : Nominal

Dt : Mechanic

Dt : Hydraulic

Dt : Reaction

Nominal Mechanics Hydraulics Reaction

0.7 0.1 0.1 0.1

0.0008 0.8319 0.1665 0.0008

0.0006 0.1428 0.7138 0.1428

0.001 0.001 0.001 0.997

The conditional dependence from D to S is used to describe the probability of a violation about the preset confidence level of the time scales in S, given the fault origin belongs to D. The idea is to assign higher conditional probability values to a fault signature whose time scales are closer to the dominant time constant of the given domain, but lower conditional probability values for time scales that are further away. Considering that t ∈ (0, + ∞ ], a single-tailed, log-normal distribution is selected to determine the conditional probabilities. Further, due to the dyadic nature of the wavelet transform (Rioul & Vetterli, 1991), a base 2 logarithm is used. The conditional probability of violating a confidence level at time scale  corresponding to T = 2 Ts , where Ts is the sample time, given that the fault origin domain Dt has a time constant T can be estimated as ℘(l|Dt )

= ℘(log2 Tl−1 ≤ x ≤ log2 Tl ) = ℘(x ≤ log2 Tl ) − ℘(x ≤ log2 Tl−1 )

(4)

= ℘(log2 T ; log2 T, ) − ℘(log2 Tl−1 ; log2 T, ) where ℘(x ; , ) is the normalized cumulative distribution function at x,  is the distribution mean which is derived from the dominant time constant of Dt , and  is the standard deviation of the distribution. When there is no fault signature detected,  = 0, and ℘(0|Dt ) = 0.

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Fault propagation depends on the causal links among the physical domains. The causality embodied in the bond graph provides a structure to investigate the conditional dependence among domains, that is if there are power bonds between two neighboring domains, which indicates bi-directional power exchange, the conditional probability for fault to propagate from one to the other has a high probability. If the connection is an information bond which is uni-directional (for example, the modulating effect imposed on the hydraulics by the mechanical domain), the conditional probability for a fault to propagate from the origin to the terminal is highly likely, whereas the reverse has zero probability.

5.5. FDI procedure 5.5.1. Creating the calibration model and reference signals (1) Decompose the system based on its bond graph network and domain boundaries. (2) Convert the system sensors into bond graph elements, De or Df, based on sensor type and sensor location. (3) Obtain a calibration model about the normalized nominal data by an application of PCA. It is expected that the first few eigenvectors are able to explain the majority of the variance in the data. Using these eigenvectors, the nominal system information is transformed into scores (by projection of the nominal data onto the eigenvectors) that are orthogonal to each other. (4) Apply DWT to each score. The number of recursive filtered passes is dependent on the slowest dynamic time constant present in the system. The highpass filter results at each scale are used to reconstruct the reference signals for on-line FDI. (5) Calculate the variance–covariance matrix of the Mahalanobis distance for the purposes of determining a violation (see Section 5.3).

Fig. 4. Example A: single slow drift in the mechanical domain. Top left panel: friction coefficient. Top right panel: measured pump speed. Bottom left panel: measured hydraulic pressure. Bottom right panel: measured product concentration.

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Fig. 5. CSTR: single slow drift in the mechanical domain. Top panel: DWT applied to first column of scores for 9 scales. Bottom panel: DWT applied to the second column of scores for 9 scales. Observe the violation of the confidence interval at the first three scales.

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Fig. 6. CSTR: probability of a faulty domain—single slow drift in mechanical domain.

Fig. 7. CSTR: single intermittent fault in the mechanical domain. Observe the violation of the confidence interval at the first four scales.

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Fig. 8. CSTR: probability of a faulty domain—single intermittent fault in the mechanical domain.

5.5.2. On-line FDI procedure (1) Using the calibration model find the score vector, tk that corresponds to a new measurement vector yk at time k (see Fig. 1). (2) Apply DWT to a window of score data whose length is based on the slowest time constant but also includes the most recent score data. If there are r scores then there are r datasets to be analyzed. Reconstruct the signal at each scale. (3) Using the reference score set found in step Section 5.5.1 of the off-line procedure, calculate the MD for the reconstructed signal at each scale. Violation of MD about a pre-specified confidence level at any scale is classified to be a fault. Further, if a fault is detected in the most dominant score data, the fault is readily classified as an intermittent fault as opposed to a slow drift. This work excludes abrupt faults. If a fault is found with less dominant score data, it can be classified as a slow drifting fault. In the case of no faults, the process is proclaimed to be fault free; the procedure returns to step 1 to repeat the online procedure for the next measurement vector y(k + 1) at time k + 1. (4) Any violations are classified as a fault signature. By comparing the time resolution at which a violation is found to the natural time constants in the domains of the bond graph network, the location of the fault can be isolated. The fault signature at each time is an input to the Bayesian network, while the outputs of the network are the posterior probability for that domain to be faulty (see Fig. 1). Depending on the complexity of the domain, the resolution of the isolation may be coarse, but the search space has been narrowed down to that domain.

Remark 1. The defining feature between an intermittent and a slow drifting fault is their temporal characteristics. The former is characterized by sharp pulses (varying amplitude and phase) over a short duration of time, thus a technique such as PCA can detect this variability from normal data. In contrast, the variability found with a slow drifting fault is slow to develop and may be captured in the less dominant score data.

6. Case studies 6.1. Continuous-stirred tank reactor An idealized adiabatic, continuous-stirred tank reactor (CSTR) with an irreversible, enzyme catalyzed biochemical reaction is used to demonstrate the FDI approach (see Fig. 3). The biochemical reaction denoted by R19 in Fig. 3 consumes reactant A and produces the product B in the presence of enzyme according to Monod kinetics (Rittmann & McCarty, 2001): dCB KCA = Cx Ks + CA dt

(5)

where CA , CB , and Cx are concentrations of A, B and x, respectively; K and Ks are coefficients about the reaction kinetics (see Table 2). It should be noted that the quantity of enzyme x changes not only by the TF-R combination representing biomass growth, but also by biomass decay which is described by the RS term introduced in (Thoma & Bouamama, 2000). Recently, there have been several publications that are concerned with using bond graph networks to model a process and then apply qualitative or quantitative methods for fault diagnosis. In Lo, Wong, Rad, and Chow (2002) work, the problem of fault diagnosis via integration of genetic algorithms and qualitative bond graphs is addressed. Also, Bouamama and coworkers applied bond graph-based FDI extensively in industrial thermal system (steam generator) (Bouamama, Medjaher, Samantaray, & Staroswiecki, 2006; Djeziri, Bouamama, & Merzouki, 2009; Medjaher, Samantaray, Bouamama, & Staroswiecki, 2006). Following the procedures presented in Djeziri et al. (2009), Thoma and Bouamama (2000) and Zhang (2008), a bond graph network is developed (see Fig. 3). The bond graph network naturally inherits the mechanical, hydraulic, and reaction subsystems. A transformer element, TF, indicates the boundary between the mechanical and hydraulic domains, while the dotted signal bonds link the hydraulic and reaction domains. Process measurements

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Fig. 9. CSTR: single drift in the reaction domain. Observe the violation of the confidence level at scale 9.

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Fig. 10. CSTR: probability of a faulty domain. Single drift in the reaction domain.

Fig. 11. Schematic of the wastewater treatment plant.

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Fig. 12. Bond graph network of the WWTP. Three CSTRs (dash-dot) in series are used to represent the two tubular bioreactors; pumps are denoted by dotted lines and the PB bioreactor by dashed lines. Piping is the other instrumentation.

are converted into bond graph elements De and Df to represent the measured effort and flow variables, respectively. All measurements have been investigated in the presence of Gaussian noise. The operating condition and parameters of the bond graph elements are listed in Table 2. The reaction occurs in a mixed solution system where the feed pump (denoted by Se0 in the mechanical domain, bottom of Fig. 3) delivers component A, and a ball valve (denoted by R8 in the hydraulic domain, bottom of Fig. 3) regulates the outlet flowrate (denoted by fout ). Three process variables, the pump speed ω (Df), the hydraulic pressure P at the bottom of the CSTR (De), and the product concentration CB at the outlet (De), are measured at a sampling interval of 15 s with signal-to-noise ration (SNR) = 100. In this example, the dynamic time constant of the reaction domain is approximately 120 min. To abstract the slow dynamics from the original data requires nine applications of the wavelet filter onto the data (0.25(29 ) min = 128 min). Estimates (see Section 5.4) of the conditional probability of a fault signature (St ) on a faulty domain (Dt ) are listed in Table 3. Due to the principle of indifference discussed in Section 5.4, the probability of a fault originating from any one domain (mechanic, hydraulic and reaction) is considered to be equal (a value of 0.1). Further, the conditional probabilities for fault propagation between the mechanic and hydraulic domains are equal and assigned a value of 0.2 considering the bi-directional causality based on power exchange. The conditional probability between the hydraulic and

the reaction domains is asymmetric due to the uni-directional causality based on modulating bonds. The conditional probabilities for the fault propagation from Dt to Dt+1 are listed in Table 4. 6.1.1. Slow drift in the mechanical domain The friction coefficient of the pump’s motor is decreased (−5%/min for 2 min) from its nominal value at t = 200 min (Fig. 4). In the bond graph network there would be a change in the value of element R3 . Applying the aforementioned procedures in Section 5, an indication of a fault is found in the analysis of the second score data at t0 = 201.25 min (Fig. 5). In this work, the threshold value is 99% (p = 0.01). This finding confirms the presence of a slow drift. As stated in step 4 of Section 5.5.2, the fault signature at t0 = 201.25 min is determined to be St0 = [1 2], since evidence of MD violation is detected at scale levels 1 and 2. Using the fault signature as the input to the dynamic Bayesian network (see Fig. 2), the inference about the current faulty domain, Dt0 , which is the posterior probability, is calculated based on the known previous faulty domain Dt−1 , which is the prior probability ℘(Dt0 |Dt−1 ). The likelihood is stated as ℘(St0 |Dt0 ), due to the assumption of a first-order Markov chain between two consecutive time slices. Therefore, from Eq. (3), the posterior probability can be calculated as ℘(Dt+1 |St+1 , Dt ) =

℘(St+1 |Dt+1 )℘(Dt+1 |Dt ) ℘(St+1 )

(6)

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Fig. 13. WWTP: multiple faults in the feed pump and feed composition.

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Fig. 14. WWTP: probability of a faulty domain.

In this case, t0 is the first time slice with a fault signature. Here, Dt−1 is [0 0 0 1], where columns 1–3 correspond to the probability of a fault in the mechanics, hydraulics and reaction domains, respectively. The last column is the probability of being fault free. With Table 3 serving as the conditional probability distribution (CPD) for fault propagation from Dt to St , and Table 4 as the CPD for fault propagation from Dt to Dt+1 , Dt0 at t0 = 201.25 min is estimated as [1 0.0286 0 0]. This concludes the inference for t0 . At the next time slice, t1 = 201.50 min, due to the assumption of a first-order Markov chain, Dt0 is replaced by Dt1 . The updated fault signature is found to be St1 = [1 2 3]. Repeating the procedure above, we find that Dt1 at t1 = 201.50 min is estimated as [1 0.1237 0 0]. The inference about the fault origin by the dynamic Bayesian network is illustrated in Fig. 6. As expected, the fault is located in the mechanic domain with high certainty. 6.1.2. Intermittent fault in the mechanical domain An intermittent fault is introduced by the pump by decreasing the friction coefficient of the motor at t = 200 min from its nominal value with an amplitude of −10% and a frequency 0.1/min. The interpretation of the fault is a pulse disturbance applied to element R3 in the bond graph network. As shown in Fig. 7, the fault is detected at t = 201.75 min by violations of the MD confidence levels. The origin of the fault can be isolated to the mechanical domain as shown in Fig. 8. It is worth noting that a spike appeared in the probability level of the hydraulic domain. Without the inference scheme provided by the dynamic Bayesian network, there would have been a false alarm. 6.1.3. Slow drift in the reaction domain A feed composition disturbance is introduced at a rate of −0.1%/min for 100 min at t = 200 min. Referring to Fig. 9, the analysis about the first dominant score produces no fault features but an analysis of the second score reveals the fault feature at t = 340.75 min. The dynamic characteristics of the fault is that of

a drift. Since the time resolution of the fault feature is at scale 9 (∼128 min), it can be concluded that the fault origin is within the reaction domain, as shown in Fig. 10. 6.2. Biochemical wastewater treatment plant (WWTP) The schematic in Fig. 11 is that of a coupled biochemical wastewater treatment plant whose primary unit operations are a packed bed (PB) bioreactor in series with a set of tubular (TR) bioreactors. A full description of the system can be found in Zhang and Hoo (2008). Positive displacement pumps move the fluid through the hydraulic loop. The mechanical domain consists of the pumps and the hydraulic domain is made up of the piping and the tanks. The PB reactor is represented in the bond graph network by a CSTR, while the TR is approximated by a three CSTRs in series (Couenne, Jallut, Maschke, Breedveld, & Tayakout, 2006). The structural arrangement of the bond graph network is from the unit level to the plant level and includes the hydraulic supporting loop (see Fig. 12). Three subsystems corresponding to mechanic, hydraulic and reaction domains are found from the bond graph network by identifying the TF elements and the information bonds. The dynamic Bayesian network developed in Section 6.1 is applied to this example. 6.2.1. Multiple faults Two faults with distinct origins are introduced into the process. Fault A imposes a slow drift (decrease) at a rate of −10%/min for 2 min in the friction coefficient of the feed pump at t = 200 min. Fault B is a slow drift (increase) at a rate of +1%/min in the feed composition at t = 160 min. In this example, the dynamic time constant of the reaction domain is approximately 150 min. To abstract the slow dynamics from the original data requires nine applications of the DWT filters to the smoothed data (0.25(29 ) min = 128 min) to represent the slowest dynamic time constant. Prior to t = 204 min, there is no evidence of violation of MD at any scale. The fault signa-

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ture is detected at t = 204 min. The results of applying the proposed FDI method are shown in Figs. 13 and 14. Both faults are detected at different scales from an analysis of the second score data. The first fault can be located in the mechanical domain because the MD violation occurs at scale levels 1 and 3. The other fault is identified from a violation at scale levels 7–9. These scales correspond to a time resolution of about 100 min, which points to the slow reaction domain. The probability of identifying the domain where the fault originated is found from Bayesian inferencing, which confirms the existence of two faults with different root causes (see Fig. 14). 7. Summary This work proposed and developed an efficient approach to the tasks of on-line fault detection and isolation (FDI). The approach is efficient because it is based on a domain decomposition that is inherited from a bond graph network of the process. The approach makes use of data reduction by PCA and multi-scale analysis by wavelets. The probabilistic inference of fault origin using dynamic Bayesian network improves the reliability of isolation. Two case studies were presented to demonstrate the proposed FDI. The drift or intermittent fault is found promptly through abstraction of the fault features. Moreover, the search space to locate the faults origin is isolated explicitly due to the corresponding physical domain decomposition natural to the bond graph network. Future work will concentrate on refining the resolution within the isolated domain and evaluating the severity of the fault quantitatively by considering the propagation along causal paths of the bond graph. References Bakshi, B. (1998). Multiscale PCA with application to multivariate statistical process monitoring. AIChE Journal, 44, 1596–1610. Basseville, M. (1988). Detecting changes in signals and systems—A survey. Automatica, 24, 309–326. Basseville, M., & Nikiforov, I. (1993). Detection of abrupt changes—Theory and application. Prentice-Hall. Bouamama, B., Busson, F., Dauphin-Tanguy, G., & Staroswiecki, M. (2000). Analysis of structural properties of thermodynamic bond graph models. In Proceeding of the 4th IFAC: Fault Detection Supervision and Safety for Technical Processes, vol. 2 IFAC, Budapest, Hungary, (pp. 1068–1073). Bouamama, B., Medjaher, K., Samantaray, A. K., & Staroswiecki, M. (2006). Supervision of an industrial steam generator. Part I. Bond graph modeling. Control Engineering Practice, 14, 71–83. Chen, J., & Patton, R. (1999). Robust model-based fault diagnosis for dynamic systems. Boston, MA: Kluwer Academic Publishers. Cheng, H., Nikus, M., & Jamsa-Jounela, S. (2008). Fault diagnosis of the paper machine short circulation process using novel dynamic causal digraph reasoning. Journal of Process Control, 18, 676–691. Couenne, F., Jallut, C., Maschke, B., Breedveld, P., & Tayakout, M. (2006). Bond graph modelling for chemical reactors. Mathematical and Computer Modelling of Dynamical Systems, 12, 159–174. Daubechies, I. (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Transctions on Information Theory, 36, 961–1005. Dauphin-Tanguy, G., Rahmani, A., & Sueur, C. a. (1999). Bond graph aided design of controlled systems. Simulation Practice and Theory, 7, 493–513. Djeziri, M., Bouamama, B., & Merzouki, R. (2009). Modelling and robust fdi of steam generator using uncertain bond graph model. Journal of Process Control, 19, 149–162. Finch, F., & Karmer, M. (1988). Narrowing diagnostic focus using functional decomposition. AIChE Journal, 34(1), 25–36. Frank, P. (1990). Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy—A survey and some new results. Automatica, 26(3), 459–474. Gawthrop, P., & Bevan, G. (2007). Bond graph modeling—A tutorial introdution for control engineers. IEEE Control System Magazine, 27(2), 24–45. Gertler, J. (1991). Analytical redundancy methods in fault detection and isolation. In Proceeding of IFAC/IAMCS symposium on safe processes (pp. 91–103). Gertler, J. (1993). Residual generation in model-based fault diagnosis. Control-Theory and Advanced Technology, 9, 259–285.

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