*4.2. PHM Framework Design-Framework and Algorithms Choice*

The PHM framework was designed considering the need to keep the computational effort required to perform the diagnostics/prognostics tasks at a minimum to comply with the requirement of being installed on board the iron bird. As such, the scheme provided in Figure 15 was adopted, where the data coming from the iron bird are at first checked to verify their coherence, used to evaluate the features at any time, and then sent to the fault-detection routine.

**Figure 15.** Architecture of the proposed PHM Framework.

Before being analyzed, the features were at first downsampled (80 Hz, against the 800 Hz of the iron bird signals) to reduce the computational effort and then preprocessed by an exponential filter to improve their signal-to-noise ratio. The fault-detection algorithm was based upon a simple, purely data-driven logic, comparing at each timestep the running feature distributions against the baselines obtained for health conditions. The remaining parts of the PHM algorithms were switched off. If the running distribution of 1 (or more) features deviated excessively from their own baselines, an alarm was triggered and the remaining routines of the PHM frameworks were activated. Such a solution has the benefits of keeping the most computationally expensive routines dormant until they are needed, hence, reducing the computational burden. An example of the fault-detection algorithm behavior is highlighted in Figure 16, where the running distribution of the feature overcomes a separation threshold, triggering the corresponding alarm of a fault declaration confidence equal to 0.98. In this instance the fault-detection algorithm observes a deviation

of the feature associated with the occurrence of a turn-to-turn short in the motor windings, prompting a fault declaration once the confidence threshold is met.

**Figure 16.** Output of the fault-detection algorithm in response to fault occurrence in the winglet actuator.

If a fault is detected, the fault classification algorithm is called to assess which failure mode is occurring (or is being simulated) within the flight-control actuator. Given the preliminary nature of the study, only the case where 1 degradation may occur at any given time was considered. As such, a simple Linear Support Vector Machine, operating over the features vector, was considered. The LSVM was trained with a randomly chosen subset representative of 70% of the data generated through the high-fidelity model, while the remaining 30% was used for verification purposes.

Results of the training process are aimed at a total accuracy rate higher than 95%. Once a fault was detected and classified, the information was sent to the prognostic algorithm.

The prognostic routine was based on the particle-filtering structure, depicted in Figure 17, to infer the fault severity and forecast the degradation growth [41]. The particle filter scheme tracks the fault progression by iterating at each time stamp 2 consequential steps, the first being the "prediction" stage and the latter being the "filtering" stage. The prediction step combines the knowledge of the previous state estimate *p*(*xt*|*yt*−1) with a process model *p*(*x*0:*t*−1|*y*1:*t*−1) to generate the a priori estimate of the state probability density functions for the next time instant,

$$p(\mathbf{x}\_{0:t}|y\_{1:t-1}) = \int p(\mathbf{x}\_t|y\_{t-1})p(\mathbf{x}\_{0:t-1}|y\_{1:t-1})d\mathbf{x}\_{0:t-1} \tag{2}$$

This expression usually cannot be analytically solved; the Sequential Monte Carlo algorithms can be used in combination with efficient sampling strategies for such purpose [42]. Particle filtering approximates the state probability density function through samples or "particles" characterized by discrete probability masses, or "weights", as,

$$p(\boldsymbol{x}\_{t}|\boldsymbol{y}\_{1:t}) \approx \tilde{w}\_{t}\Big(\boldsymbol{x}\_{0:t}^{i}\big)\delta\Big(\boldsymbol{x}\_{0:t} - \boldsymbol{x}\_{0:t}^{i}\big)d\boldsymbol{x}\_{0:t-1}\tag{3}$$

where *x<sup>i</sup>* 0:*<sup>t</sup>* represents the state trajectory; thus, the fault severity while *y*1:*<sup>t</sup>* is the measurements up to time *t*. During the "filtering" stage, a resampling scheme is employed to

update the state estimates by updating the particle weights. One of the most common versions of this algorithm, the Sequential Importance Re-sampling (SIR) particle filter [43], updates the state weights using the likelihood of *yt* as:

$$w\_t = w\_{t-1} p(y\_t|x\_t) \tag{4}$$

**Figure 17.** Scheme of the employed particle-filtering algorithm.

Although widely adopted within the PHM community due to its simplicity and relatively low computational requirements, it is worth mentioning that this resampling scheme has limitations in the description of the distribution tails, and that more advanced resampling schemes, aimed at obtaining and improving representation of the distribution tails, have been proposed [44]. However, given the low-TLR nature of the study, the SIR scheme was deemed sufficient. The algorithm performs long-term predictions of the fault evolution in time by iterating the "prediction" stage. Evaluating the particle trajectories, it is then possible to estimate the probability of failure by comparing their behavior against a hazard zone, defined via a probability density function with lower and upper bounds for the domain of the random variable [45]. As shown in Figure 17, this approach is based on particle-filtering schemes in which tunable degradation models are adopted as process models. These models are then used both to estimate the current a priori state of the system, *p*(*xt*|*y*1:*t*−1), and to perform the iterative steps necessary to achieve the prognosis *p*(*xt*<sup>+</sup>*k*|*y*1:*t*). Auto-tuned models are required to describe and follow changes in the degradation process and to describe the process and measurement noise. For the case study under consideration, the particle filter employs a nonlinear process model *yt* = *f*(*xt*) + *ν*, obtained considering the dependency of the selected features on degradation growth. The process noise *ν* is then obtained through a kernel function mirroring the feature distribution around the fitted model. The state model *xt*+<sup>1</sup> = *f*(*xt*, *t*) + *σ* is a time-dependent nonlinear model whose parameters are automatically tuned through a Recursive Least Square [46] algorithm operating over the state estimates provided by the particle filter itself. The measure of noise *σ* is also estimated at each time stamp computing the state estimate variance with respect to the noiseless output of the state model itself. The particle filter code also employs the noise compensation techniques described in [47] and self-adjusts the process noise by comparing the particle distribution at a given time instant (usually a few time steps ahead of the prediction time) of a previous long-time prediction against the particle distribution obtained for the same time instant from the

prediction/filtering loop. The output of the prognostic algorithm is represented as the RUL distribution for each prediction, coupled with the evaluation of the risk of failure as detailed in [45]. An example of the results achievable with such a framework is provided in Figure 18, where its response against a simulated fault, the occurrence of accelerated degradation of the motor's permanent magnets, is depicted for a single prediction step highlighting the trajectories of the long-term prognosis on both the estimated fault size (hidden state) and feature. Figure 19, instead, provides an example of the RUL distributions obtained for different prediction steps for the very same failure mode. It can be noticed that the RUL uncertainty estimate tends to decrease along with the degradation process. However, when the degradation severity has grown close to the failure declaration threshold, the prediction uncertainty tends to increase again. Such behavior is expected since the closer the degradation process approaches the failure status, the more it becomes susceptible to small variations of the physical variable responsible for its progression, thus, leading to increased uncertainty over its future expected behavior.

**Figure 18.** Example of the particle filter framework output.

**Figure 19.** Example of RUL estimate for 5 consequent prediction steps for one of the degradation patterns associated with the demagnetization of the motor permanent magnets.
