**System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling**

G. G. Kulikov, V. Yu. Arkov and A.I. Abdulnagimov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54443

#### **1. Introduction**

2009: Power for Land, Sea, and Air (GT2009),June 8–12, 2009, Orlando, Florida, USA,

[47] Volponi AJ, Brotherton T, Luppold R. Development of an information fusion system for engine diagnostics and health management. AIAA 1st Intelligent Systems Techni‐ cal Conference20 - 22 September 2004, Chicago, Illinois, AIAA Paper 2004-6461.

[48] Mathioudakis K, Stamatis A, Tsalavoutas A, Aretakis N. Performance analysis of in‐ dustrial gas turbines for engine condition monitoring. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy. 2001;215(2):173-84.

[49] Tsalavoutas A, Mathioudakis K, Smith MK. Processing of circumferential tempera‐ ture distributions for the detection of gas turbine burner malfunctions. ASME Paper

[50] Loukis E, Wetta P, Mathioudakis K, Papathanasiou A, Papailiou A. Combination of different unsteady quantity measurements for gas turbine blade fault diagnosis. ASME paper, 91-GT-201, International Gas Turbine and Aeroengine Congress and

ASME Paper no. GT2009-59942

Exposition, June 3-6 1991, Orlando.

96-GT-103.

212 Progress in Gas Turbine Performance

Reliability, safety and durability represent important properties of modern aircraft, which is necessary for its effective in-service use.

The reason of the main hazard for aircraft are both random and determined negative influ‐ ences rendering the controlled object during its use. Faults, failures, disturbances, noises, in‐ fluences of environment and control errors represent the objectively existing stream of random negative influences on the object.

Statistically, in the recent years the majority of aircraft incidents are connected with the hu‐ man factor and late fault detection in plane systems. In this regard, requirements to flight safety which demand development of new methods and algorithms of control-and-condi‐ tion monitoring/ diagnostic for complex objects raise every year. The analysis of modern gas turbine engines has shown that most faults appears in the engine itself and its FADEC (40-75% for FADEC, Figure 1).

The percentage of faults for FADEC depends on the achieved values for no-failure operation indicators of the engine and FADEC.

During the development of FADEC, it is necessary to adhere to the principles and methods guaranteeing safety and reliability of aircraft in use to guarantee proper responses in all range of negative influences.

Full information on its work is necessary for complete control of a condition of the engine:

**1.** Reliable detection of a fault cause providing decision-making on a technical condition of gas turbines;

**2.** Reliable diagnosis and localization of faults and negative influences are necessary for definition of technical condition of gas turbines for the purpose of providing a reconfi‐ guration and functioning of its subsystems [1, 2]. 2 Book Title 2. Reliable diagnosis and localization of faults and negative influences are necessary for definition of technical condition of gas turbines for the purpose of providing a

formed into independent subsystems with various levels of hierarchy on structural and

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

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215

The hierarchy analysis allows to utilize the state model on the basis of faults development

*S* = <*G*, *F* , *L* , *R* > ,

which enables the system state to be estimated at each level of the hierarchy.

The depth of hierarchy *G* is referred to as *h*, and *h* = 0 for the root element of *G*.

*<sup>ω</sup><sup>x</sup>* : *<sup>x</sup>* −→ 0, 1 ; where ∑

can be defined for all *hk*, if it equals to zero for all faults in *hk*+1 which do not belong to *x* <sup>−</sup>

*y*∈*x* <sup>−</sup>

are the hierarchy levels, and function *ωx* is a function of fault priority of one level

concerning the state of the power-plant *x*. Notice that if *<sup>x</sup>* −⊄*hk* +1 (for some level of *hw*), then *ω<sup>x</sup>*

*ωx*(*y*)=1.

.

functional features in the following way:

**•** Hydro mechanical system (actuators);

**•** Fuel system;

**•** Start-up system;

**•** Lubricant oil system; **•** Drainage system, etc.

where *S* is state vector,

The sets of *hi*

**•** physical structure;

**•** Control and monitoring system (FADEC);

The mathematical model of states is represented as

*G* is hierarchy of system faults,

For *G* the following conditions are satisfied:

*F* is quantitative estimate of faults, *L* is set of fault influence indexes,

*R* is mutual influence system of faults.

**1.** There is splitting of *G* into subsets of *hk*, *k* = 1 … *n*.

**3.** From *<sup>x</sup>* <sup>∈</sup> *<sup>L</sup> <sup>k</sup>* follows that *<sup>x</sup>* <sup>+</sup>⊂*hk* <sup>−</sup>1, *<sup>k</sup>* =2, …, *<sup>n</sup>*.

For every *x* ∈*G* there is a weight function such as:

The hierarchical FADEC model integrates:

**•** functional structure (block diagram);

**2.** From *<sup>x</sup>* <sup>∈</sup> *<sup>L</sup> <sup>k</sup>* follows that *<sup>x</sup>* −⊂*hk* +1, *<sup>k</sup>* =1, …, *<sup>n</sup>* <sup>−</sup>1.

reconfiguration and functioning of its subsystems [1, 2].

Figure1. Faults percentage for engines and FADEC **Figure 1.** Faults percentage for engines and FADEC

The hardware for condition monitoring of measurement channels in many cases allows to detect only catastrophic (breakage or short circuit) faults, i.e. their stochastic properties on time of the process observed in one object and on a set of objects are not distinguishable [3]. The criteria of warning messages on faults appearance are based mainly on determined logic operations and distinguish between only two conditions: "operational" (fully operational) or "fault". In this chapter, hierarchical fuzzy Markov models for quantitative estimation of system safety of gas turbines taking into account the monitoring of cause-effect relations are considered. Transition from two-valued to fuzzy logic for estimation of degradation indexes and the analysis of fault developments for the gas turbine and its FADEC is considered for this purpose. **2. Hierarchical model of faults development processes in gas turbines**  The hardware for condition monitoring of measurement channels in many cases allows to de‐ tect only catastrophic (breakage or short circuit) faults, i.e. their stochastic properties on time of the process observed in one object and on a set of objects are not distinguishable [3]. The crite‐ ria of warning messages on faults appearance are based mainly on determined logic opera‐ tions and distinguish between only two conditions: "operational" (fully operational) or "fault". In this chapter, hierarchical fuzzy Markov models for quantitative estimation of system safety of gas turbines taking into account the monitoring of cause-effect relations are considered. Transition from two-valued to fuzzy logic for estimation of degradation indexes and the analy‐ sis of fault developments for the gas turbine and its FADEC is considered for this purpose.

#### using the hierarchy analysis method [4, 5]. First, decomposition into independent subsystems of various hierarchy levels is carried out on structural features. Similarly, the power plant and its systems are represented in the form of hierarchy of elements and blocks. This approach enables cause-effect relationships to be identified on the hierarchy structure **2. Hierarchical model of faults development processes in gas turbines**

Complex diagnostics of the power plant is proposed to be carried out on elements and units,

of a system. In Figure 2, the hierarchical structure of states of the power-plant is shown. The power plant is represented in the form of a hierarchical structure as the complex system consisting of subsystems and elements (units) with built-in test/monitoring functions, according to the Complex diagnostics of the power plant is proposed to be carried out on elements and units, using the hierarchy analysis method [4, 5]. First, decomposition into independent subsys‐ tems of various hierarchy levels is carried out on structural features. Similarly, the power plant and its systems are represented in the form of hierarchy of elements and blocks.

This approach enables cause-effect relationships to be identified on the hierarchy structure of a system.

In Figure 2, the hierarchical structure of states of the power-plant is shown. The power plant is represented in the form of a hierarchical structure as the complex system consisting of subsystems and elements (units) with built-in test/monitoring functions, according to the distributed architecture. For this purpose, the power plant decomposition might be per‐ formed into independent subsystems with various levels of hierarchy on structural and functional features in the following way:


**2.** Reliable diagnosis and localization of faults and negative influences are necessary for definition of technical condition of gas turbines for the purpose of providing a reconfi‐

2 Book Title

2. Reliable diagnosis and localization of faults and negative influences are necessary for definition of technical condition of gas turbines for the purpose of providing a

**Russian engines Worldwide statistics**

**FADEC 43 – 55%** 

> Engine Units 39 – 50%

Hydromechanics 5 – 8%

> **FADEC 39 – 70%**

Electronics 34 – 62%

> Other Up to 5%

The hardware for condition monitoring of measurement channels in many cases allows to detect only catastrophic (breakage or short circuit) faults, i.e. their stochastic properties on time of the process observed in one object and on a set of objects are not distinguishable [3]. The criteria of warning messages on faults appearance are based mainly on determined logic operations and distinguish between only two conditions: "operational" (fully operational) or "fault". In this chapter, hierarchical fuzzy Markov models for quantitative estimation of system safety of gas turbines taking into account the monitoring of cause-effect relations are considered. Transition from two-valued to fuzzy logic for estimation of degradation indexes and the analysis of fault developments for the gas turbine and its

The hardware for condition monitoring of measurement channels in many cases allows to de‐ tect only catastrophic (breakage or short circuit) faults, i.e. their stochastic properties on time of the process observed in one object and on a set of objects are not distinguishable [3]. The crite‐ ria of warning messages on faults appearance are based mainly on determined logic opera‐ tions and distinguish between only two conditions: "operational" (fully operational) or "fault". In this chapter, hierarchical fuzzy Markov models for quantitative estimation of system safety of gas turbines taking into account the monitoring of cause-effect relations are considered. Transition from two-valued to fuzzy logic for estimation of degradation indexes and the analy‐ sis of fault developments for the gas turbine and its FADEC is considered for this purpose.

**2. Hierarchical model of faults development processes in gas turbines** 

Complex diagnostics of the power plant is proposed to be carried out on elements and units, using the hierarchy analysis method [4, 5]. First, decomposition into independent subsystems of various hierarchy levels is carried out on structural features. Similarly, the power plant and its systems are represented in the form of hierarchy of elements and blocks. This approach enables cause-effect relationships to be identified on the hierarchy structure

**2. Hierarchical model of faults development processes in gas turbines**

In Figure 2, the hierarchical structure of states of the power-plant is shown. The power plant is represented in the form of a hierarchical structure as the complex system consisting of subsystems and elements (units) with built-in test/monitoring functions, according to the

Complex diagnostics of the power plant is proposed to be carried out on elements and units, using the hierarchy analysis method [4, 5]. First, decomposition into independent subsys‐ tems of various hierarchy levels is carried out on structural features. Similarly, the power plant and its systems are represented in the form of hierarchy of elements and blocks.

This approach enables cause-effect relationships to be identified on the hierarchy structure

In Figure 2, the hierarchical structure of states of the power-plant is shown. The power plant is represented in the form of a hierarchical structure as the complex system consisting of subsystems and elements (units) with built-in test/monitoring functions, according to the distributed architecture. For this purpose, the power plant decomposition might be per‐

guration and functioning of its subsystems [1, 2].

Figure1. Faults percentage for engines and FADEC

Engine Units 35 – 55%

**Figure 1.** Faults percentage for engines and FADEC

Hydromechanics 10 – 20%

214 Progress in Gas Turbine Performance

FADEC is considered for this purpose.

of a system.

of a system.

reconfiguration and functioning of its subsystems [1, 2].

Electronics 35 – 55%

> Other Up to 3%


The hierarchy analysis allows to utilize the state model on the basis of faults development which enables the system state to be estimated at each level of the hierarchy.

The mathematical model of states is represented as

$$S = \langle G\_{\prime\prime}, F\_{\prime\prime}, L\_{\prime\prime}, R \rangle\_{\prime\prime}$$

where *S* is state vector,

*G* is hierarchy of system faults,

*F* is quantitative estimate of faults,

*L* is set of fault influence indexes,

*R* is mutual influence system of faults.

The depth of hierarchy *G* is referred to as *h*, and *h* = 0 for the root element of *G*.

For *G* the following conditions are satisfied:


For every *x* ∈*G* there is a weight function such as:

$$
\omega\_{\boldsymbol{\pi}} \colon \boldsymbol{\pi}^- \to \mathtt{[0, 1]}; \text{ where } \sum\_{\boldsymbol{y} \in \boldsymbol{\pi}^-} \omega\_{\boldsymbol{\pi}}(\boldsymbol{y}) = 1.
$$

The sets of *hi* are the hierarchy levels, and function *ωx* is a function of fault priority of one level concerning the state of the power-plant *x*. Notice that if *<sup>x</sup>* −⊄*hk* +1 (for some level of *hw*), then *ω<sup>x</sup>* can be defined for all *hk*, if it equals to zero for all faults in *hk*+1 which do not belong to *x* <sup>−</sup> .

The hierarchical FADEC model integrates:


**Fault levels Faults State Fault handling**

Fault of control function (B)

Fault of control loop (B)

Fault of actuator control circuit (B)

Fault of control function (А)

Fault of actuator control circuit (А) (fault of communication lines of sensors or fault of FADEC hardware)

FADEC fault

Fault of lane B

Fault of control loop Fault of control loop (А)

Sensors Actuators

On the hierarchical model, the system of faults interference *R* with logical operations of a disjunction and a conjunction is applied. Such a system of faults interference allows to ana‐ lyze the state of all power-plant, both from the bottom up to the top, and from the top down to the bottom and to carry out deeper analysis on various levels of decomposition of the

The degradation is understood as "package/complex of degradationary changes of the sys‐ tem" and the degradationary change is "a separately considered irreversible change of a struc‐ ture of the system, worsening its properties, changing the parameters and characteristics".

Define the main faults of FADEC, the priorities of their elimination and the state they belong to. In Table 1, an exemplary of a fragment of the hierarchical classification of faults for sen‐

Integrated fault of measurement for alternative control law

Integrated fault of measurement (fault of same measured parameters in channels A)

Fault of measurement in channel (break or short circuit)

**Table 1.** Fragment of hierarchical classification of faults for FADEC

**•** tree states (state structure) of elements and units;

control system using an intermediate state: degradation.

sors and actuators of FADEC is presented.

**•** tree of failures influence indexes.

Level 7 Fault of lane A

Level 1

Levels 2-4

Levels 5-6

Level 8

Level 9

Level 10

**priority**

217

http://dx.doi.org/10.5772/54443

Immediate

Short-term

Long-term

Fault

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

Degradation

**Figure 2.** Hierarchical state structure of power plant


**Table 1.** Fragment of hierarchical classification of faults for FADEC


Power-plant State

State of monitored

*ω*14k

parameters:

metallic cuttings in

lubrication oil; oil

overheat in turbine

pedestals,

compressors; oilfilter obstruction;

etc.

**. . .** 

Lubricant oil

*ω*14

system state

**. . .**

*ωmk*

State of Gas Turbine

Engine *m*

*ωm*

State of other

*ω*15

systems

*ω*15k

State of monitored

*ωm*1

parameters

FADEC state

*ω*11

216 Progress in Gas Turbine Performance

*ω*11k

State of

State of

circuits of

sensors and

actuators

*ω*11n

control-andconditionmonitoring

functions

State of Gas Turbine

Engine 1

*ω*13

Start-up

system

state

State of monitored

*ω*13k *ω*1

parameters

Fuel system

*ω*12

state

**Figure 2.** Hierarchical state structure of power plant

State of

*ω*12k

circuits of

sensors and

actuators

*ω*12n

State of monitored

parameters

On the hierarchical model, the system of faults interference *R* with logical operations of a disjunction and a conjunction is applied. Such a system of faults interference allows to ana‐ lyze the state of all power-plant, both from the bottom up to the top, and from the top down to the bottom and to carry out deeper analysis on various levels of decomposition of the control system using an intermediate state: degradation.

The degradation is understood as "package/complex of degradationary changes of the sys‐ tem" and the degradationary change is "a separately considered irreversible change of a struc‐ ture of the system, worsening its properties, changing the parameters and characteristics".

Define the main faults of FADEC, the priorities of their elimination and the state they belong to. In Table 1, an exemplary of a fragment of the hierarchical classification of faults for sen‐ sors and actuators of FADEC is presented.

Fault levels 1 through 6 demand immediate handlings and correspond to the "catastrophic" and "critical" states by FAA classification (Federal Aviation Administration, U.S. Depart‐ ment of Transportation), given in [6]. The emerging of such states requires immediate land‐ ing of the aircraft. Fault levels 7 through 9 are classified as a "marginal" state and demand operative handling after landing. In this case, it is possible to continue the flight, but post‐ flight repair on the ground is required. Faults at the 10th level demand their handling in long-term prospect.

Catastrophic

**Fault states classification**

**1 …**

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

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219

**0,5 …**

**Degradation**

**degree**

**0**

*t*

**, time**

Critical

Marginal

Negligible

critical situation and the speed of approximation to it (Figure 4).

Catastrophic

Negligible

Critical

Marginal

*Sh***(***x***)**

**States**

**Figure 4.** Trend of state dynamics during flight

Consider an example of correspondence of degradation degree and the operational state. At the degradation degree of 0,25, the system is capable to carry out 75% of demanded func‐ tions (50% at 0,5 degradation, 25% at 0,75 and 0% at 1, which is the unavailable state). Such scale allows to define a "threshold" state, below which further operation is not allowed for safety reasons. Using the degradation degree, it is also possible to estimate the distance to a

**Figure 3.** Estimation of "degradation" state

The degradation process for FADEC starts at the 10th level of hierarchy. From the 4th level of hierarchy, the system starts to approach the system crash that can be regarded as «a criti‐ cal situation».

Note that development of such faults in certain cases can be detected in advance by estimat‐ ing the states of elements not only at the level of "0-1" (fully operational, operational/work‐ ing, fault), but also by considering their gradual degradation.

The state of an element or a system is proposed to be represented in the form of three pa‐ rameters { operational, degradation, fault }, see Tab. 2.

In the operational state *S* = 0, while during fault *S* = 1. The degradation degree range from "0" to "1". Thus, the extreme values "0" and "1" are defined according to the determined log‐ ic, which is realized in the conventional FADEC (according to the design specifications for the system). The introduction of this intermediate state of "degradation" expands the infor‐ mativity of the conventional condition monitoring algorithms.


#### **Table 2.** Fuzzy representation of state

Based on the faults analysis and the hierarchy of states of the system at each level, the degree of degradation of each item or sub-unit is determined (Figure 3). Fault states are classified via degradation degree as "Negligible", "Marginal", "Critical" and "Catastrophic" [6]. The estima‐ tion of the degradation degree is defined on the membership function *S* which takes values in the range of*S* ∈ 0, 1 .If the degradation degree is closer to "1", the distance to a critical situa‐ tion will be closer. If the analysis of a system showed that the state vector is { 0,1 0,6 0,3 }, it is possible to ensure that there is a "distance" before complete fault (a critical situation). As soon as the system state will worsen with the appearance of new faults and will give the following state vector { 0 0,3 0,7 }, then there will be a distance of 0,3 to a system crash. Thus the most in‐ formative indicator will be a tendency of faults appearance (trend), not the existence of degra‐ dation itself. Visual trend analysis provides an estimate of time before the critical situations develop and, thus, for early planning of the crew actions [7, 8].

**Figure 3.** Estimation of "degradation" state

Fault levels 1 through 6 demand immediate handlings and correspond to the "catastrophic" and "critical" states by FAA classification (Federal Aviation Administration, U.S. Depart‐ ment of Transportation), given in [6]. The emerging of such states requires immediate land‐ ing of the aircraft. Fault levels 7 through 9 are classified as a "marginal" state and demand operative handling after landing. In this case, it is possible to continue the flight, but post‐ flight repair on the ground is required. Faults at the 10th level demand their handling in

The degradation process for FADEC starts at the 10th level of hierarchy. From the 4th level of hierarchy, the system starts to approach the system crash that can be regarded as «a criti‐

Note that development of such faults in certain cases can be detected in advance by estimat‐ ing the states of elements not only at the level of "0-1" (fully operational, operational/work‐

The state of an element or a system is proposed to be represented in the form of three pa‐

In the operational state *S* = 0, while during fault *S* = 1. The degradation degree range from "0" to "1". Thus, the extreme values "0" and "1" are defined according to the determined log‐ ic, which is realized in the conventional FADEC (according to the design specifications for the system). The introduction of this intermediate state of "degradation" expands the infor‐

**Operational Degradation Fault**

*S* = 0 0 < *S* < 1 *S* = 1

Based on the faults analysis and the hierarchy of states of the system at each level, the degree of degradation of each item or sub-unit is determined (Figure 3). Fault states are classified via degradation degree as "Negligible", "Marginal", "Critical" and "Catastrophic" [6]. The estima‐ tion of the degradation degree is defined on the membership function *S* which takes values in the range of*S* ∈ 0, 1 .If the degradation degree is closer to "1", the distance to a critical situa‐ tion will be closer. If the analysis of a system showed that the state vector is { 0,1 0,6 0,3 }, it is possible to ensure that there is a "distance" before complete fault (a critical situation). As soon as the system state will worsen with the appearance of new faults and will give the following state vector { 0 0,3 0,7 }, then there will be a distance of 0,3 to a system crash. Thus the most in‐ formative indicator will be a tendency of faults appearance (trend), not the existence of degra‐ dation itself. Visual trend analysis provides an estimate of time before the critical situations

ing, fault), but also by considering their gradual degradation.

mativity of the conventional condition monitoring algorithms.

develop and, thus, for early planning of the crew actions [7, 8].

rameters { operational, degradation, fault }, see Tab. 2.

long-term prospect.

218 Progress in Gas Turbine Performance

**Table 2.** Fuzzy representation of state

cal situation».

Consider an example of correspondence of degradation degree and the operational state. At the degradation degree of 0,25, the system is capable to carry out 75% of demanded func‐ tions (50% at 0,5 degradation, 25% at 0,75 and 0% at 1, which is the unavailable state). Such scale allows to define a "threshold" state, below which further operation is not allowed for safety reasons. Using the degradation degree, it is also possible to estimate the distance to a critical situation and the speed of approximation to it (Figure 4).

**Figure 4.** Trend of state dynamics during flight

Thus, the hierarchical model of fault developments allows to decompose the power-plant on hierarchy levels for obtaining quantitative estimates of the degradation state and gradual faults. The hierarchy analysis allows to utilize the state model and to estimate the system state at each level of hierarchy. The state is represented in the form of a vector with parame‐ ters { operational, degradation, fault }. Depending on the degradation degree it is possible to make an estimation of operability of object and system safety.

breaking the limit doesn't prove to be true during the set period of time, the monitoring al‐

Operational (B1) Fault (B2)

0 4 *xn*

9

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221

1 2 3

*T T T*

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

1 1 1

*n n n*

( ) 0,1 ( ) 0,8 ( ) 0,1

123

In Figure 5 two membership functions are shown: state B1 corresponds to the function *μB*<sup>1</sup>

The way of creating fuzzy rules is presented in Figure 6. This rule base is represented by the

In Figure 5 two membership functions are shown: state B1 corresponds to the function *μ<sup>B</sup>*<sup>1</sup>

*<sup>R</sup>* (1) :*IF* (*xn* <sup>=</sup> *<sup>A</sup>*<sup>1</sup> *AND xn* <sup>=</sup> *<sup>B</sup>*1) *THEN <sup>y</sup>* <sup>=</sup> *<sup>T</sup>*<sup>1</sup>

T3

A1 A2 B1 B2

*x*2

The way of creating fuzzy rules is presented in Figure 6. This rule base is represented by the

11 1 : ( ) *R xA xB n n* **IF AND THEN** *y T*

gorithm cancels the measurement: *B*1 = 1, *B*2 = 0.

1

0,7

0,3

Running Title

*x*1

**Figure 5.** Membership functions of "sensor state"

A1

A2

(*xn*), *B*2 is described by the function *μB*2 (*xn*).

table, which is filled in with fuzzy rules as follows [9]:

table, which is filled in with fuzzy rules as follows [9]: (1)

> 1 2 3

*T T T*

B2

Figure6. Example of fuzzy rule base

**Figure 6.** Example of fuzzy rule base

1 1 1

 

*n n n*

( ) 0,2 ( ) 0,7 ( ) 0,1

B1 T1

(*xn*), *B*2 is described by the function *μB*2 (*xn*).

Figure5. Membership functions of "sensor state"

*<sup>B</sup>*(*xn*)

#### **3. Fuzzy technique of determination of state parameters of gas turbine and its systems**

The built-in monitoring system (BMS) is a subsystem of monitoring, diagnosis and classifi‐ cation of faults of the gas turbine and its systems. The fault existence corresponds to a logic state of "1", the absence does to a logic state of "0". Such state classification doesn't allow to establish a "prefault" state, to trace faults' development, and to define degradation of the system and its elements. For more detailed analysis, the estimation of the intermediate state of degradation is proposed. For this purpose, the use of fuzzy logic is considered. Signals from sensors, and also logic state parameters from BMS will transform to linguistic variables during fuzzification to a determined value arrives to the input of the fuzzifier. Let *x* is the state parameter of an element (for example, the sensor). It is necessary to define fuzzy spaces of input and output variables, and also terms for FADEC sensors. All signals from sensors and actuators will transform into linguistic variables by fuzzification.

Consider an example of fuzzy representation of a state of the two-channel sensor of rotation‐ al speed of a high pressure turbocompressor rotor. For monitoring of operational condition of communication lines of the FADEC sensor, a linguistic variable is introduced in the fol‐ lowing way:

$$
\Omega = \{ \infty\_{n'} \: \ B = (\infty\_n)\_{\prime} \: \ U, \ G, \ M >\_{\prime} \}
$$

where Ω*n* is the sensor state,

*xn* is the number of events when *n* is beyond the allowed limit band;

*B is* { operational, fault };

*U is* [0,4]

*G* is the syntactic rule generating terms of set B,

*M* is the semantic rule, which to each linguistic value *x* associate with its sense of *M*(*xn*), and *M*(*xn*) designates a fuzzy subset of the carrier *U*.

Say that the sensor is considered failed after the fourth appearance of the shaft speed meas‐ urement beyond the allowed boundary, therefore the membership function is formed as shown in Figure 5.

At a single appearance out of limit (*xn* = 1), membership function *B*1 takes the value 0,7, and *B*2 = 0,3. The degradation degree takes the value of membership function *B*2. If the repeated

9

breaking the limit doesn't prove to be true during the set period of time, the monitoring al‐ gorithm cancels the measurement: *B*1 = 1, *B*2 = 0.

**Figure 5.** Membership functions of "sensor state"

(*xn*), *B*2 is described by the function *μB*2 (*xn*).

table, which is filled in with fuzzy rules as follows [9]: (1)

Thus, the hierarchical model of fault developments allows to decompose the power-plant on hierarchy levels for obtaining quantitative estimates of the degradation state and gradual faults. The hierarchy analysis allows to utilize the state model and to estimate the system state at each level of hierarchy. The state is represented in the form of a vector with parame‐ ters { operational, degradation, fault }. Depending on the degradation degree it is possible to

**3. Fuzzy technique of determination of state parameters of gas turbine**

The built-in monitoring system (BMS) is a subsystem of monitoring, diagnosis and classifi‐ cation of faults of the gas turbine and its systems. The fault existence corresponds to a logic state of "1", the absence does to a logic state of "0". Such state classification doesn't allow to establish a "prefault" state, to trace faults' development, and to define degradation of the system and its elements. For more detailed analysis, the estimation of the intermediate state of degradation is proposed. For this purpose, the use of fuzzy logic is considered. Signals from sensors, and also logic state parameters from BMS will transform to linguistic variables during fuzzification to a determined value arrives to the input of the fuzzifier. Let *x* is the state parameter of an element (for example, the sensor). It is necessary to define fuzzy spaces of input and output variables, and also terms for FADEC sensors. All signals from sensors

Consider an example of fuzzy representation of a state of the two-channel sensor of rotation‐ al speed of a high pressure turbocompressor rotor. For monitoring of operational condition of communication lines of the FADEC sensor, a linguistic variable is introduced in the fol‐

*Ω* = < *xn*, *B* =(*xn*), *U* , *G*, *M* > ,

*M* is the semantic rule, which to each linguistic value *x* associate with its sense of

Say that the sensor is considered failed after the fourth appearance of the shaft speed meas‐ urement beyond the allowed boundary, therefore the membership function is formed as

At a single appearance out of limit (*xn* = 1), membership function *B*1 takes the value 0,7, and *B*2 = 0,3. The degradation degree takes the value of membership function *B*2. If the repeated

*xn* is the number of events when *n* is beyond the allowed limit band;

make an estimation of operability of object and system safety.

and actuators will transform into linguistic variables by fuzzification.

*G* is the syntactic rule generating terms of set B,

*M*(*xn*), and *M*(*xn*) designates a fuzzy subset of the carrier *U*.

**and its systems**

220 Progress in Gas Turbine Performance

lowing way:

where Ω*n* is the sensor state,

*U is* [0,4]

shown in Figure 5.

*B is* { operational, fault };

In Figure 5 two membership functions are shown: state B1 corresponds to the function *μB*<sup>1</sup> (*xn*), *B*2 is described by the function *μB*2 (*xn*).

The way of creating fuzzy rules is presented in Figure 6. This rule base is represented by the table, which is filled in with fuzzy rules as follows [9]: Figure5. Membership functions of "sensor state" In Figure 5 two membership functions are shown: state B1 corresponds to the function *μ<sup>B</sup>*<sup>1</sup>

$$\mathcal{R}^{(1)} \colon \text{IF} \left( \mathfrak{x}\_n = A\_1 \, \mathbf{AND} \, \mathbf{x}\_n = B\_1 \right) \, \text{THEN} \quad y = T\_1$$

11 1 : ( ) *R xA xB n n* **IF AND THEN** *y T*

*x*2

**Figure 6.** Example of fuzzy rule base

Figure6. Example of fuzzy rule base

The values *μT*1, *μT*2, *μT*3 are set in the cell at the row "Operational" (*A*1) and the column "Fault" (*B*2).

Running Title

*consistent*.

*consistent*.

close to zero.

to the matrix order *n*.

*α* =∑ *i*=1 *n ωi* close to zero.

to the matrix order *n*.

Figure7. Continuous (a) and discrete (b) scale of degradation

**Figure 7.** Continuous (a) and discrete (b) scale of degradation

classification is obtained, along with vector *ω* satisfying the criterion:

classification is obtained, along with vector *ω* satisfying the criterion:

After the representation of quantitative judgements about the fault pairs (*Fi*, *Fj*) in a numerical expression with the numbers *aij*, the problem is reduced to that *n* possible faults *F*1, *F*2, …, *Fn* will receive a corresponding set of numerical weights *ω*1, *ω*2, …, *ωn*, which would reflect the fixed judgements about the condition of the gas turbine subsystem.

If the expert judgement is absolute at all comparisons for all *i*, *j*, *k*, then matrix *A* is called

If the expert judgement is absolute at all comparisons for all *i*, *j*, *k*, then matrix *A* is called

If the diagonal of matrix A consists of units (*aij* = 1) and *A* is the consistent matrix, then at small changes in *aij* the greatest eigenvalue λmax is close to *n*, and the other eigenvalue are

Based on the matrix of pair comparison values of faults *A*, the vector of priorities for fault

*Aω* =*λ*max*ω*,

where *ω* is the eigenvector of matrix *A* and λmax is the maximum eigenvalue, which is close

As it is desirable to have the normalized solution, let's slightly change *ω*, considering

and replacing *ω* with (1/*α*) *ω*. This provides uniqueness, and also that∑

If the diagonal of matrix A consists of units (*aij* = 1) and *A* is the consistent matrix, then at small changes in *aij* the greatest eigenvalue λmax is close to *n*, and the other eigenvalue are

Based on the matrix of pair comparison values of faults *A*, the vector of priorities for fault

where *ω* is the eigenvector of matrix *A* and λmax is the maximum eigenvalue, which is close

*A* max , 11

223

http://dx.doi.org/10.5772/54443

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

*i*=1 *n*

*ω<sup>i</sup>* =1.

Consider a fragment of the rule base for estimates of the sensor state of the low pressure shaft speed. Formulate the first rule: if the 1st coil (*n*11) of the sensor is operational (*A*1) and the 2nd coil (*n*12) of the sensor is operational (*A*1), then the sensor is in the operational condi‐ tion with the following membership functions:

$$\mathsf{L}\mu\_{\text{operational}}(n\_1) = \mathbf{1}; \quad \mu\_{\text{degradation}}(n\_1) = 0; \quad \mu\_{\text{fault}}(n\_1) = \mathbf{0} \mathbf{1}.$$

Write down this rule as follows:

$$\begin{aligned} \mathsf{R} \ ^{(1)}: \mathsf{IF}(n\_{11} = A\_1 \, \mathsf{AND} \, n\_{12} = A\_1) \, \mathsf{THEN} \quad y = T\_1 \Rightarrow\\ \mathsf{L} \mu\_{T\_1}(n\_1) = 1; \quad \mu\_{T\_2}(n\_1) = 0; \quad \mu\_{T\_3}(n\_1) = 0 \, \mathsf{L} .\end{aligned}$$

Other rules are created in the similar way.

Given a greater number of possible conditions (for example, greater number of the duplicat‐ ed coils of the sensor), one can develop a discrete-ordered scale of state parameters (Figure 7).

For further analysis of the system, enter the faults influence indexes at each level of hierar‐ chy, using a method of pairwise comparison as it is carried out in the hierarchy analysis method.

Quantitative judgements on the importance of faults are performed for each pair of faults (*F*i , *Fj* ) and these are represented by matrix *A* of the *n*×*n* size.

$$A = (a\_{ij})\_\prime \quad \text{(i. } j = 1 \text{ 2, 3)}\text{.}$$

where *aij* is the relative importance of fault *Fi* in regard to *Fj* . The value *Aij* defines the impor‐ tance (respective values) *Fi* of faults in comparison with *Fj* .

Elements *aij* are defined by the following rules:


Thus, a back-symmetric matrix A is obtained:

$$A = \begin{bmatrix} 1 & a\_{12} & \cdots & a\_{1n} \\ 1/a\_{12} & 1 & \cdots & a\_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ 1/a\_{1n} & 1/a\_{2n} & \cdots & 1 \end{bmatrix} . \tag{1D2}$$

After the representation of quantitative judgements about the fault pairs (*Fi* , *Fj* ) in a numeri‐ cal expression with the numbers *aij*, the problem is reduced to that *n* possible faults *F*1, *F*2, …, *Fn* will receive a corresponding set of numerical weights *ω*1, *ω*2, …, *ωn*, which would reflect the fixed judgements about the condition of the gas turbine subsystem.

11

After the representation of quantitative judgements about the fault pairs (*Fi*, *Fj*) in a **Figure 7.** Continuous (a) and discrete (b) scale of degradation

to the matrix order *n*.

Figure7. Continuous (a) and discrete (b) scale of degradation

Running Title

The values *μT*1, *μT*2, *μT*3 are set in the cell at the row "Operational" (*A*1) and the column

Consider a fragment of the rule base for estimates of the sensor state of the low pressure shaft speed. Formulate the first rule: if the 1st coil (*n*11) of the sensor is operational (*A*1) and the 2nd coil (*n*12) of the sensor is operational (*A*1), then the sensor is in the operational condi‐

*μ*operational(*n*1)=1; *μ*degradation(*n*1)=0; *μ*fault(*n*1)=0 .

*<sup>R</sup>* (1) :*IF* (*n*<sup>11</sup> <sup>=</sup> *<sup>A</sup>*<sup>1</sup> *AND <sup>n</sup>*<sup>12</sup> <sup>=</sup> *<sup>A</sup>*1) *THEN <sup>y</sup>* <sup>=</sup>*T*1<sup>⇒</sup>

(*n*1)=0 .

.

. The value *Aij* defines the impor‐

, then *aij* =1, *aji*=1, in particular *aii*=1 for

, *Fj*

(ID2)

) in a numeri‐

(*n*1)=0; *μT*<sup>3</sup>

Given a greater number of possible conditions (for example, greater number of the duplicat‐ ed coils of the sensor), one can develop a discrete-ordered scale of state parameters (Figure 7). For further analysis of the system, enter the faults influence indexes at each level of hierar‐ chy, using a method of pairwise comparison as it is carried out in the hierarchy analysis

Quantitative judgements on the importance of faults are performed for each pair of faults

*A*=(*aij*), (*i*, *j* = 1, 2, 3).

12 1

L L

L

*a a*

*n n*

12 2

1/ 1 .

L L LL

é ù ê ú

1/ 1/ 1

ë û

cal expression with the numbers *aij*, the problem is reduced to that *n* possible faults *F*1, *F*2, …, *Fn* will receive a corresponding set of numerical weights *ω*1, *ω*2, …, *ωn*, which would reflect

1 2

*a a*

After the representation of quantitative judgements about the fault pairs (*Fi*

the fixed judgements about the condition of the gas turbine subsystem.

*a a <sup>A</sup>*

*n n*

1

=

of faults in comparison with *Fj*

"Fault" (*B*2).

222 Progress in Gas Turbine Performance

method.

tance (respective values) *Fi*

**1.** If *aij* =*α*, *aji* = 1/*α*, *α*≠ 0.

all *i*.

(*F*i , *Fj*

tion with the following membership functions:

*μT*1

Other rules are created in the similar way.

(*n*1)=1; *μT*<sup>2</sup>

) and these are represented by matrix *A* of the *n*×*n* size.

where *aij* is the relative importance of fault *Fi* in regard to *Fj*

**2.** If fault *Fi* has identical relative importance with *Fj*

Elements *aij* are defined by the following rules:

Thus, a back-symmetric matrix A is obtained:

Write down this rule as follows:

If the expert judgement is absolute at all comparisons for all *i*, *j*, *k*, then matrix *A* is called *consistent*. If the expert judgement is absolute at all comparisons for all *i*, *j*, *k*, then matrix *A* is called *consistent*.

If the diagonal of matrix A consists of units (*aij* = 1) and *A* is the consistent matrix, then at

numerical expression with the numbers *aij*, the problem is reduced to that *n* possible faults *F*1, *F*2, …, *Fn* will receive a corresponding set of numerical weights *ω*1, *ω*2, …, *ωn*, which would reflect the fixed judgements about the condition of the gas turbine subsystem.

small changes in *aij* the greatest eigenvalue λmax is close to *n*, and the other eigenvalue are close to zero. Based on the matrix of pair comparison values of faults *A*, the vector of priorities for fault classification is obtained, along with vector *ω* satisfying the criterion: If the diagonal of matrix A consists of units (*aij* = 1) and *A* is the consistent matrix, then at small changes in *aij* the greatest eigenvalue λmax is close to *n*, and the other eigenvalue are close to zero.

*A* max , where *ω* is the eigenvector of matrix *A* and λmax is the maximum eigenvalue, which is close Based on the matrix of pair comparison values of faults *A*, the vector of priorities for fault classification is obtained, along with vector *ω* satisfying the criterion:

$$A\omega = \lambda\_{\text{max}}\omega\_{\text{\textdegree}}$$

where *ω* is the eigenvector of matrix *A* and λmax is the maximum eigenvalue, which is close to the matrix order *n*.

As it is desirable to have the normalized solution, let's slightly change *ω*, considering *α* =∑ *i*=1 *n ωi* and replacing *ω* with (1/*α*) *ω*. This provides uniqueness, and also that∑ *i*=1 *n ω<sup>i</sup>* =1.

$$A = \begin{bmatrix} F\_1 & F\_2 & \cdots & F\_n \\ F\_1 \blacksquare \omega\_1 / \omega\_1 & \omega\_1 / \omega\_2 & \cdots & \omega\_1 / \omega\_n \\ F\_2 \blacksquare \omega\_2 / \omega\_1 & \omega\_2 / \omega\_2 & \cdots & \omega\_2 / \omega\_n \\ \vdots & \vdots & \vdots & \vdots \\ F\_n \blacksquare \omega\_n / \omega\_1 & \omega\_n / \omega\_2 & \cdots & \omega\_n / \omega\_n \end{bmatrix} \begin{bmatrix} \omega\_1 \\ \omega\_2 \\ \vdots \\ \omega\_n \end{bmatrix} = \lambda\_{\max} \begin{bmatrix} \omega\_1 \\ \omega\_2 \\ \vdots \\ \omega\_n \end{bmatrix}$$

The use of the hierarchical representation allows a small amount of "short" fuzzy rules to

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

http://dx.doi.org/10.5772/54443

225

A promising approach to constructing intelligent systems of control, diagnosis and monitor‐ ing could be the stochastic modelling on the basis of Markov chains combined with the for‐

Within a fuzzy hierarchical model, consider fault development processes with the use of Markov chains. Such dynamic models allow to investigate the change of elements' states in time. Fault development can include not only single faults and their combinations, but also

During FADEC analysis, classification, formalization and representation of processes of con‐ dition monitoring and fault diagnosis for the main subsystems of gas turbines (control, mon‐ itoring, fuel supply etc.) is carried out. These processes are represented in the form of

The transition probability matrix of a Markov chain for modeling faults and their conse‐

**Single faults Faults hierarchy**

The hierarchical Markov model is built in the generalized state space where physical param‐ eters and binary fault flags are used for the estimation of a state vector of the element, unit

Elements

Units

Power-plant

Markov chains which allow to analyze the state dynamics of the power-plant.

quences, has a universal structure for all levels of system decomposition (Figure 8):

adequately describe multidimensional dependencies between inputs and outputs.

**4. Fuzzy hierarchical Markov state models**

sequences (chains) of so-called "consecutive" faults [10, 11].

malized hierarchy theory.

**•** system as a whole (power plant);

**Figure 8.** Hierarchical structure of Markov model of fault

**•** constructon units;

**•** elements.

Note that small changes in *aij* cause small change in λmax, then the deviation of the latter from *n* is a coordination measure. It allows estimating proximity of the obtained scale to the basic scale of relations. Hence, the coordination index

$$(\lambda\_{\text{max}} - n)/(n-1)$$

is considered to be an indicator of "proximity to coordination". Generally, if this number is not greater than 0.1 then it is possible to be satisfied with the judgements about the faults importance.

At each level *hi* of the hierarchy for *n* elements of the gas turbine and its subsystems, the state vector {operational, degradation, fault} is determined, taking into account the influence coefficients of failures:

$$S\_{h\_i}(\mathfrak{x}\_n) = \mu\_{h\_i}(\mathfrak{x}\_n) \cdot \omega\_{i'}.$$

where*μh <sup>i</sup>* (*xn*) is the membership function value of the element *xn* (degradation degree). To determine the element/unit state of the hierarchy at a higher level *Sh <sup>i</sup>* (*xn*) for the input states of low-level *Sh <sup>i</sup>*−<sup>1</sup> (*xn*) one stage of defuzzification is performed.

The output value *Sh <sup>i</sup>* (*xn*) is presented in the form of the determined vector of state with pa‐ rameters { operational, degradation, fault }.

The state estimation begins with the bottom level of hierarchy. The description of a state set obtained by means of fuzzification and deffuzification with the use of the logic operations of disjunction ∨ (summing), and conjunction ∧ (multiplication), which are designated as fol‐ lows:

In performing operation "AND" in the inference system of fuzzy logic, the terminal tops are summed in order to determine the general state at one level of hierarchy that is presented as follows:

$$\mathbf{X}\_{\Sigma} = \mathbf{x}\_1 \lor \mathbf{x}\_2 \lor \dots \lor \mathbf{x}\_n$$

In performing operation "OR", the "worst" state vector is chosen, with the maximum param‐ eters of degradation *μ*degradation(*x*) or faults *μ*fault(*x*). The selector of maximum chooses from the fault influence indexes the one that has the maximum value.

The use of the hierarchical representation allows a small amount of "short" fuzzy rules to adequately describe multidimensional dependencies between inputs and outputs.

#### **4. Fuzzy hierarchical Markov state models**

A promising approach to constructing intelligent systems of control, diagnosis and monitor‐ ing could be the stochastic modelling on the basis of Markov chains combined with the for‐ malized hierarchy theory.

Within a fuzzy hierarchical model, consider fault development processes with the use of Markov chains. Such dynamic models allow to investigate the change of elements' states in time. Fault development can include not only single faults and their combinations, but also sequences (chains) of so-called "consecutive" faults [10, 11].

During FADEC analysis, classification, formalization and representation of processes of con‐ dition monitoring and fault diagnosis for the main subsystems of gas turbines (control, mon‐ itoring, fuel supply etc.) is carried out. These processes are represented in the form of Markov chains which allow to analyze the state dynamics of the power-plant.

The transition probability matrix of a Markov chain for modeling faults and their conse‐ quences, has a universal structure for all levels of system decomposition (Figure 8):


*A*=

scale of relations. Hence, the coordination index

importance.

where*μh <sup>i</sup>*

lows:

follows:

At each level *hi*

of low-level *Sh <sup>i</sup>*−<sup>1</sup>

The output value *Sh <sup>i</sup>*

rameters { operational, degradation, fault }.

coefficients of failures:

224 Progress in Gas Turbine Performance

*F*1 *F*2 ⋮ *Fn*

*ω*<sup>1</sup> / *ω*<sup>1</sup> *ω*<sup>1</sup> / *ω*<sup>2</sup> ⋯ *ω*<sup>1</sup> / *ω<sup>n</sup> ω*<sup>2</sup> / *ω*<sup>1</sup> *ω*<sup>2</sup> / *ω*<sup>2</sup> ⋯ *ω*<sup>2</sup> / *ω<sup>n</sup>* ⋮ ⋮⋮⋮ *ω<sup>n</sup>* / *ω*<sup>1</sup> *ω<sup>n</sup>* / *ω*<sup>2</sup> ⋯ *ω<sup>n</sup>* / *ω<sup>n</sup>*

Note that small changes in *aij* cause small change in λmax, then the deviation of the latter from *n* is a coordination measure. It allows estimating proximity of the obtained scale to the basic

(*λ*max −*n*)/ (*n* −1)

is considered to be an indicator of "proximity to coordination". Generally, if this number is not greater than 0.1 then it is possible to be satisfied with the judgements about the faults

state vector {operational, degradation, fault} is determined, taking into account the influence

The state estimation begins with the bottom level of hierarchy. The description of a state set obtained by means of fuzzification and deffuzification with the use of the logic operations of disjunction ∨ (summing), and conjunction ∧ (multiplication), which are designated as fol‐

In performing operation "AND" in the inference system of fuzzy logic, the terminal tops are summed in order to determine the general state at one level of hierarchy that is presented as

*X<sup>Σ</sup>* = *x*<sup>1</sup> ∨ *x*<sup>2</sup> ∨ … ∨ *xn*

In performing operation "OR", the "worst" state vector is chosen, with the maximum param‐ eters of degradation *μ*degradation(*x*) or faults *μ*fault(*x*). The selector of maximum chooses from the

(*xn*) =*μh <sup>i</sup>*

*Sh i*

(*xn*) one stage of defuzzification is performed.

determine the element/unit state of the hierarchy at a higher level *Sh <sup>i</sup>*

logic operation "AND" logic operation "OR"

fault influence indexes the one that has the maximum value.

of the hierarchy for *n* elements of the gas turbine and its subsystems, the

(*xn*) is presented in the form of the determined vector of state with pa‐

(*xn*) ⋅*ω<sup>i</sup>* ,

(*xn*) is the membership function value of the element *xn* (degradation degree). To

*ω*1 *ω*2 ⋮ *ωn*

=*λ*max

*ω*1 *ω*2 ⋮ *ωn*

.

(*xn*) for the input states

*F*<sup>1</sup> *F*<sup>2</sup> ⋯ *Fn*

**Figure 8.** Hierarchical structure of Markov model of fault

The hierarchical Markov model is built in the generalized state space where physical param‐ eters and binary fault flags are used for the estimation of a state vector of the element, unit and power-plant. The state vector includes three parameters { operational, degradation, fault } which allow to track the fault development and degradation process of the system. During FADEC diagnosis, the area of single faults is mostly considered. The proposed Mar‐ kov model enables to present the system with multiple faults and their sequences. The top state level of a system reflects in the aggregated form the information on faults at the lower state levels.

essary to have the model of the system behavior in various states with various flight condi‐

As the basic mathematical model of the controlled plant, the description in the state space is

where *X* ∈*R <sup>s</sup>* is the *s*-dimensioned state vector; *U* ∈*R <sup>s</sup>* is the *s*-dimensioned control vector;

ent random variables. Thus, the dynamic object described by this finite-difference equation, with input coordinate (control variable) *U* and output coordinate (state variable) *X*, in the closed scheme of the automatic control system is the controlled Markov process [12, 13].

The level of quantisation allows the Markov process to be converted into the Markov chain. Provided *ξ*(*t*) is a stationary process, the Markov chain will be homogeneous. Such chain is described by the means of the stochastic transition probability matrix *P* with the dimensions (*m*×*m*), where *m* is the number of the chain states. Each element of the matrix *Pij* represents

, *X* (*t* + 1)= *Xj*

1 ; ; 1, 1, . 2 2 *m*

*j*

Condition (2) means that the matrix *P* should be stochastic and define the full system of

The size of the matrix *P* is defined by the prior information on the order of the object model (1) and the number of the sampling intervals Δ*x* and Δ*u*. The transition probabilities are then

The statistical estimation of the transition probabilities for the controlled Markov chain is performed as the calculation of the frequencies for the corresponding events during obser‐

*ijk*

*N* =

where the numerator *Nijk* is the number of the following events:

, *U* (*tn*) =*Uk* }, and the denominator corresponds to the number of

*ijk*

vation and the subsequent calculation of the elements of matrix *P* using the formula:

*ijk m*

=

*P*

1

*N*

*j*

=

*x x Xx x P i m*

events. The sum of elements in each row of the stochastic matrix should equal 1.

}, ∀*n* ∈ *N* ,

\_\_\_\_\_

å (3)

Î- + = = ê ú ë û <sup>å</sup> (2)

x

(1)

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

into the condition *Xj*

is the vector of independ‐

http://dx.doi.org/10.5772/54443

227

during

tion to guarantee system safety, reliable localization and accommodation of faults.

*Xt Xt Ut t* ( 1) ( ) ( ) ( ), += + + **ABF**

considered in the form of stochastic difference equations:

A, B and F are (*n*×*n*), (*n*×*s*) and (*n*×*r*) matrices accordingly; *ξ* ∈*R <sup>s</sup>*

the probability of the system transition from the condition *Х<sup>i</sup>*

*Pij* =Prob{*X* (*t*) = *Xi*

*ii i ij*

é ù D D

estimated as relative frequencies of the corresponding discrete events.

the time interval Δ*T*:

{*X* (*tn*) = *Xi*

, *X* (*tn*+1) = *Xj*

The elements' state at the levels of the hierarchy depends on the previous values of state pa‐ rameters of the elements, values of membership functions and fault influence indexes.

For the estimation of transition probabilities between the states of a Markov chain, it is re‐ quired to calculate relative frequencies of events such as *Si* →*Sj* for a given interval of time. In particular, at the top level, the number of events during one flight (Figure 9) can be of interest.


**Figure 9.** Transition probability matrix of power-plant during one flight

The most important events during flight are the emergency turning off of the engine stop (shutdown) and the possibility of its restart. For the probability estimation of such events, it is required to use statistics on all park of the same type engines. For realization of such esti‐ mation methods, it is required that flights information was available on each plane and power-plant. Such information should be gathered and stored in a uniform format and should be available for processing. Modern information technologies open possibilities for such research. To analyze fault development processes of one FADEC, it is possible to use results of the automated tests at the hardware-in-the-loop test bed with modeling of various faults and their combinations. In any case, to receive reliable statistical estimates one needs a representative sample of rather large amount of data.

In the analysis of the Markov model, the relation of the transition probability matrix with state of elements and subsystems at each level of hierarchy is considered. Therefore it is nec‐ essary to have the model of the system behavior in various states with various flight condi‐ tion to guarantee system safety, reliable localization and accommodation of faults.

and power-plant. The state vector includes three parameters { operational, degradation, fault } which allow to track the fault development and degradation process of the system. During FADEC diagnosis, the area of single faults is mostly considered. The proposed Mar‐ kov model enables to present the system with multiple faults and their sequences. The top state level of a system reflects in the aggregated form the information on faults at the lower

The elements' state at the levels of the hierarchy depends on the previous values of state pa‐ rameters of the elements, values of membership functions and fault influence indexes.

For the estimation of transition probabilities between the states of a Markov chain, it is re‐

particular, at the top level, the number of events during one flight (Figure 9) can be of interest.

*P***<sup>11</sup>**

*P***<sup>21</sup>**

*P***<sup>31</sup>**

The most important events during flight are the emergency turning off of the engine stop (shutdown) and the possibility of its restart. For the probability estimation of such events, it is required to use statistics on all park of the same type engines. For realization of such esti‐ mation methods, it is required that flights information was available on each plane and power-plant. Such information should be gathered and stored in a uniform format and should be available for processing. Modern information technologies open possibilities for such research. To analyze fault development processes of one FADEC, it is possible to use results of the automated tests at the hardware-in-the-loop test bed with modeling of various faults and their combinations. In any case, to receive reliable statistical estimates one needs a

In the analysis of the Markov model, the relation of the transition probability matrix with state of elements and subsystems at each level of hierarchy is considered. Therefore it is nec‐

**123**

*P***<sup>12</sup>**

*P***<sup>13</sup>**

*P***<sup>23</sup>**

*P***<sup>33</sup>**

*P***<sup>22</sup>**

*P***<sup>32</sup>**

**Engine restart in flight**

for a given interval of time. In

quired to calculate relative frequencies of events such as *Si* →*Sj*

**1. Operational** 

Prob{ } *P SS ij* = *i j*

**2. Degradation**

**3. Fault leading to engine stop**

**Figure 9.** Transition probability matrix of power-plant during one flight

representative sample of rather large amount of data.

**state**

state levels.

226 Progress in Gas Turbine Performance

As the basic mathematical model of the controlled plant, the description in the state space is considered in the form of stochastic difference equations:

$$X(t+1) = \mathbf{A}X(t) + \mathbf{B}lI(t) + \mathbf{F}\xi(t),\tag{1}$$

where *X* ∈*R <sup>s</sup>* is the *s*-dimensioned state vector; *U* ∈*R <sup>s</sup>* is the *s*-dimensioned control vector; A, B and F are (*n*×*n*), (*n*×*s*) and (*n*×*r*) matrices accordingly; *ξ* ∈*R <sup>s</sup>* is the vector of independ‐ ent random variables. Thus, the dynamic object described by this finite-difference equation, with input coordinate (control variable) *U* and output coordinate (state variable) *X*, in the closed scheme of the automatic control system is the controlled Markov process [12, 13].

The level of quantisation allows the Markov process to be converted into the Markov chain. Provided *ξ*(*t*) is a stationary process, the Markov chain will be homogeneous. Such chain is described by the means of the stochastic transition probability matrix *P* with the dimensions (*m*×*m*), where *m* is the number of the chain states. Each element of the matrix *Pij* represents the probability of the system transition from the condition *Х<sup>i</sup>* into the condition *Xj* during the time interval Δ*T*:

$$\begin{aligned} P\_{\vec{\eta}} &= \text{Prob}\{X(t) = X\_{\cdot i} \mid X(t+1) = X\_{\cdot j}\} & \quad \forall \ n \in \mathbb{N}\_{\prime} \\\\ X\_{i} &\in \left[ \left. \infty\_{i} - \frac{\Delta x}{2} ; \ge \varepsilon\_{i} + \frac{\Delta x}{2} \right| ; \quad \sum\_{j=1}^{m} P\_{\vec{\eta}j} = 1 ; \ i = \overline{1, m} \right. \end{aligned} \tag{2}$$

Condition (2) means that the matrix *P* should be stochastic and define the full system of events. The sum of elements in each row of the stochastic matrix should equal 1.

The size of the matrix *P* is defined by the prior information on the order of the object model (1) and the number of the sampling intervals Δ*x* and Δ*u*. The transition probabilities are then estimated as relative frequencies of the corresponding discrete events.

The statistical estimation of the transition probabilities for the controlled Markov chain is performed as the calculation of the frequencies for the corresponding events during obser‐ vation and the subsequent calculation of the elements of matrix *P* using the formula:

$$P\_{ijk} = \frac{N\_{ijk}}{\sum\_{j=1}^{m} N\_{ijk}} \tag{3}$$

where the numerator *Nijk* is the number of the following events: {*X* (*tn*) = *Xi* , *X* (*tn*+1) = *Xj* , *U* (*tn*) =*Uk* }, and the denominator corresponds to the number of events such as{*X* (*tn*) = *Xi* , *U* (*tn*) =*Uk* }. Thus, for any combination of state *Xi* and control *Uk*, a full system of events will consist of the set of the state transitions *Xj* .

greater "intelligence" and is designed independently in the assumption of ideal system sta‐

Engine

Actuator P11 P12

2

> 2

**. . . . . . . . . . . . . . . . . . .**

Elements

Elements

P1 1 P12

> Pn n

a1

a2

.

.

.

a*k*

Oil system

P1 1 P12

> Pn n

a1

a2

.

.

.

a*k*

and units

and units

**. . .**

Pnn

a1

a2

.

.

.

a*k*

*k*

*n*

**. . .**

a1

a2

.

.

.

a*k*

Pn n a1

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

http://dx.doi.org/10.5772/54443

229

a2

.

.

.

a*k*

P11 P12

Pn n

P11 P12

bility of the lower level.

**Level 1**

State of gas

turbine

**Level 2**

State of gas

turbine

subsystems

**Level** *h*

State of sensors

and actuators

Intermediate states

Elements

FADEC P11 P12

> P11 P12

> > Pnn

a1

a2

.

.

.

a*k*

Pn n a1

a2

.

.

.

a*k*

1

1 and units

**Figure 11.** Hierarchical fuzzy Markov model of gas turbine states

The normalisation of Equation (3) makes matrix *P* stochastic. As a result, the set of probabili‐ ties in each row *Pij* describes the full system of events for which the sum of probabilities is equal to unit:

$$\sum\_{j=1}^{m} P\_{ij\vec{k}} = 1\dots$$

The estimation of a transition probability matrix of the Markov model consists of creation of multidimensional histograms which represent an estimate of joint distribution [14, 15].

The use of the hierarchical Markov model allows to "compress" information which has been recorded during one flight, and to present it in a more compact form. In this case, the possi‐ bility of analysis and forecast of dynamics of degradation degree (Figure 10) opens. It is pos‐ sible to analyze the state dynamics of elements and functions at each level of hierarchy in time for decision-making support. The use of the hierarchical Markov model allows to "compress" information which has been recorded during one flight, and to present it in a more compact form. In this case, the possibility of analysis and forecast of dynamics of degradation degree (Figure 10) opens. It is possible to analyze the state dynamics of elements and functions at each level of hierarchy in time for decisionmaking support.

Figure 10.Dynamics of state parameters during flight **Figure 10.** Dynamics of state parameters during flight

The analysis of fault information and state change can be carried out over flight data for the whole duration of maintenance and the whole "fleet" of engines and their systems (Figure 11). Such analysis will assist to increase efficiency for processes of experimental maintenance development and monitoring system support. Given statistics on all park of engines within several years, it is possible to build empirical estimates of probabilities of the first and The analysis of fault information and state change can be carried out over flight data for the whole duration of maintenance and the whole "fleet" of engines and their systems (Figure 11). Such analysis will assist to increase efficiency for processes of experimental maintenance development and monitoring system support.

second type errors. Thus, possibilities of application of hierarchical Markov models for the gas turbine and its FADEC for compact representation of Given statistics on all park of engines within several years, it is possible to build empirical estimates of probabilities of the first and second type errors.

information on flight and for the assessment of "sensitivity" of the monitoring system according to actual data are considered. The levels of hierarchy differ with the ways of introducing redundancy and realization of system safety with use of intellectual algorithms of control and diagnosis. Each higher level of hierarchy has greater "intelligence" and is designed independently in the assumption of ideal system stability of the lower level. Thus, possibilities of application of hierarchical Markov models for the gas turbine and its FADEC for compact representation of information on flight and for the assessment of "sensi‐ tivity" of the monitoring system according to actual data are considered. The levels of hier‐ archy differ with the ways of introducing redundancy and realization of system safety with use of intellectual algorithms of control and diagnosis. Each higher level of hierarchy has

greater "intelligence" and is designed independently in the assumption of ideal system sta‐ bility of the lower level.

events such as{*X* (*tn*) = *Xi*

228 Progress in Gas Turbine Performance

time for decision-making support.

Elements

maintenance development and monitoring system support.

development and monitoring system support.

estimates of probabilities of the first and second type errors.

**Figure 10.** Dynamics of state parameters during flight

assumption of ideal system stability of the lower level.

Figure 10.Dynamics of state parameters during flight

equal to unit:

making support.

second type errors.

, *U* (*tn*) =*Uk* }. Thus, for any combination of state *Xi*

The normalisation of Equation (3) makes matrix *P* stochastic. As a result, the set of probabili‐ ties in each row *Pij* describes the full system of events for which the sum of probabilities is

*Pijk* =1.

The estimation of a transition probability matrix of the Markov model consists of creation of multidimensional histograms which represent an estimate of joint distribution [14, 15].

The use of the hierarchical Markov model allows to "compress" information which has been recorded during one flight, and to present it in a more compact form. In this case, the possi‐ bility of analysis and forecast of dynamics of degradation degree (Figure 10) opens. It is pos‐ sible to analyze the state dynamics of elements and functions at each level of hierarchy in

The use of the hierarchical Markov model allows to "compress" information which has been recorded during one flight, and to present it in a more compact form. In this case, the possibility of analysis and forecast of dynamics of degradation degree (Figure 10) opens. It is possible to analyze the state dynamics of elements and functions at each level of hierarchy in time for decision-

The analysis of fault information and state change can be carried out over flight data for the whole duration of maintenance and the whole "fleet" of engines and their systems (Figure 11). Such analysis will assist to increase efficiency for processes of experimental

The analysis of fault information and state change can be carried out over flight data for the whole duration of maintenance and the whole "fleet" of engines and their systems (Figure 11). Such analysis will assist to increase efficiency for processes of experimental maintenance

Given statistics on all park of engines within several years, it is possible to build empirical estimates of probabilities of the first and

Thus, possibilities of application of hierarchical Markov models for the gas turbine and its FADEC for compact representation of information on flight and for the assessment of "sensitivity" of the monitoring system according to actual data are considered. The levels of hierarchy differ with the ways of introducing redundancy and realization of system safety with use of intellectual algorithms of control and diagnosis. Each higher level of hierarchy has greater "intelligence" and is designed independently in the

Thus, possibilities of application of hierarchical Markov models for the gas turbine and its FADEC for compact representation of information on flight and for the assessment of "sensi‐ tivity" of the monitoring system according to actual data are considered. The levels of hier‐ archy differ with the ways of introducing redundancy and realization of system safety with use of intellectual algorithms of control and diagnosis. Each higher level of hierarchy has

Given statistics on all park of engines within several years, it is possible to build empirical

∑ *j*=1 *m*

full system of events will consist of the set of the state transitions *Xj*

and control *Uk*, a

.

**Figure 11.** Hierarchical fuzzy Markov model of gas turbine states

Consider an example. In Figure 12, the FADEC state estimation with faults is presented on the basis of the degradation degree of the elements.

( ) ( ) ( )

fuel feed α is also good working.

not influencing the system operability.

or in maintenance on the ground.

FADEC is shown.

sumption.

( )( ) ( )( ) ( )( )

0,2 0,5 1 0,5 0,60 0,7 0,5 0 0,5 0,35 0,1 0,5 0 0,5 0,05

ê ú ´ +´ ë û ë û

temperature *T* is "operational", therefore, the sensor *T* state equals

é ù ´ +´ é ù ê ú ê ú = ´ +´ = ê ú

( ) ( ) ( )

ê ú <sup>ê</sup> = = úê ú ë û <sup>ë</sup> ûë û

é ù é = = ùé ù ê ú ê úê ú

, 0,2 1

, 0,1 0

11 12 1 4 o 11 12 d 1 4 11 12 f 1 4

*Sn n n T Sn n n T Sn n n T*

m

m

m

1 1 2 2 3 3

*T T T T T T*

, 0,7 0 0,5; 0,5

ê ú = = =´ = <sup>ê</sup> úê ú é ù ë û ê ú <sup>ê</sup> úê ú

 m

 m

 m

.

The state of an element of a higher level is calculated by multiplication of the current state to fault influence indexes of the fault elements. The state of both measurement channels of

At the 8th level the state of two sensors *n* and *T* after similar calculations becomes equal { 0,78; 0,19; 0,03 } that indicated the system degradation in the part of control of fuel con‐

For the estimation of a state of the fuel consumption control function, the "OR" operation is also used. The state of the actuator of fuel consumption control circuit is characterized by the parameters { 0,66; 0,34; 0 }. The state of FADEC is characterized by the fault of fuel con‐ sumption control function or the state of the guide vanes control function. Using the opera‐ tion "OR", the state of FADEC is detected as { 0,66; 0,34; 0 }. In this example, the whole system is considered to be operational, whereas partial degradation is observed, which is

Thus, the technique of state parameters determination for FADEC and its systems on the ba‐ sis of fuzzy logic and Markov chains is proposed. This technique can be used during flight

At the present time a necessary condition for realization of intellectual algorithms is the complete development of the distributed intellectual control models focused on control opti‐ mization, forecasting and system safety [16, 17]. In Figure 13, the scheme of the distributed

Thus, in each sensor or actuator, it is necessary to have physically built-in control system (or function) to form and monitor the fault signals in the unit, communication lines, cooperating sensors, indication devices and systems [18]. The use of the built-in monitoring control sys‐ tems working in real time allows to obtain a number of additional possibilities for improv‐

ing control quality and system operational characteristics as followings:

**•** emergency states detection of the control object and system;

**•** fault detection of elements of the control object;

( ) ( ) ( )

(ID1)

231

The sensor of

*S*(*T* ) <sup>o</sup> = 1

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

http://dx.doi.org/10.5772/54443

*S*(*T* ) <sup>d</sup> = 0

*S*(*T* ) <sup>f</sup>= 0

**Figure 12.** Example of hierarchical estimation of state parameters

At the 10th level the BMS detected a fault of measurement in the form of break of the first coil of parameter *n*11 (shaft speed sensor). On the basis of the fuzzy rule *R*(2), the parameters of measurement state of *n*1 in the channel A are characterized by the following three values

*μT*1 (*n*11)=0, 2 *μT*2 (*n*11)=0, 7 *μT*3 (*n*11)=0, 1 .

$$R^{(2)}: \text{IF} \left(\eta\_{11} = A\_2 \text{ AND } \eta\_{12} = A\_1\right) \text{ THEN } \ y = T\_2 \Rightarrow$$

$$\mathsf{L}\mu\_{T\_1}(\mathsf{n\_1}) = 0, \text{ 2: } \quad \mu\_{T\_2}(\mathsf{n\_1}) = 0, \text{ 7: } \quad \mu\_{T\_3}(\mathsf{n\_1}) = 0, \text{ 1]}.$$

The measurement state in the channel B is defined as *μT*1 (*n*12)=1 *μT*2 (*n*12)=0 *μT*3 (*n*12)=0 , because no faults were de‐

tected. At the 9th level, the sensor state of the *n* is obtained using the multiplication of the vector of state parameters and faults influence indexes:

$$\begin{aligned} \left[\begin{aligned} &S\left(n\_{11},n\_{12}\right)\_{\text{o}}\right]\_{\text{o}} = \left[\begin{aligned} &\mu\_{T\_{1}}\left(n\_{1}\right)=0,2\\ &S\left(n\_{11},n\_{12}\right)\_{\text{d}} = \left[\begin{matrix} &\mu\_{T\_{1}}\left(n\_{1}\right)=0,2\\ \mu\_{T\_{2}}\left(n\_{1}\right)=0,7\\ &\mu\_{T\_{3}}\left(n\_{1}\right)=0,1 \end{matrix}\right] \begin{aligned} &\mu\_{T\_{2}}\left(T\_{4}\right)=1\\ &\mu\_{T\_{2}}\left(T\_{4}\right)=0\\ &\mu\_{T\_{3}}\left(T\_{4}\right)=0 \end{aligned} \end{aligned} \right] \times \left[\begin{aligned} &\mu\_{T\_{2}}\left(T\_{4}\right)=1\\ &\mu\_{T\_{3}}\left(T\_{4}\right)=0\\ &\mu\_{T\_{3}}\left(T\_{4}\right)=0\\ &\mu\_{T\_{4}}\left(0,1\times0,5\right)=1\\ &\left(0,1\times0,5\right)+\left(0\times0,5\right) \end{aligned} \right] \tag{ID1}$$

The state of an element of a higher level is calculated by multiplication of the current state to fault influence indexes of the fault elements. The state of both measurement channels of

temperature *T* is "operational", therefore, the sensor *T* state equals *S*(*T* ) <sup>o</sup> = 1 *S*(*T* ) <sup>d</sup> = 0 *S*(*T* ) <sup>f</sup>= 0 The sensor of

fuel feed α is also good working.

Consider an example. In Figure 12, the FADEC state estimation with faults is presented on

**State of FADEC**

**. . .**

State of measurement of *n*<sup>11</sup> (А)

D2 (0,5) D4 (0,5)

{0,2 0,7 0,1} {1 0 0}

At the 10th level the BMS detected a fault of measurement in the form of break of the first coil of parameter *n*11 (shaft speed sensor). On the basis of the fuzzy rule *R*(2), the parameters of measurement state of *n*1 in the channel A are characterized by the following three values

*<sup>R</sup>* (2) :*IF* (*n*<sup>11</sup> <sup>=</sup> *<sup>A</sup>*<sup>2</sup> *AND <sup>n</sup>*<sup>12</sup> <sup>=</sup> *<sup>A</sup>*1) *THEN <sup>y</sup>* <sup>=</sup>*T*2<sup>⇒</sup>

tected. At the 9th level, the sensor state of the *n* is obtained using the multiplication of the

(*n*1)=0, 7; *μT*<sup>3</sup>

*μT*1

*μT*2

*μT*3

(*n*1)=0, 1 .

(*n*12)=1

(*n*12)=0

(*n*12)=0

State of sensor *n*

State of sensors *n* и *T*

A1 (1) A<sup>2</sup> (1)

{0,66 0,34 0} {1 0 0}

State of measurement of *n*<sup>12</sup> (B)

{0,6 0,35 0,05}

{0,66 0,34 0} {0,78 0,19 0,03}

C2 (0,55) C3 (0,45) Cn

State of measurement of *T* (А)

State of sensor *T*

State of actuator of fuel consumption control circuit

State of guide vanes control function А(B)

D5 (0,5) D6 (0,5)

{1 0 0} {1 0 0}

B1 (1) B2 (1)

**. . .**

State of measurement of *T* (B)

, because no faults were de‐

{1 0 0}

the basis of the degradation degree of the elements.

State of fuel consumption control function А(B)

{0,66 0,34 0}

**Level 10**

*μT*1

*μT*2

*μT*3

(*n*11)=0, 2

(*n*11)=0, 7

.

(*n*11)=0, 1

**Level 9**

State of sensor α

{1 0 0} {1 0 0}

State of measurement of α (B)

**Figure 12.** Example of hierarchical estimation of state parameters

*μT*1

The measurement state in the channel B is defined as

vector of state parameters and faults influence indexes:

(*n*1)=0, 2; *μT*<sup>2</sup>

D1 (0,5) D3 (0,5)

C1 (1)

State of measurement of α (А)

{1 0 0}

**Level 8**

**Level 1**

230 Progress in Gas Turbine Performance

**. . .**

> At the 8th level the state of two sensors *n* and *T* after similar calculations becomes equal { 0,78; 0,19; 0,03 } that indicated the system degradation in the part of control of fuel con‐ sumption.

> For the estimation of a state of the fuel consumption control function, the "OR" operation is also used. The state of the actuator of fuel consumption control circuit is characterized by the parameters { 0,66; 0,34; 0 }. The state of FADEC is characterized by the fault of fuel con‐ sumption control function or the state of the guide vanes control function. Using the opera‐ tion "OR", the state of FADEC is detected as { 0,66; 0,34; 0 }. In this example, the whole system is considered to be operational, whereas partial degradation is observed, which is not influencing the system operability.

> Thus, the technique of state parameters determination for FADEC and its systems on the ba‐ sis of fuzzy logic and Markov chains is proposed. This technique can be used during flight or in maintenance on the ground.

> At the present time a necessary condition for realization of intellectual algorithms is the complete development of the distributed intellectual control models focused on control opti‐ mization, forecasting and system safety [16, 17]. In Figure 13, the scheme of the distributed FADEC is shown.

> Thus, in each sensor or actuator, it is necessary to have physically built-in control system (or function) to form and monitor the fault signals in the unit, communication lines, cooperating sensors, indication devices and systems [18]. The use of the built-in monitoring control sys‐ tems working in real time allows to obtain a number of additional possibilities for improv‐ ing control quality and system operational characteristics as followings:


The application of hierarchical Markov models for the analysis of experimental data is also considered for control system development: as the compact representation of information of

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

http://dx.doi.org/10.5772/54443

233

The work was supported by the grants from the Russian Foundation for Basic Research (RFFI) №12-08-31279, №12-08-97027 and the Ministry of Education and Science of the Russi‐

\*Address all correspondence to: gennadyg\_98@yahoo.com; abdulnagimov@gmail.com

Automated Control and Management Systems Department, Ufa State Aviation Technical

[1] Kulikov GG. Principles of design of digital control systems for aero engines. In Cher‐ kasov BA. (ed.), Control and automatics of jet engines. Mashinostroyeniye, Moscow;

a system state change during flight, the estimation of transition probabilities.

**Nomenclature**

FADEC – Full Authority Digital Engine Control.

G. G. Kulikov, V. Yu. Arkov and A.I. Abdulnagimov\*

BMS – built-in monitoring system. *ω* – weighting coefficient of fault.

*S* – state (condition) of gas turbines.

*μ* – membership function.

*P* – probability.

an Federation.

**Author details**

University, Ufa, Russia

1988. p253-274.

**References**

*X* – state variable.

*U* – control variable.

**Acknowledgements**

**•** state diagnosis and parametrical degradation of the object.

**Figure 13.** Distributed architecture of FADEC

#### **5. Conclusion**

In this chapter, the hierarchical fuzzy Markov modeling of fault developments processes has been proposed for the analysis of an airplane system safety. The hierarchical model integra‐ tes functional, physical structure of gas turbine and its FADEC elements and units, the tree of states, a tree of fault influence indexes. This model allows to decompose the power-plant for a quantitative estimates of degradation state and gradual faults. The analysis of hierar‐ chies allows to utilize the state model on the basis of fault development processes which es‐ timates the power-plant state at each level of hierarchy. Furthermore, the technique of determination of state parameters of the gas turbine and its systems on the basis of fuzzy logic is presented. The state of each element, unit and system is represented in the form of a vector with parameters { operational, degradation, faults }. The use of the proposed indica‐ tor "degradation degree" allows to obtain an objective quantitative estimate of the current state which can be used as, the "distance" to a critical situation and the reserve of time for decision-making in-flight. This indicator is defined on the basis of the discrete-ordered scale and fault influence indexes that allows to determine about 30 % of gradual faults in gas tur‐ bine and its systems at the stage of fault development. The examples of fuzzy rules on the basis of expert knowledge are given, whereas fuzzy logic is used for interpolation.

The application of hierarchical Markov models for the analysis of experimental data is also considered for control system development: as the compact representation of information of a system state change during flight, the estimation of transition probabilities.

#### **Nomenclature**

**•** state diagnosis and parametrical degradation of the object.

Actuator Controller

**Digital bus**

Actuator

232 Progress in Gas Turbine Performance

Sensor

Sensor

**Figure 13.** Distributed architecture of FADEC

**5. Conclusion**

**Tracking system** 

**FADEC**

Sensor

In this chapter, the hierarchical fuzzy Markov modeling of fault developments processes has been proposed for the analysis of an airplane system safety. The hierarchical model integra‐ tes functional, physical structure of gas turbine and its FADEC elements and units, the tree of states, a tree of fault influence indexes. This model allows to decompose the power-plant for a quantitative estimates of degradation state and gradual faults. The analysis of hierar‐ chies allows to utilize the state model on the basis of fault development processes which es‐ timates the power-plant state at each level of hierarchy. Furthermore, the technique of determination of state parameters of the gas turbine and its systems on the basis of fuzzy logic is presented. The state of each element, unit and system is represented in the form of a vector with parameters { operational, degradation, faults }. The use of the proposed indica‐ tor "degradation degree" allows to obtain an objective quantitative estimate of the current state which can be used as, the "distance" to a critical situation and the reserve of time for decision-making in-flight. This indicator is defined on the basis of the discrete-ordered scale and fault influence indexes that allows to determine about 30 % of gradual faults in gas tur‐ bine and its systems at the stage of fault development. The examples of fuzzy rules on the

basis of expert knowledge are given, whereas fuzzy logic is used for interpolation.

**Functional unit**

Sensor

Sensor

Actuator


#### **Acknowledgements**

The work was supported by the grants from the Russian Foundation for Basic Research (RFFI) №12-08-31279, №12-08-97027 and the Ministry of Education and Science of the Russi‐ an Federation.

#### **Author details**

G. G. Kulikov, V. Yu. Arkov and A.I. Abdulnagimov\*

\*Address all correspondence to: gennadyg\_98@yahoo.com; abdulnagimov@gmail.com

Automated Control and Management Systems Department, Ufa State Aviation Technical University, Ufa, Russia

#### **References**

[1] Kulikov GG. Principles of design of digital control systems for aero engines. In Cher‐ kasov BA. (ed.), Control and automatics of jet engines. Mashinostroyeniye, Moscow; 1988. p253-274.

[2] Arkov VY, Kulikov GG, Breikin TV. Life cycle support for dynamic modelling of gas turbines. Prepr. 15th Triennial IFAC World Congress, Barcelona, Spain; 2002. p2135-2140.

[16] Vasilyev VI, Ilyasov BG, Valeyev SS. Intelligent Control Systems for gas Turbine en‐ gines. Proc. of the Second Scientific Technical Seminar on GT Engines, Turkey, Istan‐

System Safety of Gas Turbines: Hierarchical Fuzzy Markov Modelling

http://dx.doi.org/10.5772/54443

235

[17] Culley D, Thomas R and Saus J. Concepts for Distributed Engine Control, AIAA-2007-5709, 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Ex‐

[18] Kulikov GG., Arkov VYu., Abdulnagimov AI. Hierarhical Fuzzy Markov Modelling for System Safety of Power-plant : Proc. of 4th International Symposium on Jet Pro‐ pulsion and Power Engineering, September 10-12, 2012, Xi'an, China: Northwestern

bul; 1996. p71-78.

hibit, Cincinnati, Ohio, July 8-11, 2007.

Polytechnical University Press; 2012. p. 589-594.


[16] Vasilyev VI, Ilyasov BG, Valeyev SS. Intelligent Control Systems for gas Turbine en‐ gines. Proc. of the Second Scientific Technical Seminar on GT Engines, Turkey, Istan‐ bul; 1996. p71-78.

[2] Arkov VY, Kulikov GG, Breikin TV. Life cycle support for dynamic modelling of gas turbines. Prepr. 15th Triennial IFAC World Congress, Barcelona, Spain; 2002.

[4] Saaty TL. Analytic Hierarchy Process: Planning, Priority Setting, Resource Alloca‐

[5] Saaty TL, Vargas LG. Models, Methods, Concepts & Applications of the Analytic Hi‐

[6] SAE ARP 4761. Guidelines and Methods for Conducting the Safety Assessment Proc‐ ess on Civil Airborne Systems And Equipment. Aerospace recommended practice;

[7] Arkov V, Evans DC, Fleming PJ, et al. System identification strategies applied to air‐ craft gas turbine engines. Proc. 14th Triennal IFAC World Congress; 1999. p145-152.

[8] Kulikov G, Breikin T, Arkov V and Fleming P. Real-time simulation of aviation en‐ gines for FADEC test-beds. Proc. Int. Gas Turbine Congress, Kobe, Japan; 1999.

[10] Kulikov GG, Fleming PJ, Breikin TV, Arkov VY. Markov modelling of complex dy‐ namic systems: identification, simulation, and condition monitoring with example of

[11] Breikin TV, Arkov VY, Kulikov GG. On stochastic system identification: Markov

[12] Breikin TV, Arkov VY, Kulikov GG. Application of Markov chains to identification of gas turbine engine dynamic models. International Journal of Systems Science 2006;

[13] Kulikov G, Arkov V, Lyantsev O, et al. Dynamic modelling of gas turbines: identifi‐ cation, simulation, condition monitoring, and optimal control. Springer, London,

[14] Kulikov G, Arkov V, Abdulnagimov A. Markov modelling for energy efficient con‐ trol of gas turbine power plant. Proc. IFAC Conf. on Control Methodologies and Technology for Energy Efficiency, CMTEE-2010, Faro, Portugal; 2010. http://

[15] Arkov V, Kulikov G, Fatikov V, et al. Intelligent control and monitoring unit and its investigation using Markov modelling. Proc. IFAC Int. Conf. on Intelligent Control

Systems and Signal Processing ICONS-2003, Faro, Portugal; 2003. p489-493

www.ifac-papersonline.net/Detailed/42981.html

[9] Zadeh LA. "Fuzzy algorithms," Information and Control 1968;12(2): 94-102.

digital automatic control system of gas turbine engine. USATU, Ufa; 1998.

models approach. Proc 2nd Asian Control Conf ASCC'97; 1997. p775-778.

[3] Kuo BC. Automatic control systems. Prentice-Hall: Englewood Cliffs; 1995.

tion. McGraw-Hill, New York, London; 1980.

erarchy Process. Kluwer Academic Publisher; 2001.

p2135-2140.

234 Progress in Gas Turbine Performance

1996.

p949-952.

37(3) 197-205.

New York; 2004.


**Chapter 10**

**Thermal Shock and Cycling Behavior of**

Abdullah Cahit Karaoglanli, Kazuhiro Ogawa,

Additional information is available at the end of the chapter

Ahmet Türk and Ismail Ozdemir

http://dx.doi.org/10.5772/54412

**1. Introduction**

**Thermal Barrier Coatings (TBCs) Used in Gas Turbines**

Gas turbine engines work as a power generating facility and are used in aviation industry to provide thrust by converting combustion products into kinetic energy [1-3]. Basic concerns regarding the improvements in modern gas turbine engines are higher efficiency and per‐ formance. Increase in power and efficiency of gas turbine engines can be achieved through increase in turbine inlet temperatures [1,4]. For this purpose, the materials used should have perfect mechanical strength and corrosion resistance and thus be able to work under aggressive environments and high temperatures [2]. The temperatures that turbine blades are exposed to can be close to the melting point of the superalloys. For this reason, internal cooling by cooling channels and insulation by thermal barrier coatings (TBCs) is used in order to lower the temperature of turbine blades and prevent the failure of superalloy substrates [1-4]. By utilizing TBCs in gas turbines, higher turbine inlet temperatures are allowed and as a result an increase in turbine efficiency is obtained [5]. TBCs are employed in a variety of areas such as power plants, advanced turbo engine combustion chambers, turbine blades, vanes and are often used under high thermal loads [6-11]. Various thermal shock tests are conducted by aerospace and land gas turbine manufacturers in order to develop TBCs and investigate the quality control characteristics. Despite that fact, a standardized method is still lacking. The reason lies behind the difficulty of finding a testing method that can simulate all the service and loading conditions. Present testing systems developed by the engine manufacturers for simulation of real thermal conditions in engines consist of; burner rig thermal shock testing units, jet engine thermal shock testing units and furnace cycle tests [16-20]. In this study, thermal cycle and thermal shock behavior of TBC systems under service conditions are examined, and a collection of testing methods used in evaluation of performance and endur‐

> © 2013 Karaoglanli et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

> © 2013 Karaoglanli et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
