A Particle Filtering Framework for Failure Prognosis

Bayesian estimation techniques are finding application domains in machinery fault diagnosis and prognosis of the remaining useful life of a failing component/subsystem. This paper introduces a methodology for accurate and precise prediction of a failing component based on particle filtering and learning strategies. This novel approach employs a state dynamic model and a measurement model to predict the posterior probability density function of the state, i.e., to predict the time evolution of a fault or fatigue damage. It avoids the linearity and Gaussian noise assumption of Kalman filtering and provides a robust framework for long-term prognosis while accounting effectively for uncertainties. Correction terms are estimated in a learning paradigm to improve the accuracy and precision of the algorithm for long-term prediction. The proposed approach is applied to a crack fault and the results support its robustness and superiority. INTRODUCTION Prediction of the evolution of a fault or fault indicator entails large-grain uncertainty. Accurate and precise prognosis of the time-to-failure of a failing component/subsystem must consider critical state variables, such as crack length, corrosion pitting, etc., as random variables with associated probability distribution vectors. Once the probability distribution of the failure is estimated, other important prognosis attributes -such as confidence intervalscan be computed. These facts suggest a possible solution to the prognosis problem based on recursive Bayesian estimation techniques that combine both the information from fault growth models and on-line data obtained from sensors monitoring key fault parameters (observations). Prognosis or long-term prediction for the failure evolution is based on both an accurate estimation of the current state and a model describing the fault progression. If the incipient failure is detected and isolated at the early stages of the fault initiation, it is reasonable to assume that sensor data will be available for a certain time window allowing for corrective measures to be taken, i.e., improvements in model parameter estimates so that prognosis will provide accurate and precise prediction of the time-to-failure. At the end of the observation window, the prediction outcome is passed on to the user (operator, maintainer) and additional adjustments are not feasible since corrective action must be taken to avoid a catastrophic event. Particle filtering is an emerging and powerful methodology for sequential signal processing with a wide range of applications in science and engineering [1]. It has captured the attention of many researchers in various communities including those of signal processing, statistics and econometrics. Founded on the concept of sequential importance sampling and the use of Bayesian theory, particle filtering is particularly useful in dealing with difficult nonlinear and/or non-Gaussian problems. The underlying principle of the methodology is the approximation of relevant distributions with particles (samples from the space of the unknowns) and their associated weights. Compared to classical Monte-Carlo method, sequential importance sampling enables Particle Filtering to reduce the number of samples required to approximate the distributions with necessary precision, and makes it a faster and more computationally efficient approach than Monte-Carlo simulation. This is of particular benefit in diagnosis and prognosis of complex dynamic systems, such as engines, gas turbines, gearboxes, etc., because of the nonlinear nature and ambiguity of the rotating machinery world when operating under fault conditions. Moreover, particle filtering allows information from multiple measurement sources to be fused in a principled manner. A PARTICLE FILTERING FRAMEWORK FOR FAILURE PROGNOSIS Figure 1 depicts a conceptual schematic of a particle filtering framework aimed at addressing the fault prognosis problem. Available and recommended PHM/CBM sensors and the feature extraction module provide the sequential observation (or measurement) data of the fault growth process k z at time instant k. We assume that the fault progression can 1 Copyright © 2005 by ASME be explained through the state evolution model (1) and the measurement model (2) ) | ( ) , ( 1 1 − − ↔ = k k k k k k x x p x f x ω (1) ) | ( ) , ( k k k k k k x z p x h z ↔ = υ (2) where k x is the state of the fault dimension (such as the crack size), the changing environment parameters that affect fault growth, k ω and k υ are the non-Gaussian noises, and k f and k g are nonlinear functions. Figure 1. Fault Prognosis based on Particle Filtering The first part of the approach is state estimation, i.e., estimating the current fault dimension as well as other important changing parameters in the environment. This task involves two basic steps. In the Prediction Step, the a priori state estimation is generated from the knowledge of the previous state estimation and the process model (Eq. (3)). ∫ − − − − − = 1 1 : 1 1 1 1 : 1 ) | ( ) | ( ) | ( k k k k k k k dx z x p x x p z x p (3) The Update Step incorporates the new observation data k z into the a priori state estimate ( ) 1 : 1 − κ κ z x p in order to generate the posteriori state estimation ( ) κ κ : 1 z x p , as Eq. (4) shows. ) | ( ) | ( ) | ( ) | ( 1 : 1 1 : 1 : 1 − − = k k k k k k k k z z p z x p x z p z x p (4) Particle Filtering approximates the state pdf by using samples or "particles" having associated discrete probability masses. The actual distributions would be then approximated by a set of samples [1] drawn from an importance distribution ) | ( : 1 : 0 k k z x q and the corresponding normalized importance weights ) ( ~ ~ : 0 i k k i k x w w = for the i-th sample,