See Figures 16–18.

#### Figure 16.

Pipeline degradation (a) and RUL (b) under linear damage law for middle-pressure mode of excitation (mode 2).

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

#### Figure 18.

5.2 The parameter simulation in the pipeline prognostic for mode 2

Pipeline degradation (a) and RUL (b) under linear damage law for middle-pressure mode of excitation (mode 2).

Degradation and CPP parameters with Chf (a) and with MChf (b) for mode 2.

See Figures 16–18.

The complex probability vector Z in terms of t for mode 1.

Fault Detection, Diagnosis and Prognosis

Figure 15.

Figure 16.

Figure 17.

86

Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 2.

### 5.2.1 The complex probability cubes for mode 2

#### See Figures 19–21.

Figure 19. DOK and Chf in terms of t and of each other for mode 2.

#### 5.3 The parameter simulation in the pipeline prognostic for mode 3

#### See Figures 22–24.

#### 5.3.1 The complex probability cubes for mode 3

#### See Figures 25–27.

Figure 22.

Figure 23.

Figure 24.

89

Pipeline degradation (a) and RUL (b) under linear damage law for low-pressure mode of excitation (mode 3).

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

DOI: http://dx.doi.org/10.5772/intechopen.90157

Degradation and CPP parameters with Chf (a) and with MChf (b) for mode 3.

Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 3.

Figure 20.

Pr and Pm/i in terms of t and of each other for mode 2.

Figure 21. The complex probability vector Z in terms of t for mode 2.

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

Figure 22. Pipeline degradation (a) and RUL (b) under linear damage law for low-pressure mode of excitation (mode 3).

Figure 23. Degradation and CPP parameters with Chf (a) and with MChf (b) for mode 3.

#### Figure 24.

Degradation, rescaled RUL, and CPP parameters with Chf (a) and with MChf (b) for mode 3.

Figure 20.

Figure 21.

88

Pr and Pm/i in terms of t and of each other for mode 2.

Fault Detection, Diagnosis and Prognosis

The complex probability vector Z in terms of t for mode 2.

6. Final analysis: explanation and the general prognostic equations

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

machine).

91

Figure 27.

The complex probability vector Z in terms of t for mode 3.

DOI: http://dx.doi.org/10.5772/intechopen.90157

We will present in this section the original general prognostic equations, we will interpret all the achieved simulations and the obtained data, and we will do a final analysis. Also, we will illustrate the results and a detailed discussion of the all the previous simulations and figures and of the following corresponding tables.

Firstly, we have linked prognostic characterized by the degradation D(t) with probability theory characterized by the CDF F(t) by supposing that D(t) = F(t) and the justification for this assumption were given. Consequently, the deterministic D(t) computed from deterministic analytic linear prognostic becomes a nondeterministic cumulative probability distribution function. Therefore, the deterministic and discrete variable of pressure cycles time t becomes a random and discrete variable. Thus, the resultant of all the factors influencing the system which was deterministic becomes a stochastic resultant because D(t) quantifies now the random degradation of the pipeline in terms of the random cycle time t. Accordingly, all the parameters' exact values of the D(t) expression (Eq. 6) become now the mean values of the stochastic factors influencing the pipeline and are embodied by PDFs as functions of the stochastic variable of pressure cycle time t (refer to Section 3.5). As a matter of fact, this is the real-world case where randomness is omnipresent in one form or another. What we consider and judge as a deterministic phenomenon is nothing in reality but a simplification and an approximation of an actual chaotic and stochastic phenomenon and experiment due to the impact of a huge number of nondeterministic and deterministic forces and factors (a good example is a lottery

Subsequently, we do an updated follow-up of the performance of the random degradation in terms of time or cycle number, which is subject to non-chaotic and

#### Figure 25.

DOK and Chf in terms of t and of each other for mode 3.

Figure 26. Pr and Pm/i in terms of t and of each other for mode 3.

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

Figure 27. The complex probability vector Z in terms of t for mode 3.

#### 6. Final analysis: explanation and the general prognostic equations

We will present in this section the original general prognostic equations, we will interpret all the achieved simulations and the obtained data, and we will do a final analysis. Also, we will illustrate the results and a detailed discussion of the all the previous simulations and figures and of the following corresponding tables.

Firstly, we have linked prognostic characterized by the degradation D(t) with probability theory characterized by the CDF F(t) by supposing that D(t) = F(t) and the justification for this assumption were given. Consequently, the deterministic D(t) computed from deterministic analytic linear prognostic becomes a nondeterministic cumulative probability distribution function. Therefore, the deterministic and discrete variable of pressure cycles time t becomes a random and discrete variable. Thus, the resultant of all the factors influencing the system which was deterministic becomes a stochastic resultant because D(t) quantifies now the random degradation of the pipeline in terms of the random cycle time t. Accordingly, all the parameters' exact values of the D(t) expression (Eq. 6) become now the mean values of the stochastic factors influencing the pipeline and are embodied by PDFs as functions of the stochastic variable of pressure cycle time t (refer to Section 3.5). As a matter of fact, this is the real-world case where randomness is omnipresent in one form or another. What we consider and judge as a deterministic phenomenon is nothing in reality but a simplification and an approximation of an actual chaotic and stochastic phenomenon and experiment due to the impact of a huge number of nondeterministic and deterministic forces and factors (a good example is a lottery machine).

Subsequently, we do an updated follow-up of the performance of the random degradation in terms of time or cycle number, which is subject to non-chaotic and

Figure 25.

Figure 26.

90

DOK and Chf in terms of t and of each other for mode 3.

Fault Detection, Diagnosis and Prognosis

Pr and Pm/i in terms of t and of each other for mode 3.

chaotic influences, by using the quantity Prð Þ tk =ψ<sup>j</sup> due to its definition that evaluates the jumps in the stochastic degradation CDF D(t). Hence,

Moreover, since Prð Þ¼ tk ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �, this leads to the following recursive

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

In the case of general prognostic, if we possess the PDF of system failure then it can be included in Eqs. (25) and (26) and hence evaluate at any instant tk the system degradation and vice versa. Consequently, all the other CPP model parameters (DOK, Chf, MChf, Pr, Pm, Pm/i, Z, Pc) will follow. This would be our new prognostic

> Xt¼tk t¼t<sup>0</sup>

It is crucial to indicate here that the PDFfailure function of the system failure has all the mathematical characteristics and all the possible features of a probability density function whether it is a continuous or a discrete stochastic function and it can follow any imaginable probability distribution in condition only that it characterizes the failure function and the random degradation of the studied system whether it is a petrochemical pipe in the buried, unburied, or offshore case or a vehicle suspension system or any nondeterministic system under the effect of randomness and chaos. In fact, the function PDFfailure inherits all the attributes and features of the failure system function and of the nondeterministic degradation. Furthermore, by applying CPP to the pipe prognostic, and in the three simulations of pressure modes, we were successful in the original prognostic model to quantify in R (our real laboratory) both our chaos embodied by Chf and MChf and our certain knowledge embodied by DOK. These three parameters of CPP are evaluated and caused by the resultant of all the nonrandom (deterministic) and random (nondeterministic) aspects influencing the system of pipeline. Knowing that, in the novel paradigm, the factors' resultant effect on RUL and D is materialized by the jumps in their curves and is accordingly expressed and concretized in R by Pr and in M by Pm. As it was defined in CPP, M is an imaginary probability extension of the real probability set R, and the complex probability set C is the sum of both probability sets; thus, C ¼ R þM. Because Pm = i(1 – Pr), therefore it is the complementary probability of Pr in M. Hence, if Pr is identified as the failure probability of the system in R at the pressure cycle time t = tk, then Pm is identified as the corresponding probability in the set M that the system failure will not occur at the same pressure time t = tk. So, Pm is the associated probability in the set M of the system survival at t = tk. It follows that Pm/i = 1 – Pr is the associated probability but in the set R of the system survival at the same pressure cycles time. Accordingly, we know that the sum in R of both complementary probabilities is surely 1 from classical probability theory. This sum is nothing but PC which is equal to Pr + Pm/i = Pr + (1 – Pr) = 1 always. The sum in C of both complementary probabilities is the complex random number and vector Z which is equal to

Pr + Pm = Pr + i(1 – Pr). And as the complex probability cubes show and illustrate, we realize that Z is the sum in C of the real probability of failure and of the imaginary probability of survival in the complex probability plane that has the equation

D tð Þ¼ <sup>k</sup> D tð Þþ <sup>k</sup>�<sup>1</sup> Prð Þ tk =ψj; for every tk, t<sup>0</sup> ≤ tk ≤ tC: (26)

ProbðÞ¼ <sup>t</sup> <sup>X</sup>t¼tk

t¼t<sup>0</sup>

D tð Þ¼ <sup>k</sup> D tð Þþ <sup>k</sup>�<sup>1</sup> PDFfailureð Þ tk (28)

PDFfailureð Þt (27)

relation:

93

model general equation:

And the recursive relation

DOI: http://dx.doi.org/10.5772/intechopen.90157

with PDFfailureð Þ¼ t<sup>0</sup> D0.

D tð Þ¼ <sup>k</sup> Probð Þ¼ t<sup>0</sup> ≤ t≤tk

$$P\_r(t\_k) = \boldsymbol{\nu}\_j \times [D(t\_k) - D(t\_{k-1})], \text{for any pressure mode } j = 1, 2, 3.$$

Referring to classical probability theory, this makes Prð Þ tk =ψ<sup>j</sup> the system probability of failure at <sup>t</sup> <sup>=</sup> tk, with 0<sup>≤</sup> Prð Þ tk <sup>=</sup>ψ<sup>j</sup> <sup>≤</sup> 1 and <sup>P</sup>t¼tC <sup>t</sup>¼t<sup>0</sup> Prð Þ<sup>t</sup> <sup>=</sup>ψ<sup>j</sup> = [sum of all the jumps in D from t<sup>0</sup> to tC] = DC = 1, just like any probability density function (PDF).

In addition, in the simulations, a constant and very small increments in t have been taken which lead to very small increments in D and hence in Prð Þ tk =ψj. So, we have multiplied those very small jumps in D by a simulation magnifying factor that we called ψ<sup>j</sup> . Note that 1=ψ<sup>j</sup> is a normalizing constant that is used to reduce Prð Þ tk function to a probability density function with a total probability equal to one. 1=ψ<sup>j</sup> is a function of the pressure mode and conditions, and it depends on the parameters in the degradation (Eq. (6)). We have from the simulations ψ<sup>1</sup> ¼ 5082 for the highpressure mode (j = 1, mode 1), ψ<sup>2</sup> ¼ 6737 for the middle-pressure mode (j = 2, mode 2), and ψ<sup>3</sup> ¼ 9151 for the low-pressure mode (j = 3, mode 3). So we get the following: if t tends to t<sup>0</sup> = 0, then Prð Þ tk tends to 0, and if t tends to t<sup>C</sup> then Prð Þ tk tends to 1, so 0≤Prð Þ tk <sup>≤</sup>1 and <sup>P</sup><sup>t</sup>¼tC <sup>t</sup>¼t<sup>0</sup> PrðÞ¼ <sup>t</sup> <sup>ψ</sup><sup>j</sup> � DC <sup>¼</sup> <sup>ψ</sup><sup>j</sup> � <sup>1</sup> <sup>¼</sup> <sup>ψ</sup><sup>j</sup> as if Prð Þ tk was a CDF although mathematically speaking it is not at all. This, since Prð Þ tk is not cumulative, it is just ψ<sup>j</sup> times the probability of failure at t = tk. Hence, in the simulations, Prð Þ tk becomes now the probability that the system failure occurs at t = tk and is used accordingly to compute all the CPP parameters.

Therefore, D tð Þ¼ <sup>k</sup> F tð Þ¼ <sup>k</sup> Probð Þ 0≤t≤ tk = Prob(t = 0 or t = 1 or t = 2 or … or t = tk) = sum of all failure probabilities between 0 and tk = probability that failure will occur somewhere between 0 and tk. So, if tk = 0 then Probð Þ¼ t≤ 0 Dð Þ¼ 0 D<sup>0</sup>

= probability that failure will occur at t = 0 and before. If tk = t<sup>C</sup> then Probð Þ¼ 0≤t≤tC D tð Þ¼ <sup>C</sup> 1 = sum of all failure probabilities between 0 and tC = probability that failure will occur somewhere between 0 and tC. If tk > tC then Probð Þ¼ t>tC D tð Þ¼ <sup>C</sup> 1 = probability that failure will occur beyond tC. We can see that failure probability increases with the increase of the pressure cycles time tk until at the end it becomes 1 when tk ≥ tC.

Hence, if t<sup>0</sup> ¼ 0 and D tð Þ¼ <sup>0</sup> 0 then

$$D(t\_k) = P\_{rob}(\mathbf{0} \le t \le t\_k) = \sum\_{t=0}^{t=t\_k} P\_{rob}(t) = \sum\_{t=0}^{t=t\_k} P\_r(t) / \mu\_f$$

This implies that D tð Þ¼ <sup>C</sup> Probð Þ¼ 0≤t≤tC t P¼tC t¼0 ProbðÞ¼ <sup>t</sup> <sup>t</sup> P¼tC t¼0 Prð Þt =ψ<sup>j</sup> ¼ 1 and

$$D(\mathbf{0}) = P\_{rob}(t \le \mathbf{0}) = \sum\_{t=0}^{t=0} P\_{rob}(t) = \sum\_{t=0}^{t=0} P\_r(t) / \mu\_j = P\_r(\mathbf{0}) / \mu\_j = \mathbf{0}.$$

If t<sup>0</sup> 6¼ 0 and D tð Þ<sup>0</sup> 6¼ 0, then the prognostic equation in the new model is

$$D(t\_k) = P\_{mb}(t\_0 \le t \le t\_k) = \sum\_{t=t\_0}^{t=t\_k} P\_{mb}(t) = \sum\_{t=t\_0}^{t=t\_k} P\_r(t) / \mu\_j \tag{25}$$

for any mode j of pressure profile and with Prð Þ t<sup>0</sup> =ψ<sup>j</sup> ¼ D0.

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

Moreover, since Prð Þ¼ tk ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �, this leads to the following recursive relation:

$$D(t\_k) = D(t\_{k-1}) + P\_r(t\_k) / \mu\_j; \text{for every } t\_k, t\_0 \le t\_k \le t\_{\mathcal{C}}.\tag{26}$$

In the case of general prognostic, if we possess the PDF of system failure then it can be included in Eqs. (25) and (26) and hence evaluate at any instant tk the system degradation and vice versa. Consequently, all the other CPP model parameters (DOK, Chf, MChf, Pr, Pm, Pm/i, Z, Pc) will follow. This would be our new prognostic model general equation:

$$D(t\_k) = P\_{mb}(t\_0 \le t \le t\_k) = \sum\_{t=t\_0}^{t=t\_k} P\_{mb}(t) = \sum\_{t=t\_0}^{t=t\_k} PDF\_{failure}(t) \tag{27}$$

And the recursive relation

chaotic influences, by using the quantity Prð Þ tk =ψ<sup>j</sup> due to its definition that evalu-

Prð Þ¼ tk ψ<sup>j</sup> � D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �, for any pressure mode j ¼ 1, 2, 3:

jumps in D from t<sup>0</sup> to tC] = DC = 1, just like any probability density function (PDF). In addition, in the simulations, a constant and very small increments in t have been taken which lead to very small increments in D and hence in Prð Þ tk =ψj. So, we have multiplied those very small jumps in D by a simulation magnifying factor that

function to a probability density function with a total probability equal to one. 1=ψ<sup>j</sup> is a function of the pressure mode and conditions, and it depends on the parameters in the degradation (Eq. (6)). We have from the simulations ψ<sup>1</sup> ¼ 5082 for the highpressure mode (j = 1, mode 1), ψ<sup>2</sup> ¼ 6737 for the middle-pressure mode (j = 2, mode 2), and ψ<sup>3</sup> ¼ 9151 for the low-pressure mode (j = 3, mode 3). So we get the following: if t tends to t<sup>0</sup> = 0, then Prð Þ tk tends to 0, and if t tends to t<sup>C</sup> then Prð Þ tk

CDF although mathematically speaking it is not at all. This, since Prð Þ tk is not cumulative, it is just ψ<sup>j</sup> times the probability of failure at t = tk. Hence, in the simulations, Prð Þ tk becomes now the probability that the system failure occurs at

Therefore, D tð Þ¼ <sup>k</sup> F tð Þ¼ <sup>k</sup> Probð Þ 0≤t≤ tk = Prob(t = 0 or t = 1 or t = 2 or … or t = tk) = sum of all failure probabilities between 0 and tk = probability that failure will occur somewhere between 0 and tk. So, if tk = 0 then Probð Þ¼ t≤ 0 Dð Þ¼ 0 D<sup>0</sup>

> Xt¼tk t¼0

ProbðÞ¼ <sup>t</sup> <sup>X</sup><sup>t</sup>¼<sup>0</sup>

If t<sup>0</sup> 6¼ 0 and D tð Þ<sup>0</sup> 6¼ 0, then the prognostic equation in the new model is

t P¼tC t¼0

t¼0

Xt¼tk t¼t<sup>0</sup>

ProbðÞ¼ <sup>t</sup> <sup>X</sup><sup>t</sup>¼tk

ProbðÞ¼ <sup>t</sup> <sup>X</sup><sup>t</sup>¼tk

t¼t<sup>0</sup>

t¼0

ProbðÞ¼ <sup>t</sup> <sup>t</sup>

Prð Þt =ψ<sup>j</sup>

P¼tC t¼0

Prð Þt =ψ<sup>j</sup> ¼ Prð Þ 0 =ψ<sup>j</sup> ¼ 0:

Prð Þt =ψ<sup>j</sup> ¼ 1 and

Prð Þt =ψ<sup>j</sup> (25)

t = tk and is used accordingly to compute all the CPP parameters.

= probability that failure will occur at t = 0 and before. If tk = t<sup>C</sup> then Probð Þ¼ 0≤t≤tC D tð Þ¼ <sup>C</sup> 1 = sum of all failure probabilities between 0 and tC = probability that failure will occur somewhere between 0 and tC. If tk > tC then Probð Þ¼ t>tC D tð Þ¼ <sup>C</sup> 1 = probability that failure will occur beyond tC. We can see that failure probability increases with the increase of the pressure cycles time tk

Referring to classical probability theory, this makes Prð Þ tk =ψ<sup>j</sup> the system proba-

. Note that 1=ψ<sup>j</sup> is a normalizing constant that is used to reduce Prð Þ tk

<sup>t</sup>¼t<sup>0</sup> Prð Þ<sup>t</sup> <sup>=</sup>ψ<sup>j</sup> = [sum of all the

<sup>t</sup>¼t<sup>0</sup> PrðÞ¼ <sup>t</sup> <sup>ψ</sup><sup>j</sup> � DC <sup>¼</sup> <sup>ψ</sup><sup>j</sup> � <sup>1</sup> <sup>¼</sup> <sup>ψ</sup><sup>j</sup> as if Prð Þ tk was a

ates the jumps in the stochastic degradation CDF D(t). Hence,

bility of failure at <sup>t</sup> <sup>=</sup> tk, with 0<sup>≤</sup> Prð Þ tk <sup>=</sup>ψ<sup>j</sup> <sup>≤</sup> 1 and <sup>P</sup>t¼tC

we called ψ<sup>j</sup>

tends to 1, so 0≤Prð Þ tk <sup>≤</sup>1 and <sup>P</sup><sup>t</sup>¼tC

Fault Detection, Diagnosis and Prognosis

until at the end it becomes 1 when tk ≥ tC. Hence, if t<sup>0</sup> ¼ 0 and D tð Þ¼ <sup>0</sup> 0 then

D tð Þ¼ <sup>k</sup> Probð Þ¼ 0≤t≤tk

t¼0

for any mode j of pressure profile and with Prð Þ t<sup>0</sup> =ψ<sup>j</sup> ¼ D0.

D tð Þ¼ <sup>k</sup> Probð Þ¼ t<sup>0</sup> ≤t≤tk

This implies that D tð Þ¼ <sup>C</sup> Probð Þ¼ 0≤t≤tC

<sup>D</sup>ð Þ¼ <sup>0</sup> Probð Þ¼ <sup>t</sup><sup>≤</sup> <sup>0</sup> <sup>X</sup><sup>t</sup>¼<sup>0</sup>

92

$$D(\mathbf{t}\_k) = D(\mathbf{t}\_{k-1}) + DDF\_{\text{failure}}(\mathbf{t}\_k) \tag{28}$$

with PDFfailureð Þ¼ t<sup>0</sup> D0.

It is crucial to indicate here that the PDFfailure function of the system failure has all the mathematical characteristics and all the possible features of a probability density function whether it is a continuous or a discrete stochastic function and it can follow any imaginable probability distribution in condition only that it characterizes the failure function and the random degradation of the studied system whether it is a petrochemical pipe in the buried, unburied, or offshore case or a vehicle suspension system or any nondeterministic system under the effect of randomness and chaos. In fact, the function PDFfailure inherits all the attributes and features of the failure system function and of the nondeterministic degradation.

Furthermore, by applying CPP to the pipe prognostic, and in the three simulations of pressure modes, we were successful in the original prognostic model to quantify in R (our real laboratory) both our chaos embodied by Chf and MChf and our certain knowledge embodied by DOK. These three parameters of CPP are evaluated and caused by the resultant of all the nonrandom (deterministic) and random (nondeterministic) aspects influencing the system of pipeline. Knowing that, in the novel paradigm, the factors' resultant effect on RUL and D is materialized by the jumps in their curves and is accordingly expressed and concretized in R by Pr and in M by Pm. As it was defined in CPP, M is an imaginary probability extension of the real probability set R, and the complex probability set C is the sum of both probability sets; thus, C ¼ R þM. Because Pm = i(1 – Pr), therefore it is the complementary probability of Pr in M. Hence, if Pr is identified as the failure probability of the system in R at the pressure cycle time t = tk, then Pm is identified as the corresponding probability in the set M that the system failure will not occur at the same pressure time t = tk. So, Pm is the associated probability in the set M of the system survival at t = tk. It follows that Pm/i = 1 – Pr is the associated probability but in the set R of the system survival at the same pressure cycles time. Accordingly, we know that the sum in R of both complementary probabilities is surely 1 from classical probability theory. This sum is nothing but PC which is equal to Pr + Pm/i = Pr + (1 – Pr) = 1 always. The sum in C of both complementary probabilities is the complex random number and vector Z which is equal to Pr + Pm = Pr + i(1 – Pr). And as the complex probability cubes show and illustrate, we realize that Z is the sum in C of the real probability of failure and of the imaginary probability of survival in the complex probability plane that has the equation

Pr(t) = iPm(t) + 1 for ∀t : 0≤ t≤tC, ∀Pr : 0≤ Pr ≤1, and ∀Pm : 0≤Pm ≤i. What is interesting is that the square of the norm of <sup>Z</sup> which is j j <sup>Z</sup> <sup>2</sup> is nothing but DOK, as it was proved in CPP and in the new model. Moreover, since MChf = �2iPrPm = 2PrPm/i, therefore it is twice the product in R of both the probability of failure and the probability of survival, and it quantifies the magnitude of chaos since it is always 0 or positive. All the simulations show and prove all these facts.

<sup>¼</sup> <sup>X</sup>t¼tC t¼tkþ<sup>1</sup>

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

Moreover, from Eqs. (25), (26), (27), and (28) and for any mode j of pressure

Prob½ �¼ RUL tð Þ<sup>k</sup> 1 � D tð Þ¼ <sup>k</sup> 1 � D tð Þþ <sup>k</sup>�<sup>1</sup> Prð Þ tk =ψ<sup>j</sup>

¼ 1 � D tð Þþ <sup>k</sup>�<sup>1</sup> PDFfailureð Þ tk

¼ 1 � 1 � Prob RUL tð Þ <sup>k</sup>�<sup>1</sup> ½ �þ Prð Þ tk =ψ<sup>j</sup>

In the ideal case, if all the factors are 100% deterministic, then we have in R the probability of failure for tk <tC is 0 and is 1 for tk ≥tC; accordingly the probability of system survival for tk < tC is 1 and is 0 for tk ≥tC, since certain failure will occur only at tk ¼ tC. So, degradation is determined surely everywhere in R, and its CDF is replaced by a deterministic function and curve. Therefore, chaos is null, and hence Chf = MChf = 0, and DOK = 1 always for all 0≤ tk ≤ tC. Thus, Prob½RUL tð Þ <sup>k</sup> < tC � ¼ 1

Furthermore, at each instant t in the original prognostic paradigm, the stochastic RUL(t) and D(t) are predicted with certitude in the complex probability set C with Pc<sup>2</sup> = DOK – Chf = DOK + MChf maintained as equal to 1 through a continuous compensation between Chf and DOK. This compensation is from the instant t = 0 where D(t) = D<sup>0</sup> = 0.020408 ≈ 0 until the instant of failure tC where D(tC) = 1. Moreover, we can realize that DOK does not include any uncertain knowledge (with a probability less than 100%); it is the measure of our certain knowledge (probability = 100%) about the expected event. We can understand that we have eliminated and subtracted in the equation above all the random factors and chaos (Chf)

it only exists (if it does) in R; consequently, this has led to a 100% deterministic outcome and experiment in C since the probability Pc is constantly equal to 1. This is one of the advantages of extending R to M and therefore of working in

for any mode j of pressure profile.

DOI: http://dx.doi.org/10.5772/intechopen.90157

where Prob RUL tð Þ <sup>k</sup>�<sup>1</sup> ½ �¼ 1 � D tð Þ <sup>k</sup>�<sup>1</sup> .

from our random experiment when computing Pc<sup>2</sup>

The new prognostic model parameters for any pipeline internal pressure mode.

and Prob½RUL tð Þ <sup>k</sup> ≥tC � ¼ 0.

Table 2.

95

profile, we have the following recursive relations:

PDFfailureð Þt (37)

� � (39)

n o (40)

¼ Prob RUL tð Þ <sup>k</sup>�<sup>1</sup> ½ �� Prð Þ tk =ψ<sup>j</sup> (41) ¼ Prob RUL tð Þ <sup>k</sup>�<sup>1</sup> ½ �� PDFfailureð Þ tk (42)

n o (38)

; hence no chaos exists in C, and

We can conclude from all the above that since D(t) is a CDF, since the factor resultant is random, and since the jumps in D are the simulations failure probabilities Prð Þ tk , then we are dealing with a random experiment, thus the natural appearance of Chf, MChf, DOK, Z, and hence Pc. So, we get in the simulations:

$$\text{Cbf}(\mathbf{t}\_k) = -2P\_r(\mathbf{t}\_k)P\_m(\mathbf{t}\_k)/i = -2\left\{\boldsymbol{\nu}\_j[D(\mathbf{t}\_k) - D(\mathbf{t}\_{k-1})] \right\} \left\{\mathbf{1} - \boldsymbol{\nu}\_j[D(\mathbf{t}\_k) - D(\mathbf{t}\_{k-1})] \right\}.\tag{29}$$

$$\text{MChf}(\mathbf{t}\_k) = |\text{Chf}(\mathbf{t}\_k)| = 2\left\{\boldsymbol{\nu}\_j[D(\mathbf{t}\_k) - D(\mathbf{t}\_{k-1})] \right\} \left\{1 - \boldsymbol{\nu}\_j[D(\mathbf{t}\_k) - D(\mathbf{t}\_{k-1})] \right\}.\tag{30}$$

$$\begin{split} D\text{OK}(\mathfrak{t}\_{k}) &= \mathbf{1} - 2\mathcal{P}\_{r}(\mathfrak{t}\_{k})\mathcal{P}\_{m}(\mathfrak{t}\_{k})/i \\ &= \mathbf{1} - 2\left\{\boldsymbol{\mu}\_{j}[\boldsymbol{D}(\mathfrak{t}\_{k}) - \boldsymbol{D}(\mathfrak{t}\_{k-1})] \right\} \left\{\mathbf{1} - \boldsymbol{\mu}\_{j}[\boldsymbol{D}(\mathfrak{t}\_{k}) - \boldsymbol{D}(\mathfrak{t}\_{k-1})] \right\}. \end{split} \tag{31}$$

$$Z(\mathbf{t}\_k) = P\_r(\mathbf{t}\_k) + P\_m(\mathbf{t}\_k) = \boldsymbol{\psi}\_j[\boldsymbol{D}(\mathbf{t}\_k) - \boldsymbol{D}(\mathbf{t}\_{k-1})] + i \left\{ \mathbf{1} - \boldsymbol{\psi}\_j[\boldsymbol{D}(\mathbf{t}\_k) - \boldsymbol{D}(\mathbf{t}\_{k-1})] \right\}. \tag{32}$$

$$\text{Pr}^2(t\_k) = \text{DOK}(t\_k) - \text{Cly}(t\_k) = \text{DOK}(t\_k) + \text{MChf}(t\_k) = \text{1;for every } t\_k, 0 \le t\_k \le t\_{\text{C}}.\tag{33}$$

Furthermore, in the new model, we have

$$RUL(t\_k) = t\_C - t\_k \dots$$

Note that since t and D are random, then RUL is also a random function of t. Thus, we have in the set R:

$$\begin{aligned} \left[P\_{mb}\middle|RUL(t\_k)\right] &= P\_{mb}\left(\text{the system } will \text{ survive for } t\_k < t \le t\_C\right) \\ &= 1 - P\_{mb}(the \text{ system } will \text{ fail for } t \le t\_k) \\ &= 1 - D(t\_k) \\ &= \text{Rescaled } \left[RUL(t\_k)\right] \text{ in all the three pressure modes simulations} \end{aligned} \tag{34}$$

Then, we get always Prob½ �þ RUL tð Þ<sup>k</sup> D tð Þ¼ <sup>k</sup> 1 everywhere. This implies that Prob½RUL tð Þ <sup>k</sup> ¼ 0 � ¼ 1 � D tð Þ¼ <sup>k</sup> ¼ 0 1 � D<sup>0</sup> ≈1. and Prob½RUL tð Þ <sup>k</sup> ¼ tC � ¼ 1 � D tð Þ¼ <sup>k</sup> ¼ tC 1 � DC ¼ 1 � 1 ¼ 0. Hence, we reach a new and general prognostic equation for RUL. If t<sup>0</sup> 6¼ 0 and D tð Þ<sup>0</sup> 6¼ 0 then

$$\begin{split} P\_{rob}[RUL(t\_k)] &= P\_{rob}(\text{Survivalent } : t\_k < t \le t\_C) = 1 - P\_{rob}(\text{Failure } : t\_0 \le t \le t\_k) \\ &= 1 - \sum\_{t=t\_0}^{t=t\_k} P\_r(t) / \psi\_j; \quad \text{with } P\_r(t\_0) / \psi\_j = D\_0 \end{split} \tag{35}$$
 
$$\begin{split} &= \mathbf{1} - D(t\_k) = \sum\_{t=t\_{k+1}}^{t=t\_C} P\_r(t) / \psi\_j \\ &= \mathbf{1} - \sum\_{t=t\_0}^{t=t\_k} PDF\_{failure}(t); \quad \text{with } PDF\_{failure}(t\_0) = D\_0 \end{split} \tag{36}$$

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

$$=\sum\_{t=t\_{k+1}}^{t=t\_C} PDF\_{failure}(t) \tag{37}$$

for any mode j of pressure profile.

Pr(t) = iPm(t) + 1 for ∀t : 0≤ t≤tC, ∀Pr : 0≤ Pr ≤1, and ∀Pm : 0≤Pm ≤i. What is interesting is that the square of the norm of <sup>Z</sup> which is j j <sup>Z</sup> <sup>2</sup> is nothing but DOK, as it was proved in CPP and in the new model. Moreover, since MChf = �2iPrPm = 2PrPm/i, therefore it is twice the product in R of both the probability of failure and the probability of survival, and it quantifies the magnitude of chaos since it is always 0 or

We can conclude from all the above that since D(t) is a CDF, since the factor resultant is random, and since the jumps in D are the simulations failure probabilities Prð Þ tk , then we are dealing with a random experiment, thus the natural appearance of Chf, MChf, DOK, Z, and hence Pc. So, we get in the simulations:

n o

Z tð Þ¼ <sup>k</sup> Prð Þþ tk Pmð Þ¼ tk ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ � þ i 1 � ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

ð Þ¼ tk DOK tð Þ� <sup>k</sup> Chf tð Þ¼ <sup>k</sup> DOK tð Þþ <sup>k</sup> MChf tð Þ¼ <sup>k</sup> 1; for every tk, 0≤ tk ≤ tC:

RUL tð Þ¼ <sup>k</sup> tC � tk:

Note that since t and D are random, then RUL is also a random function of t.

¼ Rescaled RUL t ½ � ð Þ<sup>k</sup> in all the three pressure modes simulations

Hence, we reach a new and general prognostic equation for RUL. If t<sup>0</sup> 6¼ 0 and

; with Prð Þ t<sup>0</sup> =ψ<sup>j</sup> ¼ D<sup>0</sup>

PDFfailureð Þt ; with PDFfailureð Þ¼ t<sup>0</sup> D<sup>0</sup>

Prob½ �¼ RUL tð Þ<sup>k</sup> ProbðSurvival : tk <t ≤tCÞ ¼ 1 � Probð Þ Failure : t<sup>0</sup> ≤t≤tk

Prð Þt =ψ<sup>j</sup>

n o

1 � ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ � n o

1 � ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ � n o

n o

1 � ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ � n o :

(29)

: (30)

: (32)

(33)

(34)

(35)

(36)

: (31)

positive. All the simulations show and prove all these facts.

Chf tð Þ¼� <sup>k</sup> 2Prð Þ tk Pmð Þ tk =i ¼ �2 ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

¼ 1 � 2 ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

Prob½ �¼ RUL tð Þ<sup>k</sup> Probð Þ the system will survive for tk <t ≤tC

¼ 1–Probð Þ the system will fail for t ≤tk

Then, we get always Prob½ �þ RUL tð Þ<sup>k</sup> D tð Þ¼ <sup>k</sup> 1 everywhere. This implies that Prob½RUL tð Þ <sup>k</sup> ¼ 0 � ¼ 1 � D tð Þ¼ <sup>k</sup> ¼ 0 1 � D<sup>0</sup> ≈1. and Prob½RUL tð Þ <sup>k</sup> ¼ tC � ¼ 1 � D tð Þ¼ <sup>k</sup> ¼ tC 1 � DC ¼ 1 � 1 ¼ 0.

Prð Þt =ψ<sup>j</sup>

Xt¼tC t¼tkþ<sup>1</sup>

n o

MChf tð Þ¼ <sup>k</sup> j j Chf tð Þ<sup>k</sup> ¼ 2 ψ<sup>j</sup> D tð Þ� <sup>k</sup> D tð Þ <sup>k</sup>�<sup>1</sup> ½ �

Furthermore, in the new model, we have

Thus, we have in the set R:

D tð Þ<sup>0</sup> 6¼ 0 then

94

¼ 1–D tð Þ<sup>k</sup>

<sup>¼</sup> <sup>1</sup> �X<sup>t</sup>¼tk

<sup>¼</sup> <sup>1</sup> �X<sup>t</sup>¼tk

t¼t<sup>0</sup>

¼ 1 � D tð Þ¼ <sup>k</sup>

t¼t<sup>0</sup>

DOK tð Þ¼ <sup>k</sup> 1 � 2Prð Þ tk Pmð Þ tk =i

Fault Detection, Diagnosis and Prognosis

Pc<sup>2</sup>

Moreover, from Eqs. (25), (26), (27), and (28) and for any mode j of pressure profile, we have the following recursive relations:

$$P\_{rob}[RUL(t\_k)] = \mathbf{1} - D(t\_k) = \mathbf{1} - \left\{ D(t\_{k-1}) + P\_r(t\_k) / \psi\_j \right\} \tag{38}$$

$$=\mathbf{1} - \left\{ D(t\_{k-1}) + DDF\_{\text{failure}}(t\_k) \right\} \tag{39}$$

$$=\mathbf{1} - \left\{\mathbf{1} - P\_{rob}[RUL(\mathbf{t}\_{k-1})] + P\_r(\mathbf{t}\_k)/\mu\_j\right\} \tag{40}$$

$$\mathbf{u} = P\_{rab}[RUL(\mathbf{t}\_{k-1})] - P\_r(\mathbf{t}\_k)/\mu\_j \tag{41}$$

$$\mathbf{h} = P\_{rab}[\text{RUL}(\mathbf{t}\_{k-1})] - \text{PDF}\_{failure}(\mathbf{t}\_k) \tag{42}$$

where Prob RUL tð Þ <sup>k</sup>�<sup>1</sup> ½ �¼ 1 � D tð Þ <sup>k</sup>�<sup>1</sup> .

In the ideal case, if all the factors are 100% deterministic, then we have in R the probability of failure for tk <tC is 0 and is 1 for tk ≥tC; accordingly the probability of system survival for tk < tC is 1 and is 0 for tk ≥tC, since certain failure will occur only at tk ¼ tC. So, degradation is determined surely everywhere in R, and its CDF is replaced by a deterministic function and curve. Therefore, chaos is null, and hence Chf = MChf = 0, and DOK = 1 always for all 0≤ tk ≤ tC. Thus, Prob½RUL tð Þ <sup>k</sup> < tC � ¼ 1 and Prob½RUL tð Þ <sup>k</sup> ≥tC � ¼ 0.

Furthermore, at each instant t in the original prognostic paradigm, the stochastic RUL(t) and D(t) are predicted with certitude in the complex probability set C with Pc<sup>2</sup> = DOK – Chf = DOK + MChf maintained as equal to 1 through a continuous compensation between Chf and DOK. This compensation is from the instant t = 0 where D(t) = D<sup>0</sup> = 0.020408 ≈ 0 until the instant of failure tC where D(tC) = 1. Moreover, we can realize that DOK does not include any uncertain knowledge (with a probability less than 100%); it is the measure of our certain knowledge (probability = 100%) about the expected event. We can understand that we have eliminated and subtracted in the equation above all the random factors and chaos (Chf) from our random experiment when computing Pc<sup>2</sup> ; hence no chaos exists in C, and it only exists (if it does) in R; consequently, this has led to a 100% deterministic outcome and experiment in C since the probability Pc is constantly equal to 1. This is one of the advantages of extending R to M and therefore of working in


Table 2. The new prognostic model parameters for any pipeline internal pressure mode.


that expensive failure may in general happen unexpectedly. A novel model of analytic prognostic was established in my earlier work and publications as a counterpart of existent classical strategies of maintenance in order to take into account the evolving environment and product state and in order to make them more efficient. We have applied this model to systems of petrochemical pipes that are exposed to fatigue failure under cyclic repetitive triangular pressure. It is known that the effects of fatigue will initiate micro-cracks that can spread rapidly and hence will lead to failure. This model is founded on existing laws of damage in fracture mechanics which are the law of Palmgren-Miner of linear damage accumulation and the law of Paris-Erdogan of crack propagation. This prognostic model estimates the system RUL from a predefined threshold of degradation DC. The model of degradation established in this earlier work is founded on the damage measurement D accumulation after each cycle time of pressure. The system is judged to be in wear-out state when this measured and predefined threshold DC is reached. Moreover, to make the model more realistic and accurate, we have taken into consideration the stochastic influences afterward as well here. We have applied this model to the industry of pipelines; therefore, a prognostic study of the pipeline

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

DOI: http://dx.doi.org/10.5772/intechopen.90157

system enables us to enhance its strategies of maintenance.

the scope of my previous 12 research works on this subject.

(or bigger) failure probability Prð Þ tk =ψ<sup>j</sup>

97

In the present research work, the novel extended Kolmogorov paradigm of eight

between the remaining useful lifetime or degradation and the original paradigm was made. Therefore, the model of "complex probability" was more elaborated beyond

Although the analytic linear laws of prognostic are very well-known and deterministic in [14, 16], there are general influences and aspects that can be chaotic and stochastic (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.). Moreover, various variables in the expressions (5) and (6) of degradation which are considered as deterministic can also have a random aspect, such as the magnitude of applied pressure (due to the different conditions of pressure profile) and the length of the initial crack (potentially existing from the process of manufacturing). All those stochastic factors, embodied in the model by their mean values, influence the buried pipeline system and make its function of degradation diverge from its computed trajectory modeled by these deterministic laws. An updated follow-up of the degradation performance and behavior with cycle number or time, which is subject to non-chaotic and chaotic influences, is made possible by Prð Þ tk =ψ<sup>j</sup> due to its definition that evaluates the jumps in D. In fact, chaos modifies and affects all the environment and system parameters

included in the degradation equations (Eqs. (5) and (6)). Consequently, chaos total effect on the pipelines contributes to shape the degradation curve D and is materialized by and counted in the pipeline system failure probability Prð Þ tk =ψj. Actually, Prð Þ tk =ψ<sup>j</sup> quantifies the resultant of all the nonrandom (deterministic) and random (nondeterministic) parameters and aspects which are contained in the equation of D, which affect the system and which lead to the consequent final curve of degradation. Consequently, an accentuated influence of chaos on the pipeline can lead to a smaller (or bigger) jump in the trajectory of degradation and therefore to a smaller

. Additionally, as it was verified and shown in the novel model, when the degradation index is 0 or 1 and correspondingly the RUL is tC or 0, then the chaotic factor (Chf and MChf) is zero, and the degree of our knowledge (DOK) is 1 since the system state is totally known. During the process of degradation (0 < D < 1), we

axioms (EKA) was applied and bonded to the analytic and linear prognostic of buried petrochemical pipeline systems subject to fatigue. Hence, a tight link

#### Table 3.

The new prognostic model and the relative pipeline pressure mode comparisons for 0<Pr <0:5.


#### Table 4.

The new prognostic model and the relative pipeline pressure mode comparisons for 0:5<Pr <1.

C ¼ R þM. Thus, in the original prognostic paradigm, our knowledge of all the indicators and parameters (RUL, Prob, D, etc.) is totally predictable, always perfect, and constantly complete because Pc = 1 permanently, independently of any random factors or any pressure profile (Table 2).

Finally, we say that we have applied for pressure modes 2 and 3 the same analysis, logic, and methodology that we have used for pressure mode 1 regarding the remaining useful lifetime, the degradation, as well as all the CPP parameters (Tables 3 and 4). Therefore, we can accordingly infer that whatever the pressure conditions and environment are, then the results and conclusions are analogous. This demonstrates the strength and soundness of the novel axioms adopted and of the new prognostic paradigm developed.

#### 7. Conclusion and perspectives

The high availability of technological systems, like defense, aerospace, automobile industries, and petrochemistry, is a central major objective of previous and latest developments in the technology of system design where it is very well-known

#### Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines… DOI: http://dx.doi.org/10.5772/intechopen.90157

that expensive failure may in general happen unexpectedly. A novel model of analytic prognostic was established in my earlier work and publications as a counterpart of existent classical strategies of maintenance in order to take into account the evolving environment and product state and in order to make them more efficient. We have applied this model to systems of petrochemical pipes that are exposed to fatigue failure under cyclic repetitive triangular pressure. It is known that the effects of fatigue will initiate micro-cracks that can spread rapidly and hence will lead to failure. This model is founded on existing laws of damage in fracture mechanics which are the law of Palmgren-Miner of linear damage accumulation and the law of Paris-Erdogan of crack propagation. This prognostic model estimates the system RUL from a predefined threshold of degradation DC. The model of degradation established in this earlier work is founded on the damage measurement D accumulation after each cycle time of pressure. The system is judged to be in wear-out state when this measured and predefined threshold DC is reached. Moreover, to make the model more realistic and accurate, we have taken into consideration the stochastic influences afterward as well here. We have applied this model to the industry of pipelines; therefore, a prognostic study of the pipeline system enables us to enhance its strategies of maintenance.

In the present research work, the novel extended Kolmogorov paradigm of eight axioms (EKA) was applied and bonded to the analytic and linear prognostic of buried petrochemical pipeline systems subject to fatigue. Hence, a tight link between the remaining useful lifetime or degradation and the original paradigm was made. Therefore, the model of "complex probability" was more elaborated beyond the scope of my previous 12 research works on this subject.

Although the analytic linear laws of prognostic are very well-known and deterministic in [14, 16], there are general influences and aspects that can be chaotic and stochastic (like humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.). Moreover, various variables in the expressions (5) and (6) of degradation which are considered as deterministic can also have a random aspect, such as the magnitude of applied pressure (due to the different conditions of pressure profile) and the length of the initial crack (potentially existing from the process of manufacturing). All those stochastic factors, embodied in the model by their mean values, influence the buried pipeline system and make its function of degradation diverge from its computed trajectory modeled by these deterministic laws. An updated follow-up of the degradation performance and behavior with cycle number or time, which is subject to non-chaotic and chaotic influences, is made possible by Prð Þ tk =ψ<sup>j</sup> due to its definition that evaluates the jumps in D. In fact, chaos modifies and affects all the environment and system parameters included in the degradation equations (Eqs. (5) and (6)). Consequently, chaos total effect on the pipelines contributes to shape the degradation curve D and is materialized by and counted in the pipeline system failure probability Prð Þ tk =ψj. Actually, Prð Þ tk =ψ<sup>j</sup> quantifies the resultant of all the nonrandom (deterministic) and random (nondeterministic) parameters and aspects which are contained in the equation of D, which affect the system and which lead to the consequent final curve of degradation. Consequently, an accentuated influence of chaos on the pipeline can lead to a smaller (or bigger) jump in the trajectory of degradation and therefore to a smaller (or bigger) failure probability Prð Þ tk =ψ<sup>j</sup> .

Additionally, as it was verified and shown in the novel model, when the degradation index is 0 or 1 and correspondingly the RUL is tC or 0, then the chaotic factor (Chf and MChf) is zero, and the degree of our knowledge (DOK) is 1 since the system state is totally known. During the process of degradation (0 < D < 1), we

C ¼ R þM. Thus, in the original prognostic paradigm, our knowledge of all the indicators and parameters (RUL, Prob, D, etc.) is totally predictable, always perfect, and constantly complete because Pc = 1 permanently, independently of any random

The new prognostic model and the relative pipeline pressure mode comparisons for 0:5<Pr <1.

The new prognostic model and the relative pipeline pressure mode comparisons for 0<Pr <0:5.

Finally, we say that we have applied for pressure modes 2 and 3 the same analysis, logic, and methodology that we have used for pressure mode 1 regarding the remaining useful lifetime, the degradation, as well as all the CPP parameters (Tables 3 and 4). Therefore, we can accordingly infer that whatever the pressure conditions and environment are, then the results and conclusions are analogous. This demonstrates the strength and soundness of the novel axioms adopted and of

The high availability of technological systems, like defense, aerospace, automobile industries, and petrochemistry, is a central major objective of previous and latest developments in the technology of system design where it is very well-known

factors or any pressure profile (Table 2).

Table 3.

Fault Detection, Diagnosis and Prognosis

Table 4.

96

the new prognostic paradigm developed.

7. Conclusion and perspectives

have �0.5 ≤ Chf < 0, 0 < MChf ≤ 0.5, and 0.5 ≤ DOK < 1. Notice that during this whole process, we have always Pc<sup>2</sup> = DOK – Chf = DOK + MChf =1= Pc, which means that the phenomenon which looked to be stochastic and random in the set R is now certain and deterministic in the set C ¼ R þM, and this after the addition of the contributions of M to the phenomenon occurring in R and thus after subtracting and eliminating the chaotic factor from the degree of our knowledge. Moreover, the probabilities of the system survival and of failure corresponding to each instant t have been evaluated, in addition to the probability of RUL after a pressure cycles time t, which are all functions of the stochastic degradation jump. Consequently, at each instance of t, all the novel CPP parameters D, RUL, Pr, Pm, Pm=i, DOK, Chf, MChf, Pc, and Z are certainly and perfectly predicted in the complex probability set C with Pc maintained as equal to 1 constantly and permanently. Furthermore, using all these illustrated simulations and drawn graphs all over the whole research work, we can quantify and visualize both the certain knowledge (expressed by DOK and Pc) and the system chaos and random effects (expressed by Chf and MChf) of the pipeline system. This is definitely very fascinating, fruitful, and wonderful and proves once again the advantages of extending the five probability axioms of Kolmogorov and thus the novelty and benefits of this original field in prognostic and applied mathematics that can be called verily "The Complex Probability Paradigm."

Z the sum of Pr and Pm, complex probability number and vector

Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines…

random event and experiment

fj(t) probability density function for each pressure mode j

ψ<sup>j</sup> simulation magnifying factors for each pressure mode j 1=ψ<sup>j</sup> the normalizing constant of Pr(t) for each pressure mode j

Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences,

© 2019 The Author(s). Licensee IntechOpen. This chapter is 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,

F(t) cumulative probability distribution function

Prob[RUL(t)] probability of RUL after a pressure cycle time t.

MChf magnitude of the chaotic factor

tC pressure cycle time till system failure Pj pipelines internal triangular pressure

D degradation indicator of a system RUL remaining useful lifetime of a system

, the square of the norm of Z, degree of our knowledge of the

DOK = |Z|<sup>2</sup>

Author details

Abdo Abou Jaoude

99

Notre Dame University-Louaize, Lebanon

provided the original work is properly cited.

\*Address all correspondence to: abdoaj@idm.net.lb

Chf chaotic factor

t pressure cycle time

DOI: http://dx.doi.org/10.5772/intechopen.90157

As a prospective and future work and challenges, and concerning some applications to practical engineering, it is planned to more elaborate the original created prognostic paradigm and to implement it to a varied set of nondeterministic and dynamic systems like vehicle suspension systems and offshore and buried petrochemical pipes which are under the influence of fatigue and in the cases of nonlinear and linear damage accumulation. Furthermore, we will apply also CPP to other random experiments in classical probability theory and in stochastic processes and to the field of prognostic in engineering using the first order reliability method (FORM) as well as to the random walk problems which have enormous applications in physics, in economics, in chemistry, in applied and pure mathematics.

#### Conflict of interest

No potential conflict of interest was reported by the author.
