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1 Aerospace Science and Technology 16 (2012) Contents lists available at ScienceDirect Aerospace Science and Technology Aircraft engine health management via stochastic modelling of flight data interrelations D. Dimogianopoulos, J. Hios, S. Fassois Stochastic Mechanical Systems & Automation (SMSA) Laboratory, Department of Mechanical Engineering & Aeronautics, University of Patras, GR Patras, Greece article info abstract Article history: Received 20 November 2009 Received in revised form 1 February 2011 Accepted 2 March 2011 Availableonline5March2011 Keywords: Health management Fault detection Fault isolation Stochastic modelling Nonlinear models Engine systems Decision maing Statistical inference A novel engine health management (EHM) scheme is introduced. It uses flight-level, instead of thermodynamic, data to cost-effectively augment the onboard EHM redundancy. For a nominal healthy aircraft, fault-sensitive interrelations among flight data are globally modelled inside a flight regime via Constant-Coefficient Pooled Nonlinear AutoRegressive with exogenous (CCP-NARX) excitation representations. Single or sequential engine faults perturb these interrelations. Statistically evaluating the perturbation-induced effects draws reliable conclusions on the engine s health. Validation and comparisons with Kalman filter-based alternatives are made throughout the regime under various operational conditions Elsevier Masson SAS. All rights reserved. 1. Introduction In modern aircraft, and even more so in future pilot-less versions, high reliability and safety should be cost-effectively obtained. The multiplication of existing critical hardware (the hardware redundancy principle), coupled to voting schemes for health management (HM) purposes, has its limits due to added weight and cost [26]. Significant improvements on reliability/safety may, hence, only result from additional analytical redundancy based on the smart use of available system-level signals and hardware [18]. Introducing engine HM (EHM) analytical redundancy by modelling part of the healthy engine dynamics using physics-based principles, and then detecting fault-induced trends in model parameters (or specific functions of them) is commonly used in either on-board [6] or on-the-service-bay [34] versions. Other schemes use the gas path analysis (GPA) approach to identify highly accurate models on typical health-state-related data along the gas path (afterfan total temperature and pressure, high pressure compressor temperature and pressure, and so on): In [25] linear stochastic models (AutoRegressive with exogenous excitation * Corresponding author. Tel.: ; fax: Since March 2010, the author is Assistant Professor at the Technological Educational Institute of Piraeus, 250 Thivon Av., Egaleo Athens, Greece. Tel: addresses: dimogian@mech.upatras.gr (D. Dimogianopoulos), hiosj@mech.upatras.gr (J. Hios), fassois@mech.upatras.gr (S. Fassois). ARX, Output Error OE or Prediction Error PE) are augmented with model-error models, designed following the H principle and aiming at minimizing modelling uncertainties around an envelope location. In [27] the model augmentation uses a neural networ-based part. In both schemes fault-induced trends between the model output and that of the actual engine provides HM results. A majority of approaches rely, however, on models of the healthy engine dynamics identified via GPA measurements and coupled to fuzzy logic rules [19], or Kalman filters (KF) in standard or adaptive form [24] for estimating fault-induced changes on health parameters. An interesting comprehensive study [30] compares schemes using KFs in linear (LKF), extended (EKF) and unscented (UKF) versions. Both EKF- and UKF-based schemes show superior performance at the price of (one and two, respectively, orders of magnitude) higher online computational effort than the LKF. This is due to necessary onboard transformations (which potentially involve badly-conditioned matrices). Other EHM schemes rely on multiple engine models or observers to describe the engine dynamics in various health states, and compare the output of each model to that of the engine. Then, the correct model and the associated health state are deduced based on the model output being close to that currently measured from the engine. Multiple health-state-related models are built using either KFs [21,23], Taagi Sugeno fuzzy rules [12], or neural networs [11,28]. Multiple health-state-related observers (designed /$ see front matter 2011 Elsevier Masson SAS. All rights reserved. doi: /j.ast

2 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Fig. 1. Theenginehealthmanagementschemeanddetailoftheaircraftsimulatorwiththeenginesubsystem. for enhanced robustness) are reported in [10,29], or, in nonlinear versions in [35]. Finally, non-model based schemes use thresholds on engine health state parameters [32], pattern recognition techniques [3], or even fuzzy inference [2] on groups of engine parameters to detect fault-induced trends. These wors are comprehensively reviewed in [33]. The proposed onboard EHM scheme innovates by operating on flight-level, rather than engine thermodynamic, data. Since available data are used, the expensive/complex development of new physical sensors or hardware is avoided. The resulting (unrelated to GPA) EHM redundancy is intended to complement the existing GPA-related one, in order to enhance the available decision-maing capabilities. The approach extends the generic ideas of relevant wor in [13], and relies on stochastic nonlinear modelling of the interrelations among flight data such as acceleration, thrust and so on. Hence, the use of (often unavailable) physics-based or empirical models, as in [6], is no longer needed. These interrelations are fault-sensitive, that is, valid exclusively for a nominal healthy aircraft, and are modelled via Constant-Coefficient Pooled Nonlinear AutoRegressive with exogenous (CCP-NARX) excitation representations. Potential single or sequentially occurring engine faults perturb these nominal (CCP-NARX modelled) interrelations. Then, custom-built statistical hypothesis tests evaluate the perturbationinduced effects in order to detect abnormal engine operation, and (when possible) isolate causes of engine malfunctions. Hence, additional ability to judge the GPA-based internal engine diagnosis is provided. The paper is organized as follows: Section 2 presents the aircraft (a nonlinear simulator) and the considered engine faults. Section 3 describes the CCP-NARX modelling procedure and the design of the statistical tests. Section 4 shows some modelling and HM results, along with brief comparisons with LKF-based schemes. Finally, Section 5 presents some concluding remars. 2. The aircraft and the faults A simulator of an autonomous twin-engined small passenger aircraft is used in this study. It is a 6 degree-of-freedom nonlinear aircraft model, with the inputs (stic, wheel, pedal, throttle) provided from a specifically designed autopilot bloc according to a predetermined flight trajectory (Fig. 1). The input and the resulting attitude signals (including the angle-of-attac α, the sideslip angle β, surface moments, accelerations and angular rates) are available. The wind and turbulence effects (conforming to MIL- F-8785C) are considered as system disturbances. The considered faults affect the left engine, and are simulated by acting on selected signals fed into the engine subsystem, as shown in Fig. 1. Table 1 The faults considered. Type Description Magnitude F A Part of throttle input (% oftotal) = 25 fed to engine F B Unstable thrust & flameout = 550 F C Recurrent engine thrust reduction (% ofnominal) = 30 F D F A fault shortly followed (in t s) 1,2,t 1 1 = 25, 2 = 550, by F B fault t = 10, 20 s 2 The HM scheme monitors the engine operational level, rather than any internal components (fans, compressors and so on) as in GPAbased schemes. Thus, a relatively simple engine model is considered, since no GPA signals (other than the total thrust) have to be used. The simulated faults involve: a) Partial reduction of the throttle input signal fed into the left engine control module, with the measurements of altitude and Mach signals (designated as Alt[t] and Mach[t], respectively) being unaffected. The engine operation is substantially altered, as if the lin from throttle to engine suffered a severe, but not fatal, failure (see Fig. 2(a) from instant 150 s onwards). This fault is referred to as F A with A indicating the fault type and its considered magnitude (% of the throttle input normally entering the engine control module, see Table 1). b) Unstable left engine thrust output simulating inconsistent compressor airflow before flameout (see Fig. 2(b)), that is, engine shutdown. This situation may be due to severe successive internal incidents (such as blade loss) causing fast deterioration. This fault is referred to as F B with B indicating the fault type, and its magnitude, corresponding to the colored noise s standard deviation σ = added to the thrust output to simulate the fault. c) Recurrent left engine thrust reduction (see Fig. 2(c)). The left engine thrust repeatedly drops to lower values for short time intervals, possibly due to unstable compressor airflow. This situation may be due to either physical damage, or control software issues. The fault is referred to as F C,withC indicating the fault type and the % periodic reduction of the current left engine thrust nominal value. d) Multiple sequentially occurring faults affecting the left engine. The occurrence of F A faults is followed some seconds later by F B faults affecting the already faulty engine. Clearly, this sequential occurrence is not equivalent to superposing the effects of single F A and single F B faults, meaning that a separate F D class is necessary. These sequential faults are 1, 2,t referred to as F D, with D indicating the fault type and 1, 2,t

3 72 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Fig. 2. Total engine thrust response for faults occurring at 150 s: (a) Healthy versus F A 25 ;(b)healthyversusf B 550 ;(c)healthyversusf C 30 ;(d)healthyversusf D 25,550,10 affected aircraft. 1, 2 the magnitudes of the initial F A and the subsequent F B 1 2 faults, respectively. Finally, t indicates the time interval in s between the F A 1 and F B 2 fault occurrence (see a typical case in Fig. 2(d)). Both F A and F B are considered as modelled faults, meaning that their possible occurrence has been accounted for during the HM scheme s design. The same is valid for F D faults, since they 1, 2,t consist of single modelled but sequentially occurring ones. On the contrary, F C faults are considered as unmodelled, since many possible incidents cause a recurrent partial loss of engine thrust. As shown in Section 4.2, an abnormal engine operation engine may always be detected, but the originating fault may be isolated only if it belongs to the set of modelled faults. Remar 1. Shortly after F B or F C fault occurrence at 150 s, the affected engine s thrust in Figs. 2(b) and (c), respectively, is largely similar to that from a healthy engine. Changes are noted only after the flameout (160 s onwards) in the F B case. This is due to the choice of F B and F C magnitude, intended to produce minor fault-induced effects on the total thrust, in order to chec the EHM scheme s sensitivity to fault occurrence. 3. The engine health management scheme 3.1. Overview of the scheme s design and operation The attitude of a standard passenger aircraft is designed to be linear under the (auto) pilot s action, and may, hence, be modelled via linear representations. Nevertheless, abrupt maneuvering, significant turbulence/wind and other disturbances may induce nonlinearities in relationships between specific flight variables (see Section 4.1). Such factors are also responsible for extra noise in measured signals. Finally, aircraft operation in broad envelope areas may also induce modelling uncertainty, meaning that models with varying parameters are required to accurately represent the dynamics at each area s point. Instead of using multiple, often adaptive, local models (associated to high onboard computational burden) for globally describing the aircraft dynamics as in [21], the proposed scheme relies on a single stochastic pooled nonlinear model of the considered dynamics (see Fig. 1). Specifically, a Multi-Input Single-Output (MISO) CCP-NARX representation (Section 3.2) describes the relationship among four signals obtained from healthy aircraft; three taen as inputs (lateral acceleration A y, side-slip angle β and rudder moment) and one as output (total engine thrust). Thrust from one engine may also be used, for compatibility with vectoring strategies. Note that, instead of examining separate signals, the joint dependence between all inputs and the thrust is considered. The reason is that, for instance, changes in roll rate may be caused by either F A faults or manoeuvering, faults in control surfaces and so on. Such changes must be related to the engine behavior, in order to obtain reliable EHM results. This particular input and output signal selection is made for two reasons: first, because these input signals are random functions of time due to turbulence and other factors and have, thus, varying amplitude and rich frequency content even under reduced pilot activity. This provides persistent excitation, which is the prerequisite for effective CCP-NARX identification [20]. Second, because several fault-induced engine thrust variations may be associated to specific combined changes of the acceleration A y, β angle and rudder moment. Naturally, other signal choices are possible, provided that the previous conditions are met. The current MISO CCP-NARX model addresses all previous modelling issues, because: The model s stochastic nonlinear form explicitly accounts both for noise-in-measurement issues and for nonlinearities in the dynamics, without requiring nown disturbance bounds as when linear [9] or nonlinear deterministic models are used [35,36]. Via the pooling technique (Section 3.2) relevant recorded past data may contribute to the CCP-NARX identification (see Fig. 3). Thus, modelling uncertainties from operation in broad envelope areas are globally reduced, instead of being minimized around given points via H or other robust approaches [25]. The identified CCP-NARX model is then used during the operational (diagnostic) phase (Section 3.3), which taes place onboard an aircraft operating with engines in unnown health state. Using data collected during any such flight and the identified CCP-NARX

4 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Fig. 3. Schematic of the pooled representation s identification problem, showing the flight envelope, the excitation/response signals from different flights and the CCP-NARX structure. model, the engine s current condition is assessed as follows: For healthy engines, the obtained residual sequence e[t] (that is, the difference between the real thrust output and its CCP-NARX resulting prediction, see Fig. 1) features certain statistical properties. For faulty engines, these statistical properties are no longer valid. Then, statistical decision-maing hypothesis tests may chec and evaluate the fault-related information in e[t] for EHM purposes. Remar 2. The proposed EHM scheme is designed to operate along (and interact with) similar HM schemes, monitoring critical systems such as control surfaces (see [15]). Hence, the analytical decision-maing ability is enhanced and further complements that based on GPA or other approaches for cost-effectively promoting safety. Finally, the interaction between HM schemes enables crosschecing of HM decisions, so that faults affecting a given circuit may not influence HM decisions in other monitored systems Baseline modelling phase (on-the-ground tuning phase) The CCP-NARX model representing the interrelations among the considered input/output signals admits the form: L y j [t]= θ i p i, j [t]+e j [t] { i=0 E e j [t] e s [t τ ] } = γ e [ j, s]δ[τ ] e j [t] ( NID 0,σe 2 ( j)) (1) with t designating normalized discrete time, y j [t] and e j [t] the representation s output and prediction error (or residual, assumed to be a zero-mean uncorrelated sequence with variance σe 2( j)) signals of the j-th flight, respectively. The input signals u m, j [t] (m = 1, 2,...) are not explicitly shown in (1), but are involved in the regressors p i, j [t] (i = 0,...,L and j = 1,...,M). These are nonlinear terms such as products between lagged values of the output y j [t] and the m-th input u m, j [t], orevenpowersofthese signal values (with p 0, j [t] = 1, by definition). The nonlinearity degree of the representation, that is, the sum of powers of the signal values involved in each regressor is less than or equal to nl. The coefficient corresponding to the i-th regressor is designated as θ i and is constant for all flights. The maximum lags of signals y j [t], u m, j [t] in (1) (model orders) are n y and n um, respectively. The statistical expectation is designated as E{ }, whereas NID(.,.) stands for Normally Independently Distributed (with the indicated mean and variance), δ[τ ] the Kronecer delta (δ[τ ]=1 when τ = 0 and δ[τ ]=0 when τ 0), and γ e [ j, s] the cross covariance (see [8, pp ]). Note that, from (1), the residuals from two different flights j and s may potentially be correlated. Rewriting (1) in a linear regression form yields 1 : y j [t]=φ T j [t] θ + e j[t] (2) with φ j [t] =[p 0, j [t]...p L, j [t]] T the regressor and θ =[θ 0...θ L ] T the vector of coefficients, respectively. If M flights (each of N recorded data samples) are available for the modelling phase, the following matrix equation is obtained: y = Φ θ + e (3) where y = [y 1... y M ] T R [NM 1], Φ = [Φ 1...Φ M ] T R [NM (L+1)], e =[e 1...e M ] T R [NM 1]. The vectors y j, e j R [N 1] include the N data samples from the j-th flight, while Φ j =[φ T j [1]...φT j [N]]T R N (L+1). Finally, note that when a single flight is considered, the previous CCP-NARX representation becomes a simple NARX one. During the baseline phase, the objective is to identify the CCP- NARX(n y, n um ) model, that is: a) To choose the regressors p i, j of nonlinearity degree up to nl, which most accurately describe the interrelations between the considered signals (structure selection), and b) to compute the associated vector of CCP-NARX coefficients. The structure selection involves determination of nl for the mapping between the I/O signals, based on a ey property of linear systems: Their response to a sinusoidal input (of arbitrary amplitude but given frequency) is also a sinusoidal signal of the same 1 Lower case/capital bold face symbols designate column vector/matrix quantities, respectively.

5 74 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) frequency (and shifted phase). The ratio of the output over the input amplitudes is constant and only related to the frequency used. On the contrary, for nonlinear systems this ratio may change for a given input frequency according to the input amplitude. Hence, determining nl involves exciting the nonlinear system with input signals of different amplitudes at a given frequency, and creating a plot of the output amplitude versus that of the input: The value of nl equals the degree of the polynomial, which approximates sufficiently well the plot curve. The procedure is repeated for as many frequencies as possible inside the frequency range of interest, for finding the most representative nl with respect to the considered frequencies. The determination of the current system s nonlinearity degree is presented in Section 4.1. For a given nl value, the regressors are selected using either the Forward Orthogonal Search (FOS) algorithm [7], or bootstrapping techniques [22], or even genetic algorithms. Past experience with CCP-NARX identification [13] shows that the best results are achieved by combining the FOS algorithm with the bootstrapping technique. In the next paragraphs, this procedure is briefly outlined, with more details (along with presentations of the FOS algorithm and the bootstrapping technique) given in [13]. i) From the original data set of M flights, each data sequence (of N samples) is sectioned to a number of N s -sample long overlapping segments. ii) New pooled data sets, each one containing M N s samples, are formed and for each such set the FOS algorithm performs structure selection. The regressors identified for each such set represent the dynamics of M flights during the short time segment of N s samples. Due to the overlap, the selected regressors account effectively for changing dynamics between two successive segments. iii) The union of all previously selected regressor forms the Extended Regressor Set (ERS), which is highly representative of the dynamics of all M flights throughout the considered time period. Via the bootstrapping technique [22] the ERS is considerably reduced without noticeably deteriorating the modelling accuracy. iv) Finally, the number of regressors obtained in step iii) may be reduced again via term clustering techniques [1]. Each cluster contains regressors formed by the same combination of signals (but of different lagged values). The clusters are successively removed from the CCP-NARX model (one at each iteration) and the accuracy of the remaining CCP-NARX model part is assessed by computing the Residual e Sum of Squares over the Signal y Sum of Squares (RSS/SSS) ratio, with signals e and y found in (3). The omitted clusters causing significant RSS/SSS increases are retained, thus yielding a compact final set of regressors. The corresponding CCP-NARX model is validated via the tests in [7]: The successful CCP-NARX model must offer the best compromise between low RSS/SSS values, almost uncorrelated residuals e j [t], and limited cross-correlation among the inputs u m, j [t] and the residuals e j [t], for each flight j. For the selected structure, the estimation of CCP-NARX coefficients is made via an Ordinary Least Squares (OLS) or Weighted Least Squares (WLS) criterion [16]. Remar 3. The model orders n y and n um in (1), are determined prior to the global CCP-NARX structure selection, via a simple trialand-error procedure: For many individual flights, local NARX(n y, n um ) models are identified (as in the CCP-NARX case) with the maximum initial n y and n um values supported by the given computers. Then, local NARX(n y, n um ) models are identified and validated for reducing n y and n um values, until obtaining a low order but accurate model. Those model orders are then considered for the global CCP-NARX model. Remar 4. The identified CCP-NARX model is valid for a healthy aircraft of given weight and center of gravity position. However, due to fuel consumption the accuracy of the initial CCP-NARX modelmaybealteredbytheendoftheflight.thisproblemmay be trivially solved by creating in the baseline phase CCP-NARX models valid for weight ranges (for instance, full tan, two-thirds, and so on). The tas is trivial since the original CCP-NARX regressors are ept, and only the associated coefficients are instantly re-estimated with flight data from an aircraft operating with the designated weight. Switching among the CCP-NARX models allows for consistently obtaining accurate system representations at the price of the HM scheme being off during the switch. Similar reestimations could also deal with (healthy) engine aging issues. Note that, a definitive solution to this problem involves allowing the (currently constant) CCP-NARX coefficients to be functions of the aircraft weight. Then, the CCP-NARX model becomes a Timedependent Functional Pooled-NARX (TFP-NARX) one, since the aircraft weight is a function of time. Recently, TFP-NARX representations have been used for HM of control surface actuators, with promising results [14]. Remar 5. The computational effort required for a CCP-NARXbased HM scheme is analyzed in [15]: Compared with schemes based on multiple models [21], EKF or UKF[30] any considerable effort is related to the on-the-ground baseline phase (the CCP-NARX identification), rather than the onboard diagnostic one. Crucially, insignificant onboard effort is needed for operating the statistical tests Operational (diagnostic) phase (onboard) For a flight conducted under unnown aircraft engine state (healthy/faulty), the residuals e j [t] (Fig. 1) are collected using the CCP-NARX model identified during the baseline modelling phase. For simplicity, the index j will be dropped hereafter. If the aircraft operates with healthy engines, the residual variance σe 2 is bounded within specific limits. However, this changes when certain engine faults occur: The residual variance increases substantially. Based on this property, a statistical hypothesis test may be designed for the online detection of faults affecting the engine(s). For this purpose, an l-sample long vector [e[t (l 1)]...e[t]], referred to as sliding window, is considered at each time instant t, and the hypothesis testing problem is designed as follows [17]: H 0 : σu 2 [t] σ 2 0 [t] healthy engine H 1 : σu 2 [t] > σ 2 0 [t] faulty engine (4) with H 0 and H 1 designating the null and alternative hypothesis, respectively. The residual variances for the e[t] sequences resulting from the current (unnown) engine state and the healthy state are designated as σu 2[t] and σ 2 0 [t], respectively. The current (unnown health state) variance estimate 2 ˆσ u 2 [t] iscomputedfromthe l-sample long sliding window. Since the current flight is assumed to start with healthy aircraft engines, an estimate of the healthy state variance ˆσ 2 0 is computed using l 0 of the initial samples of the e[t] residuals. The statistic F[t]= ˆσ u 2[t] follows F distribution, and ˆσ 2 0 the following test is designed at ris level α (that is false alarm probability equal to α): F[t] F u H 0 accepted, no fault at time t Else H 1 accepted, fault at time t (5) 2 Symbols with hat designate estimators/estimates.

6 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Fig. 4. Statistical hypothesis tests used in the HM scheme: (a) F-test for checing the variance of the residuals e[t] (fault detection); (b) chi-square test for checing the correlation of the residuals e[t] (fault isolation), and (c) normal test for the PACF coefficients (fault isolation). The shaded areas correspond to accepting the null hypothesis H 0 at ris level α. with F u = F (1 α) (l 1,l 0 1) indicating the distribution s (1 α) critical point (see Fig. 4(a)). Following fault detection, isolation is also performed by means of specifically designed hypothesis tests. These statistical tests utilize the lac of correlation property of the residuals e[t], when they result from an aircraft with either healthy engines or engines affected by faults unrelated to those designated for isolation. An l-sample long sliding window [e[t (l 1)]...e[t]] T is formed at each time instant t, and a hypothesis testing problem is designed [31]: H 0 : ρ i = 0 i = 1,...,r (engine not affected by considered fault) H 1 : ρ i 0 for some i (engine affected by considered fault) (6) The terms ρ i designate the autocorrelation coefficients of the sequence e[t] [8, p. 55]. Under the null hypothesis the Q statistic below follows χ 2 (chi-square) distribution with (r s + 1) degrees of freedom: r Q = l ˆρ 2 i χ 2 (r s + 1) (7) i=s with l designating the window length and s indicating the lower sum limit, usually chosen equal to 1 as in [31, pp ]. The hypothesis test at the ris level α, is then formulated as: Q < χ 2 (r s + 1) (1 α) H 0 is accepted Else H 1 is accepted (8) with χ 2 (r s + 1) (1 α) being the value of a statistical quantity following χ 2 distribution with (r s + 1) degrees of freedom and (1 α) critical point (see Fig. 4(b)). The test (8) (referred to as Test A) utilizes(r s + 1) coefficients of the autocorrelation estimate ˆρ i (insteadofthetrueunnownρ i ). An alternative test for fault isolation, based upon the Partial Auto-Correlation Function (PACF) Φ ii of the residual sequence, is also designed. The PACF is the correlation between e[t] and e[t i] after their mutual linear dependency on the intervening variables e[t 1], e[t 2],...,e[t i + 1] has been removed (see [8, pp ] for more details). To set up the test, the variable δφ ii = Φ ii Φ 0 ii is considered, with Φ ii being the partial autocorrelation coefficient at lag i for a data sequence (that is, an l sample long sliding window of e[t]) andφ 0 the empirical mean partial autocorrelation value at lag i when the system is affected by the fault ii designated for isolation. This value is obtained from a number of flights (other than those used for testing) conducted with an aircraft with engines affected by the considered fault. The following composite hypothesis testing problem for the true (but unnown) partial autocorrelation coefficient is then designed: H 0 : δφ ii = 0 (the considered fault occurred) H 1 : δφ ii 0 (another fault occured) (9) In practice, the PACF is obtained by fitting successive AutoRegressive (AR) representations of orders 1, 2,... by OLS and eeping the last coefficient of each regression [8, pp ]. Then ˆΦ ii is equal to the last coefficient of the identified i-th order AR model. Moreover, its variance is given by the (i, i)-th element of the covariance matrix of the estimated AR coefficients. It may be shown that the AR coefficient estimates follow normal distribution [31, pp ]. Hence, treating the computed variance of ˆΦ ii as a fixed quantity, the statistic Z = δσ δ ˆΦ ii 2 follows N (0, 1) distribution under the H 0 hypothesis. Then, a test (referred to as Test B) characterized by ris level α is formed: Z α Z Z 2 1 α H 0 is accepted 2 Else H 1 is accepted (10) with δσ 2 designating the variance of δ ˆΦ ii (which is equal to that of ˆΦ ii ) and Z α the standard normal distribution s α critical point, as in Fig. 4(c).

7 76 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Engine health management results A CCP-NARX model is identified upon data from 84 flights of 300 s each, conducted in the clean flight regime under low turbulence with healthy engines. As seen in Fig. 5, these flights evolve in almost all areas of the clean flight regime. The aim is to obtain data representative of the aircraft dynamics in the majority of clean flight operating points. Thus, the identified CCP-NARX model is globally valid throughout this regime. Then, the EHM scheme is extensively tested via 1008 flights conducted in the clean flight regime as follows: For each one of the six health states 84 flights (of 300 s each) are conducted under low and again under moderate turbulence. All single faults (or the first of the sequential faults) occur at around the same time (t = 150 s), that is, well after the beginning but also well before the end of the flight. In all cases the acceleration A y, β angle, rudder moment and thrust signals are recorded at 25 Hz maing for flights of N = 7501 samples. All flights are conducted under slowly changing wind, with commanded heading slowly varying around zero degrees, and with commanded roll tracing a reference signal r[t] = sin(2π 0.01 t), corresponding to slowly changing cruising conditions. Finally, in order to simulate as realistic conditions as possible, the signal measurements are corrupted with noise with its variance being equal to ,5 10 5, and 1 for the A y, β, rudder and thrust signal measurements, respectively (values typically used in relevant studies [9]) Baseline (on-the-ground) phase modelling results Fig. 5. The aircraft clean flight envelope with the trajectories of the 84 flights used in CCP-NARX identification. When identifying a CCP-NARX model, the first tas to be undertaen is related to determining the nonlinearity degree nl. For this purpose, frequency analysis is performed on the set of 84 modelling flights, and those flights with A y, β angle, rudder moment and thrust signals featuring the most intense amplitude variations and rich frequency content are chosen. Then, following Section 3.2, plotsoftheoutput(thrust) amplitude versus that of each of the three inputs (A y, β and rudder) at each of the considered frequencies are formed: at a given frequency, the square root of the output s power spectral density G y (estimated via the Welch method) is plotted for growing values of each input s G um, m = 1, 2, 3, with G y and G um being the amplitudes of the output and the m-th input, respectively. Fig. 6 presents such a plot for the dominant thrust frequency of Hz, identified by the pre- Fig. 6. Estimation of the nonlinearity degree nl via excitation response plots at f = Hz: (a) thrust versus A y,(b)thrust versus β, and(c)thrust versus rudder moment ( 2nd order, 3rd order and 4th order polynomial fits).

8 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Table 2 Regressors of the CCP-NARX(7, 4) model with 33 terms. 1 y[t 1] y[t 4] y[t 6] y[t 2] y[t 6] y[t 6] y[t 3] u 3 [t 4] u 3 [t 4] y[t 4] u 2 [t 4] y[t 7] y[t 5] u 1 [t 4] y[t 1] y[t 1] y[t 6] y[t 3] y[t 3] y[t 3] y[t 7] y[t 3] y[t 4] y[t 4] u 3 [t 2] y[t 3] y[t 4] y[t 5] u 3 [t 3] y[t 3] y[t 4] y[t 7] y[t 1] y[t 1] y[t 4] y[t 4] y[t 4] y[t 1] y[t 2] y[t 4] y[t 4] y[t 6] y[t 1] y[t 3] y[t 4] y[t 5] y[t 5] y[t 1] y[t 4] y[t 4] y[t 5] y[t 6] y[t 1] y[t 5] y[t 5] y[t 5] y[t 5] y[t 2] y[t 5] y[t 5] y[t 5] y[t 6] y[t 4] y[t 5] y[t 6] y[t 7] y[t 7] vious frequency analysis. Since G y Gum corresponds to the modulus of the empirical transfer function estimate H y,um at a given frequency (see [31, p. 57]), had the system been linear, H y,um would have been constant for growing input amplitude levels. This is not thecaseinfig. 6 for all three input/output relationships: in fact, the curves fitted correspond equally well to third- or fourth-order polynomials. Hence, according to Section 3.2, nonlinear CCP-NARX regressors with value nl up to three may be considered. For nl 3, the model orders n y and n um (m = 1, 2, 3) are selected via the trial-and-error procedure of Section 3.2. Starting from n y = 20 and n um = 10 (the maximum supported by the given computers), tests with individual flights lead to local NARX(7, 4) models offering the best compromise between low order structure and accurate description of the system dynamics. Nevertheless, for a CCP-NARX(7, 4) with nl 3 the number of possible regressors p i, j [t] in (1), as resulting from all possible combinations of lagged input and output signals, is almost In order to reduce this clearly excessive value without significantly compromising the CCP-NARX model s ability of representing the system dynamics, the four-step structure selection procedure of Section 3.2 is used. Finally a CCP-NARX(7, 4) model with 33 regressors (shown in Table 2) is identified and successfully validated using the correlation tests in [7] On-board operational (diagnostic) phase results and discussion The on-the-ground identified CCP-NARX(7, 4) model with 33 regressors is implemented on-board and used for online obtaining the residual sequence e[t], as specified in Section 3.1. Compromised engine operation is detected by using the statistical hypothesis test (5) on the e[t] sequence. Subsequently, the faults compromising the engine operation are isolated by using either Test A in (8) (for F B faults) or Test B in (10) (for F A faults), also applied to the e[t] sequence. For the case of F D faults, each of 1, 2,t the sequentially occurring single faults are separately detected and isolated via the previous tests. A sliding window of l = 200 samples and a fixed window of l 0 = 200 samples are defined, while the ris level is set to α = The choice of the window length has to optimize the trade-off between minimal delay of detecting malfunctions and the scheme s low sensitivity to false alarms. The latter is achieved via the rather long sliding window currently selected, as minimizing false alarms is a priority. Choosing a long sliding window is possible thans to the CCP-NARX model s capability of accounting for fast emerging nonlinearities in the modelled dynamics. These may occur when the aircraft operates close to the envelope boundaries and/or under significant wind/turbulence levels. With respect Table 3 Fault detection and isolation results for low[moderate] turbulence flights. Aircraft condition Correct detections/ flights Correct detection rates (%) Correct isolations/ flights Correct isolation rates (%) Healthy 82[81]/ [96.4] n/a[n/a] n/a[n/a] F A 25 83[79]/ [94] 83/83[78/79] 100[98.7] F B [80]/ [95.2] 82/82[80/80] 100[100] F C 30 82[81]/ [96.4] n/a[n/a] n/a[n/a] F D 25,550,10 82[80]/ [95.2] 82/82[79/80] 100[98.8] F D 25,550,20 82[79]/ [94] 82/82[78/79] 100[98.7] Regime: clean flight Heading around 0 deg Turbulence: low[moderate] Flights: 504[504] Ris level α: 10 6 Commanded roll r[t] N/a: not applicable. to false alarms due to such nonlinearities, the CCP-NARX based scheme is designed to perform better than linear model based schemes, especially when long sliding windows are used. Obviously, the downside of a long sliding window is the longer fault detection or isolation time. Robustness of the HM scheme is also affected by the choice of the signal part used to form the fixed window during the initial seconds of the flight. It has been noted that the variance of e[t] changes more over the few seconds following the scheme s initialization, than it does later on. In the present case, this initial phase is limited to almost 400 samples (or 16 s). For enhanced robustness, the scheme monitors all successive 200-sample long windows during these 16 s and selects that with the largest variance for computing ˆσ 2 0. Thus, the scheme is operational 16 s after being switched on Detection phase Detection results obtained via the test (5) are presented in Table 3. Fig. 7 shows a typical case with engine malfunction-causing faults detected when the test quantity exceeds the statistical limits corresponding to the α = 10 6 ris level. The HM scheme performs quite well, since the majority of faults are detected and very few false alarms ( 3.5%) are noted. As expected, the scheme s performance for low turbulence flights is better than that obtained for moderate turbulence flights. Obviously, the turbulence increase is a source of extra noise in all signals, which mass part of the fault-induced effects on the left engine dynamics. But even then, the detection performance is not severely degraded, as shown by the only slightly worse false alarm rates in Table 3 resulting from the comparison of low versus moderate turbulence flights: In general, the CCP-NARX(7, 4) based scheme is quite accurate and robust with respect to induced noise with successful detection rates between 94% and 98.8% Isolation phase Isolation results of F A 25 faults obtained via Test B in (10) are presented in Table 3. Note that computing the Z statistic in (10) means that a typical (empirical) value for Φ 0 11 (see Section 3.3) must be nown. The latter is computed from 10 representative flights conducted with a F A 25-affected aircraft. Fig. 8(b) presents a typical case of F A 25 being isolated when the Z statistic enters the isolation zone (delimited by the horizontal dashed lines designating statistical limits at the α = 10 6 ris level). Generally, F A faults show as abrupt increases of the e[t] values followed a few samples later by equally abrupt reductions (spies). This behavior is caused by the autopilot compensating immediately after F A 25 fault occurrence. The PACF coefficient ˆΦ 11 abruptly increases when the sliding window covers the region of e[t] sequence where these spies are present. Finally, since the ˆΦ 11 values stay away from the isolation

9 78 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) Fig. 7. Fault detection for typical flights under low turbulence and commanded roll tracing of the reference signal r[t]: (a) Healthy; (b) F A 25 affected; (c) F B 550 affected; (d) F C 30 affected; (e) F D 25,550,10 affected aircraft (the horizontal lines designate statistical limits at the α = 10 6 ris level). Fig. 8. Fault isolation with Test A (Q statistic) and Test B (Z statistic) for typical flights under low turbulence and commanded roll tracing of the reference signal r[t]: (a) (b) F A 25 fault isolated via Test B; (c) (d) F B 550 fault isolated via Test A; (e) (f) F C 30 fault not isolated; (g) (h) F D 25,550,10 fault isolated via Test A and Test B (the horizontal lines designate statistical limits at the α = 10 6 ris level). zone when the engine is healthy or affected by F B 550 or F C 30 faults (see Figs. 8(d), (f)), the F A 25 faults cannot be mistaen for any other fault types. Isolation results of F B 550 faults obtained via Test A in (8) are also presented in Table 3. TheQ statistic, involving all lags from the 6th to the 9th, is used in Test A. Fig. 8(c)presentsatypicalcase of F B 550 being isolated when the Q statistic crosses the horizontal dashed line (with the isolation area being over the line designating statistical limit at the α = 10 6 ris level). Generally, F B faults show up as abrupt increases of the e[t] sequence s variance followed by its abrupt reduction samples later, when the left engine shuts down. During these samples, the values of the Q statistic cross the limits (see Fig. 8(c)), and this is exclusively due to the effects of F B fault occurrence: Indeed, for healthy, F A 25 or F C 30 affected engines the Q values are always out of the isolation area (see Figs. 8(a), (e)). Thus, F B 550 faults cannot be mistaen for other fault types. Finally, isolation results for the sequential faults F D 25,550,10 and F D 25,550,20 obtained via Tests A and B are also presented in 3, with typical cases in Figs. 8(g), (h). Note that the isolation of such multiple faults is not a trivial tas: The effects of F B 550 faults on already F A 25 -affected engines (a situation corresponding to F D 25,550,10 or F D 25,550,20 fault occurrence) are quite different than the effects of F B 550 faults on healthy engines. The reason is that the autopilot compensation follows immediately after F A 25 fault occurrence, and, thus, the system dynamics are altered. The difficulty of isolating sequential faults is reflected in the isolation rates for F D 25,550,t

10 D. Dimogianopoulos et al. / Aerospace Science and Technology 16 (2012) faults under demanding circumstances induced by moderate turbulence levels: These are lower (though only slightly) than the rates for pure F B 550 faults. Nonetheless, the CCP-NARX(7, 4) based scheme still remains quite robust leading to good isolation of sequential faults under all circumstances, with rates between 98.7 and 100%. This concludes on the EHM scheme s robustness with respect to external disturbances Comparison with Kalman filter based approaches Due to operating on cruise flight instead of GPA-derived data, the proposed scheme complements rather than replaces the traditional EHM schemes. These, furthermore, focus on detecting and isolating faults affecting specific internal engine parts, whereas the current scheme detects engine malfunctions and isolates possible causes on macroscopic level. A direct comparison between such schemes would, thus, be meaningless. However, meaningful comparisons may be done between schemes sharing the operational concept based on cruise flight data, but featuring different modelling and decision-maing approaches. Using the guidelines in [21] (except for the part of employing GPA data), such schemes may operate with Kalman filters (instead of CCP-NARX modelling) and thresholding of residuals (instead of statistical hypothesis tests). In the absence of an explicit (physics-based or other) model representing the fault-sensitive interrelation among acceleration A y, side-slip angle β, rudder moment and thrust, the standard linear version (LKF) is chosen. This is due to the fact that a linear state-space model may easily be identified via OLS upon the same flight data used for CCP-NARX identification. On the contrary, the use of Extended or Unscented Kalman filters would require a linearized approximation at each operating point of the true nonlinear model of the fault-sensitive interrelation, which, as stated, is not nown. For the same reason, using particle filters based, for instance, on the Sequential Importance Resampling (SIR) for nonlinear state and output estimation via Bayesian techniques (see [5] or [4]) are hardly applicable. Although quite effective, these methods require among others nowledge of the nonlinear (as seen in Section 4.1) state and output generating functions along with the probability density function of the noise affecting the state [4], all of which are currently unavailable. Besides, these methods rely on Monte Carlo recursive techniques to simulate N s vector samples of system state and to execute O(N s ) operations for resampling: All these operations should be carried out online at each time instant [4], and since N s is large for good accuracy, the online computational effort could be significant. In contrast, any considerable computational effort for identifying the CCP-NARX models is invested off-line, and the online effort is insignificant, since mainly associated to running the statistical hypothesis tests for obtaining HM results. Following the approach in [21], the data from the 84 modelling flights are used for identifying via OLS a linear state-space model admitting the same inputs (lateral acceleration A y, side-slip angle β and rudder moment) and covering the same envelope area as the CCP-NARX model. The state variables involve four health parameters, which are measurable sensor outputs and related to potentially isolable faults: The throttle position (related to F A faults), the thrust value (related to F B faults) and the Alt and Mach measurements. Note, that only the first and second measurements are related to modelled faults, which are those considered in the CCP- NARX based case. Equivalently, the unmodelled F C faults are not isolable. The HM scheme utilizes bans of Kalman filters, with each one predicting all state variables using all but one measured sensor output. As in [21], the filter not using the output of a faulty sensor gives the less distorted state prediction, with respect to all other bans. The sensor whose output is not used by that filter is, then, concluded as faulty. Obviously, unlie the CCP-NARX based scheme, sequential faults cannot be trivially isolated, unless the standard KF based scheme is suitably modified. In Fig. 9 the residuals between the measured and the predicted throttle/thrust value, are shown for two different aircraft missions. The first evolves at 7500 ft Mach 0.5 under low turbulence (Figs. 9(a) (d)), and the second at ft Mach 0.5 under moderate turbulence (Figs. 9(e) (h)). Figs. 9(a), (b) and (e), (f) show residuals from the first KF ban which performs state prediction not using throttle sensor measurements, whereas Figs. 9(c), (d) and (g), (h) show residuals from the second KF ban which performs state prediction not using thrust measurements. During the few critical time samples after (abrupt) F A 25 fault occurrence at t = 150 s, the second ban residuals exhibit abnormally large spies with respect to their previous time history. These admit maximum values in Figs. 9(d) and (h), with spies 9 and 5 times larger, respectively, than each signal s past maximum pea. Thus, a threshold on the spies relative magnitude as previously defined, may provide F A 25 fault isolation results under both turbulence levels. However, more turbulence leads to lower relative magnitude of F A 25-caused spies (see also Figs. 9(c) and (g)), and slightly compromised fault isolation. Following F B 550 fault occurrence, the first ban residuals become more noisy than those from the second ban. The noisy evolution is very obvious under low turbulence, as shown in Fig. 9(a). Accordingly, under such conditions F B 550 fault isolation may be achieved when an appropriately selected frequency threshold is exceeded. However, the moderate turbulence induces severe noise even before the F B 550 fault occurrence at t = 150 s, as shown in Figs. 9(e) and (f). Hence, isolating the F B 550 fault via the previous frequency thresholds may prove ineffective for moderate turbulence flights. Even if a frequency threshold specific to the turbulence level could be determined, the problem of distinguishing between noise-induced uncertainty and F B 550 fault occurrence is not trivial. This behavior shows that the LKF is affected by turbulenceinduced noise, or modelling uncertainties, which partly mas the fault effects and undermine the HM scheme s capabilities. In comparison, the CCP-NARX model is more accurate in representing the dynamics throughout the flight regime under noise or modelling uncertainty. Thus, all fault-induced information may be retrieved by the (computationally simple) onboard statistical hypothesis tests. Again, if explicit models of the fault-sensitive interrelation were available, the EKF and UKF could outperform LKF based schemes. Even then, however, the UKF operation requires the square root of the (often ill-conditioned) covariance matrix to be timely and reliably computed onboard at each time instant. Under the current circumstances, the CCP-NARX based scheme proves effective for single and sequentially occurring faults, even in the absence of explicit system model. 5. Conclusions An HM scheme dealing with single and sequentially occurring faults in aircraft engines has been introduced. The scheme uses available cruise flight data, instead of data from gas path analysis, to create additional redundancy, unrelated to that already implemented onboard. Hence, without new HM-dedicated sensors, increased onboard HM decision-maing capabilities for enhanced safety are cost-effectively achieved. The scheme relies on a CCP- NARX(7, 4) representation with 33 regressors to model the MISO relationship among the lateral acceleration A y, side-slip angle β, rudder moment and thrust signals. The CCP-NARX model is identified on-the-ground via relevant past available data. Thus, unlie schemes using extended/unscented Kalman (or even particle) filters, the scheme may use any suitable fault-sensitive relationship for HM purposes even if a (physics-based or other) model