Computational framework for real-time diagnostics and prognostics of aircraft actuation systems
CComputational framework for real-time diagnostics andprognostics of aircraft actuation systems
Pier Carlo Berri a,1, ∗ , Matteo D.L. Dalla Vedova a,2 , Laura Mainini a,b,3 a Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129 Turin (IT) b Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Abstract
Prognostics and Health Management (PHM) are emerging approaches toproduct life cycle that will maintain system safety and improve reliability, whilereducing operating and maintenance costs. This is particularly relevant foraerospace systems, where high levels of integrity and high performances are re-quired at the same time. We propose a novel strategy for the nearly real-timeFault Detection and Identification (FDI) of a dynamical assembly, and for theestimation of Remaining Useful Life (RUL) of the system. The availability ofa timely estimate of the health status of the system will allow for an informedadaptive planning of maintenance and a dynamical reconfiguration of the mis-sion profile, reducing operating costs and improving reliability. This work ad-dresses the three phases of the prognostic flow – namely (1) signal acquisition,(2) Fault Detection and Identification, and (3) Remaining Useful Life estimation– and introduces a computationally efficient procedure suitable for real-time,on-board execution. To achieve this goal, we propose to combine informationfrom physical models of different fidelity with machine learning techniques toobtain efficient representations (surrogate models) suitable for nearly real-time ∗ Corresponding author
Email address: [email protected] (Pier Carlo Berri) PhD Student, Department of Mechanical and Aerospace Engineering (DIMEAS), Politec-nico di Torino Assistant Professor, Department of Mechanical and Aerospace Engineering (DIMEAS),Politecnico di Torino Visiting Professor, Department of Mechanical and Aerospace Engineering (DIMEAS),Politecnico di Torino. Research Affiliate, Massachusetts Institute of Technology, Cambridge,MA 02139, USA
Preprint submitted to October 1, 2020 a r X i v : . [ c s . C E ] S e p pplications. Additionally, we propose an importance sampling strategy anda novel approach to model damage propagation for dynamical systems. Themethodology is assessed for the FDI and RUL estimation of an aircraft elec-tromechanical actuator (EMA) for secondary flight controls. The results showthat the proposed method allows for a high precision in the evaluation of thesystem RUL, while outperforming common model-based techniques in terms ofcomputational time. Keywords:
Multifidelity Modeling, Prognostics and Health Management(PHM), Aircraft Actuation systems, Machine Learning
1. Introduction
The steadily increasing complexity of aircraft systems results in large amountof heterogeneous components to integrate. Each component is characterized byits own set of failure modes, which can interact with those of the other com-ponents, increasing the overall system failure rate and making the fault identi-fication and isolation process difficult and time expensive. This can eventuallylead to worsen the reliability and availability characteristics of the vehicle. Thetraditional approach to system life-cycle management is based on schedulingmaintenance interventions a priori : components are replaced at the end of theirdesign life, regardless their actual health status [1, 2, 3]. This strategy leads tohigh maintenance costs and cannot guarantee that no failure will occur beforethe predicted end of life, for example as the result of an undetected manufactur-ing defect; to reduce risk on safety-related equipment, critical components areredounded [4, 5], increasing weight and further reducing basic reliability. Con-versely, latest approaches like Condition Based Maintenance (CBM) [6, 7, 8] andIntegrated Vehicle Health Management (IVHM) [9, 10, 11] aim to account foradvances in Prognostics and Health Management (PHM) disciplines, in orderto better manage the maintenance schedule, reducing costs and increasing mis-sion reliability [12, 13, 14, 15, 16]. PHM relies on continuous monitoring of theactual health status of components, to adaptively estimate the system Remain-2ng Useful Life (RUL) [17, 18, 19]. The benefits promised by CBM and IVHMmotivate the great interest in enabling next generation systems and vehicles toautonomously detect damages and faults at their early stage, and predict theassociated RUL during operations. This capability would allow to replace com-ponents only when really needed, avoid disposing systems that are still healthy,and even recalibrate systems operational envelope to guarantee a longer andsafer system life.Common approaches to PHM leverage either model-based strategies (i.e. re-lying on physics-based representations of the monitored system [20, 21, 22, 23])or data-driven methods [24, 25, 26]. A review of model-based condition moni-toring strategies to enable system prognostics is provided by Tinga and Loen-dersloot [27]. In [28] a structured residual between the system response and adigital twin is compared to a threshold in order to detect faults of industrialequipment. In [29] faults are detected online with a data-driven algorithm, andlater identified offline employing a model-based strategy. Henry et al. [30] pro-pose to compare attitude command and measurement of the inertial platformto determine failures in the attitude control system of a spacecraft. Huanget al. [31] and Zhao et al. [32] provide reviews of data-driven approaches toprognostics leveraging statistical methods and deep learning. In [33] an ExtremeLearning Machine (ELM) is employed for fault detection of wind turbines, whilein [34] feedforward networks are used for similarity-based prognostics. Autore-gressive integrated moving average (ARIMA) is applied to the RUL predictionof milling machine cutting tools in [35]. Model-based techniques usually re-quire large computational resources, and cannot be executed in real-time byon-board hardware. Data-driven methods, conversely, need large datasets fortraining, which are usually not available from field: as an example, few fielddata is available regarding the system-level effects of uncommon but criticalfailure modes.This paper proposes a computational framework for a nearly real-time es-timation of the Remaining Useful Life for dynamical assemblies from measure-ments available from installed feedback or diagnostic sensors. Those can be of3eterogeneous nature: for example, current flowing inside an electric circuit,position and speed of an actuator, pressure and temperature of hydraulic fluidat given locations of the system. The methodology combines an optimal signalcompression strategy with reduced order modeling and machine learning tech-niques; this allows to obtain a computationally efficient map from the measuredsignals to the RUL, and to reduce the storage and processing power required foron-board, time and resource-constrained computations. Our strategy learns sur-rogate models of the system offline: online, these surrogate models are employedto speed up the computational burden associated with the determination of thecurrent system health condition and with the estimation of the RUL. Addition-ally, offline we determine the location of a set of informative components of themonitored signals to store and process; those are employed online to reduce thedimensionality of the problem.As an application of our methodology, we consider the case of actuators foraircraft flight control systems (FCSs). FCSs are critical aircraft systems becausea failure can lead to the impossibility to control the vehicle, with catastrophicconsequences. Hence, health monitoring for FCSs has great potential to bringsignificant improvements in terms of mission reliability, operating costs, aircraftperformance, and eventually relax requirements on system redundancies. Theproblem is inherently challenging: the models of FCS equipment need to com-bine different disciplines, as mechanical, aerodynamic, structural, hydraulic, andelectrical/electronic subsystems operate together to achieve the required perfor-mances. The number of possible failure modes is high, and so the dimensionalityof the FDI problem. Additionally, different faults may result in similar effectson the system behavior, or particular operating conditions may be misidentifiedas faults. All these aspects make this application an interesting demonstrationcase for the proposed strategy, as they highlight the shortcomings of currentapproaches.In this manuscript, Section 2 introduces the general formulation of the prob-lem, Section 3 details the methodology we propose for the prognostic analysis,Section 4 presents the demonstration problem discussed in this paper and the4ssociated physical models, and Section 5 presents the results of our investiga-tions.
2. Prognostics and Health Management (PHM): problem formulation
The common prognostics flow includes three steps, namely signal measure-ment and storage, Fault Detection and Identification (FDI) and estimation ofRemaining Useful Life (RUL), as depicted in Figure 1. In the signal measure-ment and storage phase, an output signal is measured from the system with astandard acquisition frequency. The signal, sensitive to the system condition, isan indicator of the health status of the components, and can be used to informthe subsequent FDI phase. In the FDI phase, the system output is processed toidentify the early signs of damage and wear. Eventually, in the RUL estimationphase, the identified health condition is used to inform an estimate of the actualremaining useful life of the system.In a traditional PHM process, the signal measurement and storage phaseis the only one performed in real-time. It consists in acquiring data y ( k , t ) 𝒌𝒚(𝒌) Signal measurementand storageFDI(system identification)RUL estimation 𝑦(𝒌, 𝑡)
𝑅𝑈𝐿
Figure 1: Schematic representation of the ideal RUL estimation flow y ( k ),dependent on the health condition of the system k . Large amounts of data canbe easily produced in this phase, which can be cumbersome to store and to dealwith in the following phases of the PHM process: this motivates why the FDIand RUL estimation tasks are usually performed offline. We address this issueaiming to compress the useful information in order to reduce the dimensionalityof the FDI problem.The subsequent FDI phase estimates the current health condition of themonitored system by processing the signals acquired and compressed in theprevious step. Common approaches to FDI rely on the use of models, reliablerepresentations of the physical systems and emulators of their dynamic behavior:a system output signal, sensitive to the damage condition, is measured andcompared to the output signal computed with a numerical model. In [36], aphysics-based model of aircraft flight dynamics is evaluated to compute theresidual between the response of the physical system and its digital twin; then astatistical anomaly detection algorithm analyzes this residual to identify faultsof the aileron actuation. A similar approach is proposed in [37] to determineanomalous behavior of the flight control actuator of a UAV; the strategy analyzesthe effects of the failure at aircraft level: as a result, incipient faults are notdetectable. In [38], a dynamical observer leveraging a Kalman Filter is employedfor the model-based condition monitoring of wind turbines. Hence, the FDIproblem is a system identification problem whose solution (the current faultcondition k c ) is the one that minimizes (ideally vanishes) the discrepanciesbetween the measured signal y and the simulated one y model ( k ): k c = arg min k ( err y ( y , y model ( k )) (1)where, in the most general case, the error function err y ( y , y model ( k ) is a mono-tonically increasing function of (cid:107) y − y model ( k ) (cid:107) ; the particular norm to be used6ay vary, and usually is chosen depending on the peculiar characteristics of themeasured signals. If a purely model-based technique is employed, the computa-tion of y model ( k ) is usually expensive. The need to evaluate the error functioniteratively within an optimization algorithm leads to computational times in-compatible with real-time execution; additionally, the definition of a propererror function may be challenging. Conversely, data-driven strategies are faster,but require large datasets for training, as highlighted by Booyse et al. [39].Such amounts of field data are often unavailable, especially during the designand validation of equipment, since their collection can only be carried out withseveral thousands of hours of operation of such equipment. The health condi-tion k determined with FDI is employed for the estimation of Remaining UsefulLife.The RUL of a system is the remaining time until the system will no more beable to meet its functional or performance requirements, that is, the time whenthe system will not be able to perform its function either at all or within thedesign performance parameters [40, 41]. This definition can be formalized as:RUL = max( t )s.t. φ a ( k ( t )) = “healthy” (2)where φ a ( k ) is an assessment function. φ a ( k ) is a binary valued function as-suming the possible values “healthy” or “faulty” that determines whether thefault vector k corresponds to a healthy system or not (i.e. whether the systemis still compliant to its functional and performance requirements).With the traditional approach to life cycle management, the system usefullife is computed a priori in the design phase, solely from the probabilistic com-bination of components failure rate. This strategy does not account for the realevolution of the components health status, and then produces estimates affectedby a very large uncertainty interval [1, 2, 3, 4]. Popular approaches to RUL es-timations aim at obtaining a more precise estimate of the system life either byextrapolating the current fault propagation rate [42], or by employing a modelof damage growth until the damage condition reaches a threshold. In [43], a7tatistical approach combines a semi-markov model and the Maximum Likeli-hood Estimation (MLE) method to infer a degradation model for the equipment.Nascimento and Viana [44] discuss the use of recurrent neural networks merg-ing physics-informed and data-driven knowledge to model the time evolution ofstructural fatigue. Jacazio et al. [45, 46] propose to employ particle filtering toestimate the system RUL; in [47] particle filtering is combined with CanonicalVariate Analysis (CVA) and Exponentially Weighted Moving Average (EWMA)in order to determine the RUL of rotating equipment. However, these methodsoften require a significant computational effort, or may be highly influenced bythe effect of uncertainty in the estimation of the fault condition. Additionally,the definition of a proper critical failure threshold may be difficult: usually in-dividual thresholds are set for each considered failure mode, not accounting forthe combined effect of multiple faults. These can affect the system performancein a different way than the linear superposition of the effects of individual faults.As a result, a more general and comprehensive definition of a critical failure levelmay be needed.Active research in PHM aims to enable early estimate the RUL, much inadvance to the actual failure event, in order to allocate time for the optimalplanning of maintenance strategies and for the logistics of fleet management.This motivates the interest for advanced FDI procedures to detect incipientfaults at their early stages, before the system-level performances of the equip-ment start becoming significantly and adversely affected. To capture incipientfaults, we specifically developed an importance sampling strategy (Section 3.1)for the computation of the dataset needed for training the machine learningtools. The proposed technique is intended to get denser sampling for smallfaults, where most useful information is expected. Nevertheless, the choiceof an adequate sampling procedure is problem dependent, and other samplingstrategies can outperform the proposed one on different applications of the samemethodology.Both FDI and RUL estimation tasks imply the execution of a system emu-lator: usually this model is associated with an expensive computational effort.8herefore, most existing model-based [20, 21, 22] and data-driven [24, 48, 25]strategies are not suitable for real-time execution. Specifically, we wish to per-form the FDI and RUL estimation tasks on-board, which requires to meet thehardware resources limitations to achieve a nearly real-time process. Thereforespecific strategies are needed to achieve such computational efficiency and tomeet these constraints.
3. Methodology
Our methodology proposes specific combinations of machine learning tech-niques to address each of the three phases of the flow described in Figure 1 ina computationally efficient manner. Specifically, offline we compute surrogatemodels that are employed to speed up the online computations.For the first phase of signal acquisition and storage, we aim to reduce thedata required to store a system output signal y ( k , t ). A uniform standard ac-quisition sampling with a suitable frequency produces a vector y ( k ) whose sizeis impractical for the storage and subsequent processing. For example, moni-toring a single electromechanical actuator may imply the acquisition of currentsand voltages with frequencies in the order of tenths of kilohertz, resulting in adatarate up to several MB/s. To address this issue, offline we define an optimalsignal compression in two steps: projection based model reduction (Proper Or-thogonal Decomposition) and unsupervised machine learning (Self-OrganizingMaps) are combined to determine a set of informative components of y ( k ) tostore and process. Online, the compressed output ˆ y ( k ) is a vector contain-ing only the selected informative components of y ( k ). To improve robustnessagainst measurement uncertainty, ˆ y ( k ) is not fed directly to the subsequentphases. Online we adopt Gappy POD to reconstruct POD coefficients fromthe compressed representation of the signal ˆ y ; those are used as input for faultestimation.The second phase is the Fault Detection and Identification (FDI): this stepaims at identifying the health condition of the system, i.e. the specific fault vec-9or k , with limited computational resources. Offline, we use supervised machinelearning (Multi-Layer Perceptron) to compute a model for the fault condition k as a function of the coefficients α of the POD expansion. Online we use thesurrogate model learned offline to estimate the fault vector k from the recon-structed POD coefficients.The third phase is the RUL estimation. Here, a simple model of damagepropagation is evaluated to compute an estimate of the remaining life of thesystem. In this phase, an estimator of the computationally expensive assessmentfunction is needed as a stopping criterion for the damage propagation. To meetthe constraints in terms of time and available processing power, we proposethe use of a binary classifier, specifically a Support Vector Machine (SVM) toreplace the complete assessment function. The SVM is trained offline on thereference dataset and employed online to speed up the RUL estimation.The strategy is schematically illustrated in Figure 2: Sections 3.2, 3.3 and3.4 describe the two steps signal compression, the FDI phase and the RULestimation procedure, respectively; Section 3.1 describes the collection of thereference dataset used to learn the models (offline). Collection of reference dataset (High dimensional)High fidelity models of system dynamics(computationally expensive)Learn an informative compression map(two-step compression)
Linear projection via Proper Orthogonal DecompositionNon-Linear projection via Self-Organizing MapsLearn a model for fault identification from reduced representations of the signalLearn a model for the assessment of the health statusSurrogate assessment function via Support Vector Machine
Fault estimator via Multi Layer Perceptron
Signal acquisition and compression
Diagnostics:
Fault Detection and
IdentificationPrognostics: Remaining Useful Lifeestimation
Importance sampling to capture early stage faults
Signal acquisition with fixed acquisition frequency (High dimensional)
Signal compression 𝑛 𝑤 -dimensional compression map Reconstruction of signal reduced representationvia GappyPODEstimation of fault condition Assessment of heath statusDamage Propagation model
POD basis vectorsPOD coefficients 𝜶(𝒌)
Status Φ Signals 𝑌 Faults
𝐾 𝒌(𝜶)
Reconstructed POD coefficients 𝒂(𝒌)𝒌(𝑡) healthy
OFFLINE - Learning models ONLINE - Onboard PHM 𝑦(𝒌)
Signal 𝒚(𝒌,𝑡) 𝑅𝑈𝐿 = 𝑡 faulty
Estimator for 𝒌(𝜶)
Health assessmentModel phi(k) 𝑦(𝒌)
Figure 2: Schematic representation of the proposed prognostic flow .1. Importance sampling via particular scaled latin hypercube strategy Learning the surrogate models requires the collection of a training datasetrepresentative of the system behavior under the expected operating conditions.It can be collected according to a variety of sampling strategies, depending onthe specific problem at hand. For the application discussed in this paper wepropose a particular importance sampling method. The reference data can becollected from a variety of different sources including historical data, numericalsimulations of the systems through evaluations of high fidelity models, or exper-imental measurements. In this paper we use data from high-fidelity, accuratemodels of the systems, considered as a ground-truth reference.The data set in organized into the following quantities of interest: • Fault conditions matrix
K: K = [ k , k , ..., k n s ] (cid:62) is a n s -by- n k matrixcontaining in its rows the n s fault combinations k i collected in the dataset. k i are the n k -dimensional fault vectors that carry the information aboutthe system health condition. Each fault vector encodes in its elementsa combination of progressive damages of the system. The elements ofthe fault vectors are, in general, related to physical quantities of differentnature, such as the friction coefficient between two sliding surfaces of amechanism, the mechanical play of a transmission, or the resistance of anelectric circuit; to avoid the effect of different scales and inhomogeneousmeasurement units, we chose to normalize those quantities, in order tobound the elements of k i between 0 and 1. • Measured signals matrix
Y: Y = [ y , y , ..., y n s ] is a n e -by- n s matrixcontaining in its columns the output vectors y ( k i ) of the system. y ( k i )are the output signals of the system for each sampled fault combination k i , expressed in the form of n e -dimensional vectors by capturing them ata fixed acquisition frequency. • Assessment function matrix
Φ: Φ = [ φ a, , φ a, , ..., φ a,n s ] is a 1-by- n s matrix containing the values of the assessment function corresponding11o each fault combination. φ a,i = φ a ( k i ) is the value of the assessmentfunction for the fault vector k i .The particular sampling strategy for collecting the dataset is problem depen-dent. The application presented in this paper requires to detect the early signs ofa system damage with high accuracy in order to determine the Remaining UsefulLife in advance enough to plan corrective actions. For this reason, the trainingdataset shall be denser of health conditions close to the nominal one, that is,when either no faults are present or faults are small and do not have a significanteffect on the system performances. In this context, we implemented a form ofimportance sampling strategy through a scaled latin hypercube [49, 50, 51, 52].This technique is meant to increase the density of sampling points near thenominal condition; this allows to collect more informative samples to capturesmall and incipient faults.For example, the assessment function is expected to assign “healthy” labelsto fault conditions k near the nominal condition k , and “faulty” labels to faultconditions far from the nominal one. Being the number of parameters large, auniform distribution of points in the domain of k would result in a small fractionof points near the nominal condition (i.e. “healthy” fault combinations), and thesurrogate assessment function would be difficult to train. To give an instance,given the eight-dimensional unit hypercube, the points whose distance from theorigin is smaller than 1 account for less than 1.6% of the total volume.To have a significant fraction of sampling points associated to a healthycondition, we proceed with two steps. First, we implement a standard LatinHypercube sampling to obtain a n k -by- n t matrix J = [ j , ..., j n k ]. Each rowof the matrix encodes a sampling point of the training set, and each columnis related to one fault parameter. Then, the points are rescaled to producea uniform distribution in the distance from the origin, measured with a L ∞ metric. Assuming that the nominal condition is in the origin, we do not loseof generality, since we can define ˆ k = k − k . The matrix K = [ k , ..., k n k ] is12omputed from J by scaling its elements near the origin:K ij = (J ij ) n k (3)Then, the rows of J characterized by L ∞ (J i, : ) ≤ L are contained in an n k -dimensional hypercube of side 0 ≤ L ≤
1; being the probability distribution ofJ uniform, their number is approximately: n t L n k (4)Those points are mapped to points of K contained in 0 ≤ L n k ≤
1, then L ∞ (K i, : ) ≤ L n k . Hence, the number of points of K such that (cid:107) K i, : (cid:107) ∞ ≤ a isproportional to a for any a ∈ R , resulting in a uniform probability distribu-tion for (cid:107) K i, : (cid:107) ∞ . This sampling strategy is employed, in combination with thephysics-based models described in Section 4.2, to obtain the reference data ofthe matrices K, Y and Φ. The first obstacle that results in a computationally expensive process is thehigh dimensionality of y . Any output signal measured from the system (i.e. cur-rents and voltages of an electrical machine, hydraulic pressures, accelerationsetc.) with a uniform sampling is a vector composed by n e = ∆ tf s elements,where f s is the acquisition frequency and ∆ t is the observation time. In mostapplications, to capture the required amount of information, f s needs to be inthe order of tenths of kilohertz and n e can easily be in the order of severalthousands. This requires large storage capabilities and processing power for thesubsequent Fault Detection and Identification phase. For instance, the use ofcommon least squares methods for the subsequent FDI phase involves the QRfactorization of an n e -by- n k matrix, where n k is the number of fault parame-ters, which computational cost ( O ( n e )) makes the online/on-board executionimpractical for the large n e of common output signals y .To overcome this problem, we use a particular strategy for the optimal se-lection of a small number of the sampling points to retain, store and process. A13rst approach of this kind was introduced by Mainini and Willcox [53], whereProper Orthogonal Decomposition and Self-Organizing Maps are combined forthe optimal placement of sensors for on-board assessment of structural capa-bilites. In this work a similar approach is adopted to reduce the computationalburden associated to the FDI task by reducing the problem dimensionality to n w (cid:28) n e .The selection of the signal points to process online is computed offlinethrough a two-step procedure to learn a compression map: it combines loworder representations of high dimensional data (projection based model reduc-tion) and machine learning techniques (unsupervised machine learning) to iden-tify the most informative instants of time of the measured signal to be storedfor the subsequent online Fault Detection and Identification. We aim to determine an informative compression map for the signal y ( k , t ).Only those points of the signal will be stored and processed online, reducing therequired computational resources. The offline signal compression process takesas input a set of measurements from the system. For this purpose, we use thefault conditions in K and the associated output signals (snapshots) y assembledinto the columns of the n e -by- n s measurement matrix Y. Through the proposedtwo-steps offline compression strategy, we determine the set of informative time-locations for the signal. The compression process is articulated into the two stepsleveraging Proper Orthogonal Decomposition (POD) and Self-Organizing Maps(SOMs) respectively. Linear projection via Proper Orthogonal Decomposition (POD).
The first stepof compression employs data gathered by simulations or experimental cam-paigns, with the purpose of obtaining a reduced order representation of thesystem. This reduced model is computed through Proper Orthogonal Decom-position (POD). POD [54, 55, 56, 57, 58] is a projection based reduced ordermodeling technique commonly employed to obtain low dimensional represen-tations of high dimensional quantities, through the identification of underlying14eatures (in the form of dominant modes). One of the most employed strategy isthe method of snapshots [59]. Data points are represented in the n e -dimensionalspace, and the dominant modes are the principal directions along which thepoints are dispersed. The eigenvalues associated with each mode encode thevariance of the data set along that direction.We apply POD to the measurements matrix Y, in order to extract the modesassociated with the largest eigenvalues, that explain most of the variance of thedataset. The POD modes constitute an orthonormal basis for the measuredsignals collected in Y [60, 61]: it is optimal in the least squares sense and can becomputed through Singular Value Decomposition (SVD) of matrix Y to obtaina representation of each training signal as: y ( k ) = y + n s (cid:88) i =1 v i α i ( k ) (5)where y denotes a reference signal (in our application the system output innominal conditions), n s is the number of snapshots, equivalent to the totalnumber of POD modes, v i are the POD modes and α i ( k ) are the coefficientsof the POD expansion. The eigenvalue λ i associated to each mode v i is a mea-sure of the dispersion of the high dimensional training data along the directiondefined by the mode itself: by considering only the first n m modes of the PODexpansion (equation 5), the fraction of retained information is given by the cu-mulative sum of the eigenvalues (cid:80) n m i =1 λ i / (cid:80) n s i =1 λ i . The POD modes are orderedaccording to their associated eigenvalue, so we can truncate the expansion toretain only the first n m (cid:28) n s modes and to get a low dimensional representa-tion of the signal. If the cumulative sum of the retained eigenvalues is close to100%, the information lost in the compression is accordingly small; additionally,if the training set is statistically representative of the actual system behavior,the same compression can be applied to signals not belonging to the trainingset.Through POD we obtain a set of basis vectors v i and the associated coef-ficients α i for each column of the training set Y. Bases and coefficients of the15OD expansion are employed both offline and online in the following steps ofour procedure. Non-linear projection via Self-Organizing Map (SOM).
In the second step ofsignal compression we use the first n m POD basis vectors to find a compressedrepresentation of the basis vectors themselves through a Self-Organizing Map(SOM). This compressed representation is identified in the form of a set of n w (cid:28) n e highly informative time-locations for storing and processing the signal.A Self-Organizing Map is a single layer neural network that can be usedto identify subsets of similar data through unsupervised competitive learning[62, 63, 64]. The n w neurons of the SOM have representations in the inputspace as weight vectors whose values are updated during the training. In thiscase the input space is the n m +1 dimensional parameter space given by the timecoordinate t and the n m modes of the POD (see Section 3.2.1). The training setfor the SOM is given by the first n m modes of the POD and the correspondingtime coordinate t , arranged in an n e -by-( n m + 1) array T:T = [ t , v , ..., v n m ] (6)where t and v i are column vectors of n e elements. During training, all thepoints of the training set are presented to the network multiple times ( epochs )in a different order, to avoid a training bias. For each training point τ i (the i -throw of T), a winner neuron l is the one whose weight vector w l is the closest tothe input point: l = arg min j ( (cid:107) τ i − w j (cid:107) ) (7)where (cid:107) · (cid:107) denotes the L norm adopted as similarity metric for the studydiscussed in this work. The neighbor neurons are activated according to aneighborhood function, defined in the space of the topological representationof neurons, usually decreasing with the distance from the winner neuron andsymmetric about the winner neuron [62, 64]. One of the key characteristics ofSOMs is that during training, the weight vectors of the neurons are updated to16epresent a non-linear projection of the high dimensional training data (the first n m POD modes) onto a lower dimensional manifold, where prototype vectorsencode representative points of the POD modes [53, 65]. As a result, oncetraining is complete, the first components of all the n w weight vectors encodethe most informative time-locations ˆ t for the signal y . Online, these specific values in the time coordinate are used to store andprocess the measured signals. Those signals are acquired in real-time by sensorsinstalled on the monitored equipment, with a constant frequency high enoughto capture the information related to the considered progressive failures. Thisresults in a continuous data stream from the sensor to the acquisition electron-ics, with a rate than can reach the order of megabytes per second for a singlesensor. Data measured during the observation time ∆ t could be stored in an n e dimensional vector y . However, leveraging the two steps compression com-puted offline, we can store only the n w informative components. The resultingcompressed signal is a n w -by-1 vector ˆ y ( k ), with n w (cid:28) n e , that preserves theuseful information regarding the faults affecting the system. As such, ˆ y ( k ) isused as informative input for the subsequent phase.In principle, the compressed signal ˆ y ( k ) can be directly processed for thedetections and identification of the associated fault condition; however, ˆ y ( k )carries measurement noise and a random error on the signal would directlyaffect the identification of faults (FDI). To mitigate the effect of measurementnoise, we propose to compute the POD coefficients α ( k ) from the compressedsignal ˆ y ( k ) via Gappy POD and move the FDI task onto the reduced spaceidentified by the POD in Equation 5.Gappy POD is a procedure derived from Proper Orthogonal Decompositionand is commonly used for the recovery of incomplete data [66, 67, 68, 69].Given ˆ y , the reconstructed signal can be obtained as a linear superpositionof the first n m POD modes computed offline (Equation (5), Section 3.2.1).The expansion coefficients α j ( k ) are computed to minimize the squared error17etween the known points of the compressed signal ˆ y and its reconstruction inthe n w informative elements. The coefficients α j are reconstructed by solvingthe linear system: G α = f (8)where G = ˆ v (cid:62) ˆ v is the Gappy Matrix and ˆ v = [ˆ v . . . ˆ v n m ] is a n w -by- n m matrix whose columns contain the n w informative elements of the first n m PODmodes. The vector f is the projection of the compressed signal ˆ y along thecompressed POD modes ˆ v : f = ˆ v (cid:62) ˆ y (9)An approximation of the uncompressed signal could be recovered as thelinear combination of the first n m modes weighted by the coefficients α , as perEquation (5). However, for the purpose of this work, we are not interested inthe reconstruction of the original signal, but exclusively in recovering the PODcoefficients α ( k ). These are employed in the next step to identify an estimateof the fault condition k . The Fault Detection and Identification (FDI) phase of our strategy aimsat identifying the health condition of the system (the specific fault vector k )from the information of the compressed signal ˆ y . This task is a parameteridentification problem and is formulated as an optimization problem (Equation1). However, the use of common gradient-based or meta-heuristic optimizationalgorithms for parameter identification requires the iterative evaluation of sys-tem emulators that are frequently expensive. The accuracy level required for areliable identification of the fault condition demands for the evaluation of mod-els of the dynamical system that are usually too computationally expensive toevaluate online [70, 71].To meet the efficiency requirements of time constrained online evaluations,we employ feedforward Neural Networks to estimate the fault vector k from18 𝛼 𝛼 𝛼 n 𝑚 𝑊 𝑊 𝑊 n 𝑚 𝑎 𝑖 tansig 𝑏 . . . Σ 𝑎 𝑎 𝑎 n ℎ 𝑊 𝑊 𝑊 n 𝑘 𝑘 𝑖 𝑏 . . . Figure 3: Block diagram of the i -th sigmoid neuron of the hidden layer (left) and the i -thlinear saturated neuron of the output layer (right) the compressed representation provided by the reconstructed POD coefficients α . Specifically, a Multi-Layer Perceptron (MLP) maps the POD coefficientsto the fault vector. The FDI task is split into an offline phase, in which theNeural Network model is trained, and an online phase, in which it is evaluatedto estimate k . In order to estimate the fault vector k within a computational time suitablefor real-time evaluation, we train offline a neural network to obtain a surrogatemodel for the fault condition k from the low dimensional representations of themeasurement provided by α ( k ).The specific implementation of Multi-Layer Perceptron (MLP) adopted inthis paper is characterized by a standard feedforward architecture, with a singlehidden layer; more complex machine learning strategies may be tested in futureworks. The network receives in input the POD coefficients α ( k ) and returns thefault vector k . The hidden layer has n h neurons with sigmoid activation func-tion, while the output layer has n k neurons with a linear saturated activationfunction. The specific choices for the activation functions reflect the physicalcharacteristics of the input and output variables of the problem.19igure 3 represents the architecture of a sigmoid and a linear saturatedneuron. The n m inputs α (column vector) are weighted by the coefficients W (row vector) and summed. Then the weighted sum is fed to a sigmoid function,which returns the output a of the neuron: a = tansig( W α + b ) (10)where: tansig( x ) = 21 + e − x − b is a bias constant. The output layer is composed by saturated linearneurons, whose transfer functions are linear saturations: k i = , if W a + b < . W a + b, if 0 ≤ W a + b ≤ . , if W a + b > . (12)The saturation is introduced to account for the bounds of the output faultvector whose components are bounded between 0 and 1, as defined for ourapplication (Section 4.2). Figure 4 shows the complete network architecture.The weights W and the bias b of each neuron are determined during training, inorder to tune the network to approximate the expected output for a training data Σ 𝜶 𝑾 ℎ 𝒌tansig Σ 𝑾 𝑜 hidden layer output layer Figure 4: Architecture of the two layer perceptron employed for the FDI task. W h are theweights of the hidden layer neurons, W o are those of the output layer α i , k i ), includingthe fault vectors k collected with the sampling strategy detailed in Section 3.1and the associated POD coefficients computed as per procedure described inSection 3.2.1.During training, a Levenberg-Marquardt backpropagation algorithm [72, 73]updates the weight and bias variables to minimize a performance function, de-fined as the mean squared error between the expected and actual output of thenetwork for the training set. The training is stopped when either the maxi-mum number of epochs is reached or the performance gradient decreases belowa threshold. Once training is complete, we obtain a model to map from α to k using the n s training signals. The input of this phase are the POD coefficients α estimated via GappyPOD, as per Section 3.2.2. Those are fed to the MLP model learned offline,in order to estimate the faults k . This approach is preferred over the straightadoption of a neural network over the full dimensional dataset because the com-pression allows to significantly reduce the computational cost, both in trainingand in evaluation of the MLP [74]. The output of the FDI process is an estimateof the fault vector k , to be employed in the subsequent RUL estimation. The estimation of the Remaining Useful life is the last phase of the PHMprocess. We aim to complete it onboard, given the fault condition k estimatedthrough the FDI procedure discussed in Section 3.3.In this paper we propose a strategy for RUL estimation relying on a damagetolerant approach to system design similar to that adopted for the estimationof fatigue life in aircraft structures. Leveraging the definition introduced byEquation (2), the heath state k detected at the mesurement time t = 0 is usedas an initial value to compute the evolution of the health condition through aspecific model for damage propagation. The damage propagation model is in21he form of an Ordinary Differential Equation (ODE) whose evaluation providesthe rate of damage growth as a function of the current system health and theoperating and environmental conditions. An assessment function φ a ( k ) is em-ployed as a stopping criterion for the integration of the ODE model: accordingto the definition of Equation (2), it evaluates the system performances for eachvalue assumed in time by the fault vector, to determine whether that specificfault vector is compatible with the system operation. When a faulty system isdetected, the integration is stopped and the last time step is assumed as thesystem RUL.The damage propagation model in the form of ODE may not be computa-tionally expensive since the fault propagation rate is considered to be affectedby a limited number of factors (heat dissipation, vibration levels, degradationof surface finish). Conversely, the evaluation of the assessment function φ a ( k )usually implies multiple executions of the models of system dynamics, which iscomputationally expensive and unsuited for nearly real-time applications. Toaddress this limitations we propose to use supervised learning techniques (specif-ically Support Vector Machines, SVMs), which are trained offline on a referencedataset to obtain surrogate models of the assessment function to employ online. The assessment function φ a ( k ) is essentially a binary classifier: it analyzesthe behavior of the system in presence of the fault combination k and determineswhether in this condition the functional and performance requirements are met.This process usually involves the resolution of a dynamical model of the systemand requires a high computational effort. To meet the time constraints foron-board estimation, we adopt a standard implementation of a Support VectorMachine (SVM) as a binary classifier and a surrogate for the assessment function φ a ( k ) to run online.A Support Vector Machine [75, 76, 77] is a machine learning paradigm com-monly used for data classification and regression. According to the standardlinear formulation, given the training set K of n s fault conditions k i ∈ R n k a i = ±
1, we seek the hyperplane in R n k separating thecategories a i (Figure 5): f ( k ) = k (cid:62) β + b (13)where β ∈ R n k and b ∈ R . The training of the SVM searches for the besthyperplane which divides the classes of k i , that is, the hyperplane that pro-duces the largest margin between the classes (see Figure 5); this is equivalentto find β and b that minimize (cid:107) β (cid:107) , subject to a i f ( k i ) ≥
1. The optimizationis a quadratic programming problem, and the training algorithm implements aLagrange multipliers method.In our application, the classes of the training data set cannot be separatedby a linear boundary. For this reason, a polynomial kernel function ψ ( k ) isused to map the input points to a transformed predictor space where a linearboundary can be identified.After training, new input points are classified according to the sign of a scorefunction, that is the equation of the separating hyperplane; this quantifies the Training points (class 1)
Training points (class -1) Support
Vector
SupportVectorSupportVector
Figure 5: Graphical representation of an SVM classifier f ( k ) = (cid:88) i σ i a i < ψ ( k ) , ψ ( k i ) > + b (14)where σ i are the Lagrange multipliers computed during training. The sign of thescore function ˜ φ a ( k ) = sgn( f ( k )) constitutes our surrogate for the assessmentfunction φ a ( k ), suitable to run online and used for the RUL estimation process. The SVM trained offline is employed as a surrogate assessment function tospeed up the real-time computations involved in RUL estimation.The fault propagation rate can usually be described by an Ordinary Differ-ential Equations (ODE) model, accounting, in the most general case, for thecurrent health condition of the system, the environmental and operating con-ditions, and the expected mission profile. The evolution of the system healthstatus is computed through the numerical integration of this ODE model. Theinitial condition is set as the fault vector k estimated in the previous FDI phase.At each integration time step t i the surrogate assessment function ˜ φ a ( k ) deter-mines whether the current fault vector k ( t i ) corresponds to a healthy systemor not. Since this has to be evaluated iteratively, the use of the full model-based assessment function φ a ( k ) would result in long computational times, notsuitable for real-time evaluation. When a faulty condition is detected by the(surrogate) assessment function, the integration is stopped. At this point wecan assume: RUL = t (15)where t is the current integration time. That is, the Remaining Useful Life ofthe system is assumed to be equal to the timestep when the system transitionedfrom a healthy condition to a faulty one.24 . Diagnostics and Prognostics of Aircraft Actuation Systems We develop and demonstrate our methodology for the real-time predictionof Remaining Useful Life for aircraft actuators. Actuation systems involve theinteraction of several, heterogeneous engineering disciplines, such as electronicsand software, electrical machines, mechanical systems, hydraulics, structures,thermal dissipation problems, fluid dynamics, vibrations, and tribology. Addi-tionally, damage propagation may be affected by operating conditions that arenot completely predictable, as opposed to similar actuation devices employedfor static applications, such as for industrial automation. As a result, faultsaffecting such systems have effects on performances that are difficult to predict,and the associated detailed models are computationally expensive.Computational methodologies intended to address the open challenges of di-agnostic and prognostics for aircraft actuators are of critical interest for missionreliability and cost effectiveness of the whole fleets. Failures in such subsystemscan lead to increased down time of the vehicle and may require risk mitigation,since most of these devices are safety critical.Similar actuation technologies are employed in different fields of engineering,sharing the same open challenges regarding health monitoring and management.As an example, the failure of an actuator on a production line can require theshutdown of the whole production line for repair, with significant income losses.Given the multidisciplinary nature of the considered application, computationaltechniques developed to address the prognostic analysis of servo actuators canbe extended to deal with health monitoring of similar components not necessar-ily within the domain of actuation systems. Specifically, any dynamical systeminvolving power electronics, electrical machines, sensors, or mechanical and hy-draulic power transmission, can be a potential application of the proposed healthmonitoring strategy.
The particular application addressed in this paper is the real-time estima-tion of Remaining Useful Life for an Electromechanical Actuator (EMA) for25ircraft Flight Control Systems from on-board measurement of the motor cur-rent. EMAs [78, 79, 80, 81] exploit an electric motor coupled to a mechanicaltransmission to convert power from the aircraft electric system into mechanicalpower to move the flight control system aerodynamic surfaces. Those actuationsystems are commonly employed in small scale UAVs and for secondary flightcontrols of larger manned aircraft.An overall weight reduction, compared to the more traditional hydraulicand electrohydraulic systems, can be achieved employing electromechanical ac-tuators to power the whole flight control system of an aircraft (as highlightedby the More Electric Aircraft and All Electric Aircraft design approaches [82,83, 84, 85, 86]). The elimination of a centralized hydraulic power generationsystem is particularly advantageous for the weight budget of the smaller vehi-cles, although the power density of an electric actuator is lower than that of anhydraulic one [87, 88, 89]. Moreover, maintenance on electric systems is easierthan on hydraulic ones, since there are no issues related to fluid leakages andcontamination.On the other hand, EMAs are not yet widely employed on safety criticalfunctions for manned aircraft. This is mainly due to the presence of a complexmechanical reducer between the motor and the aerodynamic surface, that intro-duces the risk of mechanical jamming as a possible failure mode [90, 91, 92, 93].This eventuality can lead to the impossibility to control the aircraft with catas-trophic consequences. The introduction of accurate and reliable PHM tech-niques would increase the safety of operations. Then, a more widespread use ofEMAs in larger manned and unmanned vehicles would be allowed, enabling toexploit their advantages on weight and power budget [94, 95].The block diagram of Figure 6 shows the architecture of the considered EMA.This includes a BrushLess Direct Current (BLDC) electric motor along with itspower and control electronics, a reducer with a high gear ratio to increase thetorque for the user, and a Linear Variable Differential Transformer (LVDT)position sensor to close the feedback loop.For this study, we consider the effects of five different failure modes, chosen26 ontrol electronics
Power inverter
BLDC motor Gearbox UserPosition transducer (LVDT)Motor position sensor
Command
Power supply electrical connection
Motor position signal
User position signal
Figure 6: Architecture of the Electromechanical Actuator among the most common for EMAs [96, 97, 98] and characterized by a slowpropagation rate, to allow an effective estimation of the Remaining Useful Life.Those are namely variations in dry friction ( k ) and backlash ( k ), partial shortcircuit of each of the three stator phases ( k , , ), rotor static eccentricity ( k , )and controller proportional gain drift ( k ); this results in a fault vector k =[ k , k , k , k , k , k , k , k ] of n k = 8 elements. The complete definition of thefault vector k is illustrated in Table 1. We chose the stator envelope current asthe informative variable y to monitor for the prognostic analysis. The reasonfor this choice is twofold: it is highly sensitive to a number of fault modes andcan be easily measured in a physical system; in addition, in many cases statorcurrents are already measured with the purpose of closing a current feedbackloop. Table 1: Definition of the fault vector k fault parameter fault mode lower bound ( k i = 0 ) upper bound ( k i = 1 ) k dry friction nominal friction 300% of nominal friction k backlash nominal backlash 100 times nominal backlash k phase A short circuit no short circuit full short circuit k phase B short circuit no short circuit full short circuit k phase C short circuit no short circuit full short circuit k rotor eccentricity no eccentricity eccentricity equal to air gap width k eccentricity phase − o o k proportional gain drift 50% of nominal gain 150% of nominal gain .2. Physical models of system dynamics Two models of the actuator with different fidelity are employed: a High Fi-delity (HF) model (Section 4.2.1) is used only offline as the source of referencedata: for the high accuracy of the model this dataset can be used as a goodemulator of ground truth reference data [99]. A Low Fidelity (LF) model (Sec-tion 4.2.2) is used within the assessment function to determine the frequencyresponse of the actuator and compare it with its requirements. The accuracy ofthe LF model is considered suitable for the sake of this task; the computationalcost of the HF model would be impractical for the iterative evaluation requiredby the assessment function, even for offline execution. Section 4.2.3 describesthe model for the damage propagation rate, employed for the RUL estimationprocess.
The HF model is the accurate dynamical model of the EMA, simulating indetail the physical behavior of the actuator subsystems and components. Themodel accounts for the effects of Pulse Width Modulation (PWM) three-phasecurrent control logic of the motor power electronics, and includes a completelumped parameters model of the electromagnetic coupling between stator androtor. This HF model is employed as a simulated test bench, to compute thereference data in replacement of a physical system.The architecture of the model is shown in the block diagram of Figure 7.The Actuator Control Electronics block simulates the EMA controller, whichcompares the commanded and actual positions to compute a reference currentsignal I ref for the motor. The BLDC Power Electronics subsystem contains themodel of the three-phase inverter used to power the BLDC motor. It applies theneeded voltages ( V A , V B and V C for the three motor phases, respectively) on themotor windings to produce to produce the currents I A , I B and I C required bythe controller; the Hall sensors measure the rotor position θ m that is then usedto synchronize the phase commutation with the motor rotation. The BLDCElectromagnetic model contains the rotor-stator coupling model to evaluate the28 ctuator Control Electronics Commanded position 𝐼 𝑟𝑒𝑓 Evaluation of active phase 𝐼 𝑟𝑒𝑓𝐴 𝐼 𝑟𝑒𝑓𝐵 𝐼 𝑟𝑒𝑓𝐶 Motor Position Sensors (Hall
Sensors) + - + - + - HysteresisPWM Control Three-phaseInverter 𝐼 𝐴 𝐼 𝐵 𝐼 𝐶 𝑉 𝐴 𝑉 𝐵 𝑉 𝐶 torque Computationof back-EMF
Motor-Transmission
Dynamical Model 𝜃 𝑚 𝜃 𝑚 Load 𝑇 𝑚 𝜃 𝑢 BLDC Power Electronics + - + - + - BLDC electromagnetic model 𝜃 𝑚 𝜃 𝑚 𝜃 𝑚 𝜃 𝑚 𝜃 𝑢 𝐼 𝐴 𝐼 𝐵 𝐼 𝐶 𝐴 𝐶 𝐵𝐴 𝐵 𝐶 Computation of envelope currentand signalfilter 𝐼 𝐻𝐹 𝐼 𝐴 𝐼 𝐵 𝐼 𝐶 Rotor-Statorcoupling modelMagneticFlux
Figure 7: Detailed block diagram of the High Fidelity Model magnetic flux across the air gap. This is employed to compute both the counter-electromotive forces on the stator windings and the torque T m produced by themotor. The motor-transmission dynamical model is a second-order model ofthe actuator mechanics and computes the angular positions θ m and θ u of motorand user respectively, accounting for several nonlinear effects such as backlash,dry friction and mechanical endstops. The signal acquisition block computesthe envelope I HF of the three phase currents and applies a low-pass signal filterto suppress the high frequency noise produced mainly by the PWM control andobtain the output signal y employed for the PHM analysis.The HF model is implemented in the Matlab-Simulink simulation environ-ment. Its accuracy comes at the expense of a relatively high computationaleffort. The simulation of a reference 0.5 seconds test signal takes about oneminute on a common laptop PC, making this model unsuitable for real-timeapplications. 29 .2.2. Low Fidelity (LF) model The LF model is a simplified dynamical representation of the same EMA,with complex subsystems represented by simpler blocks. This model is usediteratively to compute the assessment function φ a ( k ) (see Sections 3.4 and 4.2.3).The block diagram of the model is shown in Figure 8. The most compu-tationally expensive sections of the HF model are the three-phase inverter andthe computation of the magnetic flux across the air gap. Those subsystems aredifferently handled in the LF model, replaced by a first order DC model whosegoverning equation directly relates the motor current I m , voltage V m and torque T m through the back-Electromotive Force (back-EMF) coefficient κ v : RI m + L ˙ I m = V m − κ v ω (16) T m = κ v I m (17)where R and L are the stator resistance and inductance, and ω is the motorangular speed. This simplified model does not allow accounting directly for theelectric faults of partial short circuit and rotor eccentricity. For this reason,Berri et al. [99] proposed to model those failure modes by introducing twoshape functions to modulate the motor parameters as a function of the angularposition of the rotor. The same approach is adopted in this paper: the shapefunctions are fitted to emulate the waveform of the back-EMF of the HF model PID controller 𝐼 𝑟𝑒𝑓 + - 𝐼 𝑚 sign 𝑉 𝑎𝑙 𝑉 𝑚 𝑅 𝑠 + -Back-EMF 𝐼 𝑚 𝜅 𝑣 𝑇 𝑚 Motor-User dynamics (2° order model) 𝜃 𝑚 𝜃 𝑚 Back-emf model
Commanded position Load
Signal filter 𝐼 𝐿𝐹 𝜃 𝑚 Figure 8: Block Diagram of the Low Fidelity Model
30n presence of the eccentricity and short circuit fault modes.The computational time needed for the execution of the LF model is abouttwo orders of magnitude lower than that of the HF model. The average rootmean square discrepancy between the HF and LF model is in the order of 1%.
The model of damage propagation is an ODE based model assuming thateach fault grows linearly with the others. It is expressed as:˙ k ( t ) = F∆ k ( t ) + (cid:15) (18)where ˙ k denotes the fault growth rate, ∆ k ( t ) = k ( t ) − k , k is the faultvector in nominal conditions, (cid:15) is an independent identically distributed normalnoise, and F is a square matrix whose F ij element expresses the influence ofthe j -th fault parameter on the growth rate of the i -th fault parameter. Thematrix F depends on the physics of the system, and can be identified from fielddata. The integration is stopped when the assessment function φ a ( k ) indicatesthat the fault condition k achieves damage levels that jeopardize the systemperformance. The assessment function calls the LF model iteratively with a setof input frequencies to compute the system Bode diagram (it would be veryimpractical to use the HF model, even for offline computations). Then, thephase margin, gain margin and cutoff frequency are compared to the thresholdsimposed by performance requirements to determine whether the actuator isworking correctly or not. The computational time of the assessment functionon a common laptop PC is in the order of 10 seconds: despite the use of the LFmodel, the computational cost is not compatible with real-time RUL estimation;this motivates the need of developing a surrogate model for the assessmentfunction to handle the task efficiently (Section 3.4.1).The integration of the ODE is affected by uncertainty due to the noise (cid:15) andproduces a large dispersion. In this paper, we handle uncertainty propagation byperforming the integration several times for a given starting condition. Then,a gaussian probability distribution is fitted over the RUL estimates, and the31alues corresponding to 5%, 50% and 95% probability are saved. In order toobtain a deterministic algorithm, the uncertainty component (cid:15) is replaced witha disturbance δ on the initial fault vector k ( t = 0). δ assumes three valuescomputed with a bisection method and calibrated to produce the aforementioned5%, 50% and 95% probability RUL estimates.
5. Results and Discussion
This section discusses the results obtained with the application of the offlineand online phases of the PHM methodology to the specific problem consid-ered in this paper. A training set is collected from data computed with thephysics-based models of Section 4.2. Specifically, a set of n s = 10000 faultcombinations is computed with the importance sampling strategy described inSection 3.1, to obtain the matrix K. In our application, k is usually excludedby the distribution rescaling process. This parameter encodes the phase of therotor static eccentricity with respect to the stator windings; then, its probabilitydistribution is necessarily uniform, and should not be modified. The HF modelof the actuator (Section 4.2.1) is evaluated for each fault combination; a testcommand is employed, consisting in a linear chirp characterized by a 0.5s dura-tion, 5 · − rad amplitude, 0Hz start frequency and 15Hz end frequency. Theoutput signals y ( t, k ) are acquired with a constant frequency of 20kHz (result-ing in n e = 10001), necessary to capture the information required by the FDI,and then assembled into the columns of the measurement matrix Y. The as-sessment function described in Section 4.2.3 provided the “healthy” or “faulty”labels associated for each fault combination, to be stored into the matrix Φ.Additionally, two validation sets are assembled to assess the performance of theproposed strategy. A first validation set is computed to assess the offline train-ing of surrogate models and the online compression and FDI (Sections 5.1 and5.2). This includes 500 fault combinations, sampled as per Section 3.1, and theassociated signals and values of the assessment function. A second validationset is employed to assess the online RUL estimation procedure (Section 5.3).32his dataset includes 100 fault combinations, the corresponding HF signals andassessment function values, and the RUL (with the associated uncertainty) com-puted through the damage propagation model described in Section 4.2.3. Thevalidation of the online procedure requires the initial faults to be small (i.e.near the nominal condition); otherwise, the system would be already faulty, itsremaining Useful Life would be null, and the real time RUL estimation couldnot be properly tested. Therefore, this second test set is sampled with the pro-cedure of Section 3.1 on a restricted domain in k , to include mostly “healthy”conditions.The following sections describe the application of our methodology to theproblem of RUL estimation for aerospace EMAs, described in Section 4. Inparticular, Section 5.1 shows the offline learning of surrogate representationsof the models that describe the physics of the considered problem; Section 5.2discusses the results of the online procedure. All computations were performedon a desktop PC with a i5 3330 quad-core processor @3.00GHz and 8GB ofmemory, running Windows 10 and Matlab R2016a The following paragraphs discuss the application of the offline procedure tothe specific problem addressed in this work: the results of the strategy for deter-mining the optimal time coordinates to store and compress the measured signal,derived through the procedure described in Section 3.2; the performance of themodel that maps the fault vector k ( α ) as a function of the POD coefficients,derived according to Section 3.3.1; the outcome of the surrogate model for theassessment function φ a ( k ), derived as per Section 3.4.1. Two-steps data compression.
The matrix Y of of the training dataset is em-ployed to learn the informative compression map described in Section 3.2. Yis used to compute the POD expansion in the form of Equation (5) whose first n m modes are retained and used for (i) optimal points selection for signal com-pression (determined offline) and (ii) POD coefficient reconstruction via gappyPOD (to run online). 33 time, s -0.03-0.02-0.0100.010.020.03 f i r s t P O D m ode [ A ] First POD mode Location of n w = 30 informative components Figure 9: Placement of 30 sampling points over the first POD mode
For the offline identification of the most informative signal points we use thetwo steps procedure of Section 3.2 that allows us to compute a set of n w = 30points. Those are optimally placed to capture the information of the first n m POD modes. In this paper, the number of points is chosen to retain sufficientinformation for the FDI on the base of a previous investigation presented in [71].Figure 9 shows an example of placement of the sampling points for n m = 1: thepoints are not equally spaced in time, but rather tend to be placed by thealgorithm in the most significant points to capture the shape of the mode. Learning the model for k ( α ) . For the FDI step, a Multi-Layer Perceptron withone hidden layer is used to compute the estimated fault vector k from thecoefficients α reconstructed via Gappy POD. The training set for the neuralnetwork model is composed by the POD coefficients computed in the previousstep, and the matrix K. The choice of a suitable number of neurons for thenetwork emerges from a tradeoff between accuracy and computational time. Astudy was performed on the number of neurons, by varying the neurons of thehidden layer from 5 to 100, while the number of neurons of the output layerremains constant at n k = 8. Figure 10 shows the computational time of thenetwork in training and in evaluation, as well as the mean squared error in the34
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000.050.10.150.2 e rr k E v a l ua t i on T i m e [ m s ] Gappy POD Total (POD + MLP) neurons of the hidden layer T r a i n i ng T i m e [ s ] Figure 10: Error in fault identification (top), computational time in evaluation (middle) andin training (bottom) for variable number of neurons in the hidden layer. fault identification plotted against the number of neurons in the hidden layer: err k = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i =1 w i ( k estimated i − k actual i ) (19)where the weights w i are all unitary except for w = k actual6 . The elements k and k of the fault vector encode the amplitude and phase of the rotor staticeccentricity, respectively (see Table 1). The error on eccentricity phase is thenweighted by the actual eccentricity amplitude: this permits to achieve a morephysically significant error estimate when the eccentricity is small in amplitude.By increasing the number of neurons in the hidden layer, the accuracy ini-tially increases and the mean squared error decreases down to 5% for 20 neurons.Adding more neurons does not produce significant benefits, neither on accuracy35or on computational time in evaluation; conversely, the increased complexityof the model reflects in longer computational times for training. Therefore, weconsider the network with 20 neurons in the hidden layer as the most efficientcandidate to perform the FDI for the addressed application. The computationaltime required by Gappy POD for the estimation of POD coefficients is at leastone order of magnitude shorter than that required by the MLP in evaluation;so, the use of POD coefficients instead of the signal does not carry a significantpenalty in computational time for real-time FDI.The fault vector k estimated computed by the network is employed as initialcondition for the RUL estimation. Learning the model for the assessment function.
For the surrogate modelingof the assessment function φ a ( k ), we employ a polynomial kernel SVM, trainedwith the matrix K as the input and Φ as the target. The SVM was assessed withthe 500 signals validation set, and achieved a 98.2% success rate in emulatingthe assessment function (see Table 2), with an average computational time inthe order of 1ms, thus reducing the computational effort of almost four ordersof magnitude with respect to the evaluation time required by the completeassessment function. This section presents the application of the online process to the consideredproblem. Specifically, to assess the performance of the prognostic framework,
Table 2: Performance of the SVM; test set composed of 500 fault combinations, of which 123corresponding to a healthy actuator and 377 corresponding to a system failure correctly detected 119 96,75%correctly undetected 372 98,67%missed detections 4 3,25%false positives 5 1,33% total correct 491 98,20%total wrong 9 1,80%
Signal acquisition and compression.
The signal is compressed online by storingand processing only the measures corresponding to the time coordinates deter-mined offline through the procedure described in Section 3.2. Online, GappyPOD is employed to determine the coefficients α ( k ) from the stored signal.Figure 11 shows the reconstruction of a compressed signal from the test set viaGappy POD, with an increasing number of POD modes. To assess the accuracyof the signal estimate, we evaluate the normalized root mean squared error err α of the POD coefficients α gappy estimated with Gappy POD with respect to thecoefficients α full computed using the full signal: err α = (cid:115) n m n m (cid:80) i =1 ( α gappy i − α full i )max ( α full ) − min ( α full ) (20) time, s -20020 c u rr en t s i gna l , A n m =1 original signalreconstructed signal time, s -20020 c u rr en t s i gna l , A n m =2 time, s -20020 c u rr en t s i gna l , A n m =5 time, s -20020 c u rr en t s i gna l , A n m =14 Figure 11: Signal reconstruction via Gappy POD with a variable number of retained modes n m number of retained POD modes -2 -1 e rr , Figure 12: Offline preliminary assessment of Gappy POD coefficients identification, for varyingnumber of modes retained. The blue crosses are the median error over the test set, while theamplitude of the tolerance band is set equal to the interquartile range. n m = 22 modes resultin the best accuracy, with a median error below 2% Figure 12 shows the error computed for increasing size of the gappy ma-trix G, that is, for increasing number of dominant components used for signalreconstruction through Gappy POD (according to Equation (5)).For a number of modes larger than 10, it is already possible to identify thecoefficients of the POD modes with an error in the order of 1%, comparable tothe discrepancy yielding for the LF model with respect to the HF model (SeeSection 4.2). We choose to employ the first 22 POD modes for the followingsteps, corresponding to about 97% of the snapshot information; as shown inFigure 12, for our application this number of modes yields the minimum meanvalue of err α . The use of a larger number of modes might increase the riskof including misleading information, affecting the accuracy of the signal recon-struction via Gappy POD. Additionally, with more than 30 modes (i.e. moremodes than sampling points) the gappy matrix G becomes ill conditioned; thesignal reconstruction is strongly affected by numerical noise with an increase ofcomputational time and an error the same order of magnitude of the signal. Asimilar behavior has been already observed in literature [100].38 ault Detection and Identification. This section describes the assessment of theonline performance of the two-layer perceptron for the FDI algorithm. Fig-ure 13 shows the error on parameter identification highlighting the contributionof each component of the fault vector. The average error for each variable | k estimated i − k actual i | is at most in the order of 1%, which is comparable to thediscrepancy between the HF and LF model. Then, our data driven FDI tech-nique performs comparably to a more traditional model-based technique, whichcommonly exploits an online optimization algorithm to match the HF and LFresponses [70, 101]. The FDI accuracy with respect to the individual fault pa-rameters depends on the particular application and on the sensitivity of themonitored variable to the different fault modes. Figure 13 highlights how, forthe considered application, the FDI performs better in the identification of thedry friction fault ( k ): the envelope current (i.e. the analyzed output signal y ) is highly sensitive to this failure mode, resulting in better accuracy. Onthe other hand, the rotor eccentricity ( k ) is identified with a higher error andgreater dispersion. This failure mode results in high frequency disturbances ofthe monitored signal, and the effect of information loss due to the truncationof the POD expansion is worse. The computational time is in the order of afew milliseconds, allows computations to be run onboard, and enables real-timefault detection. k(1) k(2) k(3) k(4) k(5) k(6) k(7) k(8)00.020.040.060.08 e rr k Figure 13: FDI error on each fault parameter stimate the Remaining Useful Life. To assess the accuracy of the online RULestimation procedure alone, we test the performance of our online RUL stepleaving out the error contribution associated with the FDI step. To do so, wetest the SVM based algorithm using the reference fault vector k actual as initialcondition, in place of the estimate computed in the FDI phase k estimated .Figure 14 shows the upper and lower bound of the RUL (with 90% confi-dence) as a function of the expected RUL. On the horizontal axis is reported theRUL computed by the full model (i.e. the damage propagation model of Section4.2.3 in combination with the assessment function φ a ( k )) at 50% probability,which we assume to be the actual value. The red dashed line is the bisector ofthe first quadrant, and represents an ideal RUL estimate (i.e. not affected byany error). The tolerance band represents the uncertainty interval of the fullmodel: its lower and upper bounds correspond to the RUL computed by the fullmodel respectively at 5% and 95% probability, respectively. The blue crossesare the RUL values estimated with the SVM model from the actual initial con-dition; in most cases, the estimated values fall within the uncertainty intervalassociated to the reference physics-based model. A final test assesses the whole real-time PHM flow, including the signalcompression and FDI strategy in combination with the RUL estimation method.The SVM based RUL estimation is computed with the initial condition k estimated Actual RUL, h E x pe c t ed RU L , h Uncertainty interval of the physics-based modelBisector of the first quadrantRUL estimate
Figure 14: Expected Remaining Useful Life with physics-based assessment function and SVM,assuming the FDI error to be null (i.e. starting from the actual fault condition). Actual RUL, h E x pe c t ed RU L , h Uncertainty interval of the physics-based modelBisector of the first quadrantRUL estimate
Figure 15: Predicted Remaining Useful Life with estimated fault condition and SVM surrogateassessment function estimated through the FDI algorithm described in Section 3.3; the results, shownin Figure 15, are compared to the expected RUL (with the associated uncer-tainty interval) computed through the full damage propagation model. Theglobal RUL estimate is affected by the errors introduced by both the RUL es-timation itself and the FDI; then, the uncertainty is necessarily higher thanthe previous case, resulting in a slightly greater dispersion. RUL values largerthan 4000 hours are commonly underestimated. In these cases, the initial faultparameters assume very small values, that are of the same order of magnitudeof the uncertainty associated to the FDI process. The identified fault condi-tion is commonly worse than the actual one, which results in estimating a fasterfault propagation and a shorter RUL. Larger system faults can be detected withhigher relative accuracy and the dispersion associated with the RUL estimatedecreases. Therefore, our strategy achieves a higher accuracy when it is needed,that is when a failure is about to occur. When the Remaining Useful Life is long,there is no stringent need to know its value with high precision because thereis long time ahead to plan the maintenance strategy at best, purchase sparesand schedule replacements. Additionally, in the first part of the system oper-ational life, an underestimation of the RUL is preferable to an overestimation,for obvious safety reasons, and does not trigger an unnecessary maintenanceintervention, since the estimated time to failure is still long.Figure 16 compares the uncertainty associated to the physics-based modelfor RUL with the relative error in RUL estimation err
RUL resulting from the41UL estimation alone (starting from k actual ) and in combination with the FDI(starting from k estimated ). The error is defined as follows: err RUL = | RU L estimated − RU L actual | RU L actual (21)and the uncertainty of the physics-based model is:∆
RUL = | RU L − RU L | RU L actual (22)where
RU L and
RU L are the upper and lower bounds of the RUL with90% confidence. The error of the RUL estimation is the same order of magnitudeof the uncertainty interval of the physics based model. The error associated withthe whole PHM process is still the same order of magnitude, but its distributionis more skewed to the right compared to the error of the RUL estimation alone.This reflects in a higher median error, confirming the underestimation of verylong RULs that emerges from the comparison from Figures 14 and 15.The absolute variance of the error is relatively high, being usually in theorder of 20% of RUL, and in some cases almost comparable to the RUL itself.This behavior is partly due to the inherent uncertainty in the rate of damagepropagation, which depends on a multitude of uncontrollable and unpredictablevariables, such as environmental conditions and the particular time-history ex- -2 -1 uncertainty of physics-based model nu m be r o f o cc u r en c i e s -2 -1 error on the RUL estimate -2 -1 error for the entire PHM information flow Figure 16: Comparison between the uncertainty of the physics-based model for RUL evaluation(left), the error of the real-time RUL estimation process (middle) and the error of the entirePHM online information flow (right). a priori estimate [1, 2, 4, 3], and adds virtually nocost for the implementation, since no dedicated hardware is required. The com-putational time required online by the whole PHM process is in the order ofmilliseconds and allows the FDI and RUL estimation to be performed in real-time. In contrast, traditional model based approaches require computationaltimes that range from minutes to tenths of minutes, which would be completelyunsuited for on-board applications. These strategies [102, 103] imply solvingnumerically systems of ODEs with very small integration timesteps, and theindentification is performed over the full dimensional dataset with empirical orsemi-empirical optimization algorithms.
6. Concluding remarks
A comprehensive methodology for real-time fault detection and prognos-tics of dynamical assemblies has been proposed. Our methodological frameworkleverages a combination of projection-based model reduction and machine learn-ing strategies to achieve reliable and timely estimates of the system useful life.The method has been developed and assessed for the overall Prognostics andHealth Management (PHM) process applied to an electromechanical actuatorfor aircraft flight control systems. In addition, a simple model for estimatingthe fault propagation rate has been proposed, and a custom sampling techniquehas been employed to capture sufficient information from the system with alimited number of samples, which resulted particularly effective for our specificapplication.The strategy permits to achieve an accuracy in FDI and RUL estimationthat is comparable to computationally-intensive physics-based methods; at thesame time, it requires few online data storage and processing resources, allowing43or a fast and reliable on-board, real-time execution: the online computationaltime for fault detection is reduced by several orders of magnitude with respectto standard computationally expensive, physics-based methods. The availabil-ity of RUL estimate in real-time would permit to efficiently inform adaptivemaintenance planning, allowing for significant cost reduction with respect tothe standard periodical inspections and replacements, based on the analysis ofthe average failure rate of the components. The results show that our strategyfor the real-time estimate of Remaining Useful Life allows to achieve high predic-tion accuracy when the monitored components are approaching the end of theiroperative life: this permits a dynamic and informed scheduling of maintenanceinterventions and an adaptive delivery of supplies and spares; additionally, themission can be dynamically reconfigured to avoid overstressing faulty subsys-tems. When the system behaves nominally, our strategy tends to underestimateits useful life; however, this occurrence is safe and does not result in planningthe unnecessary replacement of healthy components, since both the actual andestimated RUL are long.Future developments include the assessment of alternative machine learn-ing strategies for the FDI and RUL estimation, a more exhaustive study onuncertainty propagation, and the experimental validation of the models andalgorithms.
Acknowledgments
The authors thank Prof. Paolo Maggiore at Politecnico di Torino for hissupport. Additional acknowledgments to the Visiting Professor Program ofPolitecnico di Torino for the support to Dr. Laura Mainini.
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