Rotor Fault Detection and Identification on a Hexacopter Based on StatisticalTime Series Methods

Airin DuttaPhD Student

Michael McKayPhD Student

Fotis KopsaftopoulosAssistant Professor

Farhan GandhiRedfern Professor

DirectorCenter for Mobility with Vertical Lift (MOVE)

Rensselaer Polytechnic Institute, Troy, NY

ABSTRACTThis work introduces the use of statistical time series methods to detect rotor failures in multicopters. A conciseoverview of the development of various time series models using scalar or vector signals, statistics, and fault detectionmethods is provided. The fault detection methods employed in this study are based on parametric time series repre-sentations and response-only signals of the aircraft state, as the external excitation is non-observable. The compara-tive assessment of the effectiveness of scalar and vector statistical models and several residual-based fault detectionmethods are presented in the presence of external disturbances, such as various levels of turbulence and uncertainty,and for different rotor failure scenarios. The results of this study demonstrate the effectiveness of all the proposedresidual-based time series methods in terms of prompt rotor fault detection, although the methods based on VectorAutoRegressive (VAR) models exhibit improved performance compared to their scalar counterparts with respect totheir robustness and effectiveness for different turbulence levels and ability to distinguish between healthy and faultcompensated condition after rotor failure.

NOTATION

α : Type I risk levelβ : Type II risk levelγ : Autocorrelationτ : Lagσ2 : Residual varianceΣ : Residual covariance matrixARMA : AutoRegressive Moving AverageE· : Expected valuePE : Prediction ErrorARX : AutoRegressive with eXogenous excitationPSD : Power Spectral DensityBIC : Bayesian Information CriterionRSS : Residual Sum of SquaresFRF : Frequency Response FunctionACF : Auto-Covariance Functioniid : identically independently distributedSPP : Samples Per ParameterLS : Least SquaresSPRT : Sequential Probability Ratio TestSSS : Signal Sum of SquaresAR : Scalar AutoRegressive modelVAR : Vector AutoRegressive model

Presented at the Vertical Flight Society 75th Annual Forum &Technology Display, Philadelphia, Pennsylvania, May 13–16, 2019.Copyright c© 2019 by AHS - The Vertical Flight Society. All rightsreserved.

INTRODUCTIONMulticopters, being capable of hovering and vertical take-offand landing, have attracted the interest of the community withrespect to both commercial and defense applications over thelast decade. Given the increasing interest and widespread useof these vehicles in a number of important arenas, early faultdetection and identification of such systems are critical in or-der to ensure and improve their overall safety and reliability.Rotorcraft are complex systems that exhibit strong dynamiccoupling between rotors, fuselage, and control inputs, as wellas time-varying and cyclo-stationary behavior. As a result,they face certain system modeling and fault detection andidentification challenges that are not present in fixed-wing air-craft. These issues, as well as potential solutions, have beenexplored in the recent literature.An algorithm for online detection of motor failure usingonly inertial measurements and control allocation by an ex-act redistributed pseudo-inverse method for octacopters hasbeen demonstrated by Frangenberg et al. (Ref. 1). Herediaand Ollero (Ref. 2) have addressed sensor fault identifica-tion in small autonomous helicopters using Observer/KalmanFilter identification. Fault tolerant control for multi-rotors(Refs. 3, 4), as well as various fault diagnosis methods suchas analytical models, signal processing, and knowledge-basedapproaches for helicopters have also been proposed (Ref. 5).Statistical time series methods have been used to detect var-ious fault types in aircraft control systems due to their sim-plicity, efficient handling of uncertainties, no requirement ofphysics based models, and applicability to different operat-ing conditions (Refs. 6–9). Dimogianopoulos et al. (Ref. 10)

1

Fig. 1. Schematic representation of a regular hexacopterhave demonstrated the effectiveness of two statistical schemesbased on Pooled Non-Linear AutoRegressive Moving Aver-age with eXogenous excitation (P-NARMAX) to detect andisolate faults for aircraft systems under different flight condi-tions, turbulence levels, and fault types and magnitudes. Thefirst method models the pilot input and aircraft pitch rate re-lationship, while the second approach models the relationshipbetween horizontal and vertical acceleration, angle of attackand pitch rate signals in fixed-wing aircraft.

The objective of the present study is the development of a ro-bust framework for fault detection in multicopters using sta-tistical time series methods to achieve early detection of rotorfailures in the presence of external disturbances, such as tur-bulence, and uncertainty. This information will be extremelyuseful for control allocation redistribution and reconfigurationof the vehicle to accomplish safe flight.

HEXACOPTER MODEL AND DATAGENERATION

Physics-Based Modeling of Multicopter System

A flight simulation model has been developed for a regularhexacopter (Fig. 1) using summation of forces and momentsto calculate aircraft accelerations. This model is used as themain source of simulated data under varying operating and en-vironmental conditions, as well as different fault types. Rotorloads are calculated using Blade Element Theory coupled witha 3×4 Peters-He finite state dynamic wake model (Ref. 11).This model allows for the simulation of abrupt rotor failureby ignoring the failed rotor inflow states and setting the out-put rotor forces and moments to zero.

A feedback controller is implemented on the nonlinear modelto stabilize the aircraft altitude and attitudes, as well as trackdesired trajectories written in terms of the aircraft veloci-ties. This controller is designed at multiple trim points, withgain scheduling between these points to improve performancethroughout the flight envelope.

The state vector consists of the 12 rigid body states and isdefined in Eq. 1.

x =

X Y Z φ θ ψ u v w p q rT (1)

The input vector is comprised of the first four independentmultirotor controls for collective, roll, pitch and yaw and is

Fig. 2. Controller Block Diagram

defined in Eq. 2:

u =

Ω0 ΩR ΩP ΩYT (2)

The control architecture is illustrated in Fig. 2 and detailed inRef. 3. This control design has been demonstrated to performwell even in the event of rotor failure, with no adaptation inthe control laws themselves.

Data Generation for Model Identification

A continuous Dryden wind turbulence model (Ref. 12) hasbeen implemented in the flight simulation model. The Drydenmodel is dependent on altitude, length scale, and turbulenceintensity and outputs the linear and angular velocity compo-nents of continuous turbulence as spatially varying stochasticsignals. The proper combination of these parameters deter-mines the fit of the signals to observed turbulence.

In this system, altitude is taken as 5m and the length scale asthe hub-to-hub distance of the hexacopter, which is equal to0.6096 m (2 ft). The data sets for aircraft states are generatedthrough a series of simulations for different turbulence levels(light, moderate and severe) both for healthy aircraft and dif-ferent fault types, such as failure of front and side rotors. Fora summary of the generated data sets seen Table 1. The timeseries (signals) of the hexacopter attitudes (aircraft states) forthe healthy state, as well as for different fault types, i.e. com-plete failure of front rotor or side rotor, provide useful insightinto the dynamics of the system. The rotor failures addressedin this work are: front rotor (rotor 1), right-side rotor (rotor2), and left-side rotor (rotor 6).

Workframe of Statistical Time Series for Fault Detection

Let Zo designate the aircraft under consideration in its healthystate, and ZA,ZB, . . . the aircraft under fault of type A,B, . . .

Table 1. Simulation DataAircraft state number of datasets for turbulence levels

Light Moderate SevereHealthy 20 20 20Rotor failure (1) 20 20 20Rotor failure (2) 20 20 20Rotor failure (6) 20 20 20Sampling frequency: fs = 1000 HzSignal length in samples: 60000 (60 s)

2

and so on. Zu designates the unknown (to be determined) stateof the aircraft. Statistical time series modeling is based ondiscretized response signals y[t]1 (for t = 1,2, . . . ,N) whichare the aircraft states. N denotes the number of samplesand the conversion from discrete normalized time to analogtime is based on (t − 1)Ts, with Ts being the sampling pe-riod. The response signals are represented by Z and subscript(o,A,B, . . . ,u) is used to denote the corresponding state of theaircraft that produced the signals. The sampling frequency(Fs) for the signals is chosen such that the frequency range ofinterest is 0−500 Hz.

The signals generated from simulation are analyzed by para-metric or non-parametric statistical time series methods andproper models are fitted and validated. Such models areidentified for the cases Zo,ZA,ZB, .. in the baseline phase.Fault detection is based on binary statistical hypothesis test-ing (Ref. 13) that compares the residual properties generatedfrom Zu in each inspection phase with that available frombaseline models. The design of a binary statistical hypothe-sis test is generally based on the probabilities of type I (falsealarm) and type II (missed faults) error probabilities, repre-sented by α and β respectively.

The general workframe for fault detection and identificationvia statistical time series modeling is illustrated in Fig. 3. Notethat in the current study only the task of fault detection isaddressed. The task of fault identification (classification) isthe topic of current research and will be presented in a futurestudy.

BASELINE MODELING OF THE HEALTHYAIRCRAFT

The aircraft signals for roll, pitch and yaw attitudes gener-ated via a series of simulations of forward flight of the hexa-copter under turbulence (light, moderate and severe levels) forhealthy and different faulty states are used for model identifi-cation and fault detection.

In the present scenario, the signals obtained are response onlysignals with the excitation x[t] assumed to be a white (uncor-related) signal induced by atmospheric turbulence. That isγxx[τ] = 0 for τ 6= 0, where γxx denotes the AutoCorrelationFunction (ACF) and τ the ACF time lag, given as:

γxx[τ] = Ex[t] · x[t + τ] (3)

Non-parametric Identification and Fault Detection

Short-Time Fourier Transform or Spectrogram

The Short-time Fourier Transform (STFT) of a discrete-timesignal is defined via a moving window as follows:

Y [n,ω] =L−1

∑m=0

y[n+m] ·w[m] · e−i2πkm/N (4)

1A functional argument in parentheses designates function of a real vari-able; for instance x(t) is a function of analog time t ∈ R. A functional argu-ment in brackets designates function of an integer variable; for instance x[t]is a function of normalized discrete time (t = 1,2, . . .).

where n denotes the location in time, L denotes the lengthof the window (w), and ω the frequency. The square of themagnitude of the STFT yields the spectrogram:

S[n,ω] = |Y [n,ω]|2 (5)

Spectrogram gives the power intensity of the frequenciespresent for a particular time window. With sliding windowsover time, it is possible to detect changes in the frequencyof the signal, which can provide a preliminary idea on thedynamic content and transient effects due to the existence offaults.

Parametric Identification via Time Series Models

Scalar AR Identification Method

A single signal obtained from a healthy flight simulation isparametrized to form a scalar (univariate) AutoRegressivetime series model (Ref. 14):.

y[t]+na

∑i=1

ai · y[t− i] = e[t], e[t]∼ iid N(0,σe2) (6)

with ai and na designating the AR parameters and modelorders respectively, iid stands for identically independentlydistributed, and N(·, ·) denotes a univariate normal distribu-tion with the indicated mean and variance, respectively. InEq. 6, e[t] coincides with the one-step-ahead-prediction errorand is also referred as the model residual sequence or innova-tions (Refs. 14, 15).

The identification of parametric time series models is com-prised of two main tasks: parameter estimation and modelorder selection. The parameters for the AR model can beenestimated by minimization of the Least Squares (LS) criterion(Refs. 15, 16), whereas the model order selection is achievedbased on the examination of the Bayesian Information Crite-rion (BIC) (Refs. 15,16) (Eq. 7) and Residual sum of Squaresover Signal Sum of Squares Criterion (RSS/SSS) (Eq. 8). Theformer is a statistical criterion that penalizes model complex-ity (order, and hence the number of free parameters) as a coun-teraction to a decreasing model fit criterion. The latter deter-mines the predictive capability of the model.

BIC = lnσ2e +(d× lnN)/N (7)

RSS/SSS =∑σ2

e

∑y[t]2(8)

In Eq. 7, σ2e is the variance of the residuals, d denotes the

number of parameters to be estimated for the model and Ndenotes the number of samples used for estimation.

Vector AR Identification Method

Vector AutoRegressive (VAR) models employ s-dimensionalsignals, i.e. the aircraft states in the present study, for multi-variate (s-variate) time series modeling (Refs. 17,18). Thoughthey bear striking resemblance to their univariate or scalar

3

Fig. 3. General workframe of statistical time series methods for fault detection and identification.

counterparts, they have a much richer structure and typicallyrequire multivariate statistical decision making procedures.The univariate response signal y[t] of Eq. 6 is replaced by an s-variate vector2, hence the VAR(na) model is of the followingform:

y[t]+na

∑i=1

Ai ·y[t− i] = e[t], with

e[t]∼ iid N(0,Σ), Σ = Ee[t] · eT [t](9)

with Ai (s× s) designating the i-th AR matrix, e[t] (s× 1)the model residual sequence characterized by the non-singularand generally non-diagonal covariance matrix Σ, n the ARorder, and E· statistical expectation. Given the attitudesignal measurements y[t] (t = 1,2, . . . ,N), the estimation ofthe VAR parameter vector θ comprising all AR matrix ele-ments (θ = vec([A1A2 . . .Ana]) and the residual covariancematrixΣ is accomplished via linear regression schemes basedon minimization of the Ordinary Least Squares (OLS) orthe Weighted Least Squares (WLS) criterion (Refs. 15, 16).The modeling procedure involves the successive fitting ofVAR(na) models for increasing AR order n, until an adequatemodel is achieved. The model order is chosen by replacingthe variance of residuals for scalar case by the trace of theresidual covariance matrixΣ (Ref. 7).

RESIDUAL BASED FAULT DETECTIONModel residual based methods use functions of the residualsequences (known as characteristic quantity) for fault detec-tion, which are obtained by driving the current signal(s) (Zu)through the model(s) estimated in the baseline phase for thehealthy aircraft, denoted by Mo. The key idea is that the resid-ual sequence obtained by a healthy model that truly reflectsthe healthy aircraft properties possesses certain distinct prop-erties which are distinguishable from the faulty states of theaircraft.

2Bold–face upper/lower case symbols designate matrix/column–vectorquantities, respectively. Matrix transposition is indicated by the superscript T .

The residual series obtained by driving the current signal(s)through the aforementioned model is denoted as eou[t] and ischaracterized by variance, σ2

ou. The residual series obtainedusing healthy baseline data records are designated as eoo[t],characterized by variance σ2

oo.

Residual Variance Method

In this method, the characteristic quantity used for fault detec-tion is the residual variance (Ref. 6). Fault detection is basedon the fact that the residual series eou[t], obtained by drivingthe current signals, Zu through the model, Mo (correspond-ing to the healthy state) should be characterized by varianceσ2

ou = σ2oo which becomes minimal if and only if the current

state of the aircraft is healthy (Zu = Zo). Fault detection isbased on the following hypothesis testing procedure:

H0 : σ2ou ≤ σ

2oo (null hypothesis – healthy aircraft)

H1 : σ2ou > σ

2oo (alternate hypothesis – rotor failure)

(10)

Under the null (Ho) hypothesis, the residuals eou[t] are (justlike the residuals eoo[t]), iid Gaussian with zero mean andvariance σ2

oo. Hence the quantities Nu · σ2ou/σ2

oo and (No −d) · σ2

oo/σ2oo follow central χ2 distribution with Nu and No−d

degrees of freedom, respectively (as sums of squares of in-dependent standardized Gaussian random variables)3. No andNu designate the number of samples used in estimating theresidual variance in the healthy and current cases, respectively(typically No = Nu = N), and d designates the dimensionalityof the estimated model parameter vector. Nu and No shouldbe adjusted to Nu− 1 and No− 1, respectively, if each esti-mated mean is subtracted from each residual sequence. Con-sequently, the following statistic follows a Fischer distribu-tion (denoted by F) with (Nu,No− d) degrees of freedom as

3A hat designates estimator/estimate of the indicated quantity; for in-stance σ is an estimator/estimate of σ.

4

the ratio of two independent and normalized χ2 random vari-ables (Ref. 6):

Under H0 : F =

Nu σ2ou

Nu σ2oo

(No−d) σ2oo

(No−d) σ2oo

=σ2

ou

σ2oo

(11)

The following hypothesis test is thus constructed at the α typeI (false alarm) risk level:

F ≤ f1−α(Nu,No−d) ⇒ H0 accepted (healthy aircraft)Else ⇒ H1 accepted (rotor failure)

(12)where, f1−α(Nu,No−d) designates the corresponding Fischerdistribution’s (1−α) critical point.

Residual Uncorrelatedness Method

This method is based on the fact that the residual series eou[t],obtained by driving the current signals (Zu) through the model(M0), is uncorrelated (white) if and only if the aircraft is cur-rently in its healthy condition (Ref. 6). Fault detection is per-formed by the following hypothesis testing:

H0 : ρ[τ] = 0 (null hypothesis – healthy aircraft)H1 : ρ[τ] 6= 0 (alternate hypothesis – rotor failure)

(13)where ρ[τ] is the normalized autocorrelation function(ρxx[τ] = γxx[τ]/γxx[0]) of the residual sequence eou[t].

Therefore, the characteristic quantity for fault detection bythis method is

[ρ[1] ρ[2] ρ[3] . . . ρ[τ]

]T . For this method,r is the design variable for the statistical test, which denotesthe maximum lag in time (τ) for which the normalized ACFsare being accounted for. Under the null hypothesis (H0), theresiduals eou[t] are iid Gaussian with zero mean and the teststatistic χ2

ρ follows a χ2 distribution with r degrees of free-dom, given as:

Under H0 : χ2ρ = N(N +2) ·

r

∑τ=1

(N− τ)−1 · ρ[τ]2 ∼ χ2(r)

(14)where ρ[τ] denotes the estimator of ρ[τ].

Statistical decision making is achieved by the following testfor α (false alarm) risk level:

χ2ρ ≤ χ2

1−α(r) ⇒ H0 is accepted (healthy aircraft)

Else ⇒ H1 is accepted (rotor failure)(15)

where χ21−α

(r) denotes the χ2 distribution’s 1− α criticalpoint.

Sequential Probability Ratio Test

This method employs the Sequential Probability Ratio Test(Refs. 19, 20) in order to detect a change in the standard de-viation σou of the model residual sequence eou[t]. The SPRT

allows for the specification of two values σo and σ1 for thestandard deviation, such that the aircraft is determined to behealthy if and only if σ ≤ σo, and in faulty state if and onlyif σ ≥ σ1. The zone between σo and σ1 constitutes an un-certainty zone, if any σ is found in this range the decision ispostponed and data or residual collection continues. The val-ues of σo and σ1 are user defined and express the increase ofthe standard deviation in terms of the ratio q = σ1

σofor which

the system is considered to be faulty. For example, a ratio ofq = 1.1 means that the aircraft is considered faulty wheneverthere is an increase of 10% in the standard deviationof thecurrent residual sequence compared to a threshold value σo.

The SPRT based method leverages both α (false alarm) andβ (missed faults) error probabilities in its design. Fault detec-tion is based on the SPRT of strength (α,β ) for the followinghypothesis testing problem:

H0 : σou ≤ σ0 (null hypothesis – healthy aircraft)H1 : σou ≥ σ1 (alternate hypothesis – rotor failure)

(16)

where σou denotes the standard deviation of the residual sig-nal eou[t] obtained by driving the current signal(s) through thehealthy aircraft model, and σ0, σ1 or their ratio q are user de-fined values. The basis of the SPRT is the logarithm of thelikelihood ratio function based on n(n≤ N) samples:

L (n) =n

∑t=1

lnf (eou[t]|H1)

f (eou[t]|H0)= n · ln σ0

σ1+

σ20 −σ2

1

2σ20 σ2

1·

n

∑t=1

eou[t]2

(17)with f (eou[t] |Hi) designating the probability density functionof the residual sequence under hypothesis Hi (i = 0,1).

Statistical decision making is then based on the following testat the (α,β ) risk levels:

L (n)≤ B ⇒ H0 is accepted (healthy aircraft)L (n)≥ A ⇒ H1 is accepted (rotor failure)B < L (n)< A ⇒ no decision made (continue the test)

(18)where:

A = ln1−β

αand B = ln

β

1−α. (19)

Once a decision is made at a stopping time n, the test is contin-ued by resetting L (n+1) to zero and continuing by collectingadditional residual samples.

RESULTS AND DISCUSSION

Data Generation

Flight simulation for the hexacopter was performed at 5 m/sforward speed with severe turbulence according to the Drydenmodel. Figures 4 through 6 show attitude time histories for thehexacopter at 5 m/s forward flight, for cases of rotor 1 failure(red) and rotor 2 failure (green). For the simulation resultspresented, rotor failure (of either rotor 1 or 2) occurs at t = 10s, indicated by the vertical dashed line.

From Fig. 4, it may be observed that rotor 1 failure resultsin a larger deviation in the pitch attitude than in the case of

5

0 10 20 30

Time (s)

-20

-15

-10

-5

0

Pitch a

ngle

(deg)

Healthy

Front rotor failure

Side rotor failure

Fig. 4. Indicative pitch attitude signals for an airspeed of 5m/s. The dashed vertical line indicates the time instant ofthe fault initiation.

rotor 2 failure. In the case of front rotor failure, the hex-acopter pitches down without any substantial change in rollangle (Fig. 5), because the loss of rotor 1 thrust does not sig-nificantly affect the aircraft roll equilibrium. However, in thecase of side rotor (rotor 2) failure both the pitch (Fig. 4) androll (Fig. 5) attitudes change and the roll attitude compensa-tion is observed to be underdamped. In Fig. 6, the heading ofthe aircraft is observed deviating in different directions withthe failure of rotor 1 compared to rotor 2. This is due to thedifferent rotor spin directions, and consequently the directionof the hub torque generated by each rotor.

Due to different controller effort for the various levels of tur-bulence, the aircraft state signals do not show discerniblechange in characteristics with respect to the healthy dynam-ics, and transients due to failure and failure compensation un-der light and moderate levels of turbulence. Similar trends areobserved for a flight speed of 10 m/s; the fault detection pro-cess follows the same steps for any other speed. In this paperwe demonstrate indicative results for a single flight speed of5 m/s while results from various speeds will be presented in asubsequent publication.

Non-parametric Identification

The STFT-based non-parametric identification of the aircraftdynamics was based on 60 s (N = 6000 samples; original sig-nal downsampled to 100 Hz) signals obtained from the aircraftstates, namely the translational and rotational position and ve-locities at 5 m/s forward speed under severe levels of turbu-lence. Indicative results for pitch signal for healthy and ro-tor failure (front) analyzed by the Spectrogram or Short TimeFourier Transform (STFT) technique are depicted in Fig. 7.The sampling frequency is taken as 100 Hz and Hammingwindow of 300 samples with an overlap of 90% is used. Itcan be observed that there is a sharp change of frequency atthe time of rotor failure (t = 10 s), distinguishable from thespectrogram of the healthy aircraft.

0 10 20 30

Time (s)

-20

-15

-10

-5

0

5

10

15

Roll

angle

(deg)

Healthy

Front rotor failure

Side rotor failure

Fig. 5. Indicative roll attitude signals for an airspeed of 5m/s.

0 10 20 30

Time (s)

-15

-10

-5

0

5

10

15

Yaw

angle

(deg)

Healthy

Front rotor failure

Side rotor failure

Fig. 6. Indicative yaw attitude signals for an airspeed of 5m/s.

Scalar AR Identification and Fault Detection

Scalar (univariate) parametric identification of the aircraft dy-namics has been based on 20 s (N = 20000 samples) of pitchsignal obtained from healthy aircraft flight at 5 m/s under se-vere levels of turbulence. In the present case, the response-only signals have been obtained from ambient excitation dueto atmospheric turbulence (assumed to be uncorrelated basedon the Dryden specifications). The model parameters andmodel order, ai and na, respectively (Eq. 6) need to be es-timated so that the model properly represents the dynamicsof the system under healthy conditions. The modeling strat-egy consists of successive fitting of AR(na) models until asuitable model with least amount of complexity (number ofparameters) and best fit is selected.

Scalar AR Identification Results

Model order selection is based on a combination of BayesianInformation Criteria (BIC) (Eq. 7) and Residual sum of

6

Healthy aircraft

5 10 15 20 250

5

10F

req

ue

ncy (

Hz)

-150

-100

-50

Pow

er/

frequ

ency (

dB

/Hz)

Rotor failure

5 10 15 20 25

Time (s)

0

5

10

Fre

qu

en

cy (

Hz)

-150

-100

-50

Pow

er/

frequency (

dB

/Hz)

Fig. 7. Spectrogram of the pitch signals under the healthyand front rotor faulty states for a speed of 5 m/s.

squares normalized by Signal sum of squares (RSS/SSS)criteria (Eq. 8) as shown in Fig. 8. A model order of na = 6yields the minimum BIC and this model is represented asAR(6). Monitoring the stabilization of RSS/SSS criteriagives the point where increasing model order does not resultfurther in reduction of prediction errors. This order exhibitsa very low RSS/SSS value of 1.6× 10−10% demonstratingaccurate identification and excellent dynamics representationof the healthy aircraft pitch signal at 5 m/s and under severeturbulence. The number of parameters estimated for theAR(6) model results in a Samples per Parameter (SPP) ratioof 3333.33 ( N

d ).

The model was validated based on the fact that the modelmatching the current state of the system should generate awhite (uncorrelated) residual sequence. Therefore, a healthypitch signal has been generated from a different realization ofsevere turbulence. The autocorrelation function of the residualsequences obtained from driving the current signal (healthy)through the model has been observed to be white with 95%confidence (confidence intervals shown in blue), as shownFig. 9. Next, pitch signals generated under front and siderotor failures have been passed through the same model togenerate residual sequences. Figure 9 shows that the resid-ual sequences for different failure cases are serially correlated,demonstrating that the dynamics of the aircraft have changedfrom that of the healthy state, due to failure.

A similar study has been repeated with the roll and yaw sig-nals to estimate scalar AR models for the healthy aircraft, thedetails of which are given in Table 2. A comparison of thesemodels using various residual-based fault detection methodswill be addressed with respect to their accuracy and ability indetecting faults, as well as their robustness to false alarms.

4 6 8 10 12 14 16 18 20-32.2

-32.1

-32

-31.9

BIC

Bayesian Information Criterion (BIC)

4 6 8 10 12 14 16 18 20

AR(n)

1.6

1.8

2

2.2

RS

S/S

SS

(%

)

10-10 RSS/SSS criterion

Fig. 8. Scalar AR model order selection criteria.

Residual Based Fault Detection

The current (unknown) pitch signals (5 m/s under se-vere turbulence) were driven through the identified AR modelto generate residual sequences. Fault detection was attemptedthrough the characteristic quantities which are functions ofthe residual sequences, as previously discussed.

Residual Variance Method

Real-time fault detection is achieved through taking 5 s (N =5000 samples) windows of the current pitch signal at a timewith the window being updated every 0.1 s. Then the win-dowed data are filtered through the estimated healthy model(baseline model) to generate a residual sequence of the samelength. The variance of the generated residuals is statisti-cally compared to the corresponding baseline residual vari-ance. The critical limit is determined from the F distribu-tion’s (1−α) critical limit for 5000, 5000-6 degrees of free-dom. The hypothesis test is conducted at the α (false alarm)risk level of 10−12 and the results for different states of theaircraft (healthy, front, and side rotor failure) are presentedin Fig. 10. In all the figures, the vertical black and the hori-zontal red dashed lines represent the time of rotor failure andthe critical limit (threshold), respectively. A fault is detectedwhen the F statistic exceeds the critical limit at the designatedtype I risk level.

Since a minimum number of 5000 samples are required so thatthe test is statistically significant with minimum false alarms,the test starts at 5 s. For the current pitch signals coming froma healthy flight, the test statistics (Eq. 10) always remain be-

Table 2. Model identification summary results.Model Signals Model Parameters SPPType used order estimated

roll AR(6) 6 3333.33Scalar AR pitch AR(6) 6 3333.33

yaw AR(6) 6 3333.33Vector AR roll,pitch,yaw AR(4) 36 166.67

7

-0.2

0

0.2

Healthy aircraft

0 10 20 30 40 50

-0.2

0

0.2

Re

sid

ua

l A

uto

co

rre

latio

n

Front rotor failure

0 10 20 30 40 50

-0.2

0

0.2Side rotor failure

0 10 20 30 40 50

Lag

Fig. 9. Autocorrelation function of the pitch residual forthe healthy and considered faulty rotor cases.

low the critical limit for the test, hence the null hypothesis(that it is a healthy case) is accepted. In the case of signalsgenerated from an aircraft with rotor failure, the fault detec-tion at the time of failure is immediate, within 0.1 s, which isthe window update interval, showing a violation of the criticallimit of the test. Thus, the alternate hypothesis of fault is ac-cepted. As observed from the signals in Fig. 4, the pitch signalis affected more for the case of front rotor failure than side ro-tor failure. Hence, the residual variances for pitch in the caseof front rotor failure change more compared to the case siderotor failure, that is also evident from the extent of limit vi-olation of the test statistics. Also, it should be noted that thecontroller compensates for the fault, returning the pitch dy-namics to almost the original state after front rotor failure, asalso demonstrated by the test limits receding within the criti-cal limit after the transient signal has passed. However, in thecase of side rotor failure, the test statistics continue to crossthe critical limit marginally, even after the transient signal haspassed, denoting that the pitch dynamics are have been mod-ified with respect to the healthy case. This may be due to thefact that there is strong coupling between pitch and roll con-trol post failure, as side rotor failure causes a large change inthe roll axis relative to the front rotor (Fig. 5).

Residual Uncorrelatedness Method

Online fault detection via the residual uncorrelatednessmethod is performed with the same window length and up-date interval as discussed in the residual variance method. Inthese methods, the test statistics (Eq. 13) based on the auto-correlation function of the residuals generated from the cur-rent pitch signals are statistically compared to the autocorre-lation function of the baseline residuals obtained during thebaseline phase. After a preliminary investigation of the ef-fect of the maximum lag τ on the method’s performance, avalue of 20 has been chosen as adequate. Hence, the (1−α)

0 10 20 30 40 50 60

Time (s)

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1

2

3

4

5

6

7

8

9

10

Test sta

tistics b

ased o

n r

esid

ual variance o

f pitch s

ignal

Healthy aircraft

Front rotor failure

Side rotor failure

Fig. 10. Indicative pitch residual variance based fault de-tection results. The dashed red horizontal line indicatesthe statistical threshold at the α = 10−12 risk level. A faultis detected when the test statistic exceeds the threshold.

critical point of a χ2 distribution with 20 degrees of freedomdenotes the critical limit for the statistical hypothesis testing.Figure 11 shows the results for different states of the aircraftfor healthy, front and side rotor failure cases, respectively, atthe 10−3 risk level α . The test statistics crossing over the crit-ical limit (dashed red line) denote rejection of the null hypoth-esis, thus declaring fault detection. Due to the fault-inducedsharp transients in the signals, the fault detection is immedi-ate. Similar to the previous method, due to differences in thepitch signals for front and side rotor failure, the test statisticsexceed the critical limit by a greater amount in the former casecompared to the latter. Also, due to the controller compensa-tion, the test method is not able to distinguish between healthyand fault compensated pitch signals, as evident from the teststatistics going back under the limit in Fig. 11. It should benoted that fault detection is solely based on the response sig-nals; in future work, the controller signals will be also takeninto account to differentiate between healthy and fault com-pensated states of the aircraft and enhance the performance ofthe methods while taking into account the fault compensationcharacteristics of the controller.

Sequential Probability Ratio Test

The Sequential Probability Ratio Test (SPRT) method enablesonline fault detection based on residuals generated from driv-ing the current (unknown) pitch signals through the aforemen-tioned AR model obtained for the healthy state. To implementthis technique, an appropriate sampling approach needs to se-lected. This involves determination of the following three as-pects: (i) the nominal residual standard deviation σo for whichthe aircraft is considered healthy, (ii) the standard deviationratio q= σ1

σo, which establishes the standard deviation increase

under which the aircraft will be considered faulty, and (iii) the

8

0 10 20 30 40 50 60

Time (s)

0

50

100

150

200

250

300T

est sta

tistics b

ase

d o

n r

esid

ua

l w

hiten

ess o

f pitch

sig

na

l

Healthy aircraft

Front rotor failure

Side rotor failure

Fig. 11. Indicative pitch residual uncorrelatedness basedfault detection results. The dashed red horizontal line in-dicates the statistical threshold at the α = 10−3 risk level.

SPRT strength (α, β ).

The residual standard deviation σo for which the aircraft isconsidered healthy is determined based on the 20 availabledatasets for healthy forward flight at 5 m/s under severe tur-bulence. From examination of the values of q and (α, β ),for minimum false alarms without missing any fault detectionthese values have been taken as q = 1.1 and (α = 10−12, β =10−2) and the results for different aircraft states have beenpresented in Fig. 12. The test statistic (blue line) (Eq. 17)exceeding the upper limit denotes a rotor fault, while exceed-ing the lower limit denotes a healthy aircraft. When the teststatistic lies between the two limits, no decision is made andthe test continues with using new samples (Eq. 18).

In this method, the number of samples to make a decisionis not constant and after a decision is made the SPRT col-lects new samples until the next decision is made, i.e. thetest runs online and decisions are constantly being made withthe availability of new data samples. In the case of front rotorfailure compensation, the test statistics predict healthy aircraftwhereas in the case of side rotor failure, the test continues todetect fault even after the transient has passed. Therefore, theSPRT based on pitch signal is unable to distinguish the air-craft healthy and fault compensated states in some cases. Italso signifies that the pitch signal is better compensated tomatch the healthy aircraft dynamics in the case of front rotorfailure compared to side rotor failure.

Vector AR Identification and Fault Detection

Vector (multivariate) parametric identification of the healthyaircraft has been based on 20 s (N = 2000 samples at sam-pling frequency 100 Hz) data sets for the roll, pitch, and yawsignals, without any external excitation (ambient excitation

0 10 20 30 40 50 60-10

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30

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0 10 20 30 40 50 60-10

0

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Te

st

sta

tistics Front rotor failure

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-10

0

10

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30

Side rotor failure

Fig. 12. Indicative results for SPRT based fault detectionwith pitch signal.The dashed red horizontal lines indicatesthe critical limits at the α = 10−12 and β = 10−2 risk levels.

due to turbulence assumed to be white) generated from for-ward flight simulation at 5 m/s under severe turbulence. Themodel identification follows the same procedure as the scalarmodel to estimate the parameters ai and select a model orderna which can accurately represent the dynamics of the healthyaircraft.

Model Identification

The model order selection based on the BIC and RSS/SSScriteria yields yields a model order of 4, represented asVAR(4). The SPP for the model is 166.67 as the number ofestimated parameters for VAR(4) is 36 (as the Ai parametermatrix is a 3× 3 matrix with i = 1,2,3,4 for model order of4). The roll, pitch and yaw signal of the healthy aircraft flyingat 5 m/s for different severe turbulence realization has beendriven through the model estimated to generate residuals.The autocorrelation and cross-correlation functions of thethree residual sequences generated are observed to be whitewith 95% confidence. For signals generated for differentfaulty states, the residuals are found to be correlated. Thisprovides validation of the model, it is capable of representingthe dynamics of a healthy aircraft under the considered flightconditions.

VAR Residual Based Fault Detection

The current (unknown) signals (roll, pitch and yaw inthat order), when driven through the VAR(4) model estimatedin the previous section, yield three sets of residual sequences.The residual based fault detection is performed by the statis-tical comparison of each characteristic quantity obtained viathe current residual sequence with the corresponding quantityobtained via the use of the baseline signals (signal used to

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estimate the healthy VAR model) and corresponding residualseries through the baseline VAR model. In other words, thecharacteristic quantity obtained from the current roll residualsequence is compared to the baseline quantity obtainedfrom the roll residual sequence. Therefore, the statisticalhypothesis testing is performed thrice for a particular timewindow (duration of signal measured in number of samples).

Residual Variance Method

The current 5-second-long signals (N = 500 samples sampledat 100 Hz frequency), where the window is updated every0.1s, are driven through the VAR(4) model to generate threesets of residual sequences. Fault detection is achieved throughthree parallel statistical hypotheses testing of the variance ofthe current residuals to the variance of the baseline residuals(Eq. 10). The critical limit is determined from the F distribu-tion’s (1−α) critical limit for 500, 500-36 degrees of freedom(because the total number of estimated parameters for the 3-variate VAR(4) model is 36). The statistical hypothesis test isconducted at the α (false alarm) risk level of 10−12 to mini-mize the false alarms and the results for different states of theaircraft are presented in Fig. 13.

To collect a minimum of 500 samples for testing, the test startsfrom 5 s. For current signals obtained from the healthy flight,the test statistics for each attitude signal fall below the criticallimit for the test, denoting acceptance of the null hypothesis.With front and side rotor failure at 10 s, the fault detection,which is evident from all the three test statistics exceeding thecritical limit, is immediate. Here, it is interesting to note thatthe extent of violation is greater in the case of side rotor fail-ure than that for front rotor failure, contrary to the observationin case of scalar AR model of pitch signal only. In the caseof side rotor all the three signals (roll, pitch and yaw) changedrastically, while in front rotor failure the roll attitude doesnot change much. As the model captures the correlation be-tween the different signals, the prediction errors are larger forside rotor failure than front rotor failure, as evident from theirhigher level of crossing the critical limit.

It is also observed that the test statistics for the pitch residualvariance recede below the critical limit after the controller hasstabilized the system, while the roll and yaw residuals con-tinue to violate the critical limit, which indicates a faulty sys-tem. This suggests that the controller compensates for thepitch signal better than the other two attitudes and returnsthe longitudinal dynamics to a state comparable to that of thehealthy aircraft.

Residual Uncorrelatedness Method

Online fault detection by residual uncorrelatedness is per-formed with 5 s (N = 500 samples) window length and updateinterval of 0.1 s for each residual sequence. In this method,the characteristic quantity is the autocorrelation function ofthe residuals with a maximum lag τ = 20. The critical limitof the statistical hypothesis testing is given as the (1−α) crit-ical point of a χ2 distribution with 20 degrees of freedom.Figure 14 shows three parallel hypothesis tests on roll, pitchand yaw residuals for different current states of the aircraft:

0 10 20 30 40 50 60

100

Healthy aircraft

Roll Pitch Yaw

0 10 20 30 40 50 60

100

Te

st

Sta

tistic b

ase

d o

n r

esid

ua

l va

ria

nce

Front rotor failure

0 10 20 30 40 50 60

Time (s)

100

Side rotor failure

Fig. 13. Indicative residual variance based fault detectionresults. The dashed red horizontal line indicates the sta-tistical threshold at the α = 10−12 risk level.healthy aircraft, front, and side rotor failure, respectively, atthe risk level α of 10−3.

In the healthy case, the test statistic for all the three signals islower than the the critical limit, correctly declaring the systemas healthy. For the front, and side rotor failure at 10 s, faultdetection is fast. The test statistics for side rotor failure exceedthe critical limit by a greater amount than in case of front ro-tor failure due to significant change of all three signals in theformer compared to only two in the latter, as discussed. Here,post fault compensation, all three test statistics continue to vi-olate the critical limit, indicating a faulty system. Hence, thetest is able to distinguish a healthy system from a fault com-pensated system from the changes in the three attitude signalsof the aircraft. It should be noted that the test statistic for rollin the case of side rotor failure shows the maximum devia-tion, as the controller takes longer time to compensate for it,evident in Fig. 5.

Sequential Probability Ratio Test

The SPRT runs online with the current residuals for roll, pitchand yaw generated from driving the corresponding attitudesignals through the VAR(4) model. The appropriate samplingplan to implement this test procedure is different for each ofthe signals and is determined from the scalar AR modeling ofeach signal. The value of the variance ratio q is taken to be1.5, 1.1 and 1.2 for roll, pitch, and yaw, respectively. Thesevalues have been decided through rigorous testing to improvefalse alarms without missing any faults. The nominal residualstandard deviation for the attitude signals for which the air-craft is considered healthy are determined based on 20 base-line data sets (generated from simulation of healthy aircraftat 5 m/s under severe turbulence) driven through the baselinemodel. The strength of SPRT (α, β ) is taken as 10−12 and10−3, respectively.

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0 10 20 30 40 50 60100

102

Healthy aircraft

Roll Pitch Yaw

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102

Te

st

Sta

tistic b

ase

d o

n r

esid

ua

l u

nco

rre

late

dn

ess

Front rotor failure

0 10 20 30 40 50 60

Time (s)

100

102

Side rotor failure

Fig. 14. Indicative residual uncorrelatedness based faultdetection results. The dashed red horizontal line indicatesthe statistical threshold at the α = 10−3 risk level.

0 10 20 30 40 50 60-10

0102030

Roll

0 10 20 30 40 50 60-10

0102030

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st

sta

tistics Pitch

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-100

102030

Yaw

Fig. 15. Indicative results for SPRT method based on theVAR(4) model for healthy aircraft. The dashed red hori-zontal lines indicate the critical limits at the α = 10−12 andβ = 10−2 risk levels.

Indicative fault detection results are presented in Figs. 15through 17. The aircraft is determined to be in its healthystate when the test statistic exceeds the lower critical point(Fig. 15). Conversely, a fault is detected when the test statistic(vertical axis) exceeds the upper critical point. After a criticalpoint is exceeded, a decision is made, consequently the teststatistic is reset to zero and the test continues. Hence, duringtesting multiple decisions may be made. Evidently, correctdetection is obtained in each test case, as the test statistic isshown to exceed multiple times (multiple correct decisions)

0 10 20 30 40 50 60-10

0102030

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0 10 20 30 40 50 60-10

0102030

Te

st

sta

tistics Pitch

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Time (s)

-100

102030

Yaw

Fig. 16. Indicative results for SPRT method based on theVAR(4) model for front rotor failure. The dashed red hor-izontal lines indicate the critical limits at the α = 10−12

and β = 10−2 risk levels.

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st

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tistics Pitch

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20

40

Yaw

Fig. 17. Indicative results for SPRT based on the VAR(4)model for side rotor failure. The dashed red horizontallines indicate the critical limits at the α = 10−12 and β =10−2 risk levels.the lower critical point in all three signals for the healthy case.

The test statistics also exceed the upper critical point multi-ple times (multiple correct fault detections) in the rotor failurecases, and several observations can be made for the region ofpost failure failure compensation. The roll and yaw residualscontinue to predict faulty state even after the passing of thetransient due to rotor failure in all cases. However, the teststatistics based on pitch residual settles to the healthy case infront rotor failure, once the fault has been taken care of bythe controller (Fig. 16), similar to the results of the scalar AR

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Table 3. Parameters used for various fault detection methodsModel type Fault detection method Parameters

α β qResidual Variance 1e−12 - -

Scalar AR Residual Whiteness 1e−3 - -Roll SPRT 1e−12 1e−2 1.5

Residual Variance 1e−12 - -Scalar AR Residual Whiteness 1e−3 - -Pitch SPRT 1e−12 1e−2 1.1

Residual Variance 1e−12 - -Scalar AR Residual Whiteness 1e−3 - -Yaw SPRT 1e−12 1e−2 1.2

Residual Variance 1e−12 - -Vector AR Residual Whiteness 1e−3 - -Roll,Pitch,Yaw SPRT 1e−12 1e−2 1.5,1.1,1.2

α: Type I (false alarms) risk levelβ : Type II (missed faults) risk level

q: Ratio of residual standard deviation for the system to be considered faulty

pitch signal case (Fig. 12). However, in the case of side rotorfailure, this test based on pitch residual is able to distinguishbetween healthy and fault compensated states (Fig. 17). It isobserved that the SPRT method based on the yaw residuals isbest suited for distinguishing post failure compensation by thecontroller as it continues to indicate that the system is faultyunambiguously as opposed to the roll and pitch signals whichmay indicate a healthy state in this region.

Comparative Assessment of Different Models and FaultDetection Methods

The scalar AR model estimated using the pitch signal of ahealthy aircraft flying at a forward speed of 5 m/s under severelevel of turbulence exhibits excellent fault detection resultswith the residual variance and SPRT methods, for current sig-nals obtained from flight at 5 m/s and all considered levels ofturbulence (light, moderate and severe) (Table 4). The resid-ual uncorrelatedness method is rendered ineffective when thelevel of turbulence is changed from severe (level for which themodel has been estimated) to moderate and light levels. Thismethod shows 100% false alarms for healthy datasets underdifferent levels of turbulence than that used for the model es-timation. It should be noted that the forward speed of flighthas been held constant at 5 m/s for modeling and validationphase for all cases.The 3-variate VAR(4) model estimated from the roll, pitch,and yaw signals of the healthy aircraft under the same forwardspeed and level of turbulence as above, exhibits superior per-formance than the scalar AR model. The VAR-based methodsachieve remarkable results for all levels of turbulence consid-ered in this study (Table 4). In addition, the VAR-based meth-ods have the ability to distinguish between healthy and faultcompensated conditions of the aircraft, after the transient dy-namics have been compensated by the controller.With the exception of the scalar AR model estimated usingthe roll signal, all the fault detection methods can detect rotor

failures within 0.1 s of the event of failure. The scalar meth-ods based on the estimated with roll signal, show a somewhatdelayed fault detection (maximum delay being 1.2 s) usingthe residual variance method and several missed faults usingresidual uncorrelatedness method for front rotor failure. Thisis probably due to the fact the roll does not change signifi-cantly in this case. On the other hand, the SPRT method doesnot miss nor shows a delay in detecting front rotor failuresusing the same scalar model.

The SPRT-based method is able to achieve faster fault de-tection that is better suited for online application as it usesa smaller number of testing samples than the other methods.In addition, the detection time and number of sample neededfor fault detection can be theoretically established and inves-tigated via a well-defined statistical framework (Ref. 9).

Table 3 outlines the different parameters (α,β ,q) selectedfor each residual based fault detection method based on theachieved effectiveness and robustness. If α is not properlyadjusted, the method loses efficiency with respect to the prob-ability of false alarms and missed faults. Hence, it is advisedto make an initial investigation on the number of false alarmsfor different levels of turbulence using several healthy datasets. Then, missed fault errors may be checked with data cor-responding to various rotor failure states. Moreover, for ro-bust performance of parametric methods, a very small valueof the type I risk (α) is often required. This is due to the factthat the stochastic time series models (like AR, ARMA, ARX,state space, etc.) used for modeling the dynamics are still in-capable of fully capturing the experimental, operational andenvironmental uncertainties that the aircraft may be subjectedto. Therefore, to “compensate” for the lack of effective un-certainty modeling, a very small α is often selected. Anotherimportant factor is the number of samples needed for hypoth-esis testing, since the chance of missing faults (β ) dependsupon sample size. For the SPRT method, the values of type Irisk (α), type II risk (β ), and ratio of residual standard devi-

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Table 4. Accuracy of different methodsModel type Fault detection method False Alarms % Missed Rotor Failure

Rotor (1) Rotor (2) Rotor (6)Residual Variance 28.98/15.48/14.57 0/0/0 0/0/0 0/0/0

Scalar AR Residual Whiteness 3.81/-/- 17/-/- 0/-/- 0/-/-Roll SPRT 0.077/0.0025/0 0/0/0 0/0/0 0/0/0

Residual Variance 0.082/0/0 0/0/0 0/0/0 0/0/0Scalar AR Residual Whiteness 0.61/-/- 0/-/- 0/-/- 0/-/-Pitch SPRT 0.34/0/0 0/0/0 0/0/0 0/0/0

Residual Variance 15.35/2.92/3.59 0/0/0 0/0/0 0/0/0Scalar AR Residual Whiteness 10.93/-/- 0/-/- 0/-/- 0/-/-Yaw SPRT 2.68/1.03/1.09 0/0/0 0/0/0 0/0/0

Residual Variance 0.045/ 0.49/ 0.25 0/0/0 0/0/0 0/0/0Vector AR Residual Whiteness 0.16 / 0.018 / 0.24 0/0/0 0/0/0 0/0/0Roll,Pitch,Yaw SPRT 0.065/0/0 0/0/0 0/0/0 0/0/0

False Alarms % for Severe/ Moderate/ Light levels of turbulenceMissed Faults for Severe/ Moderate/ Light levels of turbulence out of 20 datasets each

ation (which gives the increase in standard deviation requiredto consider the system to be faulty) have been selected care-fully to minimize false alarms in healthy signals, but withoutmissing any faults in case of various rotor failures.

CONCLUSIONS

• Statistical time series methods for rotor fault detectionin multicopters achieve effective detection based on (i)ambient (white) excitation and aircraft state (scalar orvector) signals, (ii) statistical model building, and (iii)statistical decision making under uncertainty.

• Both scalar and vector statistical time series methodshave shown remarkable results in effectively detectingfaults, with the vector methods achieving improved per-formance with respect to false alarms, missed faults anddistinguishing between healthy and faulty compensatedstates.

• Parametric time series methods are more elaborate andrequire higher user expertise compared to generally sim-pler non–parametric approaches like the Spectrogram,but have greater sensitivity to faults and accuracy of de-tection when fault occurs in real time.

• Vector methods based on a multivariate model are moreelaborate, but have the potential of further enhanced per-formance such as distinction between healthy and faultcompensated states from the aircraft states alone, with-out the knowledge of controller effort. Also, using thedifference between autocorrelation and cross-correlationfunctions of the signal residuals in case of different rotorfailures, fault identification is possible.

• The knowledge of controller effort for different levelsof turbulence can be used along with the aircraft outputto match the performance of simple scalar models withmore complex vector models. This can improve their ap-plicability in all levels of turbulence as well as the ability

to distinguish between healthy and post-failure controllercompensated states.

• In the future, the modeling will be expanded to Func-tionally Pooled (FP) model based methods that will in-clude various forward flight speeds and levels of turbu-lence into a single integrated framework for complete ro-tor fault detection (abrupt failure and continuous degra-dation), identification and quantification.

AUTHOR CONTACTAirin Dutta duttaa5@rpi.eduMichael McKay mckaym2@rpi.eduFotis Kopsaftopolous kopsaf@rpi.eduFarhan Gandhi fgandhi@rpi.edu

ACKNOWLEDGEMENTThis work is carried out at Rensselaer Polytechnic Insti-tute under the Army/Navy/NASA Vertical Lift ResearchCenter of Excellence (VLRCOE) Program, grant numberW911W61120012, with Dr. Mahendra Bhagwat and Dr.William Lewis as Technical Monitors.

REFERENCES1M. Frangenberg, J. Stephan, and W. Fichter, “Fast Actua-

tor Fault Detection and Reconfiguration for Multicopters,” inAIAA Guidance, Navigation, and Control Conference, AIAA,Jan. 2015.

2G. Heredia and A. Ollera, “Sensor Fault Detection in SmallAutonomous Helicopters using Observer/Kalman Filter Iden-tification,” in IEEE International Conference on Mechatron-ics, Malaga, Spain, IEEE, Apr. 2009.

3M. McKay, R. Niemiec, and F. Gandhi, “Post-Rotor-Failure-Performance of a Feedback Controller for a Hexa-copter,” in American Helicopter Society 74th Annual Forum,Phoenix, AZ, AHS, May 2018.

13

4V. Stepanyan, K. Krishnakumar, and A. Bencomo, “Identi-fication and Reconfigurable Control of Impaired Multi-RotorDrones,” in AIAA Science and Technology Forum and Exposi-tion, Jan. 2016.

5X. Qi, D. Theillol, J. Qi, Y. Zhang, and J. Han, “A Lit-erature Review on Fault Diagnosis Methods for Manned andUnmanned Helicopters,” in 2013 International Conference onUnmanned Aircraft Systems, May 2013.

6S. Fassois and F. Kopsaftopoulos, New Trends in StructuralHealth Monitoring, ch. Statistical Time Series Methods forVibration Based Structural Health Monitoring, pp. 209–264.Springer, Jan. 2013.

7F. P. Kopsaftopoulos and S. D. Fassois, “Scalar and VectorTime Series Methods for Vibration Based Damage Diagnosisin a Scale Aircraft Skeleton Structure ,” Journal of Theoreticaland Applied Mechanics, vol. 49, no. 4, 2011.

8P. A. Samara, G. N. Fouskitakis, J. S. Sakellariou, and S. D.Fassois, “A Statistical Method for the Detection of SensorAbrupt Faults in Aircraft Control Systems,” IEEE Transac-tions on Control Systems Technology, vol. 16, pp. 789–798,July 2008.

9F. P. Kopsaftopoulos and S. D. Fassois, “A vibration modelresidual-based sequential probability ratio test framework forstructural health monitoring,” Structural Health Monitoring,vol. 14, no. 4, pp. 359–381, 2015.

10D. G. Dimogianopoulos, J. D. Hios, and S. D. Fas-sois, “FDI for Aircraft Systems Using Stochastic Pooled-NARMAX Representations: Design and Assessment,”vol. 17, pp. 1385–1397, Nov. 2009.

11D. Peters and C. He, “A Finite-State Induced Flow Modelfor Rotors in Hover and Forward Flight,” in American Heli-copter Society 43rd Annual Forum, St. Louis, MO, AHS, May1987.

12T. M. I. Hakim and O. Arifianto, “Implementation of Dry-den Continuous Turbulence Model into Simulink for LSA-02Flight Test Simulation,” in Journal of Physics: ConferenceSeries 1005(2018) 012017, Aug. 2018.

13E. L. Lehmann and J. P. Romano, Testing Statistical Hy-potheses. Springer, 3rd ed., 2008.

14G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time SeriesAnalysis: Forecasting & Control. Prentice Hall: EnglewoodCliffs, NJ, third ed., 1994.

15L. Ljung, System Identification: Theory for the User.Prentice–Hall, 2nd ed., 1999.

16S. D. Fassois, “Parametric identification of vibrating struc-tures,” in Encyclopedia of Vibration (S. Braun, D. Ewins, andS. Rao, eds.), pp. 673–685, Academic Press, 2001.

17T. Soderstrom and P. Stoica, System Identification.Prentice–Hall, 1989.

18H. Lutkepohl, New Introduction to Multiple Time SeriesAnalysis. Springer-Verlag Berlin, 2005.

19A. Wald, Sequential Analysis. New York: Dover Publica-tions Inc., 2004.

20B. K. Ghosh and P. K. Sen, eds., Handbook of SequentialAnalysis. New York: Marcel Dekker, Inc., 1991.

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