The paradigm of complex probability and analytic nonlinear prognostic for vehicle suspension systems

What is it about?

Adding three novel axioms to Andrey Nikolaevich Kolmogorov’s five established probability axioms which were laid in 1933 extends classical probability theory to encompass the imaginary set of numbers. Therefore, all random experiments will be executed in the complex plane and set C which is the sum of the real dimension and set R with its real probability component, and the corresponding imaginary dimension and set M with its imaginary probability component. The purpose of extending Kolmogorov’s axioms is to add supplementary new imaginary dimensions to any random event that occurs in the “real” set and laboratory and hence to be able to evaluate the associated complex probability in the complex plane C which is always equal to one. My purpose in this current work is to link the complex probability paradigm to the vehicle suspension system analytic prognostic in the nonlinear damage accumulation case. In fact, what I called the system failure probability which is derived from prognostic will be included in and applied to the complex probability paradigm. This will lead to the novel and original prognostic model illustrated in this paper. Hence, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the complex probability, the system failure and survival probabilities, and the RUL probability, after that N load cycles have been applied to the suspension, and which are all functions of the system degradation subject to chaos and random effects.

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Why is it important?

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, the probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities C. It is of great importance that the stochastic system, like the problem considered here, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature for example in statistical mechanics or in all stochastic processes. Therefore, 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 which will make the event in C = R +M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and prognostic is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This is what I called ‘The Complex Probability Paradigm (CPP)’, which was initiated and elaborated in my previous papers.

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