Ind. Eng. Chem. Res. 1995,34, 1688-1702
1688
Use of Fuzzy Cause-Effect Digraph for Resolution Fault Diagnosis for Process Plants. 1. Fuzzy Cause-Effect Digraph Ruey-Fu Shih and Liang-Sun Lee* Department of Chemical Engineering, National Central University, Chungli, Taiwan, 32054, R.O.C.
A new model graph called fuzzy cause-effect digraph (FCDG) is proposed. This model expresses quantitative deviations of variables from the normal values with fuzzy set. It uses dynamic constraints (confluences) which are converted to dynamic fuzzy relations to express the dynamic gain between the variables in a chemical process. This replaces the steady-state gain between
the variables originally expressed with a “+”, “-”, or “0” by signed directed graph (SDG). Using this FCDG model would eliminate spurious interpretations attributed to system compensations and inverse responses from backward loops and forward paths in the process. The basic idea and development of this proposed method are described in this paper. Moreover, this method can apply fuzzy reasoning to estimate the states of the unmeasured variables, to explain fault propagation paths, and to ascertain fault origins. The algorithm of fault diagnosis and its application proposed in this paper are described in part 2.
Introduction In order t o remain efficiently functioning, chemical factories make heavy use of automated systems, such as warning systems and instrumentations, t o monitor process variables and to control deviations within an allowable range in production processes. A process abnormality occurs when process variables (such as temperature/pressure) or process parameters (such as catalyst activity) deviate from the designed allowable ranges. Deviation of any individual variable or parameter is known as a process fault. Faults often directly or indirectly lead to process failure. Here, the definitions of fault and failure are the same as Himmelblau’s (1978). When a process fault occurs, the controllers, the alarms, and the display devices may be activated, and the operator must rely on his past operating experience, assessing numerous alarm device locations, activating times, and deviation values, to discover the causes of the process fault so that an appropriate action may be taken. This process is known as fault diagnosis, and the causes and locations of process fault are called as fault origins. When a fault occurs, the operator must swiRly and accurately diagnose the fault-a high-tension situation. There is thus a real need to develop automated fault diagnosis systems to ensure product quality and reduce the incidence of factory accidents. The techniques of fault diagnosis had been systematically discussed before in books by Himmelblau (1978), Pau (19811, and Patton et al. (1989). There have also been a number of authors who have addressed the methods in periodicals. These approaches may be categorized into qualitative and quantitative methods (Kramer, 1987; Yu, 1991). Another detailed classification (Wilcox, 1994) might be (1) process modeling, parameter estimation; (2) cluster analysis, statistical analysis; (3)artificial neural networks; (4) knowledgebased methods; (5) qualitative reasoning; and (6) signed digraph. Wilcox (1994) mentioned that no single diagnosis method or technique is superior to others at present, and each technique has its advantages and disadvantages depending on its usage. Here we focus on the signed directed graph (SDG) approach for fault diag-
* Correspondence
should be addressed to this author.
nosis, since SDG can clearly express qualitative relationships between variables, and only a minimum of process data is required to perform quick diagnosis. However, the current available SDG diagnosis methods focused on qualitative information of the process, so that unavoidable diagnostic restrictions and difficulties t o increase resolution exist. The reasons why it is impossible to improve SDG resolution are (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 appropriate quantitative data of process variables. Because SDG is easy to understand, and facilitates the subsequent maintenance, there are many researchers who are using SDG as a foundation, initiating further research on improvements based on the above three problems. The currently available SDG diagnosis methods (Iri, 1979; Tsuge, 1985; Shiozaki, 1985; Hwang and Lee, 1991; Han et al., 1994) used the initial direction of deviation as the basis for diagnosis, but did not include complex dynamic system variable activities such as nonsingle transitions and ultimate responses caused by compensatory and inverse responses. In order to account for ultimate response phenomena, Oyeleye and Kramer (1988) applied de Kleer and Brown’s (1984) qualitative physical concepts and the property that the system will eventually achieve a steady state. They added a nonphysical forward path to SDG to form an extended SDG (ESDG) for explaining the propagation paths of feedback loops attributed to their compensatory and inverse responses. Kramer and Palowitch (19871, in their treatment of certain loops and forward paths, preserved the dominant path of fault propagation and eliminated nondominant ones to simplify SDG and improve resolution. However, experience, skill, and understanding are required to judge which is the dominant of fault propagation. Chang and Yu (1990) proposed a method more modular than ESDG, by categorizing various continuous process response data into different states and constructing a simplified SDG and truth table for each state to increase resolution and improve modularity. They also proposed a method of handling control loops to explain nonsingle transitions caused by compensatory and inverse responses of dynamic feedback control loop systems. As with Kramer
0 1995 American Chemical Society 0888-5885/95/2634-1688$09.00~0
Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 1689 and Palowitch (19871, the diagnosis method establishes a rule base through SDG, but does not suggest a way to handle negative feedback loops and negative forward paths on the uncontrolled loops, nor can it handle multiple negative feedback or positive feedback loops. Umeda (1980) proposed using the dynamic SDG t o describe cause-effect relations of process variables t o explain the nonsingle transitions caused by feedback control loop, dynamic system compensatory, and inverse responses. He described these responses by using the discrete form of the differential and algebraic equations on the time lag for each process module. This makes it necessary to know the state value for each variable; in other words, there must be an indicator for each variable. Ulerich and Power (1988) developed a causal fault tree for on-line hazard calculation and fault detection from digraphs. They expanded verification gates with dynamic cause of delay to illustrate fast and slow responses; they did not describe how those verification gates were derived merely from digraphs. Due to cost consideration and the numerous instruments that would be required in a plant, it is impossible t o install indicators for all variables. The existence of unmeasured variables causes insufficient information either making diagnosis impossible or restricting the resolution for the diagnosis methods. In (19791, Shiozaki (1985) and Han et al. (1994) used trial and error to estimate the deviation tendency of the unmeasured variables to accord with its cause-effect rooted graph. A fault origin is derived from the root, but this requires a much longer diagnosis time. Chang and Yu (1990) and Kramer and Palowitch (1987) proposed combining the unmeasured variables of nonpotential root nodes with the neighboring measured variables to create a simplified SDG. The goal is to increase diagnostic resolution. The principle of the above combination is to eliminate the erroneous diagnosis, but once again, skill and experience of process behavior are necessary t o judge whether the diagnosis result is erroneous. Hwang and Lee (1991) combined the continuous nonbranching unmeasured variables into a simplified SDG with the goal of reducing search space, but ran into trouble with confluence variables, which could not be combined when unmeasured. Nor did they propose a method to handle this in fault diagnosis. Some researchers make an effort by using certain quantitative data to improve the diagnostic resolution of SDG. Kokawa et al. (1983)used SDG’s fault propagation time, fault propagation probabilities, and failure rate to improve resolution. However, these numerical data must be obtained through expert experience, and experts themselves are often a t a loss to clearly state the magnitude of these quantities. Therefore, Qian (1990) used fuzzy sets to express the fault propagation time, fault propagation probabilities, and fault rate of signed directed graphs, aiming to overcome the uncertainty of expert experience in furing these values. Because of the inaccuracy of the measured data in chemical engineering processes, it is difficult to make judgment based on three states of “positive deviation”, “normal”, and “negative deviation”. Han et al. (1994) used fuzzy sets to express deviation in SDGs to solve the difficulty in resolving these three states at any sampling time. In some methods of building a rule base from SDG, fuzzy numbers are used to express process variables and gains. Here, the fuzzy numbers expressing the degree of veracity of the deviation are inferred to fault origin. However, the use of fuzzy number
calculation for each variable makes the possibility distribution bandwidth of these variables even wider than it was before (Kandel, 1986);that is, they are even fuzzier. In addition, the plant dynamic gain is then very difficult to be accurately determined. The above SDG-based methods are limited in diagnostic resolution since they incompletely use quantitative process data. When SDGs are used in combination with graph theoretic calculation methods, however, there is no longer a need to expend a huge amount of labor and apply vast professional experience to formulate diagnostic rules, as is the case of expert system development. Nor are the extensive calculation times of parameter estimation methods required-another unique attraction. The goal of this research is to discover the fault origins in an early stage and to reduce the number of spurious interpretations. Because it is difficult to obtain the expert experience needed to compile a rule base or a knowledge base for a fault diagnosis expert system, and because also of the vast labor required to establish these bases, this research will not consider expert systems in its discussion of fault diagnosis. Instead, this research uses a SDG to express process knowledge, applying graph theory (Swamy, 1981) in concert with a depth-first search to partition the process SDG into maximum strongly-connected components, which serves as a basic structure t o narrow the search space. To reduce spurious interpretations, we try to replace the original SDGs qualitative expression , “+,” “0,”or “-”, of variables’ states to quantitative information by fuzzy sets. We also substitute the static gain between variables expressed qualitatively as or “-” to the dynamic gain with a dynamic fuzzy relational model transformed from dynamic-state constraints (confluences) in the original SDG. The model obtained from the above replacing is called the fuzzy cause-effect digraph (FCDG) model. This is described in this paper (part 1). This model can eliminate spurious interpretations attributed to system compensations and inverse response from backward loops or forward paths. In part 2 (Shih and Lee, 1995) we show how to apply fuzzy reasoning proposed in part 1 to estimate the states of unmeasured variables and to explain fault propagation paths and ascertain fault origins. The proposed method can also diagnose the fault origin in single or multiple loops in the early stages of process fault. By the proposed method, if the spurious interpretations still remain after diagnosis has been performed, they can be ranked by evidence degree to find the most likely fault origin. Finally, the research in part 2 uses a CSTR as an example to explain the proposed diagnosis method and compare the results to those obtained through other methods extended also from the original SDG.
“+”
Original SDG and the Limitations for Fault Diagnosis A chemical process model may be expressed by a signed directed graph (SDG) composed of vertices (or nodes) and links (or edges). Similar to the original SDG, this paper uses vertices to represent process variables or parameters, link arrows to represent variables’ cause-effect relationships, and a plus or minus sign to illustrate the relationship of positive gain or negative gain existing between two variables (see Figure 1). Constructing a SDG. We shall now define a SDG model graph that incorporates the summary above, and explain how to use a mathematical model of chemical
1690 Ind. Eng. Chem. Res., Vol. 34,No. 5,1995
i \W
Figure 1. Signed directed graph (SDG).
process to express the cause-effect relationships of the variables on the model graph. Definition 1: Defining the model graph (MG) for a SDG.
MG
(V,E,A,A)
Vis the set of all vertices in a given SDG, E is the set of all directed links, A, called the link influence function, is defined on E , A:E {+,-}, and represents the positive or negative gain influence of the links (for example, A(ek) = means that the link ek is a positive gain influence). A represents the fault pattern of the vertices, as explained in definition 2. Definition 2: Fault pattern. The fault pattern for any vertex A(u,), u, E V, is defined as follows:
-
+
if /Xu, - XuLl< E”,, then A(uJ = “0” IfX, - X’,, > cut, then A(v,) = i
IfX, - X’, i
1
“+”
< -cut, then A(uJ = “-”
Here, XUirepresents the process variable value of vertex ui,X Uis i the set point of ui,and E,, is the allowable error. The function defined here is known as a three-valued logic gate, as A: Ui {+,O,-}. To determine whether an influence function A exists for a link ek, and whether this function is positive or negative, one can go by either (1)process operation data or operating experience, or (2) a mathematical model. With the former, determinations are based on whether a variable has a direct influence on the other; the latter is expressed through a differential or algebraic equation, such as
-.
dxi/dt = fi(xi,x2,...,xJ xj = f;.(x+:2,
...&J
Suppose vertex ui represents variable x i , and uj represents x j . If &Jxj > 0, then the link ek exists, and the symbols ek- and ek+ express the initial point vj and terminal point vi of the link. Then, A(ek) = and x i , xj E {x1,.xz, ...,.xx,). On the other hand, if &J&j < 0, then N e d = -; if &J&j = 0, then there exists no link influence function between variable xj (the vertex uj) and the variable xi (the vertex vi). Applying Graph Theory To Search for the Fault Origin on a SDG. In order to reduce the search space
+,
and increase diagnostic efficiency in searching for the SDG fault origin, previous researchers applied graph theory to decompose the SDG into strongly-connected components (Iri, 1979). One can thereby obtain a rooted tree (see the Appendix) from the SDG; the roots of the rooted tree are called maximum strongly-connected components (MSCC) (Umeda, 1980). The MSCC is a strongly-connected component which might contain the fault origin. Iri (1979), Shiozaki (1985), Tsuge (19851, and Hwang and Lee (1991)offered a degree of resolution that could find the fault origin when the MSCC contains a single node or a single loop, but did not offer a diagnostic capability to handle multiple loops. Any node on a rooted tree is a strongly-connected component. If each strongly-connected component contains only one node on a given rooted tree, then it is very easy to find the fault origin on the graph’s MSCC; however, if some of the strongly-connectedcomponents contain more than one node, and there is more than one node on the MSCC, it is then difficult to ascertain which node is the true fault origin. One must test the consistency of the propagation path of each possible fault origin and estimate the state of each unmeasured variable, so that the propagation path can become a unique tree graph and the true fault origin may be found. However, this research takes the position that when the propagation path has backward or forward paths, and thus inverse or compensatory responses, the above method of seeking the propagation path through consistency testing to locate the fault origin will generate spurious or false interpretations. In order to eliminate these difficulties, we will first categorize six types of fault propagation patterns in which the strongly-connected components and the MSCCs could produce spurious or false interpretations. We will then discuss what in the diagnosis process causes spurious or false interpretations, how to facilitate further examination of the difficulties in determining the true fault origin on the MSCC, and how to resolve these difficulties. Those six fault propagation patterns will form different paths or loops, such as negative forward propagation branch (NFPB), negative back propagation loop (NBPL), positive backward propagation loop (PBPL), and three other patterns developed from the different combinations of NFPB, NBPL, and PBPL. All six fault propagation patterns are given and discussed in detail in the Appendix. We may summmarize here that purely qualitative analyses of SDGs run into problems when applied to the propagation paths containing NFPBs or PFPBs, or to MSCCs containing NFPBs, PFPBs, NBPLs, or PBPLs. Iri’s (1979) propagation path consistency criteria (see the Appendix) for diagnosis of SDGs lead to spurious or false interpretations as in above patterns A and B. If the fault origin is in a maximum strongly-connected component, and this MSCC contains more than one node, spurious interpretations as in patterns C and D would result. If the fault origin is in an MSCC containing unmeasured variables, diagnostic difficulty is compounded and spurious and false interpretations would result. Moreover, the use of a fault tree for diagnosis requires quantitative process data (Chang, 1992). Therefore, when a propagation path has an NFPB or PFPB, or an MSCC with a NFPB, PFPB, NBPL, or PBPL, the process will exhibit system compensatory or inverse responses that cause spurious or false interpretations in diagnosis. The main reason why these difficulties arise is a lack of relevant quantitative
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1691 variable data for diagnosis. For example, if the quantitative influence of ancestor variables on the confluence variables such as u4 in Figure 16a and u1 in Figure 17a were known, it would be possible to ascertain the correctness of the propagation path. In Figure 17a, for example, we need to know the quantitative influence of u6 on u1, of u4 on u1, and of u1 on up. Previous research on this subject has not made any progress in solving these problems; thus, this research is an attempt to quantify all these variable data. This research is a further extension from the use of SDG propagation network to search for the fault origin. The deviation of each variable is quantified with a fuzzy set, and the mutual interactions between variables can be expressed by converting dynamic-state constraints (confluences)into a dynamic fuzzy relational model. The further explanation of this follows.
The Reason for Using Fuzzy Set for Fault Diagnosis The states of variables at the neighborhood of threshold always confuse operators or engineers to judge whether the states are real or fake, because some uncertain factors exist in the plant which mainly come from measuring procedure and operator‘s understanding of the plant. Measurement of a chemical process variable is often influenced by the environmental noises and the uncontrolled disturbances that result in measurement error. Because of the imprecise theoretical model or the performance degradation of physical and chemical properties, a true normal state of a process variable is difficult to estimate from the theoretical model, but the normal process condition is often taken from the theoretical process model during the engineering design stage. It is difficult to include the uncertainties for most of the theoretical process model. For example, we often use an algebraic or a differential equation to express material balance and energy balance which do not clearly include uncertain heat loss quantity, equipment efficiency, etc. Therefore, a true normal state of a process variable cannot be represented by a design value, and the node (or state of process) in SDG cannot easily be discriminated absolutely as normal or abnormal by the measured and the designed values if SDG is constructed from a theoretical process model. Thus, the fault diagnosis either quantitatively or qualitatively has this inherent uncertainty, if we merely consider the discrepancy between the designed and measured values. Fortunately, the mathematical uncertainty can be described by dynamic randomness and fuzzy concept. The former exhibits the deviation from the normal values which distributes randomly, while the latter is used to quantitatively express variables in a range. A fuzzy value always provides both quantitative and qualitative information, but cannot be treated statistically. For example, how does an operator adjust the cooling water in an emergency if he does not ensure that the seriousness of the reactor temperature is beyond the allowable limit, and how much is the confidence he has for his reaction? To handle the problem of confidence, the fuzzy set theory is very helpful. Regarding the fuzzy set applications, Kramer (1987) has used the fuzzy set to the rule-based approach for evaluating the certainty factor of the fault. Fuzzy Reasoning Fuzzy reasoning is a deductive inference with fuzziness that extends the concept of classical logic and fuzzy
sets (Zadeh, 1983). Now it has been applied to many control systems, for example, automation of washing machines, air conditioners, and refrigerators. In the classical logic, the statement of a proposition is only identified as either true or false. The simplest proposition, called the atomic proposition, cannot be decomposed. We denote the atomic proposition by P, Q etc. The atomic propositions can be combined by connectives to form a compound proposition. These connectives are OR, AND, NOT, IF-THEN, EQUAL TO, etc. logic operators. The compound proposition “If P then Q plays a very important role in the cognitive science. This is called an inference rule in artificial intelligence. In logic, we call P the premise and Q the conclusion. The “If P then Q can be expressed by “P Q ; the is an implication. Example 1. In the original SDG, the nodes correspond to the process variables and the branch represents the immediate influence between the nodes. The direction of deviation of process variables relative to their nominal steady-state values are divided into three states: high (+), normal (0), and low (-). Branch signs and indicate values of the cause and effect variables tend to change in the same or opposite directions. Then, the causal relationships between U and X can be described as the following:
-
“4”
+
Its meaning in implication rules is the following: If deviation of the U is “+”, then deviation of the X i s a-n
If deviation of the U is “-”, then deviation of the X is
(‘+”.
The causal relationships are represented by the ordinary relation, R . In this work, we will provide fuzzy sets to nodes and branches of the SDG in the illustrative application for representing quantitative deviation of these variables and quantitative immediate influence between the nodes respectively. These fuzzy sets are “+”, “-”, and “0” with its membership function; we shall call that kind of propositions and implications fuzzy implication rules. We thus may describe a system’s behaviors with multiple fuzzy implication rules that are often expressed with fuzzy relation. Therefore, the SDG is no longer qualitative, but becomes quantitative with fuzzy sets. Fuzzy reasoning is easy to apprehend from many publications (Kaufmann, 1975; Dubois and 1980; Kandel, 1986; Pedrycz, 1989; Kosko, 1992). Here, we only introduce the parts of fuzzy set, fuzzy relation, and fuzzy reasoning which are related to this work. Fuzzy Sets. The fuzzy set theory was developed by Zadeh (1965) for dealing the inaccuracy of human expression. For instance, the sentence “heater temperature is too high” is only a vague expression and is dificulty to manipulate mathematically, since it has different meaning for different persons’ consideration of seriousness. The fuzzy set theory, distinguished from the precise mathematical expression, is suitable to the event without distinct bound. Definition 3. Let U represent the universe of discourse, R 3 U , and X represent the elements of a population with characteristic value x , x E X then the fuzzy set is defined as
a = Ux,p‘&)),
x
E
x)
(1)
1692 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 R
m-¤
u x
Where R is the casual relationships between U and X, and U and X are the states of U and X, respectively. Consider the multiinput, single-output system
X = U10U20...oUnoR Figure 2. Fuzzy set of the heater temperature.
where p&) is the characteristic function of x , also called the membership function or belongingness, in set A. It gives the characteristic value x a numerical value. Equation 1implies that X is mapped to a membership space M through x subregion. If M E (0,l) then it is nonfuzzy, and belongs to the conventional Booleen set. Zadeh (1978) defined M E [0,11; 0 and 1 represent nonbelongingness and complete belongingness, respectively. In this paper, the set of elements of the population X is the fuzzy set of the process variables or parameters of temperature, pressure, time constants, etc. The set point of a variable is given as fsp. The membership function, denoted as piSp(x)represents the “closeness of x to fa,or belongingness degree of x to Zap)),and pa&) E [0,11. Example 2. We specified the fuzzy sets of the heater temperature, which are too high (TH), medium high (MH), normal (N), medium low (ML), and too low (TL). Fuzzy membership functions of those fuzzy subsets are shown in Figure 2. If the heater temperature has membership values [0.1,0.8,0.1,0,01in the fuzzy set which represents the possibility of the fuzzy sets, respectively, these numeric values mean the temperature is about medium high, which is not only qualitative, but also approximate quantitative information. Fuzzy Relational Equation. The casual relationships between U and X in example 1 that can be modeled as fuzzy systems of the form:
X = UoR
(2)
where o is a composition operator (e.g., max-min). Within X E F(X), U E F(U), R E F(U x X), x is the Cartesian product. X, and U are the universes of discourse. F stands for the family of fuzzy sets defined on a particular universe, i.e., X E F(X), XX [0,11. Each fuzzy set is defined in terms of a complete set of fuzzy reference sets defined on the underlying universe of discourse. For example, {f1$2,...31,9, E F(X), i = 1, ..., n, and V x E X 3 1 5 i 5 n such that pa,(x) > 0). Where f , are the fuzzy subsets of X, and let piz(x)= p,. Any fuzzy set X E F(X) can then be represented by its vector of membership degree [Pl,p2,...,p,I related t o the fuzzy subsets of X. The fuzzy relation, R,can be written as a set of fuzzy rules in terms of the reference sets defined on each universe of discourse. For eq 2, suppose the universe X defined n reference sets, and the universe U defined m reference sets. Then R can be written as an m x n matrix, R(ij) = pv. Each element of the matrix can be interpreted as specifying the rule
-
If ii, then f J withp, For each ii,, there are n rules in the relation, R, representing the relationship between U and X. The SDG in the example one can be represented by
(3)
Suppose the universe X defined n reference sets, and the universe Uidefined mi reference sets. Then R can be written as an ml x m2 x ... x m, x n matrix, R(ii,h,...,i&) = pil,iz,...,ink. Construct Fuzzy Relation for a Cause-Effect Digraph. The gain matrix R describes the causal relationship between variables for eq 3. We use U X to express how to calculate its fuzzy relation. In this work, we use Mamdani’s method to calculate the fuzzy relation of R.
-.
R:U
-X
Express it with the mathematical equation
(4) where A is a conjunction operator. Each state of a system has its own fuzzy relation, Rk. We have to composite these fuzzy relations, Rk into a fuzzy relation, R,that must be adequate to all observed data, u k and xk.
R =U Rk k
(5)
For minimizing the errors between the influenced value and the observed data, we use the modified Newton iteration approach (Pedryzc (1983),see the Appendix) to obtain the relation, R, in this work. Defuzzification. The cause-effect fuzzy variables are estimated from the relation, R, on which the algebraic operations cannot be performed. Therefore, we must converse the estimated fuzzy variables to the quantitative value in a real number set. The centerof-area (COA) method is used for this work. This method selects the element corresponding to the center of area under the curve described by the fuzzy set membership function. This is mathematically expressed as
X =
where xi is the center of area of the ith reference set, and x is in the universe X. Example 3. Suppose that the SDG R
m-¤
u x
in example 1 is represented by eq 2, and_th_e fuzzy and X = (XN,XZ,XP}; the subsets are U = (UN,UZ,UP) causal relation between variables U,and X is described by the following rules:
Ind. Eng. Chem.Res., Vol. 34, No. 5,1995 1693
$ N
Z
1.o 0.8
U and X are given by
Ul= [l 0.6 01
P
0.6
0.3
Xl = [l 0.6 0.11 -100
X, = [0.2 0.8 0.91
x,
From eq 4, the fuzzy relations, R1 and R2 are
0.6 0.6
0.1 0.1
1
0.2
0.8 0.0
0.9 0.0
1
o.o[o.o
From eq 5,
R = R, U R,
1.0
0.6
0.1
0.2
0.8
0.9
=
If U is given, U1= [0.8 0.3 01, and Xwill be estimated through eqs 2 and 6, The final row vector represents
(0.8
A
(0.8
A
(0.8
A
= [ 0 . 8 v 0.3
=[0.8
0.9
1.0) A (0.3 A 0 . 6 ) 0 . 6 ) A (0.3 A 0 . 8 ) 0.1) A (0.3 A 0 . 8 )
0.6
V
0
V
(0
A
0.2),
V
(0
A
0.8),
V
(0
A
0.91,
0 . 6 v 0.3 v 0
%
also the complicated, dynamic variable activities such as nonsingle transitions and ultimate responses caused by compensatory or inverse responses. Constructing Fuzzy Cause-Effect Digraph (FCDG) Model from a Process. The ordinary SDG diagnostic methods might lead the diagnosis into erroneous results in the case of the existence of a propagation path, NFPB or PFPB in SDG, or NFPB, PFPB, NBPL, PBPL in MSCC, as mentioned before. This is because those methods are merely qualitative. At present, quantitatively estimating the deviated state of the fault origin candidates or the unmeasured variables in a SDG having NFPB, PFPB or MSCC having NFPB, PFPB, NBPL, or PBPL could reduce the number of spurious or erroneous interpretations by fuzzy reasoning based on the FCDG model. The quantitative model of a chemical process consists of simultaneous algebraic and differential equations. Linearizing these equations around the steady state is needed for constructing the FCDG model. The following linearized equations can represent the relationship among chemical process variables and its dynamic behaviors:
dvlldt = cllvl
0.3 0 1 0 0.2 0.8
100
+ c12v2+ ... + clnvn dvddt = c21v1+ c2,v2 + ... + cZnvn
1.0 0 . 6 0.1
X=[0.8
50
Figure 3. The inferred value of X in example 3.
1.0 Pg,(Ul) 1.0 11.0
Pop,)
0
-50
1
0.1 v 0.3
dvnldt = cnlvl V
01
0.31
pxN,pxz, and exp,respectively. Suppose the membership functions of X are shown in Figure 3. The COA in the shallow area corresponding to the universe X along the x-axis is the deduced value in a real number set.
+ cn2v2+ ... + cnnvn
(7) (8)
(9)
Equations 7-9 may be written as
dvldt = (cy)
(10)
where c = (~1,~2,...,~n)~ and v = ( ~ 1 , ~ 2 , . - . , v n ) ~ . In order to formulate the SDG at each dynamic state, the equations are rewritten in the following discrete form:
The Fuzzy Cause-Effect Digraph (FCDG) Model Usually, the information provided by SDG is qualitative and will limit fault diagnosis resolution. Therefore, fuzzifjmg the nodes and the branches of SDG is a better way for upgrading diagnosis resolution. We call the fuzzified SDG a fuzzy cause-effect digraph model. This section will describe how to fuzzify an ordinary SDG into a fuzzy cause-effect digraph model. In this work, we divide the continuous process into a time-discrete process for the purpose of converting to a dynamic fuzzy process model. The dynamic fuzzy process model will be capable of offering the dynamic, quantitative, fuzzy relationships between variables and
Therefore, each variable is expressed as the sum of the other variables,
We call vi the confluence variable, and eq 11is viewed as the transient confluence equations. The transient
1694 Ind. Eng. Chem. Res., Vol. 34, No. 5,1995
unmeasured confluence variable were caused by the adjacent ancestor measured variables. Each component in fuzzy set could be calculated from the derived fuzzy relation, R, and then defuzzified to the real number through eq 15. Adding them together will give the estimated value of the unmeasured confluence variable.
v2
(15)
Vn Figure 4. SDG of the transient confluence equation, eq 12.
V2 w3
\
where p-l is a defuzzifying operator converting fuzzy set to a quantitative value. Define Fuzzy Set of Variable. A chemical process always includes many different types of variables with different magnitude orders by physical measuring. Normalizing all the different magnitude orders of the variable deviations would be convenient to define reference fuzzy sets of variables and to implement fuzzy inference. After normalizing the variable deviations, all the absolute values of discourse of the reference fuzzy sets range from 0 to 100.
AXN = XN - Xssv= (X- Xs)/scalex = AX/scalex (16) Figure 6.
u1
scalex =
equals the addition of u2, ..., un.
confluence equation can be represented by a SDG. For example, eq 12, shown in Figure 4, can be modified to Figure 5 , and has the meaning that the unsteady-state confluence variable u1 equals the sum of u2, u3, u4, and
XMAX - XMIN 100
where AXN: -100 to +lo0
un.
A chemical process containing measured and unmeasured variables is represented by a SDG, in which the unmeasured variables are divided into three types: (1) unbranched variables, such as the relation of v6, u7, and U g shown in Figure 11;(2) branched variables, such as the relation between u5 U6, and u7 shown in Figure 13; (3) confluence variables, as observed from Figure 11in which vertex u4 is affected by u3 and us. A fuzzy relation is used to express relationship between the measured and unmeasured variables or between the measured and the estimated variables while a chemical process is translated to a fuzzy causeeffect digraph model. The gain, R, is identified by converting the unmeasured variables and its adjacent ancestors or descendant measured variables to fuzzy relation. Calculating fuzzy relations of each causeeffect pair with eq 13 is necessary and is used to deduce
R,, = Ut x X , x AXt
Equation 17 is rewritten in the following discrete form: AunN At
+
-- - Cn1AuyV Cn2Avm + ... + CnnAu,N (18) Substituting eq 18 into eq 11 gives
AunN -scale, At
= c,,(scale1)Avw
Comparing eqs 18 and 19 gives the coefficients before and after normalization:
Cnl = cn1(scalel/scale,)
(13) cnn
an unmeasured variable's state along the same path from a measured variable with their fuzzy relations. If a path contains consecutive unmeasured variables and one measured variable, it is illustrated in Figure 6. The vertex u2 can be deduced from eq 14 with composition operation. u2
= u40Ru,u30Ru3"2
=7J40Ru4u2
+ cn2(scale2)Avw+ ... + cnn(scalen)AunN(19)
(14)
Constructing a fuzzy relation for the confluence variable, one needs only consider the causally adjacent ancestor measured variables one by one. When the unmeasured confluence variable is to be estimated, such as V I shown in Figure 7, the components acting on the
= cnn
(20)
All the deviated values will lie in the universe of discourse, U,and U = [-100,1001, after normalizing the variable deviations by eqs 16-20. Defining the membership function of reference fuzzy sets for the measured and unmeasured variables is required a t the first step in the constructing process fuzzy model. The vertex method (Dong and Shah, 1987) is used to calculate the membership functions according to the relations of the measured and the unmeasured variables in process model. In the calculations, the membership functions have to satisfy each measured or unmeasured variable for which the fuzzy subset N is [-100,01, 2 is [-loo, 1001, and P is [0,1001, and satisfy another one for which the upper bound of 2 and P is less than 100 and the lower bound ofN and Z is larger than -100 at the same
Ind.Eng. Chem. Res., Vol. 34, No. 5, 1995 1695
0
: unmeasured vertex
0
: measured vertex
Figure 6. SDG of u1, ..., us.
w n
@
Rv3vi
\ RVZVI
\ tc
DefuuifY
@ . R7
time. Membership functions of the reference fuzzy subsets, N,2,and P,are shown in Figure 3. Manipulation Procedure. Considering the above discussions, the following steps can be summarized for constructing the fuzzy cause-effect digraph model: 1. Prepare a mathematical model which is not necessarily rigorous for the chemical process we are interested in. In other words, a simplified mathematical model can be used, too. 2. Calculate the coefficients of the set of equations after normalization by eqs 16-20. Then, all the deviated values of variables lie in the universe of discourse, U,and U = [-100,100]. 3. Construct SDG from the chemical process topology and the mathematical model. 4. Locate all the unmeasured variables and its adjacent ancestors or the descendant measured variables. 5. List all the relations related to each pair of the unmeasured and measured variables from the chemical process topology. 6. Define the reference fuzzy sets and the membership functions for all variables. 7. Identify the gain, R (Pedrycz, 1983), which is the fbzzy relation between the unmeasured variables and its adjacent ancestor or the descendant measured variable. A priori fbzzy model can be identified from the mathematical model which has been constructed in step 1.
Application Here is an example to illustrate the manipulation of the procedure mentioned above. A heat transfer system removes heat by a cooling medium. The system behavior can be described by energy balances as follows:
The second term on the right side of eq 21 represents the heat source. Suppose values of FO, VR,A, e, K,CA, Cp, Uo,A, TjO, Vj, Cj, and ej are given and substituted them into eqs 21 and 22, then eqs 23 and 24 are obtained. dT/dt = 0.8333(530 - T ) - 1.1328 x 1013e15075/T 20.8333(7' - Tj) (23) dTj/dt = 0.2597(Fj)(530 - Tj)
+ 156.3445(T - Tj) (24)
The ranges of T, Tj, and Fj are given and shown in Table 1. We follow the above procedure to construct the fuzzy cause-effect digraph model of this system. 1. The equations of mathematical model are linearized (Chang, 1990) as
+ 20.833Tj (25) - 169.3055Tj + 156.34457' (26)
dT/dt = -21.66677' dTj/dt = -16.7792
2. Calculate the coefficients of equations after normalization: dTN/dt= -21.66671"
+ 27.5634TjN
(27)
3. Build the SDG from eqs 27 and 28, as shown in Figure 8. Suppose that Tj is an unmeasured variable, while T and Fj are the measured variables. 4. List all the partial order pairs of the measured and unmeasured variables from Figure 8. partial order pair Fj Tj Tj-T T-Tj
-
cause
effect
Fj
Tj T Tj
Tj T
5. Define all the reference hzzy sets for the measured and the unmeasured variables of SDG. 6. Vertex method is used to calculate the membership h c t i o n s according to the relations of the measured and unmeasured variables in a process model. In the
1696 Ind. Eng. Chem. Res., Vol.. 34, No. 5, 1995
a
Z
P
P
0 0
: unmeasured vertex
-100
Figure 8. SDG of Tj, T, and Fj.
:.:r-r 4
Table 1. Ranges of Variables
P
operating condition
Ti
100
FjN, %
: measured vertex
variable Fj
50
0
-50
max 60 660 660
normal 40 593 600
min 0 488 530
0.0
unit
-11.2
b P
Conclusion The rule-based approach is oRen used for fault diagnosis, and is derived from SDG in previous work, because SDG is able to clearly represent the relations among the process variables. Unfortunately, the SDGbased model is limited in its representation of a system's behavior by its incomplete use or expression of quantitative process data. Therefore, the proposed FCDG model conceptually provides fuzzy sets to nodes and branches of the SDG for representing the quantitative deviation of the variables and quantitative immediate influence between the nodes respectively from a simple mathematical model, which is needed to construct a priori FCDG model.
5.6
~
11.2
:.:l 4
Z
N
P
~
0.0
-100
-50
0
50
100
: r - r TN, %
A
P
0.0
z
N
-90
-45
45
0
P
~
90
TJN,% C
A
; : . :
-
P
0 TN, %
-5.6
ft3h
"R T "R calculations, the membership functions have to satisfy each measured or unmeasured variable for which the fuzzy subset N is [-100,Ol and 2 is [-100,100], and satisfy another one for which the upper bound of 2 and P is less than 100 and the lower bound of N and 2 is larger than 100 a t the same time. Membership functions of the reference fbzzy subsets, N , 2, and P,are shown in Figure 9. 7. Identify the gains, R, which is a set of fuzzy relations, consisting of Fj, T, and Tj in this example, as listed in Table 2. Test of the Fuzzy Cause-Effect Digraph Model Quality. Figure 1Oc showed the simulated value, Tj, of eqs 23 and 24, if T and Fj have inverse responses as showed in Figure 10a,b. It should be noted that T,Tj, and Fj had different directions of inverse response in this case. Moreover, Tj and T influence each other. The trend of the estimated Tj,est can describe inverse response satisfactorily (Figure 1Oc) in the above situation. Note that Tj,est is deduced from the fbzzy causeeffect digraph model. From this example, using the fbzzy cause-effect digraph model could explain the system's compensation or inverse responses that do not need to judge which is the dominant or nondominant path of fault propagation (Chang and Yu, 1990), and do not require the condition branch to represent different process states (Kramer and Palowitch, 1987). Furthermore, the pathway T TJ is inconsistent (Figure 10a,c) in fact a t 0.12 h since the system has a time lag, though the pathway T Tj is the dominant obtained from the comparison of eqs 23 and 24. We have confidence that the FCDG model can improve the representatives of the SDG describing the system's behavior fairly enough, though some inaccuracy on the deduced values exists which cannot be avoided because of the nature of the "fuzzy" model.
-
Z
N
N
Z
P
Y
~
0.0
-100
Y
-50
0
50
100
:E, 0.0
-94 -9..
-47
0
TN, %
47
94
-
Figure 9. Reference fuzzy set definitions. (a) Fj Tj relation. Top: FjN. Bottom: T ~ N(b) . T Tj relation. Top: TN.Bottom: T~N. (c) Tj T relation. Top: T ~ NBottom: . TN.
-
---c
- -
-
Table 2. Fuzzy Gains between Variables: (a) Fuzzy Relation of Fj Tj;(b) Fuzzy Relation of T !lJ (c) ; Fuzzy Relation of T (a)RTj,dTj/dt,Fj 0.00 0.00 0.02 0.00 0.00 0.37 0.00 0.00 0.02 0.03 0.03 0.03 0.50 1.00 0.38 0.02 0.02 0.02 0.03 0.00 0.00 0.50 0.00 0.00 0.02 0.00 0.00
(b) &'j,dTj/dt,T 0.02 0.00 0.00 0.50 0.00 0.00 0.03 0.00 0.00 0.03 0.03 0.03 0.50 1.00 0.38 0.02 0.03 0.02 0.00 0.00 0.03 0.00 0.00 0.37 0.00 0.00 0.02
(c)RT,dT/dt,Tj 0.02 0.00 0.00 0.50 0.00 0.00 0.02 0.00 0.00 0.03 0.03 0.03 0.50 1.00 0.37 0.02 0.02 0.02 0.00 0.00 0.03 0.00 0.00 0.38 0.00 0.00 0.02
The proposed FCDG model has the following merits when compared to rule-based model based on SDG (Chang and Yu, 1990; Kramer and Palowitch,l987): 1. Skill and experience of process behavior are not necessary to judge (1)whether the removal of unmeasured variables may lead to spurious diagnosis and (2)
,
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1697
0.30 0.25 a d, a,
U
0.20
0.15
-
0.10
'=
0.05
c 0
(d
1
a
- 1 1
\
-0.05 -0.10
0.3
0.2
0.1
Time, Hr
.4
1
-20 -lo
-30 --f
1 0.1
0.3
0.2
Time, Hr 0.5 0.4
$"
0.3 0.2
0.1 0 .o -0.1 -0.2
-0.3
1I
1
I
I
0.1
I
I
I
I
I
I
0.2
Time, Hr Figure 10. Responses of variables. (a) T response. (b) Fj response. (c) Estimated Tj,est and Tj responses.
which of the paths are dominant or nondominant of fault propagation. 2. FCDG can describe a system's compensation or inverse responses that the dominant pathway has time lag, as the example demonstrated in the Application
section. (In other words, the dominant pathway has no dominant effects a t initial time.) 3. Variable states and their relations can be quantitatively expressed by the fuzzy sets that could provide more information than the original SDG.
1698 Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995
We have confidence that the FCDG model can improve the resolution of the previous SDG-based diagnosis methods, though some inaccuracy exists in the deduced value, which cannot be avoided because of characteristics of the “fuzzy”model. The application of the FCDG model for fault diagnosis is described in part 2.
Acknowledgment This work is supported by the National Science Council of the ROC under Grant NSC83-0421-P-0080012.
Nomenclature = fuzzy set A c = coefficient of equation c = vector c Cj = heat capacity of cooling water Cp = heat capacity of process liquid E = a set of all directed links ek = kth link ER = activation energy FO = inlet flow rate F( ) = a family of fuzzy sets FCDG = fuzzy cause-effect digraph Fj = cooling water flow rate G = directed graph K = reaction rate constant M = membership space MG = model graph MSCC = maximum strongly-connectedcomponent mi = number of fuzzy subsets for the ith universe NBPL = negative backward propagation loop NFPB = negative forward propagation branch PGk = element of R PBPL = positive backward propagation loop PFPB = positive forward propagation branch R = ordinary relation, or a set of real numbers R = fuzzy relation R, = gas constant scalex = (X- - X~~)/100 SDG = signed directed graph TO = feed temperature Tj = cooling water outlet temperature TJO = cooling water inlet temperature U = variable U, or a universe of discourse U = fuzzy set of U U,, = overall heat transfer coefficient V = vertex set v = vector v ui = ith vertex or variable Vj =jacket volume VR = reactor liquid volume X = variable X,or a set of objects X = fuzzy set of X x = an element of the set X Zi = fuzzy subsets ZSp = fuzzy value for set point of a variable Xui= process variable value of vertex ui Xu: = set point of vertex ui Subscripts MAX = maximum point MIN = minimum point t = time Greek Letters ,ux= membersip function of fuzzy set Z A = link influence function A = fault pattern of vertices
= allowable error of ui ,p = process liquid density ,pj = cooling water density
Other Symbols -“+” an impliction = positive deviation of a vertex, or positive gain =
= negative deviation of a vertex, or negative gain “0”= normal state of a vertex o = composition operator x = Cartesian product A = conjunction operator “-”
Appendix
Six Types of Fault Propagation Patterns. Fault propagation pattern A: The propagation path of fault origin u1 passes through a negative forward propagation branch (NFPB). In the example in Figure 11, u4 changes dynamically among three states: -1, 0, +l. When the deviation of u1 is “+”: 1. The positive influence of u2u3u4 is overwhelmed by the negative influence of hence, the deviation of u4 is “-”. Therefore, path U ~ L J ~ U does ~ U ~ not U ~ fulfill the branch consistency criterion. 2. The positive influence of u2u3u4 and the negative influence of cancel each other out; hence, the deviation of u4 is 0. Therefore, paths ~ 1 ~ 2 ~ and 3 ~ 4 u1u2u6u7U8u4u5 cannot fulfill the branch consistency criterion. 3. The positive influence of u2u3u4 overwhelms the negative influence of u2u6u7rh3u4;hence, the deviation of u4 is “+”. Therefore, path u1u2u6u7~8u4u5cannot fulfill the branch consistency criterion. When the deviation of u1 is “-”, the fault propagation paths also do not fdfill the branch consistency criterion. If the influence of u8 on u4 does not vary with time, the SDG may be simplified by eliminating all but the dominant path (Kramer and Palowitch, 1987; Chang and Yu, 1990; Yu and Lee, 1991). For example, in (1) above, the path u2u3u4 may be eliminated. However, if the influence of w3 on u4 increases with time, then u4 will be subject to the situations of inverse response and compensatory response. In this case, a simplification of the SDG through elimination of the dominant path will generate spurious or false interpretations. Fault propagation pattern Type B: The propagation path of fault origin u1 passes through a negative backward propagation loop (NBPL) such as portrayed in Figure 12a. In this case, u 2 changes dynamically among three states: -, 0, When the deviation of u1 is “+”: 1. In the early stages of process fault, the positive influence of path u1u2 is overwhelmed by the negative
+.
I+
J”’”f
4
I+
: fault origin
0
: measured vertex
Figure 11. SDG of fault propagation pattern A. The propagation path of fault origin u1 passes through a negative forward propagation branch (NFPB).
~ 5
Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 1699
I+
4
I+
I+ :fault origin
0:
measured vertex
Figure 13. SDG of fault propagation pattern C. Fault origin u3 is on a positive backward propagation loop (PBPL) which is a strongly-connected component with more than two nodes.
I+
+
+
+
: fault origin
0
: measured vertex
+
Figure 12. SDG of fault propagation pattern B. "he propagation path of fault origin u1 passes through a negative backward propagation loop (NBPL).
influence of path u2~6u7u8u4u3u2;hence, the deviation of u2 is "-". Therefore, path u1u2u6u7u@4u5 does not fulfill the branch consistency criterion. 2. When a compensatory response occurs, the positive influence of path u1u2 and the negative influence of path u2u6u7u@4u3U2 cancel each other out; hence, the deviation of u2 is 0. Therefore, path u1u2u6u7ugu4u5 does not fulfill the branch consistency criterion. 3. When an inverse response occurs, the positive influence of path u1u2 overwhelms the negative influence of path U2u6u7usu4u3~2;hence, the deviation of u2 is Therefore, path u1u2r&u7u8u4u5 does not fulfill the branch consistency criterion. The situation is identical to the deviation of u1 being negative. When the confluence variable u2 is 0, then Iri's (1979) diagnosis method obtains two rooted trees, shown in Figure 12b, from which two possible fault origins u1 and u6 are derived. Since one of which is spurious, it must be assumed that u2 is either or - to Mfill the consistency requirement for the propagation path. In the methods of Chang and Yu (1990),Yu and Lee (1991), and Hsu and Yu (1992), one must simplify the tree into a different SDG for the 0 state of each variable to formulate a set of rules. This method is difficult to apply because it relies on an experienced person to judge which variables will sometimes be in the 0 state. Fault Propagation Pattern C: Fault origin u3 is on a positive backward propagation loop (PBPL), which is a strongly-connected component with more than two nodes as depicted in Figure 13. All the nodes on the loop conform to the branch consistency requirement; but the fault origin still cannot be found by using the present branch consistencyjudgment principles for SDG diagnosis (Iri, 1979). Although PBPLs hardly ever occur in the chemical processes, we assume their existence. PBPLs usually occur together with NBPLs, preserving the operational stability of the process. This is the
+
I+
"+".
+
+
+ . .. .,..(.,....... ........,.
.................... :.:.>..+
+
+
+
+
i+ I
+
+
:fault origin
0
: measured vertex
Figure 14. SDG of fault propagation pattern D. The fault origin u3 is located in a MSCC. This MSCC is composed of NBPLs and PBPLs.
situation in a vapor recompression evaporator (Nisenfeld, 1985). When there is a sudden increase in the steam supplied to the vaporizer, causing an increase in evaporation rate, then the evaporator's vapor pressure rises and the steam sent by the compressor to the vaporizer increases even more, resulting in operational deterioration. This is a PBPL. Thus, a feedback control loop is usually installed to control the amount of steam supplied by the steam boiler to the vaporizer and compensate for the deterioration. Fault Propagation Pattern D: The fault origin u3 is located in a MSCC. This MSCC not only contains more than one node of strongly-connectedcomponent, but also more than one loop. These loops are composed of NBPLs and PBPLs, as depicted in Figure 14a. Following the current SDG diagnoses (Iri, 1979), we first assume the deviation of u3 is so that the loop
"+",
1700 Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995
+
I+
+
+
t+ +
+
+
+
+
+
+
+
0
+
+
+
+
I+
+
+
I+
+
+
t+
+
I+
v +
a 0
+
+
+
: fault origin
:measured vertex
Figure 15. SDG of fault propagation pattern E. The fault origin u g is located on a NBPL which is a strongly-connected component containing more than two nodes. u4ugu6V1u2u3U4 fulfills the branch consistency requirement in the early stages of process fault. In spite of following the judging criterion for the branch consistency, we are still unable to find the fault origin. If we proceed to assume u3 is “0” or “-”, we find paths u4ugu6u 17.12and u3 fhlfill the branch consistency requirement in the early stages of process fault, as in Figure 14b. Following branch consistency criteria, we find nodes u3 and u4 are the possible fault origins that are thus a spurious solution. Using Yu’s methods mentioned earlier, one must consecutively assume each variable to be the fault origin, and simplifjrits propagation path into a separate SDG for each “0” state, which will have its own separate propagation rules. This method is difficult to apply because it relies on an experienced person to judge which variables might coincidentlybe in the 0 state and to simplifytheir SDGs. We have an idea that using quantitative data could avoid the difficulty of formulating different fault propagation rules. If nodes u1, u5, and have the quantitative information to be used to check what the deviations of u3 and u4 should be and select the most unreasonable deviation, that may be a fault origin. Fault Propagation Pattern E: The fault origin u3 is located on a NBPL which is a strongly-connected component containing more than two nodes, as depicted in Figure 15a. Following the branch consistency criteria for SDG diagnosis, we can obtain the propagation path graphs that are shown in Figure 15b,c. Thus, nodes u4 and u3 are the possible fault origins.
0
: unmeasured vertex
0
: measured vertex
Figure 16. SDG of fault propagation pattern F. “he confluence point is an unmeasured variable that is in a backward propagation path.
Fault Propagation Pattern F: On a backward propagation or forward propagation path in a propagation pattern, sometimes the confluence point is an unmeasured variable, as in Figures 16a and 17a. Following the current SDG diagnosis (Iri, 1979), we would first assume u3 on Figure 16a to have a “+” deviation, and find the largest unique rooted tree, which in the example is the PBPL u4ugu6u1u2u3u4. Then assuming u4 to be again, we are only able to find the largest unique rooted tree: PBPL ~4u5ufju1~2u3u4.We cannot find the fault origin. Assuming u4 to be “0”or “-”, we can divide the SDG into two subgraphs’as in Figure 16b or 16c. None of these results meet the criteria of Iri’s diagnostic algorithm. In Figure 17a, u3 is first assumed to be a t the beginning of process fault, and the largest unique rooted tree can be found; the same holds true when u1 is assumed to be “+”. In either case, this PBPL is u4u5u6u1u2u3u4, but no fault origin can be found. When u1 is assumed to be “0”or “-”, the SDG is a unique rooted tree as in Figures 17b,c, and u2 is found to be the fault origin-a false interpretation. In the work of Yu as mentioned earlier, the unmeasured variable u4 cannot be combined, as this would cause spurious or false interpretations. Consequently, one must rely on skill, experience, and understanding of the behavior of the process to judge whether spurious interpretations have arisen from combining variables.
“+”,
“+”
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1701
+
+
+
+ 0
@
+
+
+
+
+
+
+
+
4
I+
+
+
+
+
where l/f'(xn) is replaced by a simple scalar multiplier andepending on the number of iteration.
Literature Cited
@4-@-b@
+
where flxn) is the first derivative of flx) at x = xn, and x is an element of R . Because Newton's method has some disadvantages, Pedrycz proposed a modified Newton method as follows:
+
+
:fault origin
0 0
: unmeasured vertex
: measured vertex
Figure 17. SDG of fault propagation pattern F. The confluence point is an unmeasured variable that is in a forward propagation path.
Rooted Tree. Definition: A rooted tree G is a directed tree if G is a tree and has a root. A vertex u in G is called a root of G if there are directed paths from u to all the remaining vertices of G. Branch Consistency Criterion. Definition: A branch ek is said to be consistent if A(ek+) Mek) A(ek-) = and A(uk) f 0, where A(uk) is the sign of vertex uk, A(ek) is the sign of ek, and A(ek+) and A(ek-) denote separately signs of initial and terminal vertices of uk (Iri, 1979). Modified Newton Iteration Approach. We obtain eq 2, when R has been identified from U and X,such that the error function Q is minimized with respect to all elements of R , that is, aQ / aR = 0, where Q is treated as a distance between respective fuzzy set u, u ', viz,
+
Q = g(u,u') u ' is calculated according to eq 2.
Pedrycz(1983)suggested a modified Newton iteration approach to obtain the relation R. The Newton iteration approach for finding the solution of aQ/aR = 0 can be given as
Chang, C. C.; Yu, C. C. On-line Fault Diagnosis Using the Signed Directed Graph. Ind. Eng. Chem. Res. 1990,29,1290. Chang, C. T.;Hwang, H. C. New Developments of the DigraphBase Techniques for Fault-Tree Synthesis. Ind. Eng. Chem. Res. 1992,31,1490. de Kleer, J.; Brown, J. S. A Qualitative Physics Based on Confluences. Artif Intell. 1984,24,7. Dong, W. M.; Shah, H. C. Vertex Method for Computing Function of Fuzzy Variables. Int. J. Fuzzy Sets Syst. 1987,24,65. Dubois, D.; Prade, H. Fuzzy sets and Systems; Academic Press: New York, 1980;Part 3,p 151. Han, C. C.; Shih,R. F.; Lee, L. S. Quantifying Signed Directed Graphs with the Fuzzy Set for Fault Diagnosis Resolution Improvement. Ind. Eng. Chem. Res. 1994,33,1943. Himmelblau, D. M. Fault Detection and Diagnosis in Chemical and Petrochemical Processes; Elseviern: Amsterdam, 1978. Hsu, Y. Y.; Yu, C. C. A Self-Learning Fault Diagnosis System Based on Reinforcement Learning. Ind. Eng. Chem. Res. 1992, 31,1937. Hwang, G. W.; Lee, L. S. Automated Diagnosis of Process Failure. J. Chin. Inst. Chem. Eng. 1991,22,195. Iri, M. K.; Aoki, K.; O'Shima, E.; Matsuyama, H. An Alogrithm for Diagnosis of System Failures in the Chemical Process. Comput. Chem. Eng. 1979,3,489. Kandel, A. Fuzzy Mathematical Techniques with Applications; Addison-Wesley: Reading, MA, 1986;Chapter 5,p 102. Kaufmann, A. Introduction to the Theory of Fuzzy Subsets; Academic Press: Orlando, FL, 1975;Vol. 1, Chapter 2,p 41. Kokawa, M.; Miyazaki, S.; Shingai, S. Fault location Using Diagraph and Inverse Direction Search with Application. Automatica 1983,19,729. Kosko, B. Neural Networks and Fuzzy Systems; Prentice Hall: Englewood Cliffs, NJ, 1992;Chapter 1,p 26. Kramer, M. A. Malfunction Diagnosis Using Quantitative Models with Non-Boolean Reasoning in Expert Systems. AIChE J. 1987,33,130. Kramer, M. A.; Palowitch, B. L., Jr. Rule-Based Approach to Fault Diagnosis using Signed Directed Graph. MChE J. 1987,33, 1067. Nisenfeld, A. E. Industrial Evaporators Principles of Operation and Control; The Instrument Society of America: Research Triangle Park, NC, 1985;Chapter 6,p 109. Oyeleye, 0.0.;Kramer, M. A. Qualitative Simulation of Chemical Process Systems: Steady-state Analysis. AIChE J. 1988,34, 1441. Patton, R.; Frank, P.; Clark, R. Fault Diagnosis in Dynamic Systems; Prentice Hall: Englewood Cliffs, NJ, 1989. Pau, L. F. Failure Diagnosis and Performance Monitoring; Dekker: New York, 1981. Pedrycz, W. Numerical and Applicational Aspects of Fuzzy Relational Equations. Fuzzy Sets Syst. 1983,11,1. Pedrycz, W. Fuzzy Control and Fuzzy Systems; John Wiley & Sons Inc.: New York, 1989;Chapter 3,p 81. Qian, D. Q. An Improved Method for Fault Location of Chemical Plants. Comput. Chem. Eng. 1990,14,41. Shih, R.-F.; Lee, L.-S. Use of Fuzzy Cause-Effect Digraph for Resolution Fault Diagnosis for Process Plants. 2. Diagnostic Algorithm and Applications. Ind. Eng. Chem. Res. 1995,34, 1703. Shiozaki, J.; Mafsuyama, H.; O'shima, E.; Iri, M. K. An Improved Algorithm for Diagnosis of System Failure in the Chemical Process. Comput. Chem. Eng. 1985a,9,285. Shiozaki, J.; Matsuyama, H.; O'shima, E. Fault Diagnosis of Chemical Processes by the Use of Signed Directed Graph. Extension to Five Range Patterns of Abnormality. Int. Chem. Eng. 1985b,25, 651.
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+
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* Abstract published in Advance ACS Abstracts, April 1, 1995.
