An architecture for fault detection and isolation based on fuzzy methods

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Expert Systems with Applications Expert Systems with Applications 36 (2009) 1092–1104 www.elsevier.com/locate/eswa

An architecture for fault detection and isolation based on fuzzy methods q L.F. Mendoncßa a,b,*, J.M.C. Sousa a, J.M.G. Sa´ da Costa a a

Center of Intelligent Systems/IDMEC, Instituto Superior Te´cnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal b Department of Marine Engineering, Escola Na´utica Infante D. Henrique, Av. Eng. Bonneville Franco, Lisbon, Portugal

Abstract Model-based fault detection and isolation (FDI) is an approach with increasing attention in the academic and industrial fields, due to economical and safety related matters. In FDI, the discrepancies between system outputs and model outputs are called residuals, and are used to detect and isolate faults. This paper proposes a model-based architecture for fault detection and isolation based on fuzzy methods. Fuzzy modeling is used to derive nonlinear models for the process running in normal operation and for each fault. When a fault occurs, fault detection is performed using the residuals. Then, the faulty fuzzy models are used to isolate a fault. The FDI architecture proposed in this paper uses a fuzzy decision making approach to isolate faults, which is based on the analysis of the residuals. Fuzzy decision factors are derived to isolate faults. An industrial valve simulator is used to obtain several abrupt and incipient faults, which are some of the possible faults in the real system. The proposed fuzzy FDI architecture was able to detect and isolate the simulated abrupt and incipient faults. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Fault detection; Fault isolation; Model-based fault diagnosis; Fuzzy decision making; Fuzzy modeling

1. Introduction The development of model-based fault diagnosis began in the early 1970s. This method of fault detection in dynamic systems has been receiving more and more attention over the last two decades. Fault detection and isolation methods are used to detect any discrepancy between the system outputs and model outputs. It is assumed that this discrepancy signal is related to a fault. However, the same difference signal can correspond to model-plant mismatches or noise in real measurements, which are erroneq This work is partially supported by the ‘‘Programa do FSE-UE, PRODEP III, accßa˜o 5.3, no aˆmbito do III Quadro Comunita´rio de apoio” and by the Project POCI/EME/59522/2004, co-sponsored by FEDER, Programa Operacional Cieˆncia e Inovacßa˜o (POCI) 2010, FCT, Portugal. * Corresponding author. Address: Center of Intelligent Systems/ IDMEC, Instituto Superior Te´cnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal. E-mail address: [email protected] (L.F. Mendoncßa).

0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.11.009

ously detected as a fault. The availability of a good model of the monitored system can significantly improve the performance of diagnostic tools, minimizing the probability of false alarms. The inconsistency between the data from the system measurements and the corresponding signals of the model is the residual. The residual generation is then identified as an essential problem in model-based FDI, since if it is not performed correctly, some fault information could be lost. The use of FDI is due to an increasing demand of man-made dynamical systems to become safer and more reliable. These requirements extend to process industry plants, which are basically controlled by servoactuated flow control valves. Taking into account that a valve malfunction in many hazardous applications can cause serious consequences, the fault diagnosis of industrial servo-actuated valves is a very important task. When the malfunction is detected and isolated, a quick response might prevent the monitored system from expensive damages and loss of efficiency and productivity.

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Models with good accuracy are necessary to improve the correct diagnostic of faults. However, in practice it is very difficult to achieve accurate models for complex nonlinear plants. Sometimes, it is even impossible to model nonlinear systems by analytical equations. Fuzzy modeling has been extensively used to model complex nonlinear systems. Thus, FDI can benefit from nonlinear fuzzy modeling. The use of fuzzy models increase the capability of FDI to work with systems without complete information and noisy. The key advantage of fuzzy logic is that it enables the system behavior to be described by ‘‘if–then” relations (Koscielny & Syfert, 2003). The main trend in developing fuzzy FDI systems has been to generate residuals using either parameter estimation or observers, and allocate the decision making to a fuzzy logic inference engine. By doing so, it has been possible to combine symbolic knowledge to quantitative information and, thereby, minimize the false alarm rate. This paper proposes a model-based FDI architecture combining synergetically fuzzy modeling and fuzzy decision making. First, fuzzy models for normal operation and for each fault are identified. The underlying idea is to predict the system outputs from the available inputs and outputs of the process, thus identifying a fuzzy model directly from data. The detection is made using the residual computed using the comparison of real data with the fuzzy model of the system running in normal operation. When a fault is detected, each faulty model output is compared to the real outputs of the process. After the detection, a fault must be isolated. The residuals are aggregated using a fuzzy decision making (FDM) approach and a fault can be isolated by evaluating fuzzy decision factors, which are built based on residuals. The proposed fuzzy FDI approach is applied to an industrial servo-actuated valve, which is a benchmark in the FDI field (Bartys, Patton, Syfert, de las Heras, & Quevedo, 2006). The paper is organized as follows. Next section presents a brief overview of classical methods for residual generation. Soft computing, and more specifically fuzzy methods, for FDI are also presented in this section. Section 3 presents the FDI architecture proposed in this paper to detect and isolate the faults, which is based on fuzzy methods. This section describes in detail the fuzzy decision making approach to isolate faults. Fuzzy modeling, which is an important tool in the proposed FDI scheme, is presented in Section 4. Section 5 presents the physical model and the description of the servo-actuated valve. The accuracy of the obtained fuzzy models and the FDI results are presented in Section 6. Finally, the conclusions are drawn in Section 7. 2. Fault detection and isolation Different approaches have been developed in FDI. One of the first was the failure detection filter, which is applied to linear systems (Beard, 1971). After that, different methods and approaches were developed such as the application

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of identification methods to the fault detection of jet engines (Rault, Richalet, Barbot, & Sergenton, 1971) and the correlation methods applied to leak detection (Siebert & Isermann, 1976). Isermann introduced process fault detection methods based on modeling parameters and state estimations (Isermann, 1984). Model-based methods for fault detection and diagnosis applied to chemical processes were presented in Himmelblau (1978), the first book about this approach. In the frequency domain, FDI is applied using the frequency spectra as criterion to isolate the faults (Ding & Frank, 2000). Other FDI approaches are based on residual generators, including physical or hardware redundancy methods, or analytical or functional redundancy methods (Chen & Patton, 1999).

2.1. FDI redundancy methods Physical or hardware redundancy methods are a traditional approach to fault diagnosis, which use multiple sensors, actuators and components to measure and control a particular variable. The major problems encountered with these methods are the extra equipment and maintenance cost, as well as the additional space required to accommodate this equipment (Isermann & Balle´, 1997). These disadvantages increase the necessity of using other methods, easier to use and with smaller costs. Analytical or functional redundancy methods can be used instead. These methods use redundant analytical relationships among several measured variables of the monitored system (Chen & Patton, 1999; Kinnaert, 2003). These variables are measured signals with estimated values, generated by a mathematical model of the considered system. In the analytical redundancy scheme, the resulting difference generated from the comparison of different variables is called residual or symptom signal. When the system is in normal operation the residual should be zero, and when the fault occurs the residual should be different from zero. This property of the residual is used to determine whether or not faults have occurred. Some examples of residual generators based on the analytical redundancy scheme are the Kalman filter, Luenberger observers, state and output observers or parity relations (Chen & Patton, 1999). The model-based FDI method can be defined as the detection and isolation of faults on a system by means of methods that extract features from measured signals. The first operation is used to generate residuals by means of the available inputs and outputs from the monitored system. The residuals can be generated by the comparison of measured, y, and estimated, ^y, outputs. The residual evaluation is obtained by a logic device. This device processes the signals computed by the residual generation, e, to detect when a fault occurs. For a simple fault that can be detected by a single measurement, a conventional threshold check may be appropriate (Chen & Patton, 1999). The general principle of model-based fault detection is presented in Fig. 1, where the two main stages are: residual

Input

Faults

System Faults

Faults

Actuators

Process

Sensors

Model

Residual generation

Output

Fault range

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Fault range

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Abrupt fault

Incipient fault

time [s]

Model-based fault detection

time [s]

Residual evaluation

Fig. 2. Abrupt and incipient faults behavior.

Fault

using model-based FDI is sometimes very difficult. This situation is consequence of a small effect in residuals when the faults are incipient, because they can be hidden by the uncertainty. The increasing interest in incipient fault detection demands the use of new approaches. In the last years, both academia and industry have considered the problems introduced by model uncertainties, disturbances and noise in model-based FDI (Gertler, 1998). This approach increased the capability of FDI techniques to work with small intensity faults. In Isermann and Balle´ (1997), some basic FDI methods are evaluated and the trends in the application of model-based FDI to technical processes are presented. When information about the relations between symptoms and faults is available in the form of diagnostic models, various methods of reasoning can be applied. Typical approximate reasoning methods are:

Fig. 1. Diagram of model-based fault detection.

generation and residual evaluation, which can be described as follows: (1) Residual generation generates residual signals using available inputs and outputs from the monitored system. (2) Residual evaluation examines residuals for the likelihood of faults and the decision rule is then applied to determine if any fault occurred. The accuracy of the model describing the behavior of the monitored system is crucial in model-based fault detection. The second operation of model-based FDI consists of a decision making process. The value of a residual signal allows to obtain the qualitative results: faulty or normal working conditions. A number of residuals can be designed, where each must be sensitive to individual faults occurring in different locations of the system. The analysis of each residual, once the threshold is exceeded, leads to the fault isolation. The evaluation of the residuals via a set of statistical tests was made in Kinnaert, Vrancic, Denolin, Juricic, and Petrovcic (2000). However, the impossibility of obtaining complete knowledge and understanding of the monitored process increases the uncertainty in the model. The reduction of sensitivity to modeling uncertainty can be used in FDI. This sensitivity reduction, sometimes, does not solve the problem since the sensitivity reduction may be associated with a reduction of the sensitivity to faults (Chen & Patton, 1999; Gertler, 1998). Thus, the main reliability problem of FDI is the modeling uncertainty, which is unavoidable in real industrial systems. The design of an effective and reliable FDI scheme for residual generation should take into account the modeling uncertainty with respect to the sensitivity of the faults. The generation of symptoms is therefore the main issue in model-based fault diagnosis. The faults used in this paper are abrupt and incipient. Abrupt faults are faults modelled as stepwise function and incipient faults are faults modelled by using ramp or other signals, see Fig. 2. Considering the problems to model industrial processes, the diagnose of incipient faults

(1) probabilistic reasoning; (2) possibilistic reasoning with fuzzy logic; (3) reasoning with artificial neural networks. Methods like neural networks, expert systems, fuzzy systems and neuro-fuzzy systems have been used with success in model-based FDI (Calado, Korbicz, Patan, Patton, & Sa´ da Costa, 2001). From the several possibilities described, fuzzy logic is a natural tool to handle complicated and uncertain conditions, considering that the characteristics of the systems are not precisely known (Mendoncßa, Sa´ da Costa, & Sousa, 2003; Mendoncßa, Sousa, & Sa´ da Costa, 2005). Next section presents some fuzzy methods used in FDI. 2.2. Fuzzy methods in FDI Fuzzy FDI methods support in a natural way the direct integration of human operators in the fault detection and isolation process. This approach can be an important way of taking account of modeling uncertainty. Sometimes the residuals in fault-free conditions are affected by the noise contamination and uncertainty effects. The consequence of this influence is the residual variation around the zero value, which can hide the faulty effects. The interesting capability to describing vague and imprecise facts and work with systems when the complete information is not available makes fuzzy logic a powerful tool in this case.

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Fuzzy systems are useful in any situation in which the measurements taken are imprecise or their interpretation depends strongly on the context or on human opinion. The application of fuzzy methods in FDI can be made in different ways. The use of expert knowledge in the form of a rule-based knowledge format is one of them (Patton, Frank, & Clark, 2000). Another approach is presented in Mechefske (1998), where fuzzy logic is used to classify the frequency spectra of various rolling element bearing with faults. Fuzzy sets can also be used to locate and identify the type of faults (Insfran, da Silva, & Lambert Torres, 1999), or for residual evaluation (Frank & Koppen-Seliger, 1997; Schneider & Frank, 1996). Industrial applications of fuzzy logic in FDI can be found in Bartys and Syfert (2002), and in Koscielny and Syfert (2003). The detection of faults can be performed by using fuzzy decision making (FDM), which avoids false alarms, as presented in Kuipel and Frank (1997). Usually, the faults are detected by a model-plant mismatch. A possible solution to this problem was proposed in Schneider and Frank (1994), where an adaptation of a fuzzy logic based threshold is used to detect faults in robots. Model-based fuzzy methods use the residuals generation to detect and isolate faults (Dexter & Benouarets, 1997; Isermann, 1998). Dexter and Benouarets (1997) proposed the use of fuzzy reference models describing faulty and normal operation. The diagnosis is made by a classifier based on fuzzy matching. In the scope of an European project, a benchmark has been developed for FDI, and several soft computing techniques have been developed. Note that the industrial valve used as example in this paper is part of this benchmark. Next section presents a brief description of the developed FDI methods. 2.3. FDI methods developed under the scope of DAMADICS Development and Application of Methods for Actuator Diagnosis in Industrial Control Systems (DAMADICS) was an European project under which several FDI methods have been developed. A benchmark for the project was developed based on industry requirements, and is described in Bartys et al. (2006). This benchmark is used with different approaches with the purpose of providing a training facility for both industry and academia and gain an understanding for the way in which the various FDI methods can perform in a realistic control engineering application setting. Different analytical FDI approaches have been developed. In Supavatanakul, Lunze, Puig, and Quevedo (2006) the problem of fault diagnosis in discrete-event systems represented by timed automata is discussed. Passive robustness in fault detection using intervals observers is presented in Puig et al. (2006). The use of signal modelbased fault detection using squared coherency functions is introduced in Previdi and Parisini (2006). An actuator fault distinguishability study is presented in Koscielny, Bartys, Rzepiejewski, and Sa´ da Costa (2006).

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Several soft computing FDI methods have been developed under the scope of DAMADICS. A data-driven method in FDI is presented in Bocaniala and Sa´ da Costa (2006), where a novel classifier based on particle swarm optimization was developed. Group method of data handling (GMDH) neural networks have been used in Witczak, Korbicz, Mrugalski, and Patton (2006) for robust fault detection. A computer-assisted FDI scheme based on a fuzzy qualitative simulation, where the fault isolation is performed by a hierarchical structure of the neuro-fuzzy networks is presented in Calado, Sa´ da Costa, Bartys, and Korbicz (2006). A neuro-fuzzy modeling for FDI, involving a hybrid combination of neuro-fuzzy identification and unknown input observers in the neuro-fuzzy and decoupling fault diagnosis scheme (NFDFDS), have been proposed in Uppal, Patton, and Witczak (2006). Please note that all these methods have been proposed very recently. This paper proposes a new fuzzy model-based FDI scheme that combines fuzzy modeling and fuzzy decision making. The proposed fuzzy FDI architecture is presented in the next section. 3. Proposed architecture for FDI This paper uses a straightforward architecture to detect and isolate faults. The two steps, fault detection and fault isolation, are presented next. 3.1. Fault detection The fuzzy FDI system is based on fuzzy models identified directly from data. Note however, that it is possible the use of any type of white or black-box model (fuzzy, neural, etc.) as only the outputs of the models are used in the proposed architecture. The model-based technique proposed in this paper uses a fuzzy model for the process running in normal operation, and one model for each of the faults to be detected. Suppose that a process is running, and n possible faults can be detected. The fault detection and isolation system proposed in this paper for these n faults is depicted in Fig. 3. The multidimensional input of the system, u, enters both the process and a model (observer) in normal operation. The vector of residuals e is defined as

Fig. 3. Fault detection and isolation scheme.

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e¼y^ y;

ð1Þ

where y is the output of the system and ^ y is the output of the model in normal operation. When any component of e is bigger than a certain threshold, the system detects a fault.

example of a system with two outputs (m ¼ 2) is presented in Fig. 4. The m membership functions lei1 ; . . . ; leim must be aggregated using a conjunction operator, which assures that a fault is isolated only when all the residuals eij are close to zero. The aggregation can be given by

3.2. Fault isolation

ci ¼ tðlei1 ; . . . ; leim Þ;

In the proposed FDI architecture (see Fig. 3), fault isolation is obtained by residual evaluation of each of the n models, one for each fault. At each time instant k, a residual ei is computed for each fault

where t is a triangular norm, which is in this paper the minimum operator (Klir & Yuan, 1995). Other t-norms could be used, but the min is used for the sake of simplicity. The function ci resulting from aggregating the membership functions leij will be used to isolated the faults.

ei ðkÞ ¼ yi  ^ yi ;

ð2Þ

where ^yi is the output of the observer for the fault i, with i ¼ 1; . . . ; n. Note that the residual ei is a vector with dimension m, which is equal to the number of outputs. Each scalar residual can be noted as eij ðkÞ ¼ y ij  ^y ij ;

ð3Þ

where ^y ij is the output of the observer for the fault i, and output j, with j ¼ 1; . . . ; m. The fuzzy fault isolation based on FDM proposed in this paper is described in the following. 3.2.1. Membership functions based on the residuals A membership function leij is derived for each residual eij . The membership functions are trapezoidal, as this type revealed to be the most appropriate to describe the residuals in a simple and effective way. The spread of these membership functions is chosen based on the maximum and minimum variations of the residuals, which can be obtained experimentally. The core of each membership function indicates the possible isolation of a fault, i.e., if eij ðkÞ is zero, then the value of the membership function leij should be one. To accommodate process noise, disturbances and model-plant mismatches, the core is a small interval around zero. The size of this interval is again determined based on experimental data. Note that this method to derive membership functions is common in various fuzzy approaches (Mendoncßa, Sousa, & Sa´ da Costa, 2004). An

γi μεi1

μ εi2

y2

y1 εi1= 0

εi2= 0

Fig. 4. Residual evaluation of fuzzy model of fault i.

ð4Þ

3.2.2. Fuzzy decision factors At each time instant the outputs yi ðkÞ are read from a sensor. The outputs of each faulty fuzzy model ^ yi ðkÞ are computed, as well as the residuals eij ðkÞ. Note that these computations are simple and fast (multiplications and additions). This characteristic can be very important for real-time implementations. Further, at each time instant the value of each function ci can also be computed in an easy and fast way. Let d i ðkÞ 2 ½0; 1, i ¼ 1; . . . ; n be the value of ci at time instant k. We called d i ðkÞ a fuzzy decision factor. A vector of fuzzy decision factors can be computed as DðkÞ ¼ ½d 1 ðkÞd 2 ðkÞ    d n ðkÞ;

ð5Þ

i.e., one fuzzy decision factor for each fault. A fuzzy decision factor d i ðkÞ is high only if all the residuals are close to zero. To isolate a fault i, the value of d i ðkÞ must be higher than a threshold T, which must be close to one. Note that the threshold T is equal for all the faults, because the fuzzy decision factors are already normalized in the interval ½0; 1. The threshold is obtained experimentally and defines the regions of fault and no fault. In practice, the definition of this value revealed to be relatively easy, and a value around T ¼ 0:7 isolated the faults properly. This value can suffer a slight change in others processes. Note that several d i ðkÞ can be above the threshold at a certain time k. Therefore, a fault i is isolated only when the remaining faults are below T. However, if only one fault is above the threshold at a certain time instant, it can occur due to noise or model errors. Therefore, our approach considers that a fault i 2 f1; . . . ; ng is isolated when  d i > T and; ð6Þ d l < T 8l–i; for tk consecutive time instants; i.e., when d i is above the threshold T and the remaining d l decision factors are below the same threshold for tk consecutive time instants. The fuzzy isolation scheme proposed in this paper is summarized in the next algorithm:

L.F. Mendoncßa et al. / Expert Systems with Applications 36 (2009) 1092–1104

(1) Build the membership functions leij for each fault i, i ¼ 1; . . . ; n, and for each output j, with j ¼ 1; . . . ; m, based on simulated or experimental values of the residuals eij . (2) Define the threshold value T and the number of consecutive time instants to isolate a fault tk . (3) REPEAT at each time instant k. (a) compute the residuals eij ðkÞ using (3); (b) calculate all the fuzzy decision factors d i ðkÞ by aggregating the membership functions leij using (4); (c) compare the fuzzy decision factors with the threshold T. (4) UNTIL the conditions defined in (6) are satisfied. Note that the fault isolation method based on fuzzy decision making proposed in this paper is general and can be applied to both abrupt and incipient faults.

with i ¼ 1; 2; . . . ; K. Here, Ri is the ith rule, Ai1 ; . . . ; Ain are T fuzzy sets defined in the antecedent space, x ¼ ½x1 ; . . . ; xn  T is the antecedent vector, ai ¼ ½ai1 ; . . . ; aim  and bi are model parameters to be determined, and y i is the rule output variable. K denotes the number of rules in the rule base, and the aggregated output of the model, ^y , is calculated by taking the weighted average of the rule consequents: PK bi y i ^y ¼ Pi¼1 ; ð8Þ K i¼1 bi where bi is the degree of activation of the ith rule: bi ¼ Pnj¼1 lAij , i ¼ 1; . . . ; K and lAij : R ! ½0; 1 is the membership function of the fuzzy set Aij in the antecedent of Ri . The nonlinear identification problem is solved in two steps: structure identification, and parameter estimation. 4.1. Structure identification The structure of the model must be determined first. To identify the model (7), the regression matrix X and an output vector y are constructed from the available data:

4. Fuzzy modeling Fuzzy modeling often follows the approach of encoding expert knowledge expressed in a verbal form in a collection of if–then rules, creating a model structure. Parameters in this structure can be adapted using input–output data. When no prior knowledge about the system is available, a fuzzy model can be constructed entirely on the basis of system measurements. In the following, we consider data-driven modeling based on fuzzy clustering (Babusˇka, 1998). This approach avoids the well-known bottleneck of knowledge acquisition. Fuzzy model is acquired from sampled process data, using the functional approximation capabilities of fuzzy systems. Assume that data from an unknown system y ¼ F ðxÞ is observed. The aim is to use this data to construct a deterministic function y ¼ f ðxÞ that can approximate F ðxÞ. The function f is represented as a collection of fuzzy if– then rules. The system to be identified can be represented as a multiple-input multiple-output (MIMO) nonlinear auto-regressive with exogenous input (NARX) model. These MIMO system can be decomposed into several multiple-input single-output (MISO) models without loss of generality, when certain conditions are fulfilled, i.e., when all state variables are considered as inputs or outputs of the system. In theoretical terms, this means that all the inputs are observable and all the outputs are controllable, see Sousa and Kaymak (2002). Only MISO models are considered in the following for the sake of simplicity. In this paper, we consider rulebased models of the Takagi–Sugeno (TS) type (Takagi & Sugeno, 1985), but recall that any other type of white or black-box model can be used in the FDI architecture. The TS fuzzy model consists of a set of rules with the following structure: Ri : If x1 is Ai1 and . . . and xn is Ain then y i ¼ ai x þ bi

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ð7Þ

XT ¼ ½x1 ; . . . ; xN ; yT ¼ ½y 1 ; . . . ; y N :

ð9Þ

Thus, the matrix to be clustered is given by ZT ¼ ½X; y. The parameter N  n is the number of samples used for identification. In this step, the significant inputs are chosen. Very recently, a decision tree search approach have been proposed for input selection in fuzzy modeling (Mendoncßa, Vieira, & Sousa, 2007). That paper proposed two different approaches of decision tree search algorithms: bottom–up and top–down. The branching decision at each node of the tree is made based on the accuracy of the model available at the node. The bottom–up approach starts with only one input, and increases the number of inputs until a given performance criterion does not improve. This method leads to simple and accurate fuzzy models, as desired in modelbased FDI applications. Note that the smaller the vector x, the faster the model. Moreover, the fuzzy models for FDI must be both simple and accurate models to detect the faults as fast as possible. Therefore, the fuzzy modeling approach used in this paper uses the bottom–up approach proposed in Mendoncßa et al. (2007). 4.2. Parameter estimation The antecedent fuzzy sets, Aij , and the consequent parameters, ai ; bi , are determined in this step. The Gustafson–Kessel fuzzy clustering algorithm computes a fuzzy partition matrix U, whose ikth element uik 2 ½0; 1 is the membership degree of the data object zk in cluster i. The fuzzy sets in the antecedent of the rules are obtained from the partition matrix U. One-dimensional fuzzy sets Aij are obtained from the multidimensional fuzzy sets defined point-wise in the ith row of the partition matrix by projections onto the space of the input variables xj :

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nþ1 lAij ðxjk Þ ¼ projN ðuik Þ; j

ð10Þ

where proj is the point-wise projection operator (Kruse, Gebhardt, & Klawonn, 1994). The point-wise defined fuzzy sets Aij are approximated by suitable parametric functions to compute lAij ðxj Þ for any value of xj (Babusˇka, 1998). The consequent parameters for each rule are obtained as a weighted ordinary least-square estimate. Let hTi ¼ ½aTi ; bi , let Xe denote the matrix ½X; 1 and let Wi denote a diagonal matrix in R N N having the degree of activation, bi ðxk Þ, as its kth diagonal element as defined in (8). Assuming that the columns of Xe are linearly independent and bi ðxk Þ > 0 for 1 6 k 6 N , the weighted least-squares solution of y ¼ Xe h þ e becomes hi ¼ ½XTe Wi Xe 1 XTe Wi y :

ð11Þ

More details of this fuzzy identification method can be found in Babusˇka (1998). 5. Physical modeling of the pneumatic valve A pneumatic servo-actuated industrial control valve, which is presented in Fig. 5a, is used as test bed of the fault detection and isolation approach proposed in this paper. The valve is situated on the outlet of thick juice from the fifth section of evaporation station of the Lublin Sugar Factory in Poland that is associated to the DAMADICS project (Bocaniala & Sa´ da Costa, 2006; DAMADICS, 2000). This type of valve is an automatic equipment designed to set the flow rate in a pipe system. The specific valve that is modeled is a flow control valve whose purpose is to control the feedwater flow into a boiler and superheater assembly. The boiler is an equipment that produces saturated steam from liquid water. Saturated steam enters the superheater and is heated until the desired temperature is reached. The pipe system where the valve is placed has a parallel circuit that is depicted in Fig. 5b. This circuit is intended to allow the replacement of the flow control valve without cutting the feedwater to the boiler. When the flow control valve is replaced, the bypass valves V 1 and V 3 are closed

and the bypass valve V 2 is opened. The boiler comprises a drum, where the water is stored, and a series of pipes, where the water is heated until the state changes, from liquid to gas. The level of the drum has a very strong influence on the behavior of the boiler. If it is either too high or too low the consequences may be disastrous. Therefore, there is a control loop intended to keep the drum level at a set value. The control system has a PI controller that based on the value of the drum level changes the CV value (a control reference variable that influences the flow across the valve), to maintain the drum level at about half of its capacity. The valve, the boiler and the superheater are modeled using the physical laws that govern their behavior. This type of modeling facilitates the introduction of faults in the system. The model is used to extract data when faults occur. The actuator valve is a final control device that acts on the controlled process. Most of these valves are pneumatically actuated, consisting of three main parts: body of the valve, actuator (e.g., spring-and-diaphragm pneumatic servomotor) and positioner controller, see Fig. 5b. The description of the main parameters of the servo-actuated valve are given in Table 1. The complete valve modeling is presented in Sa´ da Costa and Louro (2003). This paper presents briefly the three main components: valve body, pneumatic actuator and positioner. 5.1. Valve body The valve body is the component that determines the flow through the valve. A change of the restricted area in the valve regulates the flow. There are many types of valve bodies. The differences between them relate to the form by which the restricted flow area changes. This paper addresses the globe valve case. However, the results expressed here can easily be applied to other types of valve bodies. Modeling the flow through the valve body is not an easy task, since most of the underline physical phenomena are not fully understood. The most common approach to determine the flow through a valve is to use dimensional analysis (White, 1994), based on the model of the flow through a sharp-edged orifice. The flow through the valve is given by

Table 1 Servo-actuated pneumatic valve parameters

(a) Industrial valve.

(b) Diagram of the valve.

Fig. 5. Servo-actuated pneumatic valve.

PSP PT FT TT ZT E=P V 1, V 2 V3 Ps CVI CV F x

Positioner of supply air pressure Air pressure transmitter Volume flow rate transmitter Temperature transmitter Rod position transmitter Electro-pneumatic converter Cut-off valves Bypass valve Pneumatic servomotor chamber pressure Controller output Control reference value Volumetric flow Servomotor rod displacement

L.F. Mendoncßa et al. / Expert Systems with Applications 36 (2009) 1092–1104

sffiffiffiffiffiffiffi DP ; F ¼ 100K v f ðxÞ q

ð12Þ

where K v is the flow coefficient (m3 =h) (given by the manufacturer), f ðX Þ is the valve opening function, DP is the pressure difference across the valve (MPa), q is the fluid density (kg=m3 ), F is the volumetric flow through the valve (m3 =h), and x is the position of the rod (m), which is the same of the plug. The valve opening function f ðxÞ indicates the normalized valve opening area. It varies in the interval [0,1], where the value 0 indicates that the valve is fully closed and the value 1 indicates that it is fully open. The value of X is defined as the percentage of valve opening. 5.2. Pneumatic actuator The flow is set by the position of the rod, which determines the restricted flow area. The actuator sets the position of this rod. There are many types of servo-actuators: electrical motors, hydraulic cylinders, spring-and-diaphragm pneumatic servomotor, etc. The most common type of actuator is the spring-and-diaphragm pneumatic servomotor due to its low cost. This actuator consists of a rod that has, at one end, the valve plug and, at the other end, the plate. The plate is placed inside an airtight chamber and connects to the walls of this chamber by means of a flexible diaphragm. The model of the pneumatic actuator is obtained through the force equilibrium in the rod. There are seven forces acting on the rod: the dynamic force F D , the viscous damping F vd , the weight F W , the spring force F S , the pressure force F P , the Coulomb friction F f , and the force caused by the flow F flow . The equilibrium is expressed by the following formula: F D þ F vd þ F W þ F S þ F p þ F f þ F flow ¼ 0:

ð13Þ

Using (13), the dynamic model for the rod/plug position is given by d2 x dx P us  P ds A0 ¼ 0; mrp 2 þ b  mrp g þ Kðx þ x0 Þ  P c A þ K m ðxÞ dt dt ð14Þ where x is the rod position (m), mrp is the mass of the moving parts, rod plus plug (kg), b is the viscous damping constant (N s/m), g is the gravity acceleration (m=s2 ), K is the spring constant (N m), x0 is the initial compression of the spring (m), P c is the pressure inside the chamber (Pa), A

CV

Filter

+

PID controller

if

is the area of the pneumatic diaphragm [m2 ], P us is the upstream fluid pressure (Pa), P ds is the valve downstream pressure (Pa), K m is the valve recovery coefficient, and Ao is the projected area of the valve plug according to the direction of the flow (m2 ). By performing a simple mass balance and considering that the volume of the chamber changes with the rod position, the pressure inside the chamber can be computed as R P 0 V 0 þ RT m_ dt ; ð15Þ Pc ¼ V 0 þ Ax where P 0 is the atmospheric pressure (Pa), V 0 is the initial volume of the chamber (m3 ), R is the perfect gas constant (m2 =ðs2 KÞ), T is the air temperature (K), and m_ is the airflow into or from the chamber (kg/s). However, during the normal functioning of the equipment, the valve is most of the time not fully open or closed. To account for this, the airflow is estimated by ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4:13  106 jif jC v ðP a1  P a2 ÞP a2 ; if P a1 =2 6 P a2 ; m_ ¼ if P a1 =2 > P a2 ; 2:06  106 jif jC v P a1 ; ð16Þ where C v is the valve flow coefficient (m3 =s), P a1 is the air upstream pressure (Pa), and P a2 is the air downstream pressure (Pa). The signal jif j is received by the electrical-pneumatic (E/P) transducer and varies in the interval [1,1]. The negative values indicate that the chamber should be deflated, and thus the valve should establish a connection with the atmosphere (in this case P a1 is the pressure inside the chamber and P a2 is the atmospheric pressure); the positive values indicate that the chamber should be filled and therefore the valve should establish a connection with the pneumatic circuit (in this case P a1 is the pressure of the pneumatic circuit and P a2 is the pressure inside the chamber). 5.3. Positioner The positioner, shown in Fig. 6, determines the flow of air into the chamber. The positioner is the control element that performs the position control of the rod. It receives a control reference signal (set-point) from a computer controlling the process, passes it through a second order filter, to get ride of noise and abrupt changes of the reference signal, prior to the PID control action that leads the rod’s position to that reference signal. The positioner comprises

E/P transducer

-

X

1099

Position sensor

Positioner

Fig. 6. Positioner controller.

Pneumatic actuator

X

Valve

F

L.F. Mendoncßa et al. / Expert Systems with Applications 36 (2009) 1092–1104

a position sensor and an electrical-pneumatic transducer. The first determines the actual position of the rod, so that the error between the actual and the desired position (reference signal) can be obtained. The E/P transducer receives a signal from the PID controller transforming it in a pneumatic valve opening signal that adds or removes air from the pneumatic chamber. This transducer is also connected to a pneumatic circuit and to the atmosphere. If the controller indicates that the rod should be lowered, the chamber is connected to the pneumatic circuit. If, on the other hand, the rod should be raised, the connection is established with the atmosphere, thus allowing the chamber to be emptied. The if signal is the output of the PID controller, which is given by   Z 1 dðCV  X Þ ðCV  X Þ dt þ T D if ¼ K p ðCV  X Þ þ ; TI dt ð17Þ where K p is the proportional gain, T I is the integration time, T D is the derivative time, and X is the measured rod position.

MSE ¼

N 1 X 2 ðy ðkÞ  ^y i ðkÞÞ ; N k¼1 i

ð18Þ

where y i is a system output and ^y i is the output i of the model. The percentile VAF is also used to measure the performance of the obtained models VAF ¼ 1 

covðy i  ^y i Þ  100%; covðy i Þ

ð19Þ

where cov is the covariance of the respective vector. 6.2. Modeling results The set of identification data used to build the valve models in normal operation and with faults contains 56

Flow (m3/h)

1100

55 54 53 52 51 0

200

400

600

800

1000

1200

1400

1600

1800

200

400

600

800

1000

1200

1400

1600

1800

From a thorough analysis of the described variables, it can be concluded that, for FDI purposes, the most relevant output variables are the flow process value, F, and the servomotor rod displacement measured in terms of valve opening, which was defined as X (Sa´ da Costa & Louro, 2003). Therefore, these variables have been considered as the outputs of the process. To measure modeling accuracy, this paper uses the mean squared error (MSE) and the variance accounted for (VAF). The MSE is defined as

0 –0.5 –1 0

Time [s]

(a) Top: Flow output. Bottom: Flow residuals. Rod displacement (%)

6.1. Modeling of the system

1 0.5

–1.5

73 72.5 72 71.5 71

0

200

400

600

800

1000

1200

1400

1600

1800

0

200

400

600

800

1000

1200

1400

1600

1800

1

Residuals (%)

The fuzzy FDI architecture proposed in this paper, which was presented in Fig. 3, was applied to the industrial servo-actuated valve, to detect and isolate faults. The valve simulator, which was presented in Section 5, is used to obtain data for each one of the faults tested in this paper. Table 2 describes the faults considered in this work, which are the most common faults that can occur in the process. Note that all the five considered faults can present abrupt or incipient behavior.

Residuals (m3/h)

1.5

6. Application example

0.5 0 –0.5 –1

Time [s]

(b) Top: Rod displacement. Bottom: Rod displacement residuals.

Table 2 Faults description

Fig. 7. Outputs of pneumatic valve without faults.

Faults

Description

F1 F2 F3 F4 F5

Valve clogging Valve or valve seat erosion Internal leakage Medium evaporation or critical flow Flow rate sensor fault

Table 3 Model accuracy of fuzzy models: system in normal operation

MSE VAF (%)

Flow

Displacement

0.31 83.9

0.17 57.6

L.F. Mendoncßa et al. / Expert Systems with Applications 36 (2009) 1092–1104

1101

Table 4 Number of clusters of fuzzy models with and without faults Without faults

Flow Disp.

Abrupt faults

3 6

Incipient faults

F 1a

F 2a

F 3a

F 4a

F 5a

F 1i

F 2i

F 3i

F 4i

F 5i

2 2

3 9

4 5

4 5

2 6

5 9

6 8

2 4

4 6

6 3

Table 5 Model accuracy of fuzzy models for abrupt and incipient faults for flow and rod displacement Faults

Abrupt faults

Incipient faults

F 1a

F 2a

F 3a

F 4a

F 5a

F 1i

F 2i

F 3i

F 4i

F 5i

VAF(%)

Flow Disp.

74.1 76.5

77.2 63.9

77.6 56.9

97.3 99.9

74.9 60.9

96.1 99.5

99.1 63.8

76.5 59.3

97.4 96.4

99.6 84.3

MSE

Flow Disp.

3.6 3.5

1.4 0.17

1.2 0.14

2000 samples. Fig. 7a and b presents both outputs of the process (flow in m3 =h and rod displacement in % of valve closing) under normal operation and the obtained residuals e. The bottom–up tree search approach in Mendoncßa et al. (2007) was used to derive the structure of the fuzzy models for the pneumatic valve without faults. The obtained results are presented in Table 3. Both accuracy measures present good results. The VAF values are not close to 100% due to the large noise present in the process. However, by observing Fig. 7 it can be seen that the residuals are very close to zero, which is the main objective of the fuzzy model. Recall that the five faults considered were presented in Table 2. Let an abrupt fault be denoted as a and an incipient fault be denoted as i. As an example, the fault F 1 can be abrupt, F 1a, or incipient, F 1i. Again, the bottom–up tree search approach in Mendoncßa et al. (2007) was used to derive the structure of fuzzy models, now for abrupt and for incipient faults. The number of clusters (rules) selected for all the identified fuzzy models are presented in Table 4. The criterion to determine the number of clusters is presented in Sugeno and Yasukawa (1993). In general, the number of clusters for rod displacement is bigger than for the flow. This is expectable, because rod displacement has much more noise.

4.0 0.17

4.3 2.8

2.1 0.16

4.7 0.17

1.2 2.1

1.0 0.16

Table 6 Incipient faults behavior Fault

Starting time (s)

Settling time (s)

F 1i F 2i F 3i F 4i F 5i

500 50 50 50 50

1700 1700 1700 260 900

Table 7 Detection and isolation times of abrupt and incipient faults (in seconds) Faults

Abrupt faults

F1 F2 F3 F4 F5

Incipient faults

Detection

Isolation

Detection

Isolation

51 51 51 51 51

155 114 115 52 133

519 114 156 51 85

750 449 394 183 125

Abrupt fault F5 60

Flow (m3/h)

2.6 0.16

40 20 0

Fault starting time

Fault settling time

Fig. 8. Generic incipient fault behavior.

Time

Rod displacement (%)

Fault intensity

0

200

400

600

800

200

400

600

800

1000

1200

1400

1600

1800

2000

1000

1200

1400

1600

1800

2000

74 73 72 71 0

Time [s]

Fig. 9. Abrupt fault F 5a. Top: flow output. Bottom: rod displacement.

1102

L.F. Mendoncßa et al. / Expert Systems with Applications 36 (2009) 1092–1104

The accuracy of the obtained fuzzy models in terms of VAF and MSE are presented in Table 5. Note that in general the models present good accuracy. 6.3. FDI results

Abrupt fault F5

Flow residuals (m3/h)

–20

time detection of abrupt fault F5

–40 –60

Rod displacement residuals (%)

Abrupt fault F1

0

200

400

600

800

1000

1200

1400

1600

–80

1800

2 1 0 –1 –2

–60

–100

0

0

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

Rod displacement residuals (%)

Flow residuals (m3/h)

To simulate the studied faults, several parameters have to be defined, namely: the fault starting time for abrupt faults; the fault starting time and the fault settling time for incipient faults. The generic incipient fault behavior is presented in Fig. 8. The fault starting time is the time at which the faulty behavior starts. The fault settling time is the time required after a change for a process to reach its steady state. The five simulated incipient faults behaviors of the servo-actuated valve, namely their starting and settling times, are presented in Table 6. The fault starting time for all abrupt faults is at fifty seconds.

The fuzzy FDI architecture proposed in this paper was able to detect and isolate all the considered faults. The detection and isolation time instants for the considered faults are presented in Table 7. Recall that a fault is detected if a residual ei for any output deviates significantly from zero. The fault detection times were computed based on this assumption. The fault isolation is made considering the fuzzy decision making approach presented in Section 3.2. Table 7 shows that the detection of abrupt faults is made in the first time instant (51 s). The isolation time of abrupt faults can vary significantly. For example, isolation time of fault F 4a is 52 s, i.e., only one time instant after the detection time. However, all the other faults are more difficult to isolate. Fault F 1a, for instance, is only isolated after 155 s. This problem is due to a small delay in the fault model response, as it can be seen in Fig. 10b.

200

400

600

800

200

400

600

800

1000

1200

1400

1600

1800

1000

1200

1400

1600

1800

50 40 30 20 10 0

Time [s]

(a) Detection of abrupt fault F 5a (model without faults).

0

Incipient fault F5 60

–20 –40

Rod displacement residuals (%)

200

400

600

800

1000

1200

1400

1600

1800

Flow (m3/h)

time isolation of abrupt fault F5

0.5 0 –0.5 –1

40 20 0 0

1

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

(b) Isolation of abrupt fault F 5a using model of fault F 5a. Fig. 10. Detection and isolation of fault F 5a. Top: flow residuals. Bottom: rod displacement residuals.

Rod displacement (%)

Flow residuals (m3/h)

Fig. 11. Residuals of abrupt fault F 1a when fault F5a occurs. Abrupt fault F5

500

1000

1500

2000

1000

1500

2000

74 73 72 71 0

500

Time [s]

Fig. 12. Fault F 5i. Top: flow output. Bottom: rod displacement output.

L.F. Mendoncßa et al. / Expert Systems with Applications 36 (2009) 1092–1104

Rod displacement residuals (%) Flow residuals (m3/h)

Flow residuals (m3/h)

–20 –40 –60 –80 200

400

600

800

200

400

600

800

1000

1200

1400

1600

1800

1000

1200

1400

1600

1800

2 1 0 –1 –2

Time [s]

0 –20 time detection of incipient fault F5

–40 –60

0

200

400

600

800

1000

1200

1400

1600

1800

1000

1200

1400

1600

1800

2 1 0 –1 –2 0

time detection of incipient fault F5

200

400

600

800

Time [s]

Incipient fault F5 5

0

–5

time isolation of incipient fault F5

200

400

600

800

200

400

600

800

1000

1200

1400

1600

1800

1000

1200

1400

1600

1800

1 0.5 0 –0.5 –1

in this case F 1a, when fault F 5a occurs is shown in Fig. 11. This figure shows clearly that fault F 1a is not isolated, as the residuals for F 1a are very far from zero. The behavior of incipient fault F 5i is presented in Fig. 12. The residuals obtained in the detection and isolation of incipient fault F 5i are presented in Fig. 13, which show both the detection and isolation times. The residuals behavior of another fault (F 1i) when fault F 5i occurs is shown in Fig. 14. This figure shows clearly that the residuals of F 1i are very far from zero, and therefore this fault is not isolated, as expected. 7. Conclusions

(a) Detection of incipient fault F 5i (model without faults).

Flow residuals (m3/h)

0

Fig. 14. Residuals of incipient fault F 1i when fault F5i occurs.

Incipient fault F5

Rod displacement residuals (%)

Incipient fault F1

Rod displacement residuals (%)

The time detection of incipient faults is more difficult as expected, because residual values in the first time instants are much smaller than in the abrupt case. Fault F 4i is rapidly detected, but fault F 3i is only detected 156 s after the beginning of the faulty behavior, see Table 7. Note however, that this fault evolve in a much slower way, and therefore it is more difficult to detect. It is important to notice that all the faults are correctly isolated before their fault settling time. In the following, several figures showing the behavior of the system and the residuals when different faults occur are presented. First, the behavior of abrupt fault F 5a is presented in Fig. 9. The detection and isolation of abrupt fault F 5a is presented in Fig. 10. Fig. 10a shows the time instant where the fault is detected. Further, Fig. 10b shows the time isolation of the fault. Note that the flow residual takes some time to become zero, which is a necessary condition to isolate a fault. The residuals behavior of another fault,

1103

Time [s]

(b) Isolation of incipient fault F 5i using model of fault F 5i. Fig. 13. Detection and isolation of fault F 5i. Top: flow residuals. Bottom: rod displacement residuals.

This paper proposes an FDI architecture based on fuzzy modeling and on a fuzzy decision making approach. Fuzzy models (observers) are used for both normal operation and for each faulty operation. The fuzzy observers are obtained from simulated data. The relevant inputs of the fuzzy models were determined using a tree search method, using a bottom–up approach. The detection and isolation of faults are based on residual evaluation. A fuzzy decision making approach was proposed to isolate incipient and abrupt faults. The isolation is performed by evaluating fuzzy decision factors that are built based on residuals. The modeling results show that the structure of fuzzy models were well determined using the tree search algorithm. The fuzzy isolation approach proposed in this paper was applied to a pneumatic servomotor actuated industrial valve, and it was able to detect and isolate 10 abrupt and incipient faults. Note that the data contains noise, which increases the difficulty to detect and isolate the faults. Future research will consider the extension of the proposed FDI scheme to a larger number of faults and the application of the proposed method to a distillation column. Further, the proposed method will be integrated in fault tolerant control.

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