^{a}, Khac Tuan Huynh and Antoine Grall

^{a}yves.langeron@utt.fr

Nowadays, the industry is looking for more and more competitiveness to face aggressive global competition. From there, the idea of a smart factory has appeared defining the ability to monitor and control machinery and all tools of production. This factory of future relies on connected sensors collecting data to monitor the health state of machinery to avoid breakdowns by triggering maintenance action.

In this context, the present work deals with dynamic systems widely encountered in the world of industry and also on a daily basis. Dynamic systems cover a wide range of manufacturing industry applications with CNC machine tools or the aeronautical and railway sectors with aircraft undercarriage, control surfaces, level crossing actuators. These systems are usually made up of a combination of components with different technologies and structural, stochastic and economic dependencies among them. A same system may embed electric, pneumatic, hydraulic, mechanic, optic and electronic parts which suffer various faults or failures with different deterioration natures. Therefore, modeling the system degradation from the deterioration processes of its parts becomes intractable.

A system is called dynamic if there exists a function describing its time-dependent behavior. Such a function is usually expressed in the form of a differential equation, or more generally in a state-space representation. We assume a second order differential equation to be representative of most of industrial systems Cairano et al. (2007). The degrading components impact the dynamic behavior of the system which is interpreted as a modification of the initial differential equation with random coefficients.

We propose a framework with four steps supported by two examples with a shock absorber and an electronic fuel-injector. The first step is to define an indicator to describe the system degradation and its failure. At given times, the system is solicited and its output - the only source of information - is measured to infer the position of differential equation roots in the complex plane. The Euclidian distance between the current and initial positions of a root is proposed as a new indicator reflecting the gradual deterioration of system performance. The initial position stands for a system in nominal mode. A constant failure threshold is defined as non-admissible system performance.

Such an indicator presents stochastic trajectories in time according to the random evolution of the root location in complex plane. More especially, these trajectories can be modeled by a univariate non-linear diffusion process when underlying degradation sources are assumed to be homogeneous Gamma processes. The modeling of this new system health indicator is the second step.

From there, the third step is to derive the remaining useful life RUL of a dynamic system. The non-linear diffusion process presents time dependent drift and diffusion leading the deriving of the lifetime distribution tricky. To this end, the diffusion process is transformed towards a standard Brownian motion for which the properties allow to obtain more easily a closed form expression of the lifetime distribution Durbin (1985). The counterpart is that this time-space transformation transforms also the constant failure threshold to a time varying failure threshold. The results from Ricciardi (1976) solve the first passage time distribution of the obtained standard Brownian motion to this time varying boundary.

Finally, this framework ends with the application of the RUL in the definition of two predictive maintenance policies. The obtained results show the feasibility to maintain easily industrial systems in the context of smart factory in an economic and sustainable manner without costly investment.