1 October 2005

Fuzzy logic clarifies operations

Statistical tool can give real-time predictive process fault analysis.

By Richard J. Fickelscherer, Douglas H. Lenz, and Daniel L. Chester

Operating processing plants is risky due to potentially abnormal situations caused by equipment or sensor failures or out of control process conditions.

When serious single or multiple simultaneous faults occur, this situation can directly lead to accidents, releases, and injuries. Process fault analyzers are computer programs that continuously monitor process sensor data to determine the current operating status of those sensors and the underlying process. Any faults or failures go to the process operators in real time in order to maximize the time available to effectively respond to those problems. If they find no problems, then users classify those sensor measurements as validated.

A fuzzy logic based real-time diagnostic method called the Method of Minimal Evidence (MOME) uses first principle or statistical models, correlations, and experiential heuristics to define relationships between particular measured sensor data and assumed unmeasured process variables that describe normal process operation.

It continuously evaluates those relationships with real-time data to determine which close and which do not. It also exhaustively combines those relationships together to form additional but novel relationships, which continuously undergo evaluation. The resulting patterns of these evaluations undergo interpretation with a general-purpose fuzzy logic diagnostic rule to determine the certainty associated with each of the potential fault hypotheses. This algorithm is general enough to diagnose single and multiple faults directly.

Below the surface

Process fault analyzers are computer programs that can monitor process operations in order to identify the underlying cause of operational problems. A method for creating process fault analyzers for processing plants has always been a topic of discussion. Just think of the potential for improving process plant operations in terms of safety and productivity. Automated process fault analysis should help process operators prevent catastrophic operating disasters such as explosions, fires, meltdowns, and toxic chemical releases; reduce downtime after emergency process shutdowns; eliminate unnecessary process shutdowns; and maintain better product quality control.

A variety of logically viable diagnostic strategies currently exist for automating process fault analysis. However, automated fault analysis is still not widely used within the processing industries. This is mainly due to prohibitively large development, verification, implementation, or maintenance costs of these programs; inability to operate a program based upon a given diagnostic strategy continuously online or in real time; and inability to model process behavior at the desired level of detail, thus leading to unreliable or highly ambiguous diagnoses. Subsequently, the industry continues its look for an effective method for producing automated process fault analyzers. There is now an algorithm and associated software that overcomes many of these shortcomings.

Evaluating models of normal process operation with current process data is the most promising means for directly identifying underlying process operating problems. Doing so generates a source to logically infer the current state of the modeled process. The model continuously evaluates and then automatically infers the underlying cause for any models that do not close. This fuzzy logic calculation automatically on-line enables these fault analyzers to perform "intelligent supervision" of the daily operations of their associated process systems.

Automating analyzers

MOME is a model-based diagnostic strategy for developing optimal automated process fault analyzers. The strategy came about after creating a real-time, online process fault analyzer for a commercial-scale Adipic Acid plant DuPont operates in Victoria, Tex. It provides a uniform framework for examining models of normal process operation and their associated assumptions. When evaluated with process data, each process model generates a residual that gives evidence of the actual process state. Significantly large residuals directly indicate the invalidity of one or more of those models' associated assumptions. It is possible to identify faults by comparing the patterns of these residuals with those expected to occur during a fault situation. Sensor data validation occurs when specific faults are not occurring, which is the flip side of the fault analysis. Fault analyzers based on MOME have highly structured knowledge bases that directly optimize their overall competence along with their diagnostic resolution and sensitivity. MOME can directly diagnose multiple fault situations to determine the optimal placement of process sensors to facilitate fault analysis and to determine the optimal division of a large process system for distributing fault analyzers.

The following is the basic logic behind MOME:

Once a user targets a process system for automated process sensor validation and fault analysis, derive as many linearly independent models describing the normal operating behavior of that process system as possible. The basis of these models should be the most fundamental understanding of normal process behavior available, limited only by the specific type and frequency of the process data. The set of models should constitute a highly accurate description of the target process system's normal operating behavior.

All unevaluated process models describing normal process operation are functions of the following quantities:

0 =ƒ(i) (model i modeling assumption variables, time)

Modeling assumption variables equal specific sensor measurements, standard or extreme values of specific parameters, or unmeasured variables.

Once the modeling assumption variables have actual process sensor data or standard or extreme values of specific parameters or unmeasured variables, a residual results, which is a function of process sensor noise and any currently occurring modeling assumption variable deviation:

r(i) = ƒ(i) (sensor noise, modeling assumption variable deviations)

If the residual of an evaluated model is significantly high or low (significantly higher or lower than zero), you can assume at least one or more of the possible modeling assumption variable deviations is occurring.

If the residual of an evaluated model is not significant, then either there are no modeling assumption variable deviations; one or more such deviations are occurring but at magnitudes or rates of change below the sensitivity of that model to discriminate such deviations; or two or more significant assumption variable deviations are interacting in an opposing fashion.

The MOME strategy for fault analysis compares patterns of residual behavior expected to occur during the various possible assumption variable deviations with the patterns of those residuals currently present in the process. It uses the minimum unique patterns required for correctly doing this analysis, allowing the user to identify the possible multiple assumption deviations. This is important because this methodology does not discern only process faults but all possible process operating events, fault and non-fault events alike. In other words, all assumptions necessary to derive a model are potential diagnoses whether they are about the absence of faults or some other non-fault event. The logic ventures a diagnosis only if it is highly certain of the underlying problem. This conservative behavior is advantageous because it should not confuse its users with incorrect diagnoses at times when the actual process operating state is in flux.

The patterns of expected residual behavior resulting from applying this method contain the minimum patterns required to diagnose each of the possible fault situations. This method directly maximizes the sensitivity of the fault analyzer for these various faults, maximizes the resolution (discrimination between various possible faults and non-fault events) of that analysis, and optimizes its overall competence when confronted with multiple assumption deviations. The strategy can also directly determine the optimal placement of process sensors for performing fault analysis and the optimal division of large process systems for distributing process fault analyzers.

Residual effect

Because the sensors that measure process variables may not be 100% accurate or provide exact readings; because the process models may not be a perfect representation of the relationship between the process variables; and because random perturbations may occur, one assumption is the residuals are not always zero. The mathematical model of sensor validation and predictive fault analysis (SV&PFA) used in the program requires all residuals be zero, on average, when a monitored process is normal. Therefore, a calculation made from historical plant data of the average value of each residual and that average value subtracts from the corresponding residual process model.

In practice, then, each function ƒ, will behave like a statistical random variable having a mean value β and a standard deviation σ. The mean value β = β 0ρ and the standard deviation σ = σ 0ρ, where β0 and σ0 are constants and ρ is either 1 or a process variable that is the definitive measure of the production level of the monitored process. Usually, β is just a constant value, but sometimes it is the product of a constant times a process variable whose value determines the level of production at which the process is operating.

You can replace the generic residual process model with a primary residual process model defined as: r =ƒ(x₁, x₂, ..., x_n) – β, which has a mean value of zero and a standard deviation of σ. The equation defining r is a primary residual process model. The program examines the values of such (adjusted) residual process models and, among other things, infers from the pattern of deviations from zero which sensors are faulty or which other parts of the process may be faulty.

In more generic terms, if a plant engineer provides the formula ƒ(...) as the formula for a residual process model under ideal conditions, and the formula mean for the average of ƒ(...) over time based on historical plant data, and the formula sigma for the standard deviation of ƒ(...) over time, the program generates the primary residual process model: r = ƒ(...) – mean, which has the property that the average of r should be zero.

Primary residual process models are different from certain linearly dependent residual process models automatically generated. Such additional models are secondary residual process models.

The MOME algorithm computes certainty factors to identify faults and/or validate underlying assumptions. When this program monitors a process, it reads real-time sensor data, computes the associated primary and/or secondary residual values and their standard deviations, and then calculates three certainty factors for each residual value, as needed.

For purposes of calculating certainty factors for faults, <v, d> signifies a fault—that is, v is a process variable and d is a direction, either high or low. To compute the certainty factor that a fault is present, examine the certainty factors for the primary and/or secondary residuals to find evidence for the fault. If r is a residual, r provides evidence for fault <v, d> when it has deviated from zero in a direction that is consistent with variable v deviating in the direction d. For example, if ∂r/∂v is greater than zero, then v and r should go high (or low) at the same time. If, however, ∂r/∂v is less than zero, then v and r can deviate in opposite directions. The certainty factor for r in the appropriate direction is then the strength to which r can provide evidence for the fault. One strong piece of evidence for the fault is enough to strongly conclude the fault is present, unless there is also strong evidence that it is not present.

The evidence for fault <v, d> is this set of certainty factors for all relevant residuals:

evidence-for-fault(<v, d>) = {cf(r, sign(∂r/∂v)d) | (∂r/∂v) ≠ 0 (12) and r is a primary residual}

The strength of the evidence for the fault is the maximum of the values in this set.

Similarly, if a residual deviates in the opposite direction from what it expects when the fault is present, that deviation is evidence against the fault being present. The evidence against fault is this set of certainty factors for all relevant residuals:

evidence-against-fault(<v, d>)={cf(r, -sign(∂r/∂v)d | (∂r/∂v)≠0}

Certainty factors for primary and secondary residuals may be in this set.

Measuring strength

The strength of the evidence against the fault is the maximum of the values in this set. If that value subtracts from one, you can then determine the strength to which this evidence is consistent with the fault being present.

An additional consideration is significant in evaluating a certainty factor for a fault. Some residuals are not functions of v and so should not deviate from zero when the fault <v, d> is present. The secondary residual process model formed by eliminating v from two primary residual process models is such a residual. It is relevant to evaluating the presence of the fault, so this secondary residual should have a high certainty factor of being satisfactory when the fault involves v. Also, if two primary residual process models combine to generate a secondary residual process model by eliminating some variable other than v, and one of these primary residuals is a function of v but the other is not, it is expected the primary residual that is not a function of v is satisfactory. This primary residual is relevant to the fault as well.

Some primary residual process models may not be functions of v and are not combined with any models that are. These are not relevant to the fault <v, d>. Another fault can be present and cause them to deviate from zero, but this will not affect the assessment for fault <v, d>. This allows a diagnosis of the presence of several single faults that happen not to interact with each other. In addition, r may be a function of v, but at the moment, (∂r/∂v) = 0. The neutral-evidence for fault <v, d> is this set of certainty factors for all relevant residuals: neutral-evidence(<v, d>) = {cƒ(r, sat) | r is relevant as neutral-evidence for v}. The strength of this evidence is the minimum of the set because if any one of the residuals that should be satisfactory is in fact high or low that weakens the evidence for the fault <v, d>.

The certainty factors in these three sets, evidence-for-fault, neutral-evidence, and evidence-against-fault, are fuzzy logic values and combine using a common interpretation of fuzzy "AND" as the minimum function, fuzzy "OR" as the maximum function, and fuzzy "NOT" as the complement function (1 minus the value of its argument). For finite sets, the quantifier "SOME" is just the "OR" of the values in the set, so it is equivalent to taking the maximum of the set. Similarly, for finite sets, the quantifier "ALL" is just the "AND" of all the values in the set, so it is equivalent to taking the minimum of the set.

Regarding the display of a fault <v, d>: If v is a measured variable, the sensor value for that variable substitutes for the variable in computing all the primary and secondary residual values. If d = high, a conclusion then is the sensor reading is higher than the true value for that process variable. If d = low, a conclusion is the sensor reading is lower than the true value for that process variable. In either case, a conclusion is the sensor is at fault. If cƒ() is about zero for both cases, d = high and d = low, then the sensor reading has been validated.

Messages apply

In the case of an unmeasured variable v, such as a leak, a high certainty factor for <v, low> means the assumed value of v, which can be viewed as the reading from a virtual sensor, is low compared to the actual value. In order to display a conclusion about the actual value of the unmeasured variable, the program displays a message that v is high in this case. Similarly, if the certainty factor for <v, high> is high, it displays a message about v being low. If neither of these cases apply, a conclusion is the real value of v is about equal to its assumed value.

You can generalize the fuzzy logic rule to sets of faults by redefining what counts as evidence for the set, evidence against the set, and neutral evidence. An inference may be drawn that a set of faults is present when no subset of them are present. In particular, this means there must be at least one residual value for each fault deviating in the direction that the fault can cause. This leads to the following general fuzzy rule of this methodology:

Let fault-set = {<v1, d1>,...,<v_n, d_n>}
Then cf(fault-set) = SOME(evidence-for-fault(<v, d>)) AND
SOME(evidence-for-fault(<v_n, d_n>)) AND
ALL(neutral-evidence(fault-set) )AND
NOT(SOME(evidence-against(fault-set)))

These models come from a fundamental understanding of normal operating behavior of the given process system. They thus generate an unimpeachable source of information for logically inferring conclusions about the process being modeled. Automatically performing this inference after each update of process sensor data allows such programs to continuously perform "intelligent supervision" of the daily operations of their associated process systems.

The rule engine eliminates the need to explicitly derive patterns of model satisfactions and violations expected to occur when the associated modeling assumption variable deviation occurs: All that is required is to derive all possible primary models and determine and classify all the various modeling assumption variables associated with those models.

Behind the byline

Richard J. Fickelscherer and Douglas H. Lenz are with Falconeer Technologies in Williamsville, N.Y., and Daniel L. Chester is with the Department of Computer & Information Sciences at the University of Delaware in Newark, Del.