Particle filters were proposed for RUL prediction by (Orchard ME, Vachtsevanos, 2009) and (Saha, Bhaskar, et al. 2009), and have the advantage of not being bound
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by the linear system or Gaussian noise assumption. (Zio and Peloni, 2011) proposes a methodology for the estimation of the remaining useful life of components based on particle filtering. The approach employs Monte Carlo simulation of a state dynamic model and a measurement model for estimating the posterior probability density function of the state of a degrading component at future times, in other words for predicting the time evolution of the growing fault or damage state. The proposed approach is applied to a crack fault, with satisfactory results.
(An, Dawn, Joo-Ho, and Nam, 2013) present a Matlab-based tutorial for model-based prognostics, which combines a physical model with observed data to identify model parameters, from which the RUL can be predicted. The particle filter is used in this tutorial for parameter estimation of damage or a degradation model. As examples, a battery degradation model and a crack growth model are used to explain the updating process of model parameters, damage progression, and RUL prediction.
(Daigle and Goebel, 2010) overcame the problem of limited sensor data by applying a model-based prognostics approach using particle filters with application on solenoid valves. (Daigle and Goebel, 2010) also presented a detailed model of the solenoid valve and extended it according to the damage evolving during the valve’s lifetime. The measurement models, which establish the mapping from measurement to the internal system state, were the system states themselves, plus measurement noise. (Lall, Lowe and Goebel, 2010) extended their work on structural damage to Ball Grid Array (BGA) interconnects. In this paper, the effectiveness of the proposed particle filter and resistance spectroscopy based approach in a Prognostic Health Management (PHM) framework has been demonstrated for electronics. The measured state variable has been related to the underlying damage state using non- linear finite element analysis. With the particle filter being used to estimate the state variable, rate of change of the state variable, acceleration of the state variable and construct a feature vector. The estimated state-space parameters have been used to extrapolate the feature vector into the future and predict the time-to-failure at which the feature vector will cross the failure threshold. RUL has been calculated based on the evolution of the state space feature vector.
69 3.6.3.1. Particle Filter Implementation
The finalised state model needs to be projected until the failure threshold is reached, the particle filter is a powerful and emerging tool for sequential signal processing that is finding application in many engineering and science problems (Arulampalam, Maskell, Gordon, and Clapp, 2002), (Zio and Maio, 2010), (Zio and Peloni, 2011), (An, 2013) and (Saha, 2011) and is used here to determine the RUL due to the ability to add non-linear components to the model if necessary.
The framework below gives a conceptual schematic of a particle-filtering framework for addressing the fault prognosis problem. System sensors and the feature
extraction module provide the sequential observation (or measurement) data of the fault growth process 𝑧𝑘 at time instant k. The fault progression can be explained
through the state-evolution model and the measurement mode
𝒙𝒌 = 𝒇𝒌(𝒙𝒌−𝟏, 𝝎𝒌) ↔ 𝒑(𝒙𝒌 𝒙𝒌−𝟏)
𝒛𝒌 = 𝒉𝒌(𝒙𝒌, 𝒗𝒌) ↔ 𝒑(𝒛𝒌 𝒙𝒌)
Figure 3.5. Particle filtering schematic
where 𝒙𝑘 is the state of the fault dimension (such as the contact resistance), the
changing environment parameters that affect fault growth, 𝜔𝑘 and 𝑣𝑘, are the non- Gaussian noises, and 𝑓𝑘 and ℎ𝑘 are nonlinear functions.
System Sensors Features and/or measurement System description
model Next state update
based on measurements System description model for p iterations Re-sampling
Current state awareness Prior Knowledge 𝑃(𝒙𝒌−𝟏 𝒛1:𝑘−1) 𝑃(𝒙𝑘 𝒛1:𝑘−1) Previous State Posterior knowledge 𝑃(𝒙𝑘+𝑝 𝑧1:𝑘)
p-step ahead prediction Sensors
ZK
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The object of non-linear filtering is to recursively estimate 𝒙𝑘 which in this case
includes a set of parameters that define the evolution of the fault condition in time from measurements 𝒛𝑘 ∈ 𝑅𝑛𝑥. The measurements are related to the target state via the
measurement equation 𝒛𝑘 = ℎ𝑘(𝒙𝑘, 𝑣𝑘) where ℎ𝑘 is a known, possibly nonlinear function and 𝑣𝑘 is a measurement noise sequence. The noise sequences 𝑣𝑘 and 𝜔𝑘−1
will assumed to be white, with a known probability density function and mutually independent.
The particle filter exists in various forms, however, in terms of usage in literature for prognostics, the Sampling Importance Re-sampling (SIR) concept is the most prominent. The SIR algorithm proposes a posterior filtering distribution denoted as 𝜋(𝑥) = 𝑝(𝒙𝑘 𝒛𝑘) is approximated by a set of N weighted particles {〈𝒙𝑘𝑖, 𝑤𝑘𝑖〉; 𝑖 = 1, … , 𝑁}
sampled from an arbitrarily proposed distribution q(x) that is somewhat similar to 𝜋(𝑥) (e.g., 𝜋(𝑥) > 0 → 𝑞(𝑥) > 0 for all 𝒙 ∈ 𝑅𝑛𝑥. The importance weights 𝑤
𝑘𝑖 are proportional
to the likelihood p(𝒛𝑘 𝒙𝑘𝑖) associated to the sample 𝒙 𝑘
𝑖 and this is then normalised to
give the following
𝑤𝑘𝑖 = 𝜋(𝒙𝑘𝑖)/𝑞(𝒙𝑘𝑖)
∑𝑁𝑗=1𝜋(𝒙𝑘𝑗)/𝑞(𝒙𝑘𝑗) (40)
such that ∑𝑁𝑗=1𝑤𝑘𝑖 = 1 and the posterior distribution (which is essentially the target distribution) can be approximated as
𝑝(𝒙𝑘 𝒛𝑘) = ∑𝑁 𝑤𝑘𝑖
𝑗=1 𝛿(𝒙𝑘− 𝒙𝑘𝑖) (41)
As with any process that uses a Bayesian update, the filtering step consists of two stages, firstly the production of a priori state density estimate which is the prediction step and secondly a update of the estimation according to the new measurement information.
The prediction step is given by
𝑝(𝒙𝑘 𝒛𝑘−1) ≈ ∑𝑁𝑖=1𝑤𝑘−1𝑖 𝛿(𝒙𝑘− 𝑓𝑘−1(𝒙𝑘−1𝑖 ) − 𝜔𝑘−1𝑖 ) (42)
and the update step is used to modify the particle weights according to the relation
𝑤̅𝑘𝑖 = 𝑤 𝑘−1𝑖 𝑝(𝒛𝑘 𝒙𝑘𝑖)𝑝(𝒙 𝑘 𝑖 𝒙 𝑘−1 𝑖 ) 𝑞(𝒙𝑘𝑖 𝒙 𝑘−1 𝑖 , 𝒛 𝑘)
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𝑤𝑘𝑖 = 𝑤̅𝑘𝑖
∑𝑁𝑗=1𝑤̅𝑘𝑗 (43)
Re-sampling may be used if the weights degenerate such that they are close to zero, as this can occur if the system state is poorly represented, it also wastes computing resources on superfluous calculations.
To enable the particle filter to be used for prognosis, a particle filtering based prognostics approach needs to project the current PDF estimate of the damage state in time. The simplest implementation that can be used to solve this problem uses the damage state equation recursively to propagate the posterior PDF estimate defined by {〈𝒙𝑝𝑖, 𝑤𝑝𝑖〉; 𝑖 = 1, … , 𝑁} in time until 𝒙𝑝𝑖 fails to no longer meet the system failure criteria at some time 𝑡𝐸𝑂𝐿𝑖 . Therefore, the RUL PDF is the distribution of 𝑝(𝑡𝐸𝑂𝐿𝑖 − 𝑡𝑝)
given by the distribution of 𝑤𝑝𝑖.
Figure 3.6. Particle filtering flowchart.
Initialize PF Parameters
Proposed Initial Population, 𝑿0 𝑊0 Propagate Particles using State Model,
𝑿𝑘−1→ 𝑿𝑘 Update Weights, 𝑊𝑘−1 → 𝑊𝑘 Measurement 𝒁𝑘 Resample? Weights degenerated? No Yes
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Fig 3.7. Particle filter for RUL prediction flowchart (Saha, 2011)