1703
Znd. Eng. Chem. Res. 1995,34, 1703-1717
Use of Fuzzy Cause-Effect Digraph for Resolution Fault Diagnosis for Process Plants. 2. Diagnostic Algorithm and Applications Ruey-Fu Shih and Liang-Sun Lee* Department of Chemical Engineering, National Central University, Chung-li, Taiwan, 32054, R.O.C.
A new model graph called the fuzzy cause-effect digraph (FCDG) model was already proposed in part 1, and its capability to eliminate spurious interpretations attributed to system compensation and inverse responses from backward loops and forward paths is to be demonstrated. In this paper we attempt to develop a new fault diagnosis algorithm based on the fuzzy cause-effect4digraph model. This method applies fuzzy reasoning to estimate the states of unmeasured variables, to explain fault propagation paths, and to locate fault origins. In particular, it can obtain the fault origin occurring in the process with single and multiple loops at the early stage of fault. This study uses a CSTR as an example to explicate this diagnosis method and compares the results with those of other methods.
Introduction
Faulty
In part 1(Shihand Lee, 1995) six fault propagation patterns possibly encountered in real processes were used t o describe the limitations of the signed directed graph (SDG) approach in detail. We mentioned the reasons why it is impossible to improve the SDG approach in diagnostic resolution for these six fault propagation patterns. These reason is due to (1)the complex dynamic action of system variables, such as compensatory and inverse responses; (2) the existence of the unmeasured variables; and (3) the lack of the appropriate quantitative data on process variables. For the above reasons, a model graph called the fuzzy cause-effect digraph (FCDG) model was proposed. This model expresses quantitative data of deviation of variables from the normal values with fizzy sets. It uses dynamic state constraints (confluences) converted to a dynamic fuzzy relational model to express the dynamic gain between variables in a chemical process. This replaces the steady-state gain between variables conventionally expressed with a “+”, “-”, or “0”by SDG. We have demonstrated that this fuzzy cause-effect digraph model can eliminate spurious interpretations attributed t o system compensation and inverse responses from backward loops and forward paths in part 1. In this paper (part 2) an efficient diagnostic algorithm is developed for improving the diagnostic resolution of the original SDG. It applies the fuzzy causeeffect digraph model and fuzzy reasoning to estimate the state of the unmeasured variables, t o explain fault propagation paths, and t o ascertain fault origins. In particular, fault origin can be obtained at the early stage of fault occurring in the process with single and multiple loops. If the spurious interpretations still remain after diagnosis has been performed, the fault origin candidates can be ranked by the evidence degree t o find the most likely fault origin. Finally, this research uses a CSTR as an example t o explain the proposed diagnostic method and compares its results with those obtained from other approaches.
Expression for the Variable Fault Judging a process is faulty or normal is the first thing an operator faces when a disturbance has occurred in a
* Correspondence should be addressed to this author.
Faulty
ltl I
I !
I
Acceptable perlormance
I
1
I
I
..
I I 4 - I
I
‘ 1
process response
A
LN
’
MN
’
SN
2
’
SP
I
MP
I LP
w
-
-2s -s 0 S 2s A Figure 1. Boundaries of faulty and nonfaulty. (a) Hard partition. (b)Fuzzy partition.
plant. It is difficult for an operator to make decision, because the boundaries between the ranges of the acceptable performance (or normal) and the faulty, as indicated in Figure l a , are not clear-cut by the stochastic aspects of classification (Himmelblau, 1978). This classification is called the hard partition. The standard deviation is usually used to represent the range of the acceptable performance (or normal) that can be defined as the following: Definition 1: Let the residual be r = x - xsp. Assume r is a normal distribution N@,s), and the deviation of a process variables, 2,is represented by the normal distribution tolerance, then x =zapf K s
(1)
where xspis the set point of a process variable, s is the standard deviation, and K is a parameter. Taking a fuzzy classification of faulty and nonfaulty may be a good way for avoiding the confusion in judging
0888-5885/95/2634-1703~09.00/0 0 1995 American Chemical Society
1704 Ind. Eng. Chem. Res., Vol. 34, No. 5 , 1995
faulty or nonfaulty near the ambiguous boundary. We use a fuzzy set which has three elements, N (negative), 2 (zero), and P (positive) to describe the corresponding deviated state, “negative”, “normal”, and “positive”, of process variables, respectively, and the membership degree of each element to indicate the quantitative value of expression. These elements are the subsets of the fuzzy set that the membership functions of the subsets are defined as
lb. In other words, this interval may represent most of the range of the acceptable performance. It is obvious that a vertex has three different memberships of fuzzy subsets at the same sampling time by fuzzy set manipulation. The discrepancy of a deviated variable is taken from one of the subsets of the largest degree of membership; even this variable is still in the range of the acceptable performance.
The Diagnostic Algorithm
Lo
where
and where p d x ) , pz(x), and p p ( x ) are the degrees of membership of x in the fuzzy subsets, N , 2, and P , respectively. If the deviation of a variable is greater than the upper limit, x,, then pup(x) = 1,and if it is less than the lower limit, x-,, then p d x ) = 1. If it is beyond these limits is considered in fault situation. The membership functions of fuzzy subsets, N , 2, P, indicating the seriousness of deviation, are clearly shown in Figure lb. We can see that the deviated state of each measured variable is mapped into three different membership functions. It will no longer be hardpartitioned as “negative (faulty)”, “zero (normal)”, and “positive (faulty)”, but is indicated with all possible discrepancies of deviation, N , 2, and P , simultaneously by the fuzzy subsets with grade of membership. Such mapping is purposely designed to handle the deviation of a variable with the triggered alarm. This mapping will give faulty information to an engineer so that the seriousness of the faulty variable may be judged. Although a fuzzy set with N (negative), 2 (zero), and P (positive) is used to represent the deviated states of variables in this work, in order to judge the faulty grade of each measured variable conveniently, the deviated state is divided to seven grades, LN, MN, SN, Z, SP, MP, and LP. The grades of LN, MN, and SN indicate the negative deviation of variables, and LP, MP, and LP indicate the positive deviation of variables. SP and SN are the grades of possible normal that the probability is 68.26% in the [(x,, - s)] to (xap s)l interval; MP and MN are the grades of possible fault; LP and LN are the grades of faulty variables; and Z is that of the variables at normal condition. The probability including grades MP, SP, Z, SN, and MN is 95.46% of the interval [(x,, - 2s)l to (xep 2s)], as shown in Figure
+
+
In addition to the improvement of diagnosis resolution with the process fuzzy model, the fuzzy quantitative data provide propagation path strength, possible deviating quantities of fault origin candidates, and the priorities of verifying the possible fault for the operator. The following algorithm combines fuzzy cause-effect digraph model and SDG for fault diagnosis: 1. Build an SDG for the process, distinguish the unmeasured variables, and label them on the SDG. 2. Construct the fuzzy cause-effect digraph model. Construct the fuzzy relations of each unmeasured variable and its adjacent ancestors or the descendant measured variables from the set of equations of the linearized mathematical model. 3. Collect the information of process variables either from computer output of simulation program or by direct measurement. Calculate the fuzzy degree of membership and the belonging area of each fuzzy subset for each measured variable. 4. Estimate the states of the unmeasured variables on SDG from its adjacent ancestors or the descendant measured variables with the constructed fuzzy causeeffect digraph model. Construct the SDG, G, for all variables by the defined fuzzy reference sets of deviation at this sampling stage. 5 . Partition the SDG, G, into a graph consisting of strongly-connected components, {Gscl, Gsc2,..., G,,,) (Iri et al., 1979; Swamy, 1981) by the method of depth-first search (Tarjan, 1972). Such a graph is a rooted tree, T,with strongly-connected components as vertices and without any cycle in between. Then, search for the common ancestor of all strongly-connected components or maximum strongly-connected component (MSCC). 6. Remove the normal strongly-connectedcomponent (defined below) which is a MSCC; thus, new MSCC switches to its adjacent descendant strongly-connected component. The final rooted tree, TD,,, does not involve any normal state of MSCC. Definition: (1)Normal strongly-connected component is a strongly-connected component in which all the variables are measurable and normal; (2) normal strongly-connected component is a strongly-connected component in which all loops are positive backward propagation loop (PBPL), and has at least one normal measured variable. 7. Search for the common ancestor, which is a MSCC from the rooted tree, TD,,,of all strongly-connected components. The common ancestor of all stronglyconnected components is called fault swarm candidate, S,. We call S, an unmeasured fault swarm candidate S,,uif it has any unknown unmeasured variable. If all the variables are unmeasured in a fault swarm candidate, adjacent descendant strongly-connected components may be another fault swarm candidate, Sj. All the fault swarm candidates form a set, Sc,, that is, Sc, = {Si}U {Sj]. All the variables in this set are called
Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 1705 CAO Fo TO
0 ____-I
4
I
I
I
FJO
FJ TJ
-I
F CA T
Figure 2. CSTR process of Chang and Yu.
fault origin candidates. This set is denoted as C, C = {Cl, C2, ..-,Cnl. 8. Select the most possible fault origin candidate from set C: (1)The beginning node of two adjacent nodes which do not obey edge consistency along the path is the most possible fault origin candidate. (2) The common ancestor of all variables is the most possible fault origin candidate that satisfies edge consistency along the propagation path. (3) A deviated confluence variable in NFPB or NBPL is the most possible fault origin candidate if it differs significantly from the estimated value. The estimated value is deduced from eq 14 of part 1. (4) The node may be the most possible fault origin candidate in NBPL such as Figure 6a, if it has an inconsistent edge along the path. ( 5 ) The earliest or the largest deviated variable is the most possible fault origin candidate when quantitative data were used to verify each variable in the case of PBPL. However, it is impossible t o back search the fault origin for PBPL along a propagation path. (6) An unmeasured variable found from steps (1)(51, is the most possible fault origin candidate. The measured variable used to deduce the unmeasured variable through the fuzzy relational model may be another most possible fault origin candidate. (7) Assume an unmeasured and unestimated variable to be a possible fault origin candidate for a fault swarm candidate. We call C' the set of the fault origin candidates obtained in step 8. 9. Verify the eligible fault origin candidate from C': Assume fault swarm candidate, Scl, has the most possible fault origin candidates, and Si+ (Si,u* Scl)are normal. Repeat steps 4-8. Installing sensors for unmeasured and unestimated variables is suggested in order to improve the resolution when step 4 still has the unmeasured and unestimated variables. Repeat steps 4-8 to calculate c12, ..., C', from i = 2 to i = n. The union, C", of the sets is the eligible fault origin candidates, C" = {Cl}u { C Z U } ... u {Cn}.
Table 1. CSTR Parameter Values bias(1) = 11.008 psi F = 40 ft3/h bias(2) = 10 psi VR = 48 R3 CAO = 0.5 mol/ft3 CA = 0.1952 mol/ft3 T = 600 "R TJ = 593 OR FJ = 66.4 ft3/h VJ = 3.85 ft3 KO = 7.08 x 1O'O h-' ER = 30 000 Btdmol R, = 1.99 Btu/(mol."R) FJMAX = 199.6 ft3/h FMAX = 96 R3/h
bias(3) = 11.008 psi U,= 150 Btu/(h*R3."R) A = 250 R2 TJO = 530 OR TO = 530 "R , I= -30 000 Btdmol Cp = 0.75 Btd(lb2R) Cj = 1.0 Btu/(lWR) e = 50 ib-m0l/~3 ej = 62.3 lb-moVR3 TS = 600 "R
10. Arrange a sequence of fault possibility membership by the degree of membership, MEM, of the eligible fault origin candidates. If some of the candidates have identical degree of memberships, then the sequence can be arranged by their credible fault membership, CFM, as defined in definition 2. If the candidates have the same degree of memberships and credible fault membership (see definition 2), then compare the frequencies, NUM, of the candidates appearing in the C". A variable with larger MEM, CFM, or NUM has a higher possibility of being a fault origin. Definition 2: Define credible fault membership, CFM, as follows (Lesmo et al., 1985): CFM = u
+ b(b - a)
(10)
where n
a = ne(MEMj) and b = min e(MEMj) j=l
lqjjan
Fault Diagnosis of CSTR A CSTR example, the same in as Chang and Yu (1990), is used t o illustrate the effectiveness of the proposed diagnostic algorithm. The process is a CSTR undergoing an irreversible first-order exothermic reaction (Figure 2). Parameters were taken from the literature (Luyben, 1973) and are listed in Table 1. The
1706 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 Table 2. Controller Parameters of CSTR controller TC FJC
gain 70 0.5 40
vc
Table 4. Equations for Mathematical Model of CSTR reset, llmin 0.17 28.3 0
Table 3. Operating Conditions of CSTR variable normal TM 600 TC 11 11 FJC
operating condition max min unit 660 530 "R 15 3 Pslg
allowable dev, % (normalized)
FV
15
1
3
0.33
PW
FJM TJM TOM
40
60 660 600
0
ft3h
2 1 1 1 5
488 520
"R "R
2 2
VM
72
0
ft3
0.42 96 40 72
3
Psig position
5 1
LV FMAX F VR CA CAO KO K FO TO
115.2 15 1 100
vc
593 530
11
0
position
100 0.6
0 50 0 0 0.17
0.87 40
1 60
0 0
530 150 593 530
600
520
170
660 660
100 488 488
FJMAX 199.6
200
150
FJ FT T
199.6
0
R3/h R3/h
15
3 530
Psig "R
U, TJ TJO
0.2 0.5
7.08 x
66.4 600
60
ft3/h
R3/h ft3
moUft3 0.6 0.4 moUft3 1O1O 8 x 10'0 3 x 1Olo h-*
660
1 5 5 5 1 1 2
2 R3/h
"R BTU/(hft3*"R) OR O R
5 5 5 2 2 5 5 1
d -V - ~ ~ - ~ dt -dCA - q C A O - CA) - K(CA) dt VR
dT -%TO -dt VR
- T ) - LK(CA) -u& (T- TJ) dCP) (VR)@(CP) dTJ - FJ(TJ0 - T J ) + -(T-TJ) dt VJ (VJ)(ej XCj) K = KO~-ERKR~T)
TC = bias(1)
FT = 12-
+ KC(1) (TS - T )+ Wt,(UJ ( T S
- T ) dt
FJ + 3 FJMAX
FJC = bias(3)
+ KC(3) (TC - FT) +
- FT) dt %3,
Fv=-FJC - 3
12 FJ = (FJMAX)(FV)
L = V/(4nD2) LC = bias(2)
+ KC(2)(LS - L)+ gq(2) J (
L S - L) dt
LC - 3 12 F = (FMAX)(LV) LV=-
2
Table 5. Linearized Equations of CSTR Model
reactor temperature, T,is controlled by the temperature controller, TC, through cascade cooling water flow control, FJC. Reactor liquid level, L , is controlled by the outlet flow rate, F. The parameters of each controller are given in Table 2. The maximum, normal, and minimum operating conditions and the ranges of the acceptable performance (or normal) of process varaibles are listed in Table 3. During the diagnosis process, the result at each stage of the proposed approach is compared with that of Chang and Yu (1990). Diagnosis Procedure. The mathematical equations of this CSTR process (Luyben, 1973, p 144)as given in Table 4,are linearized for constructing SDG. The linear equations are shown in Table 5. The SDG constructed for this process is shown in Figure 3. The measured variables and the control signals available for fault diagnosis are shown in Table 6. Two fault cases are studied to test the proposed algorithm. The fault origins are purposely designed as the deactivation of catalyst activity, KO, and the sudden decrease of maximum flow rate of cooling water, FJMAX. Both fault origins are the root nodes in SDG. In addition, €or purposely testing the resolution for the case where fault origin locates in a recycle loop whose its adjacent nodes are mostly unmeasured, we demonstrate by two cases of fault origins, the stuck cooling water control valve, FV, and the level control valve, LV. The diagnosis algorithm is programmed in Turbo PASCAL, and executed in an Intel 80486-DX66 CPU. Fault Case 1: The Maximum Flow Rate of Cooling Water, FJMAX, Suddenly Drops to 40%. The diagnosis system starts to access process variable data when the maximum flow rate of cooling water, FJMAX, suddenly drops to 40%. The dynamic changes of process variables are shown in Figure 4. The detailed diagnosis
dvldt = FO - F dCA/dt = 0.0053(FO) - O.O44(VR) 0.8333(CAO) 1.7005(CA) - 0.2450K dT = -1.4583(FO) 3.5591(vR) 0.8333(TO) - 21.66672' 196K 693.7267(CA) - 0.7500U0 20.8333(TJ) dTJ/dt = -16.7792(FJ) 12.9610(TJO) - 169.3055(TJ) 5.6284U0 156.3445T K = 1.2248 x lO-"(KO) 0.03632' TC = 70(TS - 7') 700j(TS - T) dT FT = 12(FJ - FJS)/FJMAX FJC = 0.5(TC - FT) 0.5(1700)j(TC - FT)dt FV = -FJC/12 FJ = 199.6(FV) 0.33(FJMAx) LC = 40(LS - L) 6j(LS - L)dt LV = -LC/12 F = 96(LV) 0.42(FMAX)
+ +
+
+
+ +
+
+
+
+
+
+
+ +
+
Table 6. Measured Variables and Control Signals List for CSTR measd variable feed temp reactor bulk temp cooling water outlet temp cooling water flow rate temp controller output cooling water controller output reactor liquid level level controller output reactor concn
variable in SDG TO T TJ FJ TC FJC
VM
vc
instrument legend in Figure 2 TI 1 TIC 1 TI 2 FJ TIC 1 FIC 1 LT LIC 1
CA
procedure at 14 s after process subjected to this fault is described as follows: Step 1. SDG and measured and unmeasured variables are shown in Figure 3. Step 2. Construct the fuzzy cause-effect digraph model. Step 3. Calculate the fuzzy degree of membership and the belonging area of each fuzzy subset for each measured variable.
Ind. Eng. Chem. Res., Vol. 34,No. 5,1995 1707
FMAX
E:) measured variable Vertex : unmeasured variable
:fault origin Figure 3. SDG for the CSTR example.
Step 4. Estimate the states of unmeasured variables on the SDG from its adjacent ancestor or the descendant measured variables with the constructed fuzzy causeeffect digraph model as listed in Table 7 after the degree of membership of fuzzy sets is calculated for each measured variable. Assign the corresponding deviated state on each vertex with defined hzzy reference sets of deviation on the SDG, G, at this sampling stage, as indicated in Figure 5 , which shows the propagation network. Step 5. Partition the constructed SDG, G, into a graph consisting of strongly-connectedcomponents circled by dashed lines, as shown in Figure 5. These strongly..., connected components are represented by {Gscl,Gsc2, Gsc12},respectively, so that G is obviously to be viewed " , shown in Figure 6a, where as a rooted tree, T D ~ as
Table 7. Estimated Value of Unmeasured Variables from FCDG Model unmeasd variable measd variable dev state, inferred dev value dev fuzzy sets: N , 2, P VM TOM TJM FJM TM
Fv FJ TC FJC
0.0001.000 0.000 0.0001.0000.000 0.000 0.9000.000 0.3800.6200.000 0.0000.9940.006 0.000 0.000 1.000 0.3800.6200.000 0.0000.8870.113 0.000 0.000 1.000
V TO TJ FJ FJMAX
0.00 0.00 0.69 -7.6 0.04 -98
FT
-0.55
FMAX F CAO KO
? ? ? ? ? ?
T
UO TJO
Gsc2 = {TM, TC, FJC, FV,FJM, CA,K, TJ, FJ, FT, T )
F, VR, CA, CAO, KO, K, FO, TO, Uo,TJ, TJO, FJMAX, FJ, FT, 0. Step 8. Select the most possible fault origin candidates from set C , and obtain C' = {FJ, FJM, CAO, KO, F, FO, FMAX, Uo, TJO, FJMAX, Q. Step 9. Verify the eligible fault origin candidate from
c': Gsc7= {FO}, Gs,8 = {TOM}, Gs,9 = {TO},
Step 6. Mark the unmeasured strongly-connected components on Figure 6b. Step 7. Search for fault swarm candidates, Si, from the rooted tree, 2 ' ~ ~ "We . obtain Si = {Gsc4,Gsc5, Gsc6, Gsc7,Gsc9,Gsclo,Gscll,Gsc12},and the unmeasured fault swarm candidates Si,,= {Gsc4, Gsc5, Gsc6, Gsc7, &io, Gscll}. It is possible that the adjacent descendant strongly-connected components are other fault swarm candidates, Sj, = {Gsc2,Gsc3}. SC,= {Gsc2,Gsc3,Gsc4,Gsc5, Gsc6, Gsc7, Gscg, Gsclo, Gscll, Gsc121. The fault origin candidates C = {TM, TC, FV,FJM, VM,VC, LV, FMAX,
(1)Assume fault swarm candidate SI,, = Gsc4is the most possible one, and Si,,(i f 1)are normal. Repeat steps 4-8. In step 5 a t this time, decompose the SDG, G, into a graph consisting of strongly-connected components, {Gscl,Gsc2,Gsc3, Gsc4}, as shown in Figure 7 , where
1708 Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995
a
0.5 0.4
a m 0
b ~
lo 7 -
1
-
I
+TM
0.3 -
0.2
tTJM
-
-i)-
:
1
L
E
TOM
~
a
-lo
1
l
+FJM
-0.2 -
-
-0.3
I
I
0.2 Time, Hr
0.1
'
0.2
0.1
0.3
0.3
Time, Hr
1.0
+VM .-0 .-
0.3 -
I
g
+FV
0.2
tLV
0.1
-,
0.0
~Lm---t----+
I
0.1
0.2
0.3
0.2
0.1
0.3
Time, Hr
Time, Hr
e
2E-5 1E-5 OE+O
4.0
.g 3.0 i
.-
2
-%-
TC
4 v c
1
2.0 -
-1E-5
I
7'
1.0
j,
-2E-5 -3E.5
+CA
'
.4E-5
0.0
,
-5E.5
-1.0 0.1
0.2
0.3
0.3
0.2
0.1
Time, Hr
Time, Hr
Figure 4. Responses of process variables for the CSTR example.
and obtain
C',={FMAX FJ TM)
(3) Assume the fault swarm candidate S3,u= Gsc6,is the most possible one, and Si,u(i t 3) are normal. Repeat steps 4-8, and obtain
The magnitudes of membership function of each process variable in this set are
MEM -0.0000 -0.380 -0.380 The credible failure memberships of each process variable in this set are
CFM
0.0000 0.0062 0.0018
(2) Assume the fault swarm candidate S Z ,=~ Gsc5is the most possible one, and Si,u(i t 1) are normal. Repeat steps 4-8 and obtain
C', = {CAO}
MEM
-1.0000
CFM
0.0000
C', = {KO} MEM
-0.023
CFM
0.0000
(4) Assume the fault swarm candidate S4,u= GW7is the most possible one, and Si,u (i * 4) are normal. Repeat steps 4-8, and obtain
C4= {FJ
FJMI
MEM
-0.38
CFM
0.0062 0.0018
-0.38
(5) Assume the fault swarm candidate S S ,=~ Gsc9is
1995 1709
/
I I ' 0 \
\
' , U/
,/
c) c
-
\ : stronglyconnected
-'
0
component
P I : measured variable Vertex : unmeasured variable
: fault origin Figure 5. Propagation pattern at 14th s of the example.
the most possible one, and Si,, (i Repeat steps 4-8, and obtain
MEM
-0.1490
CFM
0.0000
* 6) are normal.
(7) Assume the fault swarm candidate, S7,, = Gscll is the most possible one, and Si,, (i * 7) are normal. Repeat steps 4-8, and obtain
Gsci
C', = {TJO} MEM
-0.0750
CFM
0.0064
(8) Assume the fault swarm candidate, SB,, = Gsc1a is the most possible one, and Si,, (i f 8) are normal. Repeat steps 4-8, and obtain
Gsci
0: unmeasured strongly connected component
Figure 6. Rooted tree for the CSTR example. (a) All stronglyconnected components. (b) Unmeasured strongly-connected com-
MEM
ponents.
CFM
the most possible one, and Si,, (i Repeat steps 4-8, and obtain
f
5 ) are normal.
C', = {TO} MEM
-0.0000
CFM
0.0000
(6) Assume the fault swarm candidate S6,u = Gsc10is
+1.0000 0.0062
(9) Assume the fault swarm candidate, Sg,, = Gsc3is the most possible one, and Si,,(i f 9) are normal. Repeat steps 4-8, and obtain
MEM
-0.3800
-0.380
CFM
0.0062
0.0018
(10) Assume the fault swarm candidate Slo,, = Gsc2,
1710 Ind. Eng. Chem. Res., Vol. 34,No. 5 , 1995
/
'J
/
I
\
\
\ \
' 0 \
f/ +
\
'\
/'-
\
\
/ +
.\+
C
L
+\\
4
I
\
\
+
\
\ \
\
c
-'
\
\
--'
'1
\
U
~ F M A X1
I'
\
I T M P
F
1 +I
\
+
I
\
: strongly connected component
I
\
I
\
I I
\ \
I\lertex]: measured variable
\
I
+
\ -
\
Vertex :unmeasured variable
\
I /
/
:fault origin
Figure 7. All strongly-connected components in the case of Gsc4as the most possible fault swarm candidate for the CSTR example.
is the most possible one, and Si,,(i * 10) are normal. Repeat steps 4-8, and obtain
Clo= {FJ MEM -0.3800 CFM 0.0062
FJM} -0.380 0.0018
u
c" = {C,> {C2} KO TJO u,, -0.000 -0.023 -0.075 -0.1492 0.000 0.000 0.0064 3.53-6
c"= {FMAX TO
FJ -0.380 6.23-3
FJM CAO -0.380 -1.000 1.8E-3 5.4E-10
FJMAX} -1.000 6.23-3
The number of appearance of the variables in the set C" are
NUM
1 1 1 1 1 4 4 1 1
bmove the fault origincandidates the new C" set is
c"= {U, MEM -0.1492 CFM 3.53-6 NuM1
FJ -0.380 6.23-3 4
FJM -0.380 1.8E-3 4
process time, s
FJM CAO U, TJO FJMAX FJ MEM -0.3800 -1.0000 -0.1492 0 -1.0000 -0.3800 CFM 1.83-03 5.43-10 3.53-06 0 6.23-03 6.23-03 4 NUM4 1 1 0 1 439 MEM 0 0 -1.0000 0 0.3326 0.1902 CFM 0 3.83-09 3.83-07 0 4.33-05 0 0 NUMO 0 0 0 1 0 -1.0000 0 .605 MEM 0 0 0 CFM 0 0 0 0 1.03-07 0 0 "0 0 0 0 1 634 MEM 0 0 0 0 -1.0000 0 CFM 0 0 0 0 2.53-07 0 0 NUMO 0 0 0 1 742 MEM 0 0 0 0 -1.0000 0 0 2.33-10 0 CFM 0 0 0 0 0 1 NUMO 0 0 850 MEM 0 0 0 0 -1.0000 0 0 0 0 5.03-10 0 CFM 0 NUMO 0 0 0 1 0 0 0 0 -1.0000 0 958 MEM 0 CFM 0 0 0 0 7.8E-11 0 NUMO 0 0 0 1 0 1080 MEM 0 0 0 0 -1.0000 0 CFM 0 0 0 0 2.23-12 0 374
Step 9. Finally, the eligible fault origins obtained at the present stage are
MEM 0.000 CFM 0.000
Table 8. Diagnosis Results for a Fault in FJMAX
MEM < lo.ll; CAO -1.00 5.43-10 1
FJMAX} -1.00 6.23-3 1
Step 10. Arrange a sequence of possible fault origins by the magnitude of membership function of the fault origin candidates in set C"; we have
FJMAX > CAO > FJM > others
NUM 0
0
0
0
1
0
Table 8 showed the fault simulation for a fault origin of FJMAX during 1080 s. The proposed diagnosis system analyzed the situation a t 245 s after the process alerted to a fault as follows:
MEM
-1.000
CFM
4.OE-6
N u M 1 This means FJMAX is the most possible fault origin obtained by the proposed algorithm. As shown in Figure 8, the proposed method does not yield spurious solution.
Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 1711
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