Ind. Eng. Chem. Res. 2008, 47, 9447–9456
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Fault Diagnosis Based on Signed Digraph Combined with Dynamic Kernel PLS and SVR Ning Lu¨* and Xiong Wang Department of Automation, Tsinghua UniVersity, Beijing 100084, China
The signed digraph (SDG) method for fault diagnosis, which is one of the model-based methods, has been widely applied in the chemical industry in recent years. However, how to elicit appropriate thresholds for SDG is a very difficult problem. This study presents a new hybrid method combining SDG with dynamic kernel partial least-squares (DKPLS) and support vector regression (SVR) for fault diagnosis. Using the relationships of each variable in SDG, a series of DKPLS-SVR models are built to estimate the values of the measured variables in process. The difference between the estimation and the measured value can determine the qualitative status of the variable, and then the fault can be diagnosed by SDG reasoning. Therefore, the threshold of each measured variable does not need to be decided in advance. The method can also overcome the limited availability of using the KPLS method alone in identifying the root cause. To verify the performance of the proposed method, its application is illustrated on the Tennessee Eastman (TE) challenge process. Through case studies, the proposed method demonstrates a good diagnosis capability compared with previous hybrid methods. 1. Introduction With the increase of the scale and complexity in modern chemical industry, fault diagnosis only based on quantitative models, such as mathematical equations, is difficult, more and more because of the absence of precise mathematical descriptions to those large-scale complicated systems. Signed digraphs (SDGs), which can describe target systems as qualitative network models with nodes and branches, provide a new approach to describe large-scale systems. In this formulation, the propagation principle of fault evolution in a complicated system can be explored and inference can be applied to accomplish hazard and risk assessment or fault diagnosis. The SDG concept and method for fault diagnosis were first presented by Iri et al. in 1979.1 The SDG method can save time, manpower, and expenses. It also has good maturity, in-depth reasoning ability, and several other advantages. In recent years, the SDG method has been widely applied in the chemical industry and has already become one of the key technologies in safety engineering, such as the hazard and operability study (HAZOP)2 and fault diagnosis.3-6 To perform fault diagnosis based on SDG, each measured variable should be compared against its high and low thresholds to identify its deviation. If there are n measured variables in the SDG model, the user has to specify 2n thresholds. However, how to choose appropriate thresholds is a very difficult problem because of the large number of parameters involved and lack of any clear rules of tuning them. Usually, the thresholds are derived from the user’s experience.7 Each threshold is often a constant positive or negative bias on the basis of the average value of a variable when the system is at the normal situation. These fixed thresholds will lead to significant number of false alarms inevitably due to system noise.7 To overcome this disadvantage, a new hybrid method for fault diagnosis, combining SDG with dynamic kernel partial least-squares (DKPLS) and support vector regression (SVR), is represented here. The DKPLS-SVR model, as a kind of data-driven approach, is used * To whom correspondence should be addressed. E-mail: lun02@ mails.tsinghua.edu.cn. Tel.: +86-1381-065-1331. Fax: +86-10-62786911.
to perform fault detection. Whenever a fault symptom is indicated, the system status derived from the DKPLS-SVR model is input to the SDG to diagnose the fault origin. Therefore, there is no need to determine the thresholds for SDG in advance. The application of this proposed method is illustrated on the Tennessee Eastman (TE) challenge process. 2. Previous Studies 2.1. SDG. First, a set of descriptions accepted widely for SDG are introduced here. SDGs are developed from directed graphs by defining symbols of nodes and branches. Definition 1:8 SDG model γ ) (G, φ) is a combination of a directed graph G and a function φ. The directed graph G is composed of four portions (V, E, ∂+, ∂-), where the nodes set is V ) {V1, V2,..., Vm}; the branches set is E ) {e1, e2,..., en}; the initial node of a branch is ∂+: E f V; the terminal node of a branch is ∂-: E f V. The function φ: E f { +, - }. φ(ek) (ek ∈ E) is the sign of the branch ek. Definition 2:8 A pattern of a SDG model γ ) (G, φ) is a function ψ: V f { +, 0, - } of the statuses of all nodes. ψ(Va) (Va ∈ V) is the sign of the node Va, that is, ψ(Va) ) 0 if |XVa XVa| < εVa; ψ(Va) ) + if XVa - XVa g εVa; ψ(Va) ) - if XVa - XVa g εVa , where εVa is the threshold. A SDG model is composed of several nodes and branches. The nodes denote variables in the target system and the directed branches denote the relationships between the nodes. In a traditional 3-status SDG model, the node has three statuses, “0”, “+”, and “-”, as normal, higher, and lower compared against the threshold. If the variation of one node can result directly in the variation of another node, a directed branch will be drawn between these two nodes, from the forward node (cause variable) to the backward node (effect variable). Two kinds of relationship can exist between any two joined nodes, namely, positive effect and minus effect. If an increase in the cause variable increases the value of the effect variable, there is a positive effect. In contrast, if an increase in the cause variable decreases the value of the effect variable, there is a minus effect.
10.1021/ie8009457 CCC: $40.75 2008 American Chemical Society Published on Web 11/05/2008
9448 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
Figure 1. The process flow diagram of TE process.17 Table 1. Manipulated and Measured Variables of the TE Process17 variable
description
MV1 MV2 MV3 MV4 MV5 MV6 MV7 MV8 MV9 MV10 MV11 MV12 F1 F2 F3 F4 F5 F6 P7 L8 T9 F10 T11 L12 P13 F14 L15
D feed flow (stream 2) E feed flow (stream 3) A feed flow (stream 1) A+C feed flow (stream 4) compressor recycle valve purge valve (stream 9) separator pot liquid flow (stream 10) stripper liquid product flow stripper steam valve reactor cooling water flow condenser cooling water flow agitator speed A feed (stream 1) D feed (stream 2) E feed (stream 3) A+C feed (stream 4) recycle flow (stream 8) reactor feed reactor pressure reactor level reactor temperature purge rate (stream 9) separator temperature separator level separator pressure separator underflow (stream 10) stripper level
sampling interval (min) variable 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
For any observation of the system status in a certain moment, a SDG model can determine qualitatively whether each node has deviated from its normal state, as well as the direction of deviation, according to a set of threshold values defined in advance. By reasoning in the SDG model, all nodes which have deviated and all possible paths through which the deviations propagate in the system can be determined. The resulting paths are called fault propagation paths or consistent paths. Definition 3:8 In a SDG model γ ) (G, φ), with pattern ) Ψ, then the branch ek is consistent if φ(ek) ) ψ(∂+ ek) ψ(∂ek). If ψ(Va) * 0, the node Va is called a valid node. A consistent path consists of some consistent branches, all of which have the same direction. Faults will propagate and evolve only through such consistent paths. Therefore, the process of fault evolution in a complicated system can be explored by
P16 F17 T18 F19 J20 T21 T22 XA XB XC XD XE XF YA YB YC YD YE YF YG YH ZD ZE ZF ZG ZH
description
sampling interval (min)
stripper pressure stripper underflow (stream 11) stripper temperature stripper steam flow compressor work reactor cooling water outlet temperature condenser cooling water outlet temperature composition of A (stream 6) composition of B (stream 6) composition of C (stream 6) composition of D (stream 6) composition of E (stream 6) composition of F (stream 6) composition of A (stream 9) composition of B (stream 9) composition of C (stream 9) composition of D (stream 9) composition of E (stream 9) composition of F (stream 9) composition of G (stream 9) composition of H (stream 9) composition of D (stream 11) composition of E (stream 11) composition of F (stream 11) composition of G (stream 11) composition of H (stream 11)
1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 15 15 15 15 15
searching for all consistent paths in a SDG model. In this manner, the adverse consequences and fault causes can be determined. This is the foundation of HAZOP and fault diagnosis based on SDG. The reasoning used in a SDG model is actually a process of searching for consistent paths. The inference mechanism includes both forward inference and inverse inference. Forward inference involves searching for consistent paths from selected cause nodes to all possible effect nodes, similarly inverse inference considers paths from concerned effect nodes to all possible cause nodes. In general two methods to implement SDG inference are used at present: (i) an inference engine based on fault knowledge implemented using commercial expert system software and (ii) network iteration, based on directed graphs in graph theory. Inference using a commercial expert system
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software package is generally a kind of forward inference: a diagnosis knowledge base is first formed by off-line qualitative fault simulation (forward inference), and then pattern matching and fault diagnosis, based on the online measured data and the knowledge base, is performed. This inference is however opposite to the internal inverse mechanism of diagnosis. As a result, it will inevitably lead to the problem of combination explosion when there are multiple fault origins simultaneously present in the system. The single fault origin assumption is therefore very important for this inference engine. In this study, the SDG inference from our previous study9 is adopted, which is inverse inference combined with forward inference based on a breadth-first algorithm with consistency rules. Compared with conventional inference engines, this inference can better avoid qualitative spuriousness and combination explosion and can deal with unobservable nodes and multiple faults in SDGs more effectively. 2.2. KPLS. As one of the multivariate statistical methods used extensively, partial least-squares (PLS) was first introduced by Herman Wold.10 The PLS method projects the input and output data to a latent space and maximizes the covariance between these two subspaces. The latent space usually has fewer dimensions than the original input space. Compared to the classical principal component analysis (PCA) method, PLS has been shown to be more powerful and robust for the process modeling in systems where there are fewer observations than predictor variables.11 However, PLS is a linear method. It is inappropriate for describing the systems which exhibit significant nonlinear characteristics, such as the chemical industry. To tackle the issue of data nonlinearity, kernel PLS (KPLS) method was developed, which map the original input data nonlinearly to a feature space of arbitrary dimension where a linear PLS model is created. KPLS has been shown to perform better than linear PLS in nonlinear systems.12 Consider a set of input variables, xi ∈ Rm, i ) 1, 2,..., N and a set of output variables, yi ∈ Rq, i ) 1, 2, · · · , N. The data matrices are X ) [x1,..., xN]′ ∈ RN×m and Y ) [y1,..., yN]′ ∈ RN×q, respectively. Define Φ as the nonlinear map which projects the input variables from the original space to the feature space, Φ ) [Φ(x1),..., Φ(xN)]′ ∈ RN×M. Define a matrix K as the kernel matrix to represent ΦΦ′, where Kij ) K(xi,xj) ) 〈Φ(xi),Φ(xj)〉, i,j ) 1, 2,..., N. Just like the PLS method, the KPLS algorithm is to sequentially extract the latent vectors t, u and the weight vectors w, c from the Φ and Y matrices in decreasing order of their corresponding singular values. Therefore, the Φ and Y matrices can be represented as
(3)
B ) Φ′U(TKU)-1T′Y
(4)
Then, ˆ ) ΦΦ′U(TKU)-1T′Y ) KU(TKU)-1T′Y (5) Y 2.3. SVR. The support vector machine (SVM), which was introduced by Vapnik14 and co-workers, is a very powerful method in a wide variety of applications, such as nonlinear classification and regression. Many researchers have demonstrated that the SVM has already outperformed most other methods in nonlinear regression,15 such as least-squares regression, neural networks, and so on. The SVR model can be expressed as f(x) ) 〈w, Φ(x)〉 + b (6) where, Φ(x) represents a nonlinear mapping from the original input space to a feature space. The parameter w is a normal vector and the parameter b is the bias term. The parameters can be obtained by solving the following optimization problem: 1 minimize |w|2 subject to yi - f(xi) e ε f(xi) - yi e ε, 2 i ) 1, 2, ..., N (7) where ε is the tolerance range. By introducing the positive slack variables ξi and ξi*, the optimization problem can be changed to14 N
∑
1 minimize |w|2 + C (ξi + ξ*i ) 2 i)1
subject to yi - 〈w, Φ(xi)〉 - b e ε + ξi
〈w, Φ(xi)〉 + b - yi e ε + ξ*i ξi, ξ*i g 0, i ) 1, 2, ..., N (8) This optimization can be solved using Lagrange multipliers. The Lagrangian is defined as N
∑
1 L : ) |w|2 + C (ξi + ξ*i ) 2 i)1 N
∑ R (ε + ξ - y + 〈w, Φ(x )〉 + b) i
i
i
i
i)1 N
∑ R (ε + ξ
* i + yi - 〈w, Φ(xi)〉 - b)-
* i
i)1
N
Φ ) TP′ + E
∑ (η ξ + η ξ ) (9)
(1)
i i
* * i i
i)1
Y ) UQ′ + F
(2)
where T ) [t1,..., tA] ∈ R and U ) [u1,..., uA] ∈ R are the matrices of extracted A score vectors, P ) [p1,..., pA] ∈ Rm×A and Q ) [q1,..., qA] ∈ Rq×A are the matrices of loadings, and E ∈ RN×M and F ∈ RN×q are the matrices of residuals. The KPLS algorithm can be performed as follows:13 1 Randomly initialize u. 2 t ) ΦΦ′u ) Ku. 3 trt/|t|. 4 c ) Y′t. 5 u ) Yc, uru/|u|. 6 Repeat 2-5 until convergence. 7 Deflate K, Y matrices, Kr(I - tt′)K(I - tt′), Yr(I tt′)Y, where I is an N-dimensional identity matrix. The KPLS regression model can be expressed as: N×A
ˆ ) ΦB Y The regression coefficient B is
Problem 9 can be equivalently solved in its dual forms as:
N×A
N
maximize -
∑
1 (R* - Ri)(R*j - Rj)〈Φ(xi), Φ(xj)〉 2 i,j)1 i N
ε
N
∑ (R
* i + Ri) +
i)1
∑ y (R i
* i - Ri)
i)1
N
subject to
∑ (R - R ) ) 0 R , R i
* i
i
* i ∈ [0, C]
(10)
i)1
After training the SVR, the regression function has the following expansion: N
fSVR(x) )
∑ (R
* i - Ri)〈Φ(xi), Φ(x)〉 + b
i)1
(11)
9450 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
2.4. TE Challenge Process. The TE challenge process is a simulation of a real chemical plant provided by the Eastman Company. Downs and Vogel described the details of the TE model in 1993.16 The process has five major units: a reactor, a product condenser, a vapor/liquid separator, a recycle compressor, and a product stripper. Here, the base control scheme proposed by McAvoy and Ye17 is introduced into the process. The flow diagram is presented in Figure 1. The process produces two products G and H from four reactants A, C, D, and E. A byproduct F is also produced. In addition, there is an inert B in the C stream (stream 4), which is noncondensable. The reactions in the reactor are A(g) + C(g) + D(g) f G(l), product 1 (15) A(g) + C(g) + E(g) f H(l), product 2 (16) A(g) + E(g) f F(l), byproduct (17) 3D(g) f 2F(l), byproduct (18) The reactions are irreversible and exothermic. The reaction rate depends on the temperature in the reactor and the reactant concentrations. There are a total of 41 measurements and 12 manipulated variables available in the process. Table 1 describes these variables with their sampling interval. The TE process simulation contains 20 preprogrammed disturbances: IDV1 through IDV20. They are summarized in Table 2. The TE simulation code used in this study is available in Matlab and Simulink coded by Professor N. Lawrence Ricker. The simulation archive can be obtained from Professor N. Lawrence Ricker’s Web site (http://depts.washington.edu/control/ LARRY/TE/download.html). 3. Fault Diagnosis Based on DKPLS-SVR 3.1. SDG Modeling. By analyzing the process, the SDG model of the process can be built. There are mainly three methods for SDG modeling: the method based on mathematical equations,8,18 the method based on Piping & Instrumentation Diagram (P & ID),4,19 and the method based on experience and knowledge. Usually, a two-step procedure is performed to build the SDG model for the entire process. In the first step, the local SDGs for decomposed subsystems are built using the methods Table 2. Fault Defined in the TE Process17 fault ID IDV1 IDV2 IDV3 IDV4 IDV5 IDV6 IDV7 IDV8 IDV9 IDV10 IDV11 IDV12 IDV13 IDV14 IDV15 IDV16-IDV20
description A/C feed ratio, B composition constant (stream 4) B composition, A/C feed ratio constant (stream 4) D feed temperature (stream 2) reactor cooling water inlet temperature condenser cooling water inlet temperature A feed loss (stream 1) C header pressure loss-reduced availability (Steam 4) A,B,C feed composition (stream 4) D feed temperature (stream 2) C feed temperature (stream 4) reactor cooling water inlet temperature condenser cooling water inlet temperature reaction kinetics reactor cooling water valve condenser cooling water valve unknown
type step step step step step step step random variation random variation random variation random variation random variation slow drift sticking sticking
Table 3. Subsystems in the TE Process tag
description
S1 S2
reactor pressuresfeed flow (stream 1) control system G/H compositionsreactor levelsD, E feed flow (stream 2,3) control system reactor temperature control system recycle flow (stream 8)scondenser cooling control system purge composition of B (stream 9)s purge rate control system separator levelsseparator underflow (stream 10) control system stripper levelsstripper underflow (stream 11) control system stripper underflow (stream 11)sA+C feed flow (stream 4) control system product composition of E (stream 11)sstripper temperature control system
S3 S4 S5 S6 S7 S8 S9
discussed above. In the second step, the local SDGs are composed to derive the SDG for the entire process. The TE process can be decomposed into 9 subsystems according to the distributed control systems in the process. They are summarized in Table 3. Taking the subsystem S2 as an example, the local SDG is illustrated as Figure 2. The interactions between the local SDG of subsystem S2 and the local SDGs of the other subsystems are illustrated as Figure 3. Figure 4 shows the entire process interactions in the SDG model of the TE process. A solid line in the SDG denotes a positive effect; a dashed line denotes a minus effect. In the SDG model of the entire TE process, there are a total of 127 nodes denoting the variables and 15 rootcause nodes denoting IDV1-IDV15. The details of the entire SDG model and the modeling process will not be introduced here. 3.2. DKPLS-SVR Modeling. Using the relationships of each target variable which should be monitored in the SDG model, a series of DKPLS-SVR models12 can be built to estimate the values of the measured variables in process. For a concerned variable y, its local KPLS model is constructed with its cause variables in the SDG model as the input matrix X. To increase the accuracy of the regression, KPLS is integrated with ARMAX to become dynamic KPLS (DKPLS).20 Therefore, the resulting input of the local DKPLS model also includes the past values of the target variable as well as the cause variables. Then, the input matrix X at time t can be composed as follows:
[ ]
xi(t) xi(t - ∆t) y(t - ∆t) X) l xi(t - l∆t) y(t - l∆t)
′
(12)
where l is the number of necessary past values and ∆t is the sampling rate. The number l is usually 1 or 2, which indicates the order of the dynamic system. For example, consider the reactor level L8 in Figure 2. The cause variables of L8 are F1, F2, F3, and F5, so at time t, the input X of L8(t) is F1(t), F2(t), F3(t), F5(t), F1(t - 1), F2(t - 1), F3(t - 1), F5(t - 1) and L8(t - 1). For the entire TE process, 20 local DKPLS models are built using the relationships of each target variable in the SDG model. Table 4 shows the local models containing their target variables and cause variables. For a local DKPLS model of a target variable, the score matrix T can be calculated by the KPLS algorithm discussed above. Then, solve the optimization problem 10 with the score matrix T as input and the target variable y as output. As a result,
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Figure 2. The individual SDG model of S2. Table 4. Target Variables and Cause Variables of the Local DKPLS-SVR Models
Figure 3. The interactions between S2 and the other subsystems.
target variables
cause variables connected to the target variables
F1 F2 F3 F4 F5 F10 F14 F17 F19 ZG/ZH L8 L12 L15 P7 T9 T18 T21 T22 YB ZE
MV3 MV1 MV2 MV4 L8, T22 MV6 MV7 MV8 MV9 F2, F3 F1, F2, F3, F5 L8, F14 F14, F17 F1, F3, F4, F5 F1, F2, F3, F4, F5, MV10 MV11, F19 MV10 MV11 F4, T9, F10 T18
where τ ∈ RA is the vector that denotes the projections of the data x onto the extracted A score vectors. The residual, which is the difference between the estimated value determined by formula 13 and the measured value, is monitored to perform fault detection. ri ) yi - yˆi
Figure 4. The process interactions of TE SDG.
the final regression function of the DKPLS-SVR model has the following expansion:12 N
yˆ ) fDKPLS-SVR(τ) )
∑ (R
* i - Ri)〈Φ(τ i), Φ(τ)〉 + b
i)1
(13)
(14)
where ri is the residual of variable i, and yi and yˆi are the measured and estimated values of variable i, respectively. A qualitative state, which corresponds to ranges of possible values for the residual, will be attached to the residual. Three ranges are used similar to SDG: normal (denoted as“0”), high (denoted as “+”), and low (denoted as “-”). If a fault occurs, the qualitative state for the residual may be “+” or “-”. However, using DKPLS-SVR method alone is not enough to aid the user in identifying the root cause, because it is impossible to find the inverse mapping from the feature space to the original space so that the interpretation of measured variable contributions is
9452 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 5. Comparison between the Proposed Method and Other Methods detection delays (min)
accuracy Chiang et al. (best/worst)
IDV1 IDV2 IDV3 IDV4 IDV5 IDV6 IDV7 IDV8 IDV9 IDV10 IDV11 IDV12 IDV13 IDV14 IDV15
DKPLS-SVR
linear PLS
quadratic PLS
8 91
36 115
36 78
1 9 1 1 56
2 13 1 1 96
31 11 60 57 6
189 18 68 142 11
Chiang et al. (best/worst)
DKPLS-SVR
linear PLS
quadratic PLS
missed detection
misclassification
6/21 36/75
1 0.995
1 0.90
0.99 0.97
3 12 1 1 84
3/0/48 0/33 0/3 60/69
1 1 1 1 1
1 0.995 1 1 0.61
1 1 1 1 0.66
206 19 67 130 11
69/303 21/912 0/66 111/147 3/18 2031/-
1 1 1 1 1
1 1 0.996 1 1
1 1 0.988 1 1
0/0.008 0.010/0.026 0.981/0.998 0/0.975 0/0.775 0/0.013 0/0.486 0.016/0.486 0.981/0.994 0.099/0.666 0.193/0.801 0/0.029 0.040/0.060 0/0.158 0.903/0.988
0.013/0.880 0.010/0.441 0.734/1 0.119/1 0.006/1 0/0.834 0/0.978 0.003/1 0.773/1 0.098/1 0.118/0.989 0.005/0.988 0.208/1 0.001/0.998 0.725/1
Wrong detection
IDV1 IDV2 IDV3 IDV4 IDV5 IDV6 IDV7 IDV8 IDV9 IDV10 IDV11 IDV12 IDV13 IDV14 IDV15
Resolution
DKPLS-SVR
linear PLS
quadratic PLS
DKPLS-SVR
linear PLS
quadratic PLS
0.023(2) 0(0)
0.002(1) 0(0)
0.001(1) 0.20(1)
3.01(4) 3.96(4)
3(3) 3.97(4)
3(3) 3.65(4)
0(0) 0(0) 0(0) 0.321(3) 0(0)
0(0) 0(0) 7.594(10) 0.0004(1) 0.002(1)
0(0) 0(0) 0.293(3) 0.02(1)
3(3) 3(3) 1(1) 1(1) 3.02(5)
3(3) 7(7) 1(1) 1(1) 4.16(9)
3(3) 6.57(7) 1(1) 1(1) 3.77(4)
0(0) 0(0) 0(0) 0(0) 0(0)
0(0) 0(0) 0.116(2) 0.088(1) 0.0004(1)
0(0) 0(0) 1.345(7) 0.056(1) 0(0)
3.12(5) 2.01(3) 2.07(3) 2.06(4) 2.0(3)
5(5) 3(3) 3.49(7) 2.0(4) 3(3)
5(5) 3(3) 4.19(7) 2.44(4) 3(3)
7.77(9)
difficult. As a result, the qualitative state for the residual will be introduced into the SDG model as a fault symptom to perform our previous SDG inference to find the root cause. Here, the parameters of cumulative sum (CUSUM) are used as the thresholds to monitor the residuals to detect their qualitative change of state. As one important parameter of CUSUM, 6σ of the residual distribution, which can be defined as the minimum difference that will cause the cumulative summation to begin, is used as the minimal jump size, namely the control limit. If the monitoring residual becomes beyond the minimal jump size of CUSUM, the qualitative state for the residual will be obtained as a fault symptom and then the inference mechanism based on SDG will be triggered to start its search for root cause. The procedure of off-line model building and online diagnosis is illustrated as Figure 5. The training for DKPLS-SVR models off-line spends about 5 min, and the diagnosis interval in this study is 1 min, the same with the sampling interval. The proposed method is coded using Matlab 6.5. 4. Example and Discussion Here, some examples are illustrated to verify the performance of the proposed SDG-DKPLSSVR method. As discussed above, the DKPLS-SVR method has already outperformed the PCA method and the PLS method, so the comparisons with previous hybrid methods are also illustrated. In each example, the simulation time for the faulty data set is 48 h (2880 observations) in order to compare with previous
studies. The simulation starts with no faults, and the fault occurs from the 480th step (8 h). Example-IDV7 (C Header Pressure Loss-Reduced Availability). IDV7 induces a step change in the C feed in stream 4. As seen in Figure 6, the fault will decrease the C feed, and then the variables associated with the material balances of the reaction (level, pressure, and composition) will change correspondingly through the control loops. Figure 7 shows the residuals of some detected variables when the DKPLS-SVR model is used. The bounds of Figure 7 are the minimal jump size of CUSUM. Through the dynamics and the residual of F4, it can be shown that if the DKPLS-SVR model is adequate, the sign of the residual is always a good indication of the direction of process variable change. Using the DKPLS-SVR model, the detection sequence of symptoms are follows: F1(-), F4(-), P7(-), T9(-), T18(+), and F19(-) at 481 min, T21(+) at 483 min, T22(+) at 491 min, ZE(-) at 493 min, F14(-) at 500 min, F5(-) at 501 min, L12(-) at 502 min, F17(-) at 503 min, F3(-) at 505 min, F2(-) at 506 min, L8(+) at 514 min, L15(-) at 520 min, and YB(-) at 565 min. IDV is obtained as the solution finally, and the resolution is 1. The consistent paths found by reasoning in the SDG are T21(+)rT9(-)rF4(-)rIDV7; YB(-)rT9(-)rF4(-)rIDV7; F2(-),F3(-)rL8(+)rF1(-)rP7(-)rF4(-)rIDV7; T22(+)rF5(-)rL8(+)rF1(-)rP7(-)rF4(-)rIDV7;
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Figure 5. The procedure of the proposed method.
Figure 6. Dynamics of F4 for IDV7.
F19(-),ZE(-)rT18(+)rT11(+)rF5(-)rL8(+)rF1(-)rP7(-)r F4(-)rIDV7; F17(-)rL15(-)rF14(-)rL12(-)rL8(+)rF1(-)rP7(-)r F4(-)rIDV7. All the consistent paths point to fault IDV7. Example-IDV11 (Reactor Cooling Water Inlet Temperature). IDV11 induces a fault in the inlet temperature of the reactor cooling water in stream 12. The fault in this case is a random variation, which will result in large oscillations of corresponding variables. Figure 8 shows the residuals of the detected variables when the DKPLS-SVR model is used. The fault will only influence T9 and T21. Using the DKPLS-SVR model, the detection sequence of symptoms are follows: T21(f) at 491 min, and T9(f) at 496 min. Here, (f) means fluctuation, which denotes the signs of the residual state fluctuate between (+) and (-). IDV11 and IDV14 are obtained as the solution finally, and the resolution is 2. The consistent paths found by reasoning in the SDG are. T21(f)rIDV11; T9(f)rT21(f)rIDV11. Or T21(f)rIDV14; T9(f)rIDV14. Example-IDV7 and IDV11. This is the situation of multiple fault origins occurring simultaneously. When the proposed method is used for multiple fault diagnosis, no more other
DKPLS-SVR models need to be trained. The multiple fault causes can be obtained by SDG inference using the detected symptoms. Figure 9 shows the residuals of some detected variables when the DKPLS-SVR model is used. Using the DKPLS-SVR model, the detected symptoms are almost the same as those for IDV7 alone, except for T21 and T9. The detection sequence of symptoms are as follows: F4(-) at 481 min, T18(+) at 482 min, P7(-) at 483 min, F1(-) at 484 min, T22(+) at 491 min, ZE(-) at 495 min, F5(-) at 502 min, F19(-) at 503 min, F14(-) at 504 min, F17(-) at 507 min, F3(-) and L12(-) at 508 min, F2(-) at 509 min, T9(-) at 514 min, L8(+) at 516 min, L15(-) at 527 min, T21(+) at 531 min, YB(-) at 572 min, T21(f) at 670 min, and T9(f) at 798 min. At first, only IDV7 is obtained as the solution. When T21(f) and T9(f) are detected, IDV11 and IDV14 are put into the fault candidate set as the final solution. The consistent paths found by reasoning in the SDG are T21(+)rT9(-)rF4(-)rIDV7; YB(-)rT9(-)rF4(-)rIDV7; F2(-),F3(-)rL8(+)rF1(-)rP7(-)rF4(-)rIDV7; T22(+)rF5(-)rL8(+)rF1(-)rP7(-)rF4(-)rIDV7; F19(-),ZE(-)rT18(+)rT11(+)rF5(-)rL8(+)rF1(-)rP7(-)r F4(-)rIDV7; F17(-)rL15(-)rF14(-)rL12(-)rL8(+)rF1(-)rP7(-)r F4(-)rIDV7. T21(f)rIDV11; T9(f)rT21(f)rIDV11. (T21(f)rIDV14; T9(f)rIDV14.) Table 5 compares the diagnosis result of the proposed method with those of Chiang et al.21,22 and the PLS-SDG method.23,24 Chiang et al. compared various statistical methods, including PCA, DPCA, CVA, PLS, FDA, and DFDA. Here, four parameters are compared: accuracy, resolution, wrong detection, and detection delay.23,24 The detection delay is the time from fault occurrence to fault diagnosis. If the fault candidate set includes the true fault, the accuracy is 1. Otherwise the accuracy is 0. Wrong detection is the number of falsely detected symptoms independent of the true solution. The resolution is the number of the fault candidates. In this table, the accuracy, wrong detection, and resolution are all the average numbers every 1 min from the initial fault detection to the last. The value in
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Figure 7. Residuals of some detected variables for IDV7.
Figure 8. Residuals of the detected variables for IDV11.
brackets is the worst result obtained during all diagnostic periods. Wrong detection and the resolution are not given for Chiang et al. because these are not available. Chiang et al. considered that a fault was indicated only when six consecutive measure values have exceeded the threshold,
but the detection delay was recorded as the first time instant in which the threshold was exceeded. Therefore, the actual detection delay of their method will be more than that shown in Table 5. With our proposed method, the detections of almost all cases are faster than those of PLS-SDG method and those
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Figure 9. Residuals of some detected variables for IDV7 and IDV11.
of 5 cases (IDV4, IDV8, IDV10, IDV11, and IDV13) are faster than those of Chiang et al. The other cases show similar or slightly worse detection delays. The diagnosis for three cases (IDV3, IDV9, and IDV15) fails, because the fault sizes of these cases are so small that there are only weak variations occurring in the process variables, not enough to diagnose the faults. It indicates that the diagnostic performance of the proposed method depends on the fault size. The accuracies of all of the faults which can be detected by the proposed method are almost 1 during the diagnostic period. In the diagnosis of IDV2, the method fails for the initial 12 min because the symptom F10(+) is detected, which has no direct fault causes in the SDG model. With the exception of IDV1 and IDV7, the wrong detections of all of the faults which can be detected by the proposed method are almost 0 during the diagnostic period. In the diagnosis of IDV1, the false symptoms are T22(+), F3(-), and F14(-). In the diagnosis of IDV7, the false symptoms are T18(-), L12(+), L8(-), L15(+) and YB(+).
5. Conclusions This study presents a new hybrid method combining SDG with DKPLS-SVR model for fault diagnosis in large-scale chemical industry. Using the relationships of each variable in SDG, a series of DKPLS-SVR models are built to estimate the values of the measured variables in process. The difference between the estimation and the measured value can determine the qualitative status of the variable, and then the fault can be diagnosed by SDG reasoning. The proposed method can eliminate the choice process of thresholds for SDG-based diagnosis. Also, this method can overcome the limited availability of data-driven methods in identifying the root cause. The diagnostic performance of the proposed method is illustrated by the diagnosis case studies on the TE challenge process. Through case studies, the proposed method demonstrates a good diagnosis capability compared with previous hybrid methods. Through case studies, it can be found that the proposed method could fail to identify multiple faults that change the
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measured variables in the same direction. This is due to the qualitative nature of the SDG-based diagnosis. The possible solutions to minimize the spurious fault candidate set will be the focus of our future work. Acknowledgment This research was supported by the National High Technology Research and Development (863) Program of China (No. 2007AA04Z193). Appendix Notation b ) bias term for SVR E) residual matrix for X F ) residual matrix for Y K ) kernel matrix l ) number of time delay P ) loading matrix for X Q ) loading matrix for Y ri ) residual of variable i T ) score matrix for X U ) score matrix for Y w ) normal vector for SVR X ) input matrix Y ) output matrix yi ) measured value of variable i yˆi ) estimated value of variable i Greek Ri*,Ri,ηi*,ηi ) Lagrange multiplier ε ) tolerance range for SVR ξi*,ξi ) slack variable
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ReceiVed for reView June 16, 2008 ReVised manuscript receiVed October 4, 2008 Accepted October 7, 2008 IE8009457
