This paper considers dynamic systems widely used in industry for which the behavior can be approximated to a second order differential equation. The components of such a system suffer from random faults and failures due to wear, age or usage. These events impact the dynamic behavior which is interpreted as a modification of the initial differential equation with random coefficients. At given times, the system is solicited and its output – the only source of information – is measured to infer the position of 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. 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 an univariate non-linear diffusion process if underlying degradation sources are assumed to be homogeneous Gamma processes. Based on this model, the system remaining useful lifetime is assessed. Two predictive maintenance policies are also designed showing the feasibility to easily maintain dynamic systems solely on the system output.

本文考虑了广泛应用于工业中的动态系统，其行为可以近似为二阶微分方程。这种系统的组件由于磨损，老化或使用而遭受随机故障和故障的困扰。这些事件影响动态行为，这被解释为对初始微分方程具有随机系数的修改。在给定的时间，请求系统，并测量其输出（唯一的信息源）以推断方程式根在复平面中的位置。提出了根的当前位置和初始位置之间的欧几里得距离作为反映系统性能逐渐下降的新指标。这种指示符根据复杂平面中根位置的随机演变及时显示随机轨迹。更特别地，如果假设潜在的退化源是同质伽马过程，则可以通过单变量非线性扩散过程对这些轨迹进行建模。基于此模型，可以评估系统的剩余使用寿命。还设计了两个预测性维护策略，显示了仅在系统输出上轻松维护动态系统的可行性。