Abstract
The Andrey N. Kolmogorov's system of axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex set C which is the sum of the set R with its corresponding probability and the imaginary set M with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the real laboratory. Whatever the probability distribution of the input random variable in R is, the corresponding probability in the whole set C is always one, so the outcome of the random experiment in C can be predicted totally. The result indicates that chance and luck in R is replaced now by total determinism in C. This is the consequence of the fact that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the system. This novel complex probability paradigm will be applied to the concepts of degradation and the remaining useful lifetime of a vehicle suspension system, thus to the field of prognostic.
Highlights
This study is organized as follows: Firstly the analytic prognostic model of fatigue for vehicle suspension systems is recapitulated in the linear cumulative damage case, secondly the extended Kolmogorov’s axioms with their original parameters and interpretation are presented, thirdly the complex probability paradigm applied to prognostic is elaborated, fourthly the simulations of the new model for the three roads modes are illustrated and a comprehensive conclusion and perspectives end this research
An analytic linear prognostic model was introduced in the previous research paper (Abou Jaoude, 2015) that permits to predict the Remaining Useful Lifetime (RUL) of a dynamic suspension system
In order to have a certain prediction of any random event, it is necessary to work in the complex set C in which the chaotic factor is quantified and subtracted from the computed degree of knowledge to lead to a probability in C equal to one (Pc2 = decreases our certain knowledge (DOK)−Chf = 1)
Summary
“An intellect which at any given moment knew all the forces that animate Nature and the mutual positions of the beings that comprise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom: For such intellect nothing could be uncertain; and the future just like the past would be present before its eyes.” Marquis Pierre-Simon de Laplace “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 root of negative unity.” Gottfried Wilhelm Von Leibniz. The work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities, that makes the event in C = R + M deterministic If this is found to be fruitful, a new theory in statistical sciences and prognostic is elaborated and this to understand deterministically those phenomena that used to be random phenomena in R. This study is organized as follows: Firstly the analytic prognostic model of fatigue for vehicle suspension systems is recapitulated in the linear cumulative damage case, secondly the extended Kolmogorov’s axioms with their original parameters and interpretation are presented, thirdly the complex probability paradigm applied to prognostic is elaborated, fourthly the simulations of the new model for the three roads modes are illustrated and a comprehensive conclusion and perspectives end this research
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