63

CHAPTER 4

PROPOSED FAULT DIAGNOSIS AND IDENTIFICATION

AND FAULT TOLERANT CONTROL SCHEMES

4.1 OVERVIEW

As mentioned in Literature, modern-day chemical plants involve a complex

arrangement of processing units, connected in series and/or in parallel and highly

integrated with respect to material and energy flows through recycle streams, and to

provide information flow through tightly interacting control approaches.

Increasingly faced with the requirements of safety, reliability, and profitability,

chemical plant operation is relying extensively on highly automated process control

systems. Automation, however, tends to also increase vulnerability of the plant to

faults (for example, defects/malfunctions in process equipment, sensors and

actuators, failures in the controllers or in the control loops), potentially causing a

host of economic, environmental, and safety problems that can seriously degrade

the operating efficiency of the plant if not addressed within a time appropriate to the

context of the process dynamics. These considerations provide a strong motivation

for the development of methods and strategies for the design of suitable fault

tolerant control structures that ensure an efficient and timely response to enhance

fault recovery, to prevent faults from propagating or developing into total failures,

and to reduce the risk of safety hazards. Given the geographically-distributed,

interconnected nature of the plant units and the large number of distributed sensors

and actuators typically involved, the success of a fault-tolerant control method

requires efficient fault detection, control designs that account for the complex non-

64

linear dynamics and constraints, and a high-level supervisor that coordinates the

overall plant response to achieve fault-tolerant control.

In process control, given the complex dynamics of chemical processes (for

example, nonlinearities, uncertainties and constraints), the success of any fault-

tolerant control method [74] requires an integrated approach that brings together

several essential elements, including: (1) the design of advanced feedback control

algorithms that handle complex dynamics effectively, (2) the quick detection of

faults, and (3) the design of supervisory switching schemes that orchestrate the

transition from the failed control configuration to available well-functioning fall-

back configurations to ensure fault tolerance. The occurrence of faults in chemical

processes and subsequent switching to fallback control configurations naturally

leads to the superposition of discrete events on the underlying continuous process

dynamics, thereby making hybrid system framework, a natural setting for the

analysis and design of fault-tolerant control structures. Proper coordination of

switching between multiple (or redundant) actuator/sensor configurations provides a

means for fault-tolerant control.

In summary, a close examination of the existing Literature indicates lack of

general and practical methods for the design of integrated fault-detection and fault-

tolerant control structures for chemical plants, accounting explicitly for

actuator/controller failures, process nonlinearities, and input constraints. Motivated

by these considerations, in this research work, the problem of implementing

combined fault diagnosis and identification as well as fault tolerant control on

systems modeled in Petri net environment is proposed. With considerations such as

(i) observability and (ii) unobservability of place markings in the models developed

under the assumption that initial marking and sequence of events is known, FDI

schemes are applied for detection of actuator, sensor, and pump faults. With input

65

constraints subject to actuator, sensor, and pump failures, a method is presented to

demonstrate an approach of integrating fault detection, feedback, and supervisory

control.

To illustrate the main idea behind the proposed approach, the systems are modeled

initially in Petri net environment using discrete, continuous, and hybrid Petri nets.

For the system models under consideration, algorithms are devised to separately

detect the faults under observable conditions. Next, for the unobservable nature of

the system data available, estimation based techniques are proposed and applied to

find the place and transition markings in the models, thereby detecting the faults.

Next, depending on the nature of the faults detected i.e., whether a place fault or

transition fault or a combination of both, a family of candidate control

configurations, characterized by constraint information, is identified. For each

control configuration, an estimation based fault tolerant controller that enforces

asymptotic closed-loop stability in the presence of constraints is determined.

Finally, simulation studies are presented to demonstrate the implementation and to

evaluate the effectiveness of the proposed combined fault diagnosis and fault-

tolerant control schemes.

In this chapter, the details of the proposed FDI using estimation and observer

methods are discussed in the first and second subsections, i.e., in sections 4.2.1 and

4.2.2 of section 4.2 initially. This is followed by the proposed method adopted to

achieve FTC using controllability concepts in section 4.2.3. The proposed

algorithms to achieve combined FDI and FTC in discrete, continuous and hybrid

event systems modeled using discrete, continuous and hybrid Petri nets as discussed

in Chapter 3 are given in section 4.2.4.

66

Next, based on the details discussed to achieve combined FDI and FTC and

algorithms proposed, the applications of the methods to achieve the same on a

typical benchmark system, i.e., three tank system as discussed in Chapter 3 are

given in section 4.3. In this section, a detailed analysis to achieve FDI based on

estimation methods is given initially. Numerical analysis and results obtained to

detect faults using observer methods are given in the next subsection. Next, the

method adopted to achieve FDI using estimated evolution graph for system model

as shown in Figure 3.20 in section 3.4.2 in Chapter 3 is given. Finally, based on the

faults detected, the conditions obtained to achieve FTC along with numerical results

are presented in the final subsection.

4.2 PROPOSED METHODOLOGY

The proposed concept of achieving FDI is based on utilizing the marking estimate

from the observer to develop the observability graph [75]. From this graph, the

observability error, which is the difference between the original markings with that

of the estimated marking, is calculated. The details of the occurrence of a fault i.e.

place fault or transition fault is then eventually identified. The concept of achieving

FDI is shown in Figure 4.1.

Figure 4.1 Block diagram of the system for FDI

Status

Residue

PLC unit

INDUSTRIAL

PROCESSES

DISCRETE/CONTINUOUS

EVOLUTION

MODEL

FDI

ALGORITHM

ESTIMATOR

67

As observed in Figure 4.1, the Programmable Logic Control (PLC) unit and

industrial process blocks are shown by dotted lines, which constitute the real-time

system components. The other blocks shown by thick lines are blocks proposed to

model the system, and thereby achieve FDI. The discrete/continuous evolution and

estimator blocks are to obtain the output values and estimated output values with

the given initial status inputs (the status of each device is considered as the marking

of each place) by using the mathematical formula of marking as discussed in section

3.1.

From the model block, the initial condition of the process is known. The output of

the discrete/continuous evolution block is compared with that from the estimator

block and depending on the details obtained from the model block, the structures of

observer reachability and evolution graphs are developed. Moreover, from the

hybrid model as shown in Figure 3.20 in Chapter 3, it is seen that the firing of

continuous transitions, T1, T2, T7, T8 and T13 depends upon the marking of discrete

places, P4, P5, P6, P7, P10, P11, P12, P13, P14 and P15. Thus, any fault occurring due to

improper firing of transitions can be evaluated based on marking evolution of the

discrete places or continuous places. The methodology adopted to achieve FDI in

order to identify transition faults is based on the estimation of continuous place

markings using estimation techniques. These details are discussed in detail in

section 4.2.1, whereas FDI algorithm for identifying discrete place faults developed

from the observer coverability graph is discussed in section 4.2.2.

4.2.1 Estimation based FDI

Estimation based Fault diagnosis algorithm described in this section uses the status

signal of the devices from the PLC controller at start up to estimate the output and

predict the fault. This is otherwise called as an observer technique. The observer

does the estimation and prediction on how the output might turn out in case of an

68

error. For understanding the proposed estimation technique, an example is

considered and explained as follows:

As shown in Figure 4.2, let the initial marking and the initial observed sequence

for the Petri net be

w0 = λ, where λ is word of events,

and M0

w = [ 111 ]T.

Hence, the initial estimate is given by

0

wμ = [ 000 ]T.

Original Estimated

Sequence Sequence

Figure 4.2 Original sequence and estimated sequence of example net

Next, based on the word of events considered, i.e., λ = t1 t2 t3, first t1 is fired.

Now, based on this condition, the new marking and estimated marking are given

by

w1 = t1,

M1

w = [ 012 ]T,

and 1

wμ = [ 001 ]T.

t 2

p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3

69

The pictorial representation of the same is shown in Figure 4.3.

Original Estimated

Sequence Sequence

Figure 4.3 Original sequence and estimated sequence of example net when t1 fires

Similarly, the sequences obtained when t2 fires are obtained and shown in

Figure 4.4.

w2 = t1t2,

M2

w = [ 012 ]T,

and 2

wμ = [ 010 ]T.

Original Estimated

Sequence Sequence

Figure 4.4 Original sequence and estimated sequence of example net when t2 fires

Based on the same lines, the concept of FDI is achieved when the markings are

unknown.

4.2.2 Observer based FDI

Basile, et. al, [76] proposed a method to estimate the marking of a place/transition

(P/T) net based on the observation of transition firings, and presented a set of

t 2

p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3

t 2

p 1 p 2

p 3

t 1 t 3

t 2 p 1 p 2

p 3

t 1 t 3

70

analytical tools to determine several observability properties. The method proposed

here is similar to that proposed earlier, but in this research work, importance is

given to devise and improve the method to estimate the markings of the continuous

part of the net structure which will be highly useful when analysing a hybrid event

system modeled as hybrid Petri nets.

In the proposed method, based on the initial marking and estimated markings as

discussed in the previous section, an observer coverability graph is constructed. The

observer coverability graph is highly useful in estimating the faults occurring in the

system considered. An observer coverability graph as defined in [76], is a labelled

directed graph for a Petri net structure, 0MN, , given by G= (V, E) with transition

function given by δ :V x EV, where V is the set of all distinct labelled nodes in

the observer coverability graph, and each arc in E is labelled with a transition to

represent a firing such that δ (M/u),t)=(M'/u'), where (M/u) and (M'/u') are the

corresponding initial and new markings and the estimation errors. To understand the

concept of observer coverability graph, the example as shown in Figure 4.5 is

considered.

Figure 4.5 Example for observer coverability graph

71

As seen in Figure 4.5, the initial net structure is given by M = [ 002 ], and the

estimation bound is given by u = [ 002 ]. For the transition firing sequence, i.e.,

t1 t2 t3, the set of marking and the bound are shown in light rectangular boxes.

Likewise, the corresponding marking and its error are shown accordingly for every

transition firing within thick rectangular boxes. It can be seen that the error marking

for every instant is given by u = [ 000 ], which shows that the estimated

markings found out at every instant are ideal markings. If the same approach is

applied for faulty conditions, then depending on the estimation error value, the

corresponding fault can be easily detected. Thus, the observer coverability graph

approach is very useful in achieving FDI.

4.2.3 Controllability based FTC

In this research work, a redundant controller is identified based on the system

model and faults are identified that allow an external checker to detect and identify

conditions for stability during faulty conditions that take place in the controller.

More specifically, the approach is subjected to controller faults that lead to an

incorrect token-load of a place (place fault) or cause the token-load of either the

input or output place-set of a transition not to be properly updated following the

firing of a transition (transition fault). The methodology is based on embedding the

original controller into a separate redundant controller in a way that preserves the

state and properties of the original Petri net controller, while enabling the

development of systematic ways to achieve FTC in the redundant Petri net

controller. As a result, by performing linear parity checks on the combined

marking of the original controller places and the additional (redundant) places, the

proposed methodology is able to achieve FTC in the redundant Petri net controller

in a systematic manner.

72

An outline of how the separate fault-tolerant Petri net controller (also referred to as

the separate redundant Petri net controller) [77] achieves FTC is shown in

Figure 4.6. Given a Petri net plant, the original Petri net controller can be obtained

based on any method (e.g., a place invariant enforcing controller as in [78]). In

order to protect the controller against faults, redundant places are added in a way

that does not inhibit any transitions that would otherwise be enabled to fire by the

original controller (i.e., the redundant places retain the maximal permissiveness of

the original controller). It should be noted that the overall fault-tolerant controller

operates concurrently with the plant, and takes actions based on the activity in the

plant and the possible faults in the controller. Information about transitions in the

plant is updated by the checker, which is in charge of verifying that the internal

state of the redundant controller is consistent.

Figure 4.6 Structure of the proposed Fault tolerant controller [77]

It can also be seen that the conceptual design of the separate redundant controller

as shown in Figure 4.6 can be modified, which allows the checker to provide the

enable/disable signal for transitions. In this case, the connections from places to

transitions are only used to update the number of tokens in the controller places as

shown in Figure 4.7.

73

Figure 4.7 Modified fault tolerant controller scheme [77]

The method described in this thesis requires that the process/plant to be diagnosed

for different faults. For this purpose, the plant is modeled by a Petri net and a

corresponding Petri net controller is attached to the process net. The constraints

which must be satisfied by the process can be written as logic expressions,

inequalities or equalities.

To understand the proposed method example shown in Figure 4.8 is considered.

With the assumption that the system under analysis is modeled by a Petri net with n

places and m transitions, the structure must satisfy the following constraint:

1μμ ji , (4.1)

where iμ and jμ are the markings of places, pi and pj, respectively of the process

net. Equation (4.1) simply means that at most one of the two places pi and pj can be

marked, or, in other words, both places cannot be marked at the same time. This

inequality constraint can be transformed into equality by introducing a slack

variable, sμ , into it. The constraint then becomes

74

1μμμ sji . (4.2)

The slack variable in this case presents a new place, ps, which receives excess

tokens, thus ensuring that the sum of tokens in the set of places, iμ , and jμ , is

always less than or equal to 1. This place belongs to the controller net. The structure

of this net will be computed by noticing that the introduction of the slack variable

introduces a place invariant for the overall system defined by Equation (4.2). It is

obvious that there will be as many controller places as there are constraints of type

Equation (4.1). So, the size of the controller is proportional to the number of

constraints of type Equation (4.1). Since a new place has been added to the net, the

composite change matrix, D, of the overall controlled system is the original n x m

matrix, Dp, of the system increased by a row corresponding to the place introduced

by the slack variable. This new row belongs to the composite change matrix of the

controller, called Dc. The arcs connecting the controller place to the original Petri

net of the system will be computed by the place invariant Equation (4.3), where the

unknowns are the elements of the new row of matrix D, while the vector Xi is the

place invariant defined by Equation (4.2). These computations are described below:

X T . D=0. (4.3)

First it is to be noted that the problem can be stated in general as follows. All

constraints of Equation (4.1) can be grouped and written in matrix form

bL.μp , (4.4)

where pμ is the marking vector of the Petri net modeling the process, L is an nc x n

matrix, b is an nc x 1 vector and nc is the number of constraints of type Equation

(4.1).

75

Similarly all place invariant equations of type Equation (4.2), generated after the

introduction of the slack variables, can be grouped in a matrix form as follows:

bμL.μ cp , (4.5)

where cμ is an nc x 1 vector which represents the marking of controller places.

The place invariant defined by Equation (4.2) must satisfy the place invariant

Equation (4.3). The following matrix equation is the place invariant equation for all

invariants defined by Equation (4.5),

.DXT = IL . c

p

D

D = 0

L. Dp+Dc=0

Dc=-L. Dp, (4.6)

where I is an nc x nc identity matrix, since the coefficients of the slack variables in

the constraints are all equal to 1. The matrix, Dc, contains the arcs that connect the

controller places to transitions of the process net. So, given the Petri net model of

the process (Dp), and the constraints that the process must satisfy (nc, L and b), the

Petri net controller (Dc) is defined by Equation (4.6).

The initial marking of the controller Petri net should also be calculated. The initial

marking of the controller places, 0

cμ , must be such that the place invariant Equation

(4.5) is satisfied, and depends on the initial marking of the places of the process

Petri net which participate in the place invariants. Now here, given Equation (4.7),

.Xμ.XμT

0T , (4.7)

Equation (4.8) can be written for the initial marking vector as where 0μ is the net’s

initial marking, and μ is the net’s subsequent marking. Hence,

bμL.μ00

cp

00

pc L.μbμ . (4.8)

76

As observed in Figure 4.8, the Petri net structure has three places and four

transitions.

Figure 4.8 Example of a simple Petri net

The composite change matrix of this Petri net structure is given by

Dp =

1110

0111

1001

, (4.9)

while the initial marking is

3

2

1

p

μ

μ

μ

μ0

=

0

0

3

. (4.10)

Dp is of rank 2, thus it has one place invariant which includes the entire net, i.e.,

TpD .X = 0 where X = 111 T . The objective is to control the net so that places,

p2 and p3, never contain more than one token, i.e., one wishes to enforce the

constraint

1μμ 32 . (4.11)

77

Using the matrix notation of Equation (4.4), it can be found that

L = [ 110 ], (4.12)

and b = 1. (4.13)

The uncontrolled net does not satisfy the desired constraints, since [ 110 ] T is

not a place invariant of the net. A slack variable, sμ , is introduced and the

inequality Equation (4.11) becomes equality as given by

1μμμ s32 . (4.14)

The slack variable, sμ , denotes the marking of the place, ps, which belongs to the

controller. Equation (4.14) represents the desired invariant, X = [ 1110 ] T ,

which will be forced on the controlled Petri net. The composite change matrix of the

controller net is computed using Equation (4.6):

Dc = -L. Dp = [ 1001 ]. (4.15)

The initial marking of the controller place is computed using Equation (4.8):

00

ps L.μ1μ =1. (4.16)

The structure of the controlled Petri net is then described by the composite change

matrix,

D= c

p

D

D =

1001

1110

0111

1001

, (4.17)

while its initial marking is

78

1

0

0

3

μ

μμ

0

0

s

p0 . (4.18)

The Petri net structure of the controlled system is shown in Figure 4.9.

Figure 4.9 Controlled Petri net structure for system shown in Figure 4.8

4.2.4 Proposed algorithms to achieve FDI and FTC

In this section, the proposed algorithms to achieve FDI in discrete, continuous

and hybrid event systems, modeled using discrete, continuous and hybrid Petri nets

are presented.

79

Algorithm for FDI in discrete event systems

Step 1. With respect to the modeled control structure, the initial token content

per place is found out by determining the initial marking vector

denoted by Mi(0).

Step 2. Following this, the actual number of tokens which arise in the running

of the process is found out, and is denoted by Mi(k).

Step 3. The difference between the marking vectors are calculated, i.e.,

Mi(k) - Mi(0).

Step 4. If the difference is zero, then the system is considered to be fault free.

Step 5. If not, the corresponding place fault has occurred, and the algorithm is

developed to identify the faulty place in the structure modeled earlier.

Algorithm for FDI in continuous event systems

Step 1. Based on the Petri net model developed, the presence of observable

events is found from the reachability and coverability analysis.

Step 2. Based on the above, the initial marking vector, m0, and the value of

transition firings vector, v(t), is found out.

Step 3. The new marking me= W. v(t) is calculated.

Step 4. The error vector is calculated.

Step 5. Once the occurrence of error has been detected, the estimated error

vector is calculated, and the corresponding faulty transition, tf, or

faulty place, pf, is diagnosed.

Step 6. At every specified time instant, the estimated marking is updated by

making the current marking as estimated marking and repeating steps

1-4.

80

Algorithm for FDI and FTC in hybrid event systems

Step 1. Initialize when the initial marking is m0 and the word w are known, or

set j = 0 and goto step 8.

Step 2. For j = 1, the column of incidence matrix, C, corresponding to a

particular transition fired is added to the initial marking

mi = m0 + C(.,T).

Step 3. Select the initial estimate and initialize i=1.

Step 4. Wait until transition Ti fires and update the estimate 'im using

Q=min {mi , Pre(.,Ti)}.

Step 5. The new estimate ''im is then obtained using ''

im = Q+ C(.,Ti)} and let

i = i+1.

Step 6. The observability graph is developed by generating the estimation

error, and the fault is narrowed down by generating the residue of the

estimate error.

Step 7. Based on the type of faults, i.e., place faults, transition faults, or

combination of both, a suitable fault tolerance control law is applied to

ensure that the system is bounded (stable), goto step 10.

Step 8. For j=0, the initial estimate is found along with a bound and steps 4 to

7 are repeated.

Step 9. Initialize count and repeat steps 1 to 9 to compute the word sequence.

Step 10. Output the faulty place/transition from the estimates found.

4.3 APPLICATION OF PROPOSED METHOD IN A THREE TANK

BENCHMARK SYSTEM

In this section as discussed previously, the analysis and results obtained for

achieving FDI and FTC using estimation and observer methods are presented. For

this purpose, the system models as shown in Figure 3.10 and Figure 3.20 in sections

3.3 and 3.4 are considered and analysis is made.

81

Subsections 4.3.1 and 4.3.2 cover the analysis made for detecting faults in the

discrete part and continuous part of the system model using hybrid Petri nets as

shown in Figure 3.20. In subsection 4.3.3, the analysis made to achieve FDI in

system model using continuous Petri net as shown in Figure 3.10 is discussed.

Finally, analysis and conditions achieved for FTC in the system models considered

are presented in subsection 4.3.4.

4.3.1 Observer coverability graph for three tank system

As described in section 4.1.2, the discrete part of the system models can be

easily analyzed by developing the observer coverability graph when the markings

are unobservable in nature.

Based on the methodology adopted, the developed observer coverability graph for

the three tank system model as shown in Figure 3.20 in Chapter 3 is shown in

Figure 4.10. It can be observed from the graph that based on the initial place

marking, i.e., [ 00101 ], the subsequent markings are obtained when the

corresponding transition is fired and it can be found that at each step, the estimation

error (shown on right hand side of each marking) is zero. This means that the

markings which are obtained at each step are ideal values.

Figure 4.10 Observer coverability graph for system model shown in Figure 3.20

82

Hence, FDI can be achieved by determining the deviation in markings such that

the estimation error as shown in Figure 4.10 will not be equal to zero. Such

determination of deviations helps in detecting and analysing the faults accordingly.

4.3.2 Estimation based FDI in three tank system

The details of the pre-incidence (Pre), post-incidence (Post) and incidence matrices

(C) of the continuous part for the system model as described shown in Figure

3.20, section 3.4 are as follows:

Pre=

11000P

00100P

10000P

00010P

00001P

TTTTT

9

8

3

2

1

138721

, (4.19)

Post =

00100P

00000P

01010P

00001P

00000P

TTTTT

9

8

3

2

1

138721

, (4.20)

C=Post-Pre=

01-100P

001-00P

1-1010P

0001-1P

00001-P

TTTTT

9

8

3

2

1

138721

. (4.21)

Initially, it is considered that all places are marked. Hence, estimation is made such

that if a place is marked, its marking is replaced by 1, and 0 otherwise. Moreover,

the sequence of transitions is given by T1T2T13 or T7T8T13, where the

83

arrow represents the direction of the sequence. Original marking of the net

structure is given by

m0 = ( 1010555 )T, (4.22)

and the original estimated marking is considered to be

m'0 = ( 11111 )

T . (4.23)

Based on Equation (4.23), when T1 is fired initially, the values of pre-incidence

and incidence vectors are

Pre (., T1)= ( 00001 )T, (4.24)

C(., T1)= ( 00011 )T , (4.25)

and

Q = min[ m'0 , Pre(., T1)] = min[( 11111 )

T, ( 00001 )

T]

= ( 00001 )T. (4.26)

Hence, the newly estimated marking is

m ''0 = [Q + C(.,T1)] = [( 00001 )

T + ( 00011 )

T]

= ( 00010 )T. (4.27)

This updated marking will be used for further analysis when T2 and T13 fire. When

a fault occurs due to pump 1, the corresponding place, P1, is denoted by a null

marking, and thus the original estimated marking is

m0 = ( 11110 )T. (4.28)

Hence, by finding Q and m ''0 which are ( 00010 )

T and ( 00011 )

T

respectively, and by eventually comparing both, it can be found that the place P1 is

faulty. Similarly, the conditions for faults to occur can be detected and diagnosed.

84

The details of values for m'0 , Pre (., Ti), C(., Ti), Q and m ''

0 for various transition

firings are listed in Tables 4.1 and 4.2.

Table 4.1 Estimated values for transition sequence T1T2T13

T m'0 Pre (., Ti) C(., Ti) Q m ''

0

T1 ( 11111 )T ( 00001 )

T ( 00011 )

T ( 00001 )

T ( 00010 )

T

T2 ( 00010 )T ( 00010 )

T ( 00110 )

T ( 00010 )

T ( 00010 )

T

T13 ( 00010 )T ( 00010 )

T ( 00100 )

T ( 00100 )

T ( 00000 )

T

Table 4.2 Estimated values for transition sequence T7T8T13

T m'0 Pre (., Ti) C(., Ti) Q m ''

0

T7 ( 11111 )T ( 01000 )

T ( 11000 )

T ( 01000 )

T ( 10000 )

T

T8 ( 10000 )T ( 10000 )

T ( 10100 )

T ( 10000 )

T ( 00100 )

T

T13 ( 00100 )T ( 00100 )

T ( 00100 )

T ( 00100 )

T ( 00000 )

T

FDI scheme for transition faults is based on Equation (2.10) as discussed in section

2.3 in Chapter 2, and on the consideration that the evolution markings are

measurable and known initially.

Let the initial marking be

m0 = ( 1010555 )T. (4.29)

Based on the incidence matrix, C, using Equation (4.21), the instantaneous speed

vector is

v = ( 138643 )T. (4.30)

85

The values for new markings can be found when each transition fires. For

example,

m1 =

10

10

5

5

5

+

01100

00100

11010

00011

00001

4

0

13

8

6

4

3

dt =

20

0

5

10

0

, (4.31)

where m1 is the new marking. Now, eventually by considering a fault in transition

T1, the new value of v' due to additional noise [79] (from v*) is given by

v' = v + v* =

13

8

6

4

3

+

0

0

0

0

1

=

13

8

6

4

4

. (4.32)

Hence, the value of the marking m'1 is

m'1 =

10

10

5

5

5

+

01100

00100

11010

00011

00001

4

0

13

8

6

4

3

dt =

20

0

5

11

1

, (4.33)

and the value of m ''1 is

m ''1 = m-m '

1 1 =

20

0

5

10

0

-

20

0

5

11

1

=

0

0

0

1

1

. (4.34)

The negative value in markings of P1 and P2 in m ''1 vector eventually shows that a

transition fault in transition T1 has occurred. Likewise, similar types of faults can be

found. The limit of integration is considered from 0 to 4 sec as given in Equation

86

(4.33), since the fault in transition, T1, is valid only from 0 to 4 sec, which is also its

working range. Hence, the time of fault is also found out along with the type of fault

which has occurred in the system.

For performing FDI in the discrete part, which is eventually used as a control

signal in the model shown in Figure 3.20, the evolution and observability graphs

shown in Figure 3.21 and Figure 3.23 are considered. In this method, the graphs are

plotted based on the initial marking and incidence matrix initially. Then, based on

faulty marking, the graphs are plotted and compared. The presence of negative

values shows that faults have occurred and by similarly analyzing the values as

done earlier, faults are detected.

4.3.3 Observer based FDI in continuous system using evolution graph

For the implementation of the fault diagnosis scheme, let a fault occur at transition,

T2, i.e., the system model as shown in Figure 3.20 has a fault in transition, T2 or in

process as shown in Figure 3.19, then there is a failure in valve, V3. As explained

earlier, reachability analysis is done when the markings are observable and hence,

the same is considered here with the corresponding faulty transition, T2. Based on

this condition, the reachability graph shown in Figure 3.12 gets affected, as T2 does

not fire, and thus the marking m*3 and m*

6 cannot be reached. Hence, from earlier

statement, it can be concluded that an erroneous condition has occurred, and the

fault is due to transition, T2 or in process terms, valve, V3 failure has occurred.

Similarly, a place fault can be diagnosed from the reachability graph. For example,

let a fault in Place, P1, or in process terms, a pump fault occur. Due to this fault, as

the sequences are fired, changes occur in the macro markings, i.e., for example,

firing of transition T1 from m*1 results in m*

2 having value m*2 = ( 11111 )

T

and not m*2 = ( 11110 )

T. Hence, when both markings are compared, the

presence of non-zero elements constitutes the presence of the corresponding place

87

fault. The analysis procedure for unobservable markings is carried out by

developing the evolution graph as shown in Figure 4.11.

Figure 4.11 Evolution graph for system model shown in Figure 3.10

The evolution graph is extremely useful when the markings are unobservable,

while the transition sequence, initial marking and sequence of transitions are

observable. Based on these considerations, the current or original evolution graph is

initially developed and stored. Once the original graph is stored, the observer

estimates the evolution of markings as shown in Figure 4.12. The estimator denotes

a ‘1’ if there is a change in the original transition marking and ‘0’ if otherwise, i.e.,

at every time instant, the new transition is considered as the original transition and

based on the above method, estimation is done.

For example in the 1st and 2

nd instants (t0 to t1) for the marking vector given in the

original evolution graph by ( 25.05.015.01 )T

and ( 25.05.015.00 )T , the

88

corresponding estimated markings are ( 00000 )T and ( 00001 )

T.

Likewise, from instants t1 to t2, t1 is taken as reference and based on the changes

occurring in t2, estimation is done, i.e., estimated marking is ( 00100 )T for

the marking vector given by ( 25.05.005.00 )T (since there is change only in

3rd

element from instant t1 to t2).

Figure 4.12 Estimated evolution graph obtained from original evolution graph

as shown in Figure 4.11

Now based on the fault, the new evolution graph and corresponding estimated

graph is determined. For this purpose, let a fault in transition T1 occur. The

consideration made here is that the respective fault is shown by no change in the

value of the corresponding transition, i.e., the value of initial marking vector is

given by ( 25.05.015.01 )T, after firing results in ( 25.05.015.01 )

T and

not ( 25.05.015.00 )T , as shown in Figure 4.13.

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Figure 4.13 New estimated evolution graph obtained from new evolution graph

based on fault T1

Now by comparing the new estimated graph shown in Figure 4.13 with the original

estimation graph shown in Figure 4.12, it can be concluded that error has occurred

due to transition T1 during the period t = 0 sec to 5 sec. Likewise, similar faults can

be found using the estimation graph.

4.3.4 Controllability based FTC in hybrid event systems

In general, constraint of Type 1, i.e.,

r

1i

m'i k as discussed in section 2.5.1 in

Chapter 2, can be grouped and written in matrix form as

L m'i ≤ b, (4.35)

where m'i is the faulty marking vector of the Petri net model, L is an nc x n integer

matrix, b is an nc x n integer vector, and nc is the number of constraints of Type 1.

Now, in order to remove the inequality constraints in Equation (4.35),

90

L m'i + m'

c = b, (4.36)

where m'c is an nc x 1 integer vector that represents the marking of controller places.

So, given the Petri net model of the system, the incidence matrix (C) and the

constraints that the process must satisfy (L and b), the Petri net fault tolerant

controller, FC is defined as

FC = -LC. (4.37)

For the system model considered and shown in Figure 3.20 in Chapter 3, the

incidence matrix (discrete part) is given by

C =

01-100P

001-00P

1-1010P

0001-1P

00001-P

TTTTT

9

8

3

2

1

138721

, (4.38)

while the faulty marking is given by

m'i = [ m'm'm'm'm'

54321 ]T=[ 11110 ]

T. (4.39)

The objective function here is to control the net so that

m'1 + m'

2 ≤ 1. (4.40)

Based on the matrix notation of (4.35),

L = [ 00101 ]. (4.41)

Now, a slack variable, m'c, is added in Equation (4.40) to remove the inequality,

and thus equality is obtained as follows:

m'1 + m'

2 + m'c = 1. (4.42)

91

Based on the above constraint, the incidence matrix for the fault tolerant controller

net is computed using Equation (4.37):

FC = -LC = [ 00101 ]T

01100

00100

11010

00011

00001

= [ 11011 ]T.

(4.43)

Thus, from the result it can be observed that by incorporating FTC, the constraint

shown by Equation (4.35) can be satisfied. Likewise, the constraints for all faulty

conditions can be considered, and the corresponding FTC structure [80] can be

obtained.

In this Chapter, a complete description of the proposed methodology along with

analysis and results obtained in this research work is presented. In Chapter 5, the

applications of the proposed method to achieve FDI and FTC on real time system

models are discussed.