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### Saint-Petersburg State Technological Institute

FAULT DIAGNOSIS IN CHEMICAL PROCESSES AND EQUIPMENT WITH FEEDBACKS

L.A.Rusinov

N.V.Vorobjev

V.V.Kurkina

I.V.Rudakova

FAULT DIAGNOSIS IN OBJECTS WITH FEEDBACKS

The subject-matter of my report concerns the application of chemometrics approach to solving industrial problems relevant in particular to developing diagnostic systems for fault diagnosing in difficult cases.

Why is the diagnostics necessary?

The majority of chemical technological processes refer to the class of potentially dangerous (PDTP) or hazardous.

As a rule PDTP are characterized by:

- high level of uncertainty,
- large uncontrollable disturbances,
- essential internal nonlinearity,
- bad observability.
- lack of mathematical descriptions (often).

Why is the diagnostics necessary?

- The operation of protection systems obligatory in the PDTP is usually accompanied
- by emergency dumping of a reactionary mass,
- irreversible repressing of a reaction and
- other operations resulting in essential losses.
- The early diagnostics can define the faults at their incipient stages and thus allows undertaking necessary acts to avoid the protection systems actuating.

What is necessary for diagnostics?

Continious monitoring and fault diagnostics are carried out on the basis of a diagnostic models (DM) connecting faults in the process under control (abnormal situations) with their observable symptoms.

For this reason, mathematical descriptions of PDP cannot often be used as DM because they are usually valid only in PDP working zones and unsuitable for abnormal situations.

Structural

Fault

Trees

Functional

Statistical

Classifiers

PCA/PLS

Classification of diagnostic models

Diagnostic Models

Process

History Data Models

Quantitative

Models

Qualitative

Models

Causal

Models

Abstraction

Hierarchy

EKF

Qualitative

Quantitative

Observers

Parity

Space

Fuzzy

expert systems

Expert

Systems

Neural

Networks

Statistical

THE MASKING EFFECT FROM THE FEEDBACK

Slide 7,8

The sensor signal

Control action

The object of diagnostics

PROCESS

100%

Recycle

or Regulator

0%

Regulator has exhausted its resource

Regulator has exhausted its resource

Emergency situation

ABNORMAL SITUATION

Fault development

Normal regime

THE WAY OF SOLVING THE PROBLEM

For solving the problem, the diagnostic model (DM) should be built for the process section or equipment in the closed loop formed by feedback (further - object).

Deviations from this model can be used in detecting the fact that an abnormal situation has arisen.

However, it is impossible to identify a cause of the fault while using deviations from the model.

For this purpose, the models describing all possible abnormal situations are required.

As a result, it is necessary to have a bank of models

OF THE OBJECT DIAGNOSTIC SYSTEM

THE FLOW-CHART OF THE DIAGNOSTICS’ PROCEDURE

1. THE OBJECT MONITORING FAULT DETECTION

THE FAULT DETECTION CRITERION: i>; - threshold

2. INITIATION OF THE MODELS OFTHE BANK DESCRIBING OBJECT FAULTS

3. THE FAULT REASON (FR) IDENTIFICATION:

- Restrictions of the method:
- The necessity of the presence of diagnostic models, easy-in-use in real time.
- The necessity to obtain the knowledge of all possible faults of the object under control.

Each fault Fiis described by fuzzy rule Riof Takagi-Sugeno type:

Ri: IF x1 == Ai1 AND... ANDxn= = Ain,

THEN yi = ai1x1 +... + ainxn+ bi,

whereRi – i-thrule of fuzzy model, i[1,k];k – the number of rules;

X = {xkj} - a matrix of input variables samples [Nxn]; k [1, N]; N – number of samples; j[1,n]; n - number of inputs;

Аi= {Aij} - the fuzzy terms-sets that enter into the conditional part of each i-th rule;

yi - an output of ith fuzzy rule;

ai= [ai1, …, ai n] and bi - parameters of fuzzy model:

Ti=[aTi, bi].

FLOW-CHART OF COMPUTING THE RESULT OF FUZZY MODEL

The result of fuzzy model computing is determined by combination of contributions of all rules in a common inference:

where βi - the degree of activation of i-th rule that is determined by max-product composition:

,

[0,1] - the contribution of each term-set Aij to the conditional part of the rule Ri.

FLOW-CHART OF COMPUTING THE RESULT OF FUZZY MODEL

Coefficients of a right part of rules are determined by solution of the system equations by means of weighed МLS:

y=Xch θ ; Xch=[X,1]

The solution is:

- the diagonal matrix with the degrees of activation β on its diagonal

THE DETERMINE OF MEMBERSHIP FUNCTIONS.

By means of fuzzy clustering of the data object array, for example, by the algorithm of Gustafson-Kessel, the number of clusters сand the matrix of their fuzzy separation Uare defined.

Membership functions of fuzzy sets in the conditional part of rules are extracted from the matrix U which

(g, s)-th member mgs[0,1] characterizes the value of membership of an input-output combination in s-th column in the cluster g.

THE DETERMINE OF MEMBERSHIP FUNCTIONS.

To obtain one-dimensional fuzzy set Ggj, the multidimensional fuzzy sets, defined pointwise in g-th row of the separation matrix U, are projected into input variables space Xj.

Resulting fuzzy sets Ggj are usually nonconvex. To obtain the convex (unimodal) fuzzy sets, approximating by appropriate forms of membership functions (for example, Gaussian) is needed.

3

5

1

2

6

MEMBERSHIP FUNCTIONS OBTAINED BY CLUSTERISATION

6 clusters – by threes clusters for a forward and reverse valve strokes

MEMBERSHIP

FUNCTIONS

THE NORMALIZED VALUES OF INPUT VARIABLES

THE DIAGNOSTIC MODELBASED ON THE KALMAN FILTER

In this case DM is developed in the space of object states.

The Kalman filter is actually searching for an optimum estimate with the least-squares method.

The linearobject model for Kalman filter is of the form:

where x(k)is the state vector,

y(k)- the vector of filter output variables at the kthstep,

x(k-+1) - the predicted (extrapolated) state vector value at the (k+1)th step;

A, B, C- are known prediction, control and observation matrices;

n,w– noises.

FLOW-CHART OF COMPUTING THE RESULT OF KALMAN MODEL

The matrix of filter gain factors is given as:

where: , S(k) – correlation matrixes

The specified estimation for the system state vector is:

And finally, the specified covariance matrix of estimation of the system state vector is given in the form:

THE DIAGNOSTIC MODELBASED ON THE EXTENDED KALMAN FILTER

For nonlinear objects the model is of the form:

In this case, the filter does not use fixed matricesA(x)and C(x), but linearizes them recursively based on the previous state estimate with the use of matrices of first partial derivatives of the state equations:

These matrices are calculated at every step and then inserted into the standard Kalman filter formulas.

Case study was carried out on two types of objects:

1. The object in control loop - electropneumatic valve with the positioner

2. The object in recycle circuit - Tennessee Eastman process

OF THE ELECTROPNEUMATIC VALVE WITH THE POSITIONER

OF THE TENNESSEE EASTMAN PROCESS

*) Applicability of the method to processes with recycles is presented in the Vorobiev’s poster presentation.

THE ELECTROPNEUMATIC VALVE WITH THE POSITIONER

The POSITIONER has the mathematical model developed in the European Interuniversity Project DAMADICS. 19 various positioner faults have been considered by the model.

But it does not fulfill to the first restriction for diagnostic models of objects of the class under study: it is difficult and not easy-to-use in real time.

So, models on the basis of fuzzy logic and Kalman filtering are developed and the DAMADICS model was used for their training.

variables

Output

variable

MODELED POSITIONER FAULTS

F2

ABRUPT

FAULT

«FLUID BOILING UP IN THE VALVE CAVITY AT THE EXTREME FLOW RATE»

F1

INCIPIENT

FAULT

«SEDIMENTATION»

CV1(k), CV1(k-1),CV1(k-2) – Control signal values entered from controller at kth, (k-1)th and (k-2)th steps

ZT – position of the valve plunger at the kth step

FT – The flow rate through the valve

OUTPUT RESIDUALS OF FUZZY MODELS(INCIPIENT FAULTF1)

DETECTION

RESIDUALS,%

(MODEL OF

NORMAL REGIME)

time

RESIDUALS,%

(MODEL OF

FAULT F1

time

RESIDUALS,%

(MODEL OF

FAULT F2

IDENTIFICATION

time

OUTPUT RESIDUALS OF FUZZY MODELS(ABRUPT FAULTF2)

DETECTION

Forward plunger stroke

RESIDUALS,%

(MODEL OF

NORMAL REGIME)

time

Reverse plunger stroke

RESIDUALS,%

(MODEL OF

FAULT F1

time

IDENTIFICATION

RESIDUALS,%

(MODEL OF

FAULT F2

time

OUTPUT RESIDUALS OF MODELSBASED ON THE KALMAN FILTERS(INCIPIENT FAULTF1)

DETECTION

RESIDUALS,%

(MODEL OF

NORMAL REGIME)

time

RESIDUALS,%

(MODEL OF

FAULT F1

time

IDENTIFICATION

RESIDUALS,%

(MODEL OF

FAULT F2

time

OUTPUT RESIDUALS OF MODELSBASED ON THE KALMAN FILTERS(ABRUPT FAULTF2)

DETECTION

RESIDUALS,%

(MODEL OF

NORMAL REGIME)

time

RESIDUALS,%

(MODEL OF

FAULT F1

time

IDENTIFICATION

RESIDUALS,%

(MODEL OF

FAULT F2

time

Сhemometrics methods are very effective for execution the monitoring and diagnostics of technological processes in chemical and related industries, even in difficult cases at diagnostics of the objects in circuits with feedbacks because of feedback masking effects

Statistical methods allow constructing diagnostic models on the base of the history process data, not demanding the knowledge of process chemism and the presence of its mathematical descriptions.

For diagnosing such faults, it is suggested to use the bank of diagnostic models describing normal operation of the objects under control and their operation when faults are available.

Applicability of the method is illustrated by the example of system development with two types of diagnostic models: the model with fuzzy rules of Takagi-Sugeno type and on the basis of extended Kalman filters. Both models have demonstrated approximately equal results when diagnosing both incipient and abrupt faults.

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