2. The purpose and the advantages of the present work

Computing probabilities is the main work of classical probability theory. Adding new dimensions to the stochastic experiments will lead to a deterministic expression of probability theory. This is the original idea at the foundations of this work. Actually, the theory of probability is a nondeterministic system in its essence; that means that the event outcomes are due to the chance and randomness. The addition of novel imaginary dimensions to the chaotic experiment occurring in the set R will yield a deterministic experiment, and hence a stochastic event will have a certain result in the complex probability set C. If the random event becomes completely predictable, then we will be fully knowledgeable to predict the outcome of stochastic experiments that arise in the real world in all stochastic processes. Consequently, the work that has been accomplished here was to extend the real probabilities set R to the deterministic complex probabilities set C ¼ R þM by including the contributions of the set M which is the imaginary set of probabilities. Therefore, since this extension was found to be successful, then a novel paradigm of stochastic sciences and prognostic was laid down in which all stochastic phenomena in R was expressed deterministically. I called this original model "the Complex Probability Paradigm" that was initiated and illustrated in my 12 research publications. [20–31].

Furthermore, although the analytic linear prognostic laws are deterministic and very well-known in [14, 16], there are chaotic and stochastic influences and aspects (such as humidity, temperature, material nature, geometry dimensions, applied load location, water action, corrosion, soil pressure and friction, atmospheric pressure, etc.) that 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

intellect nothing could be uncertain; and the future just like the past would be

"The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary

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. Pipelines are the primary component of the systems of hydrocarbon transport in petrochemical industries. They are vital for human activities because they serve to transport water, natural gases, and oil from sources to all consumer sites. A novel analytic prognostic model was established in my earlier research work and applied to the case of pipelines subject to the effects of corrosion, to soil loading, and to internal pressure. These will initiate micro-cracks in the body of the tubes that can spread suddenly and can lead to failure. The increase of pipeline availability and the reduction of their global mission cost and performance necessitate to elaborate a suitable process of prognostic. Accordingly, a novel strategy based on degradation analytic laws was applied to diverse dynamic systems and was developed in my research work [1–6]. Additionally, the remaining useful lifetime (RUL) was predicted and calculated from a predefined threshold of degradation. Based on a system of a physical petrochemical pipeline, my publications developed a strategy to design a model of failure

prognostic that will be more elaborated and further enhanced in the present

ing, and analysis, based on physical measurements utilizing sensors.

systems, sliding mode control of fuzzy singularly perturbed systems with

application to electric circuits, the stabilization of quantized sampled-data neural

Moreover, prognostic is a process involving a prediction capacity. Using prognostic, we are able to evaluate the equipment remaining useful lifetime in terms of its future usage and its history of functioning. Predicting the remaining useful lifetime of industrial systems turns out to be presently a vital goal for industrialists knowing that the consequences of failure, which can occur suddenly, are usually very expensive. The traditional maintenance strategies [7, 8] founded on a static threshold of alarm are no more practical and efficient since they do not consider the instantaneous functioning state of a product. The establishment of a prognostic approach as an "intelligent" maintenance consists of the health follow-up, monitor-

Also, earlier expert studies of prognostic belong in general to three categories of technical approaches: the first category is the "experience-based prognostic" [9] which is based on measurements taken from a machine health monitoring, for example, those based on stochastic model, expert judgment, Bayesian approach, reliability analysis, Markovian process, optimization of preventive maintenance, etc. Their methodology of prognostic shows to be simple but inflexible toward changes in the environment and in the system behavior. The second category is the "estimation-based or trending prognostic" based on the statistics of vast measured data. We can cite as illustrations the work relying on the behavior of degradation expressed by abaci and utilizing a system expert description (process-missionenvironment) [10]; the work relying on artificial intelligence, machine learning [11], neural network [12], and fuzzy logic [13]; and additionally the work based on dissipativity-based fuzzy integral sliding mode control of continuous time T-S fuzzy systems, SMC design for robust stabilization of nonlinear Markovian jump singular

Marquis Pierre-Simon de Laplace.

Gottfried Wilhelm von Leibniz.

present before its eyes".

Fault Detection, Diagnosis and Prognosis

root of negative unity".

book chapter.

66

made possible by what I called the system failure probability due to its definition that estimates the jumps in the function of degradation D.

Additionally, my objective in this present work is to connect the complex probability paradigm to the buried pipeline system analytic prognostic in the case of linear damage accumulation which is subject to fatigue. In fact, the system failure probability derived from prognostic will be applied to and included in the complex probability paradigm. This will lead to the original and novel model of prognostic illustrated in this chapter. Thus, by determining the new prognostic model parameters, it becomes possible to evaluate the degree of our knowledge, the magnitude of the chaotic factor, the complex probability, the RUL probability, and the system failure and survival probabilities; after that a pressure cycle time t has been applied to the buried pipeline, which are all functions of the system degradation subject to chaotic and stochastic influences.

Accordingly, the advantages and the purpose of the current chapter are to:


Concerning some applications of the original elaborated paradigm and as a future work, it can be applied to a wide set of dynamic systems like vehicle suspension systems and offshore and buried petrochemical pipelines which are subject to fatigue and in the cases of nonlinear and linear damage accumulation. Furthermore, compared with existing literature, the main contribution of the present research work is to apply the novel paradigm of complex probability to the concepts of random remaining useful lifetime and degradation of a buried pipeline system hence to the case of analytic prognostic in the case of linear damage accumulation subject to fatigue. The following figure shows the main purposes of the complex

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

The diagram of the main purposes of the complex probability paradigm and research work.

To conclude and to summarize, in the real probability universe R, our degree of our certain knowledge is regrettably imperfect; therefore we extend our study to the complex set C which embraces the contributions of both the real probabilities set R and the imaginary probabilities set M. Subsequently, this will lead to a perfect and complete degree of knowledge in the universe C ¼ R þM (since Pc = 1). In fact, working in the complex universe C leads to a certain prediction of any random event, because in C we eliminate and subtract from the calculated degree of our knowledge the quantified chaotic factor. This will yield a probability in the universe <sup>C</sup> equal to one (Pc<sup>2</sup> <sup>=</sup> DOK � Chf <sup>=</sup> DOK <sup>+</sup> MChf =1= Pc). Many

probability paradigm (CPP) (Figure 1).

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

Figure 1.

Figure 2.

69

The EKA or the CPP diagram.


C ð Þ¼ complex set R ð Þþ real set M ðimaginary setÞ:


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

#### Figure 1.

made possible by what I called the system failure probability due to its definition

Accordingly, the advantages and the purpose of the current chapter are to:

analysis. This job was started and elaborated in my previous 12 papers.

2.Do an updated follow-up of the degradation D performance and behavior with cycle number or time which is subject to chaos. This follow-up is

therefore to link the theory of probability to the field of complex variables and

accomplished by the real failure probability of the system due to its definition that evaluates the jumps in D, therefore linking a system degradation to

3.Apply the new axioms of probability and paradigm to system prognostic; thus, I will extend the prognostic concepts to the set of complex probabilities C.

4. Show that all stochastic phenomena can be expressed deterministically in the

5.Measure and compute both the degree of our knowledge and the chaotic factor

6.Draw and illustrate the graphs of the parameters and functions of the original

7.Show that the classical concepts of random remaining useful lifetime and degradation possess a probability permanently equal to one in the complex set; hence, no randomness, no chaos, no uncertainty, no ignorance, no disorder,

C ð Þ¼ complex set R ð Þþ real set M ðimaginary setÞ:

8. Show that by adding new and supplementary dimensions to any stochastic phenomenon, whether it is a pipeline system or any other random experiment, it becomes possible to do prognostic in a deterministic way in the set C of

9.Pave the way to implement this novel model to other areas in stochastic

processes and to the field of prognostics in science and engineering. These will

of the system remaining useful lifetime and its degradation.

paradigm corresponding to a buried pipeline prognostic.

1.Extend classical probability theory to the set of complex numbers and

Additionally, my objective in this present work is to connect the complex probability paradigm to the buried pipeline system analytic prognostic in the case of linear damage accumulation which is subject to fatigue. In fact, the system failure probability derived from prognostic will be applied to and included in the complex probability paradigm. This will lead to the original and novel model of prognostic illustrated in this chapter. Thus, by determining the new prognostic model parameters, it becomes possible to evaluate the degree of our knowledge, the magnitude of the chaotic factor, the complex probability, the RUL probability, and the system failure and survival probabilities; after that a pressure cycle time t has been applied to the buried pipeline, which are all functions of the system degradation subject to

that estimates the jumps in the function of degradation D.

probability theory in a novel and original way.

set of complex probabilities C.

and no unpredictability exist in:

be the topics of my future research works.

complex probabilities.

68

chaotic and stochastic influences.

Fault Detection, Diagnosis and Prognosis

The diagram of the main purposes of the complex probability paradigm and research work.

Concerning some applications of the original elaborated paradigm and as a future work, it can be applied to a wide set of dynamic systems like vehicle suspension systems and offshore and buried petrochemical pipelines which are subject to fatigue and in the cases of nonlinear and linear damage accumulation. Furthermore, compared with existing literature, the main contribution of the present research work is to apply the novel paradigm of complex probability to the concepts of random remaining useful lifetime and degradation of a buried pipeline system hence to the case of analytic prognostic in the case of linear damage accumulation subject to fatigue. The following figure shows the main purposes of the complex probability paradigm (CPP) (Figure 1).

To conclude and to summarize, in the real probability universe R, our degree of our certain knowledge is regrettably imperfect; therefore we extend our study to the complex set C which embraces the contributions of both the real probabilities set R and the imaginary probabilities set M. Subsequently, this will lead to a perfect and complete degree of knowledge in the universe C ¼ R þM (since Pc = 1). In fact, working in the complex universe C leads to a certain prediction of any random event, because in C we eliminate and subtract from the calculated degree of our knowledge the quantified chaotic factor. This will yield a probability in the universe <sup>C</sup> equal to one (Pc<sup>2</sup> <sup>=</sup> DOK � Chf <sup>=</sup> DOK <sup>+</sup> MChf =1= Pc). Many

Figure 2. The EKA or the CPP diagram.

illustrations considering various continuous and discrete probability distributions in my 12 previous research papers verify this hypothesis and novel paradigm [20–31]. The extended Kolmogorov axioms (EKA for short) or the complex probability paradigm can be summarized and shown in the following figure (Figure 2).

model in these earlier publications and work to prove the effectiveness of my model [1–6, 13–19]. Three case studies of pipelines were taken into consideration: buried, unburied, and subsea (offshore pipes). Each one of these situations necessitates different physical parameters like friction and soil pressure, corrosion, and atmospheric and water pressure. One of these cases is elaborated here which is the system of buried petrochemical pipes where three modes of pressure profiles (mode 1 = high, mode 2 = middle, and mode 3 = low-pressure conditions) were examined and simulated. My model showed that it presented a useful tool for a prognostic analysis and that it is very convenient in such industrial systems. Furthermore, it proved that it is less expensive than other models that require a huge number of

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

The stress intensity factor was introduced to calculate the correlation between the crack growth rate, da/dN, and the stress intensity factor range, ΔK. The Paris-Erdogan's law [7] allows to evaluate the rate of propagation of the crack length a after its detection. This damage growth law is expressed by the following equation:

dN <sup>¼</sup> <sup>C</sup>ð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> (1)

da

Y að Þ, the component's crack geometry function. Δσ, the range of the applied stress in a cycle.

3.3 The modeling of linear cumulative damage

dN, the crack growth rate = the increase of the crack length a per cycle N.

m and C, the constants of materials obtained experimentally; 2ð Þ ≤ m ≤4 and

To do the prognostic of a degrading element, my approach was to evaluate and to

As a matter of fact, this law [7] is used to calculate the cumulative damage di of different stress levels σ<sup>i</sup> (i = 1, i = 2, ..., i = k) applied for ni cycles. Knowing that Ni is

predict the end of life of the element by modeling and tracking the function of degradation. My model of damage, whose progress is up to the macro-crack initiation

point, is illustrated in Figure 3 by the damage linear rule of Palmgren-Miner.

π a p , the intensity factor of the stress.

measurements and data.

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

3.2 Fatigue crack growth

<sup>Δ</sup>K að Þ¼ Y að ÞΔ<sup>σ</sup> ffiffiffiffiffiffi

where da

ð Þ 0<C ≪ 1 .

Figure 3.

71

Palmgren-Miner's linear rule of damage.
