fault detection and isolation strategy for a network of unmanned...

Fault Detection and Isolation Strategy for a Network

of Unmanned Vehicles in Presence of Large

Environmental Disturbances

Nader Meskin∗ and K. Khorasani†

Concordia University, Montreal, Quebec, H3G 1M8 Canada

C. A. Rabbath‡

Defence Research and Development Canada, Valcartier, Quebec, Canada

In this paper, the problem of designing and developing a hybrid Fault Detection andIsolation (FDI) scheme for a network of unmanned vehicles (NUVs) that are subject tolarge environmental disturbances is investigated. The proposed FDI algorithm is a hybridarchitecture that is composed of a bank of residual generators and a Discrete-Event System(DES) fault diagnoser. A novel set of residuals is generated so that the DES fault diagnoserempowered by incorporating appropriate combinations of residuals and their sequentialfeatures will robustly detect and isolate faults in the NUVs. Our proposed hybrid FDIalgorithm is then applied to actuator fault detection and isolation in a network of quad-rotorhelicopters. Simulation results demonstrate and validate the performance and capabilitiesof our proposed hybrid FDI algorithm.

Nomenclature

J Inertia matrixωi Angular speed of motor i, RPMφ Roll angle, radψ Yaw angle, radτ Angular momentum, N.m.sθ Pitch angle, radl The distance measured from the motors to the center of gravity, mfi Thrust of motor i, Ng Standard gravity, N/kgm Mass, kg

I. Introduction

Recent years have witnessed a great deal of interest and intensive activities in the area of autonomous un-manned multi-vehicle systems as in spacecraft formation flight, unmanned aerial vehicles (UAVs), unmannedground vehicles (UGVs), autonomous underwater vehicles (UUVs), etc.1,2 However, only few results on faultdetection and isolation (FDI) of a network of unmanned vehicles have been developed in the literature. InRef. 3, a decentralized fault detection filter is designed as a combination of a game theoretic fault detectionfilter and a decentralized filtering approach. The approach is applied to a platoon of cars in an advancedvehicle control system. In Ref. 4, a decentralized detection filter for a large homogeneous collection of LTIsystems is developed, with inspirations derived from platoons of vehicles with typical “look-ahead” structure.

∗Post-doct fellow, Department of Electrical and Computer Engineering, Email: n [email protected]†Professor, Department of Electrical and Computer Engineering, Email: [email protected]‡Defence Scientist, Department of National Defence, Email: [email protected]

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American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-5656

Copyright © 2009 by N. Meskin, K. Khorasani, and C. A. Rabbath. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

A distributed, model-based, qualitative fault diagnosis scheme is developed in Ref. 5 for multi robotsformation. A decentralized fault detection algorithm6 is proposed to address worst-case situation involvingconcurrent communication loss and component faults for a leader-follower formation. Recently, in Ref. 7, adecentralized non-abrupt fault detection scheme is presented for a leader-to-follower formation of unmannedairships. In Ref. 8 an FDI algorithm is developed for spacecraft formation flight in deep space. Three differentFDI architectures, namely, centralized, decentralized and semi-decentralized are developed for a network ofunmanned vehicles with relative state measurements.9 In Ref. 10, the performance of cooperative controland consensus algorithms are analyzed under different fault scenarios in a network of unmanned vehicles.In this paper, we develop a hybrid FDI algorithm for actuator fault detection and isolation problem in anetwork of unmanned vehicles consisting of multiple quad-rotor helicopters that are subject to unexpectedand large input disturbances.

Generally speaking, two approaches have been developed in the literature for design of robust FDIalgorithms with respect to disturbance inputs and modeling errors, namely disturbance decoupling anddisturbance attenuation approaches.11 In the former approaches, a residual is generated which is decoupledfrom disturbance inputs. Various methods such as Unknown Input Observer (UIO) and eigenstructureassignment11 are developed for achieving the disturbance decoupling property. One of the main limitationsof the disturbance decoupling approaches is that the number of output measurements may not be sufficientwith regards to the number of faults and disturbances which will limit the number of faults that can bedetected and isolated. With respect to the disturbance attenuation approaches, the effects of disturbanceinputs on the residuals are minimized by using optimization approaches such as the optimal parity spacemethod or the H∞-based optimization method.11 Subsequently, threshold limits are selected for the residualsby considering the worst case analysis of the bounded disturbance inputs. A fault sensitivity analysis12,13 isinvestigated for trading off between disturbance attenuation and fault sensitivity. One of the main challengesin these approaches is the selection of suitable thresholds that will not compromise the detection of theincipient faults in the system. In Ref. 14 various means for selecting the residual evaluation functions andtheir corresponding threshold values are introduced. However, in these approaches, the threshold values areselected by considering the upper limit of the bounded disturbance inputs, and this will generally lead tohigh threshold values and hence a high likelihood of missing the detection of incipient faults in the system.

In this paper, a novel hybrid FDI algorithm is developed for a network of unmanned vehicles that aresubject to large environmental disturbances. As emphasized earlier, one of the main challenges in the designof FDI algorithms is to distinguish the effects of disturbances from faults and develop a robust FDI schemewithout compromising the detection of incipient faults. In the case of unmanned vehicles such as small-scaleor even micro UAVs, this problem is more challenging due to their relatively small size and weight that makethese vehicles have higher sensitivity to external and environmental disturbances such as wind gust.

Towards these end, a hybrid architecture for a robust FDI is introduced that is composed of a bankof continuous-time residual generators and a discrete-event system (DES) fault diagnoser. First a set ofresiduals is generated based on the coding set that we have recently introduced in Ref. 9 for a family offault signatures with a given isolability index (defined subsequently). Two threshold levels are assigned tothe residuals. It is further assumed that the input disturbances can be categorized into two families, namely,the tolerable disturbance inputs and the large and unexpected disturbances.

A first level of threshold is selected such that the tolerable disturbance inputs do not generate any falsealarms by using the residuals. A complementary set of residuals is then generated by considering the effectsof the disturbances on the first set of generated residuals. A DES fault diagnoser is designed by invokingan appropriate combinations of the residuals and their sequential features to not only detect and isolatefaults and guarantee no false alarms subject to large external disturbances, but also to detect and identifythe occurrence of large external disturbances. It should be emphasized that our proposed FDI approachperforms simultaneous robust fault detection and isolation as well as large and unexpected disturbancesdetection without imposing any limitations on the total number of faults that can be detected and isolatedwhile not missing the detection of incipient faults.

A network of quad-rotor helicopters is considered as an intended target application and a case study inthis paper. Quad-rotor helicopters are emerging as a common platform for unmanned aerial vehicle researchdue to the simplicity of their construction, their hover as well as vertical take off and landing (VTOL)capabilities.15–18 These rotorcraft have four rotors in total with two pairs of counter-rotating, fixed-pitchblades located at four corners of the vehicle (refer to Figure 1). Previous work in the literature have focusedon development of different control algorithms for these vehicles. However, fault detection and isolation

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problem for quad-rotor vehicles has only been considered in Ref. 19. In Ref. 19, a fault detection (andnot a fault isolation) scheme is developed for a single quad-rotor helicopter based on the differential flatnesstheory. In that work, the effects of environmental disturbances are not considered and consequently, theproposed approach will not be robust with respect to large environmental disturbances. In this paper, ourproposed Hybrid FDI strategy is applied to a network of four quad-rotor helicopters that are subject to largeenvironmental disturbances such as wind gust. We have shown that existing method in the literature willnot be capable of robustly detecting and isolating faults under these severe circumstances.

The remainder of this paper is organized as follows. In Section II, the problem of actuator fault detectionand isolation in a network of unmanned vehicles is considered, and conventional methods in the literature forsolving this problem are investigated. In Section III, a network of four quad-rotors model is introduced andan FDI algorithm is developed. Subsequently, our hybrid FDI algorithm for a network of unmanned vehiclesthat are subject to large disturbances is developed and presented in Section IV. In Section V, our proposedrobust FDI algorithm is applied to a network of four quad-rotor vehicles. Simulation results corresponding tovarious fault scenarios are provided and a comparison is made with the specific results that were developedin Section III. Conclusions are presented in Section VI.

The following notation is used throughout this paper. B = Im B denotes the image of B; KerC denotesthe kernel of C. For any positive integer k, the bold letter face k denotes the finite set 1, 2, · · · , k. C(k, µ)denotes the number of µ combination of the set k. For a given set k, a combination is an unorder collectionof the elements of 1, ..., k. For a linear system (C, A, B), < KerC|A > denotes the unobservable subspaceof (C,A). The spectrum of A is denoted by σ(A). For a given set N, |N| denotes the cardinality of N. ⊗denotes the Kronecker product of matrices and for a given n × n matrix A and an integer number N > 0,AN denotes IN ⊗A, where IN is an N ×N identity matrix and AN denotes the (N − 1)n× (N)n matrix

AN =

A −A 0 · · · 0A 0 −A · · · 0...

. . ....

A 0 0 · · · −A

(1)

II. Actuator Fault detection and Isolation in a Network of Unmanned Vehicles

Consider a network of N homogenous (purely for sake of notational simplicity) vehicles whose lineardynamics are governed by

xi(t) = Axi(t) + Bui(t) +a∑

l=1

Llmil(t) +q∑

j=1

Bdj dij(t), i ∈ N (2)

where xi ∈ Rn is the state of the i-th vehicle; ui ∈ Ra is the control input, dij ∈ R are environmentaldisturbances, the fault signature Ll represents a fault in the l-th actuator of the vehicle, i.e, Ll is the l-thcolumn of B, and mil ∈ R is the fault signal associated to the l-th actuator. It is assumed that matrix B isfull rank (Rank(B)=a). Each vehicle is equipped with the following relative state measurements:

zij(t) = C(xi(t)− xj(t)) j ∈ Ni (3)

where the set Ni ⊆ [1, N ]\i represents the set of vehicles that vehicle i can sense and is designated as theneighboring set of the vehicle i, and zij , j ∈ Ni represent the state measurement relative to the other vehicles.It is assumed that the pair (A,C) is observable.

In this paper, we consider the semi-decentralized architecture that is proposed recently in Ref. 9 for anetwork of unmanned vehicles where it is assumed that local communication links exist between each vehicleand its neighbors and the control signals ui are communicated among them. It should be noted that in Ref.9, three different FDI architectures are considered for a network of unmanned vehicles. However, the effectsof the disturbances have not been investigated.

Let Ni = i1, i2, ..., i|Ni| represent the neighboring set of the i-th vehicle and denote zi(t) = [z>ii1(t), z>ii2

(t), · · · , z>ii|Ni|

(t)]>, so that equation (3) can be rewritten as zi(t) = Cix(t), where x(t) = [x>1 (t), x>2 (t), · · · , x>N (t)]>.Since the output measurement zi depends on the state of the neighboring vehicles, the following nodal model

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should be considered for the i-th vehicle in order to design a FDI filter,9 namely

xNi(t) = A|Ni|+1xNi(t) + B|Ni|+1uNi(t) +|Ni|∑

k=0

a∑

l=1

L(k+1)lmikl(t) +|Ni|∑

k=0

q∑

j=1

Bd(k+1)jdikj(t)

zi(t) = C|Ni|xNi(t)

(4)

where xNi(t) = [x>i (t), x>i1(t), · · · , x>i|Ni|

(t)]> ∈ Rn(|Ni|+1), uNi(t) = [u>i (t), u>i1(t), · · · , u>i|Ni|

(t)]>, L(k+1)l is

the k×a+ l column of B|Ni|+1, Bd(k+1)j is the k×d+ j column of matrix I|Ni|+1⊗Bd, Bd = [Bd

1 , ..., Bdq ], and

C|Ni| is constructed as in (1) from matrix C. Moreover, we define mi0l and di0j as the fault signatures anddisturbance inputs corresponding to the i-th vehicle, i.e. mil and dij , respectively. Therefore L1l representsthe fault signatures of the l-th actuator of the i-th vehicle in the nodal system (4).

It is easy to show that the entire state xNi is not fully observable from the relative state measurementszi, and the states of the centroid of vehicle i and its neighbors cannot be determined from zi. Since the pair(A|Ni|+1, C|Ni|) is not observable, one can first try to obtain the observable part of system (4) as

xONi

(t) = A|Ni|xONi

(t) + BO|Ni|uNi

(t) +|Ni|∑

k=0

a∑

l=1

LO(k+1)lmikl(t) +

|Ni|∑

k=0

q∑

j=1

BdO(k+1)jdikj(t)

zi(t) = C |Ni|xONi

(t)

(5)

where xONi

(t) = [x>i (t)−x>i1(t), x>i (t)−x>i2(t), · · · , x>i (t)−x>i|Ni|

(t)]> ∈ Rn|Ni|, LO(k+1)l is the k×a+ l column

of BO|Ni| and BdO

(k+1)j is the k × a + j column of BdO|Ni|, where BO

|Ni| and BdO|Ni| are constructed as in (1) from

matrices B and Bd, respectively. To avoid notational complexity by having double indices such as mikl(t)and dikj(t), system (5) is re-written as

xONi

(t) = A|Ni|xONi

(t) + BO|Ni|uNi(t) +

ν∑

l=1

LOl ml(t) +

Q∑

j=1

BdOj dj(t)

zi(t) = C |Ni|xONi

(t)

(6)

where ν = a × (|Ni| + 1), Q = q × (|Ni| + 1), ml(t)’s correspond to the fault signals mikl(t) such thatml(t) with l = 1, ..., a correspond to the fault signals of the i-th vehicle, namely mi0l(t)’s, and ml(t) withl = a + 1, ..., 2a correspond to the fault signals of the first neighbor of the vehicle i that us specified by i1,namely, mi1l(t)’s, and ml(t) with l = 2a + 1, ..., 3a correspond the fault signals of the second neighbor of thevehicle i that is specified by i2, etc. A similar association is considered between dj(t)’s and dikj(t)’s. Thefault signatures LO

l and the disturbance signatures BdOj also have a one-to-one association with LO

(k+1)l andBdO

(k+1)j , respectively.The structured fault detection and isolation approach is among one of the most common means to

accomplish the fault isolation task.11 In this approach, each residual is designed to be sensitive to a subsetof faults, while remain insensitive to other faults. In other words, the Structured Fault Detection and IsolationProblem (SFDIP)20,21 can be defined for system (6) formally as design of an LTI dynamic residual generatorthat takes the observables uNi(t) and zi(t) as inputs and generates a set of ξ residuals rk(t), k ∈ Ξ = 1, ..., ξwith the following properties:

1. When no fault is present, all the residuals rk(t) decay asymptotically to zero, and

2. The residuals rk(t), k ∈ Ωl ⊆ Ξ, are sensitive to the fault signal ml, and the other residuals rα(t),α ∈ Ξ− Ωl, are insensitive to this fault.

A prespecified family of coding sets Ωl ⊆ Ξ, l ∈ 1, ..., ν should be chosen such that by knowing whichrk(t) is zero and which ones are not, one can uniquely identify a fault. The resulting residual set whichhas the required sensitivity to specific faults and insensitivity to others is known as the structured residualset .21 For the given coding sets Ωl, l ∈ 1, ..., ν and residuals rk, k ∈ Ξ, a finite set Γk ⊆ 1, ..., ν, k ∈ Ξis defined as the collection of all fault signals ml for which the k-th residual rk is being affected, in otherwords Γk = l ∈ 1, ..., ν|k ∈ Ωl.

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An alternative approach for achieving the isolability of faults is to design a directional residual vector11

which in response to a particular fault lies in a fixed and fault-specified direction in the residual space. Witha fixed directional residual vector, the fault isolation problem can be cast as determining which of the knownfault signature directions the generated residual vector lies the closest to. The solvability conditions forgenerating a structured residual set are generally more relaxed when compared to those of the directionalresidual vector since in the later approach the design objective is to generate one residual vector with theabove isolability condition while in the former approach a set of residuals is generated and one may havemore design degrees of freedom.20 For the above reason we adopt the structured residual set approach forfault isolation in this paper. Let us now briefly review the geometric approach that was introduced in20 forsolving the SFDIP problem.

Definition 1 A subspace S is an unobservability subspace20 (u.o.s.) for system (6) if S =< KerHC |Ni||A|Ni|+DC |Ni| > for some output injection map D and measurement mixing map H.

The notation S(L) refers to the class of u.o.s. containing L ⊆ X . The class of u.o.s. is closed underintersection and is nonempty; therefore, it contains the infimal element S∗ = inf S(L). In Ref. 20, analgorithm for obtaining S∗ is presented. The solvability conditions for the SFDIP problem are also derivedas follows.20

Theorem 1 For a given family of coding sets the SFDIP problem has a solution if and only if S∗Γk∩ LO

l =0, l ∈ Γk where

S∗Γk= inf S(

∑

l/∈Γk

LOl ), k ∈ Ξ (7)

According to Ref. 20, let S∗Γkbe an u.o.s. that satisfies Theorem 1, then there exists a map Dk and Hk

such that S∗Γk=< Ker HkC |Ni||A|Ni| + DkC |Ni| >, where Hk is a solution to KerHkC |Ni| = S∗Γk

+ KerC |Ni|.Let Mk be a unique solution of MkPk = HkC, A0k = (A|Ni| + DkC |Ni| : X/S∗Γk

), Pk is the canonicalprojection of X on X/S∗Γk

, and (A|Ni| + DkC |Ni| : X/S∗Γk) denotes an induced map of A|Ni| + DkC |Ni| on

the factor space X/S∗Γkwhich satisfies the following equation

Pk(A|Ni| + DkC |Ni|) = (A|Ni| + DkC |Ni| : X/S∗Γk)Pk (8)

By construction, the pair (Mk, A0k) is observable, hence there exists a Gk such that σ(Fk) = ∆, whereFk = A0k + GkMk and ∆ is an arbitrary symmetric set. Let Ek = Pk(Dk + P−r

k GkHk) and Kk = PkBO|Ni|.

The following detection filter generates the desired residual which is only affected by the fault signals ml ∈ Γk

and is decoupled from other faults,

wk(t) = Fkwk(t)− Eky(t) + Kku(t)rk(t) = Mkwk(t)−Hky(t)

(9)

A family of fault signatures where one generates a set of residuals rk(t) with ξ = ν such that each residualis only affected by one fault and is decoupled from all other faults (dedicated residual set), is designated as astrongly detectable family. A necessary condition for a family of fault signatures of system (6) to be stronglydetectable is that there should be no dependency among the fault signatures LO

l , l ∈ 1, ..., ν.20It is shown in Ref. 9 that the fault signatures LO

l ’s in system (6) are not strongly detectable and indeeddue to the actuator redundancy that is present in the network, the nodal system (6) can be categorized asan overactuated system. New coding sets are introduced in Ref. 9 for the family of fault signatures that arenot strongly detectable. It then follows that for a family of fault signatures that is not strongly detectable,one cannot detect and isolate multiple faults in all the channels. To formalize our results, the followingisolability index is introduced next:9

Definition 2 For a given family of ν fault signatures LOl ’s, the maximum value of µ ≤ ν where one can

detect and isolate the occurrence of up to µ multiple faults is denoted as the isolability index.

It is shown in Ref. 9 that by considering the structure of matrix BO|Ni|, the fault signatures LO

l canbe categorized into a subsets FL1, ..., FLa where FLj = LO

j , LOj+a, ..., LO

j+|Ni|×a, i.e. FLj is the col-lection of the j-th actuator fault signatures of vehicle i and its neighbors. Moreover, it is shown that

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[LOj , LO

j+a, ..., LOj+|Ni|×a] is not full rank and any |Ni| combination of fault signatures in FLj is linearly

independent. Based on the above properties, let us define Γjk as the 2 combinations of the index set Ij

of FLj where Ij = j, j + a, ..., j + |Ni| × a, and the set Γk, k = 1, ..., a × C(|Ni|, 2) as Γk = Γjk for

k = (j− 1)×C(|Ni|, 2) + 1, ..., j×C(|Ni|, 2), j ∈ a. The notation C(k, j) represents the number of j combi-nation from the set k = 1, ..., k. It is shown in Ref. 9 that the corresponding coding sets Ωl = k|l ∈ Γkcan be used for fault signatures with the µ = |Ni| − 1 isolability index. Moreover, necessary and sufficientconditions for solvability of the SFDIP problem for this coding scheme are derived.9 In this work it isassumed that these conditions are satisfied and the SFDIP problem has a solution for this coding scheme.

In Ref. 9, the effects of disturbance inputs dj(t) have not been investigated and considered and onlythe ideal case with no disturbance input is studied. In this paper, our goal is to design an FDI algorithmthat is robust with respect to the disturbance inputs dj(t)’s. Generally speaking, two approaches have beendeveloped in the literature for design of a robust FDI algorithm with respect to the disturbance inputs:11

1) Disturbance decoupling approach: In this approach, a disturbance input is treated as a fault anda set of residuals is generated which are decoupled from disturbance inputs. Based on Theorem 1, necessaryand sufficient conditions for achieving both structured residual set and disturbance decoupling are existenceof unobservability subspaces S∗Γk

= inf S(∑

l/∈ΓkLO

l + BdO|Ni|) such that S∗Γk

∩ LOl = 0, l ∈ Γk, k ∈ Ξ. One of

the main limitations of this approach is that the number of output measurements may not be sufficient withrespect to the number of faults and disturbances, and consequently the disturbance decoupling requirementwill limit the number of faults that can be detected and isolated.

2) Disturbance attenuation approach: In this approach, the effects of disturbance inputs are atten-uated by using an H∞-based optimization approach. In the context of the SFDIP problem, it can be shownthat the error dynamics ek(t) = wk(t) − PkxO

|Ni|(t) corresponding to the k-th residual generator (9) can bewritten as

ek(t) = (A0k + GkMk)ek(t)− Pk

∑

l∈Γk

LOl ml(t)− Pk

Q∑

j=1

BdOj dj(t) + (PkDkDd + GkHkDd)d(t)

rk(t) = Mkek(t)−HkDdd(t)

where the output measurement disturbance Ddd(t) is also added to the output measurement zi(t) andd(t) = [d1(t), ....dQ(t)]> which includes all the disturbance signals that are present in the nodal model of i-thvehicle (5). In an H∞-based approach, it is assumed that d(t) ∈ L2[0,∞], where L2[0,∞] denotes the spaceof L2 norm bounded signals, i.e. ||d||2 < ∞. According to the bounded real lemma, the gain matrix Gl canbe found such that ||r||2 ≤ γ||d||2 by solving the following linear matrix inequality (LMI):

[XkA0k + A>0lXk + TkMk + M>

k T>k + M>k Mk ∗

(Xk(PkBdO|Ni| + PkDkDd) + TkHkDd + MT

k HkDd)> DTd HT

k HkDd − γ2I

]< 0 (10)

with the unknown Xk > 0 and Tk and the gain matrix Gk can be found as Gk = X−1k Tk.

The next crucial step in this approach is the selection of threshold limits Jthkand residual evaluation

functions Jrkfor the residuals rk(t) by considering the worst case analysis of the effects of the disturbance

input d(t) on rk(t). One of the main challenges is to select suitable thresholds without compromising missingthe detection of incipient faults in the system. In Ref. 14, the upper limit of the bounded disturbance inputsis used for obtaining the thresholds. As shown in the next section, this procedure will lead to high thresholdvalues, and hence will result in missing the detection of incipient faults in the system. It should be notedthat recently, the fault sensitivity is incorporated in design of a robust FDI algorithm12,13,22 where theminimum sensitivity of the residual to the fault is investigated based on the notion of H− index. In theseapproaches, a trade off between attenuation of the disturbance effects and enhancing the fault sensitivity isconsidered. However, for determining the threshold limits, one still needs to consider the upper limit of thedisturbances which can lead to high threshold values when the system is subjected to large disturbances.Once the residual evaluation functions Jrk

and thresholds Jthkare selected, the occurrence of faults can be

detected and isolated by using the following decision logics, namely

Jrk(t) > Jthk

,∀k ∈ Ωl =⇒ ml 6= 0, l ∈ 1, ..., νIn the next section, a robust FDI approach that is based on H∞ optimization techniques will be applied

to a network of quad-rotors to emphasize and highlight the drawbacks and limitations of the available robust

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f2

f1f3

f4

θ

φ

ψ

mg

Ex

Ey

Ez

Figure 1. Schematic of the quad-rotor rotorcraft

FDI algorithms in the literature. Our goal is to demonstrate the motivation for development of our novelFDI algorithm for the network of unmanned vehicles that are subjected to large disturbances.

III. Case Study

In this section, design of a robust FDI algorithm is presented for a network of four quad-rotor helicoptersthat is based on the structured residual set (for fault isolability) and H∞-based technique (for disturbanceattenuation).

III.A. Quad-rotor Dynamics

The quad-rotor helicopter dynamics are assumed to be governed by the following equations:15

mx = −usin(θ)my = ucos(θ)sin(φ)mz = ucos(θ)cos(φ)−mg

ψ = τψ

θ = τθ

φ = τφ

(11)

where x and y are the coordinates in the horizonal plane, z is the vertical position (refer to Figure 1), ψ isthe yaw angle about the z-axis, θ is the pitch angle about the y-axis, and φ is the roll angle about the x-axis.Each rotor generates the thrust fi = kiω

2i , where ki > 0 is a constant and ωi is the angular speed of motor

i. The control input u is the total thrust or collective input which is obtained as u = f1 + f2 + f3 + f4. Theangular moments about each axis are as follows: τψ =

∑4i=1 τMi , τθ = (f2 − f4)l, and τφ = (f3 − f1)l, where

l is the distance measured from the motors to the center of gravity and τMi is the coupling produced by themotors.

The new angular momentums τψ, τθ and τφ are defined based on the change of input variable τ =C(η, η)η + Jτ ,15 where τ = [τψ, τθ, τφ]>, τ = [τψ, τθ, τφ]>, η = [ψ, θ, φ]>, C(η, η) is the Coriolis terms andcontains the gyroscopic and centrifugal forces associated with the η dependence of J, and J is the inertiamatrix of the full-rotational kinetic energy of the quad-rotor helicopter expressed in terms of η.

For purpose of simulation and design of the residual generators, it is assumed that the quad-rotor ismoving in a plane with fixed altitude and the Euler angles ψ, θ and φ are sufficiently small. Hence, we have

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u ≈ mg and equation (11) can be re-written as follows

x(t) = −gθ(t) + Aw(t)y(t) = gφ(t) + Aw(t)

θ(t) = τθ(t) + mθ(t)

φ(t) = τφ(t) + mφ(t)

(12)

where we have used approximations sin(θ) = θ, sin(φ) = φ, and cos(θ) = 1. The term Aw that is added tothe x and y dynamics is the resulting aerodynamic force due to the wind turbulence and wind gust. Thisforce can be computed from the aerodynamic coefficients Ci as Ai = 1

2ρairCiV2 , where ρair is the air density,

and V is the velocity of the quad-rotor with respect to air,23 i.e. V = Vr − Vwind, where Vr =√

x2 + y2 isthe velocity of the quad-rotor.

The terms mθ(t) and mφ(t) represent the fault signals corresponding to the angular momentums τθ(t)and τφ(t), respectively. It is clear that mθ(t) and mφ(t) correspond to the faults in motors 2 and 4, andmotors 1 and 3, respectively. It should be noted that the z dynamics as well as the yaw angle ψ dynamicsare ignored given the fact that the quad-rotor is a hovering mode.

It is assumed that the wind is generated by two different phenomenon, namely, wind turbulence and windgust. The Dryden model24 is used here for generating a wind turbulence signal through the relationshipVturb = Gwε, where ε is a zero-mean Gaussian white noise and Vturb is the random vertical wind turbulenceprocess. The Dryden filter Gw(s) is expressed as follows

Gw(s) =

√3U0σ2

w

πLw

s + U0√3Lw

[s + U0Lw

]2(13)

where σw is the RMS vertical wind velocity (m/s), Lw is the scale for the disturbance, and U0 is thevehicle trim velocity (m/s). Moreover, a discrete wind gust model is considered here based on the MilitarySpecification MIL-F-8785C.25 The mathematical representation of the discrete gust is given by

Vgust =

0 x < 0Vm

2 (1− cos( πxdm

)) 0 ≤ x ≤ dm

Vm x > dm

(14)

where Vm is the wind gust amplitude, dm is the gust length, x is the distance traveled, and Vgust is theresultant wind gust velocity. Therefore, the total wind speed is the sum of the wind gust Vgust and the windturbulence Vturb, i.e. Vwind = Vgust + Vturb.

III.B. Robust FDI Design and Simulation Results

In this section, we consider design of a robust FDI algorithm for a network of four quad-rotor helicopterswith identical dynamics as in (12), and where each vehicle can measure its relative position with respectto its neighboring vehicles. The neighboring sets for the network studied here are given by N1 = 2, 3,N2 = 1, N3 = 4 and N4 = 1. Due to space limitations, we only consider the actuator FDI problem forquad-rotor 1 with i1 = 2 and i2 = 3 since N1 = 2, 3. It is assumed that the four quad-rotor helicopters aremaintaining a formation while following the trajectory of the first quad-rotor. The reference trajectory forthe first vehicle is considered to be x = y = t where t denotes the simulation time. A simple leader-followercontroller is designed for each quad-rotor for formation keeping.

According to the results in Section II and Ref. 9, the isolability index for the family of actuator faultsignatures in the nodal model of quad-rotor 1 is 1, i.e. µ = |N1| − 1 = 1. The fault signatures LO

l ’s can becategorized into two subsets, namely, FL1 = LO

1 , LO3 , LO

5 and FL2 = LO2 , LO

4 , LO6 and the sets Γ1

k andΓ2

k are selected as the 2 combinations of the index sets I1 = 1, 3, 5 and I2 = 2, 4, 6. Hence, we haveΓ1

1 = 3, 5, Γ12 = 1, 5, Γ1

3 = 1, 3, Γ21 = 4, 6, Γ2

2 = 2, 6 and Γ23 = 2, 4. The set Γk’s are defined as

Γk = Γ1k for k = 1, 2, 3, and Γk = Γ2

k for k = 4, 5, 6.Consequently, six residuals r1, ..., r6 are needed for fault detection and isolation. The corresponding

coding sets are as follows: Ω1 = 2, 3, Ω2 = 5, 6, Ω3 = 1, 3, Ω4 = 4, 6, Ω5 = 1, 2, and Ω6 = 4, 5,where Ω1 and Ω2 correspond to the first and second actuator faults of quad-rotor 1, Ω3 and Ω4 correspond

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to the first and second actuator faults of quad-rotor 2 (i1 = 2 in N2), and Ω5 and Ω6 correspond to the firstand second actuator faults of quad-rotor 3 (i2 = 3 in N2).

For simulations, a typical value for the wind turbulence is selected as Lw = 580m, σw = 7m/s, andU0 = 1. It is assumed that the wind gust exists between x = 50 and x = 100 with parameters dm = 10m andVm = 8m/s. Figure 2 shows the disturbance inputs d1 due to both wind turbulence and wind gust for quad-rotor 1. By considering the worst-case analysis of the residual evaluation functions Jrk

= 1T0

∫ t

t−T0||rk(τ)||22dτ

with T0 = 5 seconds, the threshold values Jthk= 0.02 are selected for the residuals rk, k = 1, ..., 6. Figure

3 shows the residual evaluation functions corresponding to concurrent fault scenarios where the fault m3 =m21 = −0.015 is injected in quad-rotor 2 at t = 120 seconds (mθ of quad-rotor 2 has a drift of −0.015) andthe fault m2 = m12 = 0.01 is injected in quad-rotor 1 at t = 150 seconds (mφ of quad-rotor 1 has a drift of0.01). It is clear that none of the faults can be detected and isolated since none of the residuals exceed theirthresholds.

Note that as expected no false alarms is generated due to the high wind gust disturbance. This resultshows that an existing robust FDI algorithm will fail to detect incipient faults (low severity faults). Oneof the main reasons for this limitation is that the threshold values are selected by considering the worstcase disturbance inputs. However, in many cases such as the wind gust the duration of large disturbancesare relatively small in comparison with the presence of small disturbances such as wind turbulence whichis persistently applied to the vehicles. Consequently, by considering the largest value of the disturbances,the threshold values are forced to be selected conservatively very high unnecessarily for all the time, whichwill lead the FDI algorithm to miss the detection of incipient faults. This motivates us to develop a hybridrobust FDI algorithm which will remain robust to large disturbances (no false alarms are generated due tolarge disturbances) without compromising the detection of incipient faults. The detail design steps of ourproposed hybrid robust FDI algorithm are presented in the next section.

IV. Hybrid Robust FDI approach

In this section, a hybrid robust FDI scheme is developed for a network of unmanned vehicles that aresubjected to both large and unexpected disturbances as well as actuator faults. Similar to system (6), thefollowing nodal model dynamics is considered for the i-th vehicle in the network, namely

xONi

(t) = A|Ni|xONi

(t) + BO|Ni|uNi(t) +

ν∑

l=1

LOl ml(t) +

Q∑

j=1

BdOj dj(t)

zi(t) = C |Ni|xONi

(t) + D(t)d(t)

(15)

where d(t) = [d1(t), ....dQ(t)]>. It is assumed that dj ∈ Lp[0,∞] for some 1 ≤ p ≤ ∞, where Lp[0,∞] denotesthe space of Lp norm bounded signals, i.e. ||dj ||p < ∞.

Assumption 1 The disturbance inputs are categorized into two groups, namely tolerable disturbance signalsD1 = dj ∈ Lp[0,∞], j ∈ Q | ||dj ||p < δ1, and large and unexpected disturbance signals D2 = dj ∈Lp[0,∞], j ∈ Q | δ1 ¿ ||dj ||p < δ2 where δ1 ¿ δ2.

Assumption 2 The actuator faults and the large disturbance inputs have not occurred simultaneously andthere exists a sufficient time separating the occurrence of a fault and the disturbance.

Our objective here is to design a Hybrid Fault Diagnoser (HFD) for detecting and isolating each faultml, l ∈ 1, ..., ν while guaranteeing that the diagnoser remains robust with respect to both types of distur-bances D1 and D2. In other words, no false alarms should be generated due to disturbance signals. Thehybrid fault diagnoser is composed of two modules, namely, a low-level bank of residual generators and ahigh-level DES diagnoser. The bank of residual generators produces first a set of residuals that are designedbased on the linear geometric FDI approach.9 It then compares, using an evaluation function, each residualto its corresponding threshold value, from which a set of residual logic units is generated.

Two levels of thresholds are needed for certain residuals (this will be discussed in more details sub-sequently). The DES diagnoser module is a finite-state automaton that takes the residual logic units asinputs and estimates the current state of the system. For designing the DES diagnoser, the combined linearplant and the bank of residual generators is modeled as a finite state Moore automaton (G). The generalarchitecture of our proposed hybrid fault diagnoser is shown in Figure 4.

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0 20 40 60 80 100 120 140 160 180 2000

0.5

1

td

1

Figure 2. Disturbance input due to wind turbulence and wind gust.

0 50 100 150 2000

0.02

0.04

t

J r 1

0 50 100 150 2000

0.02

0.04

t

J r 2

0 50 100 150 2000

0.02

0.04

t

J r 3

0 50 100 150 2000

0.02

0.04

t

J r 4

0 50 100 150 2000

0.02

0.04

t

J r 5

0 50 100 150 2000

0.02

0.04

t

J r 6

Figure 3. Residual Evaluation function corresponding to concurrent faults in both τθ and τφ.

In the next section, the procedure for designing our envisaged hybrid fault diagnoser composed of a bankof residual generators and a DES diagnoser is described in detail.

IV.A. Bank of Continuous-Time Residual Generators

In this section, a systematic approach is proposed to design a set of residual generators that provides thenecessary information required by the DES diagnoser. Towards this end, two sets of residuals are developed.The first set is generated according to the coding scheme that is introduced in9 and discussed in Section IIfor the nodal system of the i-th vehicles. The hybrid fault diagnoser (HFD) developed below is guaranteedto remain robust with respect to the tolerable disturbance inputs dj ∈ D1 by selecting appropriate thresholdvalues associated with the residuals. To ensure that the HFD is also robust to large disturbance inputs(dj ∈ D2), a second set of complementary residuals is generated so that the DES fault diagnoser by utilizingthe entire two sets of residuals will robustly detect and isolate a fault.

In the following, a family of fault signatures LOl in (15) with an isolability index of µ = |Ni| − 1 is

considered. Therefore, the SFDIP problem has a solution for the coding set Ωl, with ξ = a × C(|Ni|, 2)residuals rk. Let us denote R1 = rk, k ∈ Ξ = 1, ....ξ. Let Λk denotes the set of disturbance signals dj ’sthat affects the residual rk, i.e. Λk = dj , j ∈ Q|BdO

j ∩ S∗Γk= 0, k ∈ Ξ, where S∗Γk

is defined in (7).Assume that one can generate a set of complementary residuals R2 = rξ+k, k ∈ Ξ such that rξ+k is

decoupled from the disturbance inputs that are specified by Λk but is affected by all the faults ml, l ∈ Γk andpossibly other fault modes. Based on Theorem 1, the necessary and sufficient condition for generating the

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Figure 4. General architecture of our proposed hybrid fault diagnoser.

complementary residual rξ+k ∈ R2 is the existence of unobservability subspace S∗ξ+k = inf S(∑

j, dj∈ΛkBdO

j )such that S∗ξ+k ∩ LO

l = 0, ∀l ∈ Γk.For each disturbance input dj and the fault mode ml, the coding sets Ωd

j and Ωfl are defined respectively

as

Ωdj = k ∈ 1, ..., ξ|S∗Γk

∩ BdOj = 0 ∪ k ∈ ξ + 1, ..., 2ξ|S∗k ∩ BdO

j = 0Ωf

l = Ωl ∪Υfl

(16)

where Υfl = k ∈ ξ + 1, ..., 2ξ|S∗k ∩ LO

l = 0. In other words, the sets Ωdj and Ωf

l are the index set of thoseresiduals rk ∈ R1 ∪R2 that are affected by dj and ml, respectively.

Example 1 For our case study considered in Section III with Q = 1, we have Λ1 = Λ4 = d2, d3, Λ2 =Λ5 = d1, d3, and Λ3 = Λ6 = d1, d2 where dj denotes the wind disturbance in the j-th quad-rotor, i.e.dj = dj1 (q = 1). However, due to the structure of BdO

|Ni|, we have BdOj ∈ S∗ξ+k for all k ∈ Ξ and j = 1, 2, 3. In

other words, all the residuals in the complementary set R2 are decoupled from all disturbances. It also can beverified that these residuals are affected by all actuator faults ml in the nodal system (15) for the quad-rotor 1.Therefore, these residuals are identical and only one extra residual, namely r7 is needed for this system whichis decoupled from all disturbance inputs d1, d2 and d3 and is affected by all the actuator faults of quad-rotor1, 2 and 3. Consequently, according to (16) we have Ωd

1 = 2, 3, 5, 6, Ωd2 = 1, 3, 4, 6, Ωd

3 = 1, 2, 4, 5,Ωf

1 = 2, 3, 7, Ωf2 = 5, 6, 7, Ωf

3 = 1, 3, 7, Ωf4 = 4, 6, 7, Ωf

5 = 1, 2, 7, and Ωf6 = 4, 5, 7 and Υf

l = 7for l = 1, ..., 6.

Assumption 3 For the disturbance inputs dj such that dj /∈ ∪ξl=kΛk, it is assumed that Ωd

j = ∅.

The disturbances which satisfy Assumption 3 have no effect on the residuals, and therefore the hybriddiagnoser does not need to be robust to them. In other words, the generated set of residuals are alreadydecoupled from these disturbances and no further invoking of the DES diagnoser is required. Correspondingto each residual rk ∈ R1∪R2, an evaluation function Jrk

is now assigned. Various evaluation functions suchas Jrk

= rk, or Jrk= ||rk||p, or Jrk

= 1T0

∫ t

t−T0||rk(τ)||22dτ , have been introduced in the literature.14 For the

residuals rk ∈ R1, two different threshold values are needed as specified below

J1thk

= supdj∈D1,ml=0,l∈1,...,ν

(Jrk), k ∈ Ξ (17)

J2thk

= supdj∈D2,ml=0,l∈1,...,ν

(Jrk), k ∈ Ξ (18)

In determining the first threshold, only the tolerable disturbance inputs dj ∈ D1 are considered. However, thesecond threshold incorporates all the possible disturbance inputs. In other words, the supremum that arises

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in determining the threshold values J1thk

and J2thk

can be obtained from evaluation of Jrkcorresponding

to and during the healthy operation or simulation of the system by considering the worst case effects ofdisturbances dj in D1 and D2, respectively.

It should be noted that one may choose to only consider the threshold level given by J2thk

as a worstcase scenario associated with large disturbances. In this case, no false alarms will be generated due todisturbances. However, as shown in Section III this will lead to selection of higher threshold values thatwould unnecessarily reduce the sensitivity of the FDI algorithm to low severity faults. As will be shownsubsequently, by selecting two threshold levels and by considering the temporal and sequential characteristicsof the residuals, one can not only enhance the fault sensitivity but also design a robust FDI algorithm.

The threshold values for the residuals rk ∈ R2 are selected in the same way as the first thresholdsfor residuals in R1 in (17). For a system where the residuals rk ∈ R2 are affected by a few or even nodisturbance input channels, one can select lower threshold values for these residuals. In other words, theresiduals rk ∈ R2 are generally less sensitive to disturbance inputs than residuals rk ∈ R1. For each residualrk ∈ R1 defined at a given point in time t, we can define the corresponding two threshold logic units R1

k(t)and R2

k(t) according to

R1k(t) =

1 if Jrk

(t) > J1thk

0 otherwise, k ∈ Ξ (19)

R2k(t) =

1 if Jrk

(t) > J2thk

0 otherwise, k ∈ Ξ (20)

Similarly, for each residual rk(t) ∈ R2, the threshold logic unit is assigned in the same way as the residualin (19).

Definition 3 The fault scenarios considered for system (15) are categorized into the following three classes,namely, high severity faults, low severity faults, and non-detectable faults:

1. High severity faults correspond to faults that will affect residual logic units R1k, k ∈ 1, ..., 2ξ,

2. Low severity faults correspond to faults that will affect only R1k, k ∈ ξ + 1, ..., 2ξ, and

3. Non-detectable faults correspond to faults that do not affect any of the residual logic units R1k, k ∈

1, ..., 2ξ.

IV.B. DES Fault Diagnoser

For simplicity, let us assume that multiple faults in two actuators are possible. Furthermore, let us considerthe scenarios where only occurrence of one fault and large disturbances is allowed concurrently. This as-sumption will limit the number of all possible operational states of the DES system. However, our proposedalgorithm is easily expandable to more general cases.

First the nodal system (15) along with a bank of residual generators is modeled as a finite state Mooreautomaton26 that is specified according to G = (S,Σ, δ, s0, Y, λ), where S,Σ, Y are finite state, event andoutput sets; s0 is the initial state, δ : S × Σ → S is the transition function and λ : S → Y is the outputmap. For the nodal system (15), the state set is S = s0, s1, ..., sν , s1,2, ..., sν−1,ν , sD, s1,D, ..., sν,D, wherethe state s0 corresponds to the normal operational mode of system (no faults and no large disturbanceinputs exist), and the states sl, l = 1, ..., ν = a × (|Ni| + 1) correspond to occurrence of fault ml. Thestates sl,j , l, j ∈ 1, ..., ν, l 6= j correspond to concurrent occurrence of faults in ml and mj , the state sD

corresponds to occurrence of a large disturbance input, and the states sl,D, l ∈ 1, ..., ν correspond to aconcurrent fault of ml and a large disturbance input.

The event set is denoted by Σ = Fo1 , ...,Fo

ν ,Fr1 , ...,Fr

ν ,Do,Dr, where the events Fol and Fr

l correspondto the occurrence and removal of a fault ml, respectively, the event Do corresponds to the occurrence of alarge disturbance in one of the dj channels, and the event Dr corresponds to the removal of disturbance fromall the channels. The output set is denoted by Y = (R1

1, ..., R12ξ, R

21, ..., R

2ξ) ∈ Bκ, where κ depends on the

property of Λk’s and B = 0, 1. For instance for the case study in Section III, κ = 2ξ +1 (one extra residual

12 of 21


is required). Based on the above definitions, the transition function δ is now defined formally as follows

δ(s0,Do) = sD, δ(sD,Dr) = s0, δ(sD,Fol ) = sl,D

δ(s0,Fol ) = sl, δ(sl,Do) = sl,D, δ(sl,Fr

l ) = s0

δ(sl,Foj ) = sl,j , δ(sl,j ,Fr

j ) = sl, δ(sl,j ,Frl ) = sj

δ(sl,D,Frl ) = sD, δ(sl,D,Dr) = sl

The output map λ depends on the severity of a fault and the threshold values of the residuals. Asmentioned in the previous section, threshold values for the residuals rk ∈ R2 are usually lower than those ofrk ∈ R1. Therefore, there could be a low severity fault scenario where the residual logic unit R1

ξ+k becomesone while R1

k is zero. Thus, certain states could have outputs that may depend on the severity of the fault anddisturbances. In defining the output map λ, such scenarios are also incorporated. Moreover, non-detectablefault scenarios (refer to Definition 3) are not observable from the residual logic units, and therefore theycannot be detected and isolated. These types of faults are not considered in λ. In other words, no event isassigned to such faults. Table 1 shows the specifications associated with the output map λ.

Table 1. Output map of the plant

Output map λ

s0 (0, ..., 0)sD (R1

1, ..., R2ξ) ∈ Y |∃j ∈ Q, ∀k ∈ Ωd

j , R1k = 1

sl (R11, ..., R

2ξ) ∈ Y |∃β ∈ 1, 2, k ∈ Ωf

l , Rβk = 1

sl,j (R11, ..., R

2ξ) ∈ Y |∃β ∈ 1, 2, k ∈ Ωf

l ∪ Ωfj , Rβ

k = 1sl,D (R1

1, ..., R2ξ) ∈ Y |∃β ∈ 1, 2, k ∈ Ωf

l , Rβk = 1

∪(R11, ..., R

2ξ) ∈ Y |∃j ∈ Q, ∀k ∈ Ωd

j , R1k = 1

The objective of the DES diagnoser is to take the output sequence of the system (residual logic units) asinputs and to generate an estimate of the state of the system. In this work, a DES diagnoser is modeled asa finite state automaton H = (SH , IH , δH , z0, YH , λH), where SH , IH , YH denote the finite state, input andoutput sets, z0 is the initial state of the diagnoser, δH : SH×IH → SH denotes the transition function, and λH

is the output map. In order to eliminate any possible ambiguity in the DES model (G) output, two additionalstates with respect to the state set of G are considered for H, namely SH = S, sF , sF,D. Specifically, sF

corresponds to the faulty state when one cannot isolate the faulty channel, and sF,D corresponds to theoccurrence of a fault and a large disturbance in the system when a fault may not be isolated concurrently.The input set to the diagnoser is an output set of G (that is set Y ). The output set is the same as the stateset of the diagnoser (that is YH = SH) and the output map λH : SH → YH is an identity map.

The main step that is left in formalizing our HFD is the design of a transition map δH . First, we considerthe case when the system is in a normal operational mode s0 and find its corresponding transition function.Based on Assumption 2, three transitions are possible in the normal operation, namely transition to thestate sl which corresponds to the occurrence of a fault in the l-th component ml (event Fo

l ), transition tothe state sD which corresponds to the occurrence of a large disturbance in one of the disturbance channels(event Do), and finally the transition to the fault mode sF which corresponds to the occurrence of a lowseverity fault in one of the components that may not be isolable.

It can be shown that the hybrid fault diagnoser can easily distinguish the effects of a fault and a distur-bance by using the coding sets Ωf

l and Ωdj , and therefore the sets Ωf

l and Ωdj can be used for the transition to

states sl and sD, respectively. Indeed, let first consider the case where there exist a residual signal rk(t) ∈ R1

such that it is affected by both fault signal ml(t) and disturbance input dj(t). Moreover, the residual rξ+k

is generated such that it is decoupled from dj(t) and is affected by ml(t). Therefore, there exist a residual,namely rξ+k, such that it is decoupled from dj(t) and is affected by ml(t), i.e. Ωf

l 6= Ωdj . Next we consider

the case that there exist no residual that is affected by both ml(t) and dj(t). In this case, it is clear thatΩf

l 6= Ωdj since Ωf

l 6= ∅, i.e. there exist at least one residual, namely rk(t) that is affected by fault ml(t) andk /∈ Ωd

j .The only other remaining case of interest is when the occurrence of a low severity fault in the l-th

component will lead to changes in only rk ∈ R2. This can also be distinguished from the occurrence of

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the large disturbance by using distinct coding sets Υfl and Ωd

j since for those disturbances that satisfyAssumption 3, Ωd

j = ∅. But it is clear that Υfl 6= ∅. On the other hand, for disturbance inputs dj ∈ Q such

that dj ∈ Λk for some k ∈ Ξ one can show similar to the way shown for Ωfl and Ωd

j that Υfl 6= Ωd

j .To summarize, our proposed hybrid fault diagnoser can detect the occurrence of a fault in the system (15)

since Υfl 6= Ωd

j . However, we may have Υfl = Υf

j for some l, j ∈ 1, ..., ν, and therefore the fault cannot beisolated. In this case, the state of the fault diagnoser will change to sF . Table 2 summarizes the transitionfunction that is initiated from the state s0 when all the unspecified residual logic units are zero.

The next step is now to consider scenarios when initially a large disturbance is applied to at least one ofthe vehicles in the nodal system (15) followed by a fault that is concurrently present in one of the vehiclescomponents. Therefore, it is assumed that the diagnoser had a transition from the normal operation states0 to the disturbance state sD where we define a set D = 1 ≤ k ≤ ξ|R1

k = 1. In this state, the secondthreshold logic units R2

k are used for all the residuals rk, k ∈ D. The transition function for the state sD

is given in Table 2. It is assumed that the effects of the fault is not nullified by a large disturbance input,which is quite a reasonable consideration for practically most situations.

Table 2. Transition function of state s0, sD, and si

Current Input Nextstate (R1

1, ..., R12ξ, R

21, ..., R

2ξ) state

s0

∧k∈Ωf

lR1

k = 1 sl, l ∈ 1, ..., νs0 ∃l ∈ 1, ..., ν such that

∧k∈Υf

lR1

k = 1 sF

s0 ∃l ∈ Q such that∧

k∈ΩdlR1

k = 1 sD

sD all inputs become zero s0

sD

∧k∈Ωf

l ∩D R2k

∧k∈Ωf

lR1

k = 1 sl,D

sD ∃l ∈ 1, ..., ν such that∧

k∈Υfl

R1k = 1 sF,D

sl all inputs become zero s0

sl

∧k∈Ωf

l ∪Ωfj

R1k = 1 for the time interval τ0 sl,j

sl ∃j such that∧

k∈Ωfl ∪Ωd

jR1

k = 1 sl,D

Let us now consider a scenario when a fault is detected in the l-th component in system (15) and thestate of the fault diagnoser is sl. Generally, we should investigate three possible cases, namely 1) the removalof a detected fault, 2) the occurrence of a second fault in the j-th component, and 3) the occurrence of adisturbance dα, α ∈ Q. As it turns out, the main challenge here is to distinguish between cases 2 and 3,since the removal of a fault can be easily detected when all the threshold logic units value become zero. Thenecessary condition for distinguishing between cases 2 and 3 is governed by

Ωfl ∪ Ωf

j 6= Ωfl ∪ Ωd

α, l, j ∈ 1, ..., ν, α ∈ Q (21)

It can be shown that for any two fault modes ml, mj , l, j ∈ 1, ..., ν, l 6= j, |Ωfl ∪Ωf

j | = ξ −C(ν − 2, ν − µ).Therefore, if |Ωd

α ∩Ξ| > ξ −C(ν − 2, ν − µ), there exists at least one residual rk ∈ R1 such that k ∈ Ωdα and

k /∈ Ωfl ∪ Ωf

j and hence we have Ωfl ∪ Ωf

j 6= Ωfl ∪ Ωd

α, l, j ∈ 1, ..., ν, α ∈ Q.In the situation when Ωf

l ∪Ωfj ⊂ Ωf

l ∪Ωdα, one can potentially have a false alarm associated with the second

fault while a large disturbance input is present. To remedy this problem, the DES diagnoser will declare thedetection of a second fault after a specific waiting-time interval τ0, if all the residual threshold logics specifiedby Ωf

l ∪ Ωfj are at 1 while the remaining residual threshold logic units specified by Ωf

l ∪ Ωfj − Ωf

l ∪ Ωdα

remain at zero. Table 2 illustrates the transition function for the state associated with sl, l ∈ 1, ..., ν.Table 3 shows the remaining transitions that should be considered for the hybrid fault diagnoser. By

specifying these transitions, the design of our proposed hybrid diagnoser is completed. In the next section,our proposed FDI algorithm will be applied to the case study that was considered in Section III.

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Table 3. Transition function of states sF , sF,D, sl,D and sl,j

Current Input Nextstate (R1

1, ..., R12ξ, R

21, ..., R

2ξ) state

sF

∧k∈Ωf

lR1

k = 1 for the time interval τ0 sl

sF ∃j ∈ Q such that∧

k∈ΩdjR1

k = 1 sF,D

sF,D

∧k∈Ωf

l ∩D R2k

∧k∈Ωf

lR1

k = 1 sl,D

sF,D ∃l ∈ 1, ..., ν such that∧

k∈Υfl

R1k = 1 sF

sF,D ∃j ∈ Q such that∧

k∈ΩdjR1

k = 1 sD

sl,D ∃j ∈ Q such that∧

k∈ΩdjR1

k = 1 sD

sl,D

∧k∈Ωf

lR1

k = 1 sl

sl,j

∧k∈Ωf

lR1

k = 1 sl

sl,j

∧k∈Ωf

jR1

k = 1 sj

V. Case Study Revisited

For the case study considered in Section III, all the coding sets Ωfl and Ωd

j are found in Example 1and it was shown that only one extra residual is needed for designing our hybrid fault diagnoser. Theevaluation functions for the residuals rk, k = 1, ..., 7 are selected as Jrk

(t) =∫ t

t−T0r>k (t)rk(t)dt, k = 1, ..., 7,

where T0 = 5 seconds is the length of the evaluation window. By considering the worst case scenario ofresiduals corresponding to the healthy mode of the network subject to the measurement noise with uniformdistribution of ± 0.001 and with the wind turbulence and wind gust as given in Section III, the thresholdvalues are selected as J1

thk= 0.0035, J2

thk= 0.02, k = 1, ..., 6, and J1

th7= 0.0006. It should be pointed out

that since the residual r7 is decoupled from the disturbance inputs, one can select a lower threshold valuefor it.

The next step is to design the DES fault diagnoser H. There exist six fault signals ml, l = 1, ..., 6and hence the state modes sl, l = 1, ..., 6 are assigned to the occurrence of a single fault where s1, s3 and s5

correspond to faults in the first actuator of quad-rotors 1, 2, and 3, respectively, and s2, s4 and s6 correspondto faults in the second actuator of quad-rotors 1, 2, and 3, respectively. As shown in9 and presented in SectionII, one can detect and isolate concurrent faults in the first actuator of quad-rotor i and the second actuatorof quad-rotor j for all i, j = 1, 2, 3. Hence, the states s1,2, s1,4, s1,6, s2,3, s2,5, s3,4, s3,6, s4,5, and s5,6

are assigned to the occurrence of concurrent faults where si,j correspond to concurrent occurrence of faultsspecified by si and sj . It should be noted that the states s1,3, s1,5, s3,5, s2,4, s2,6 and s4,6 are not consideredsince they correspond to the concurrent occurrence of faults in the first actuator of two quad-rotors or theconcurrent occurrence of faults in the second actuator of two quad-rotors. These faults cannot be isolated byusing the coding scheme since the isolability index for the fault signatures in the nodal system of quad-rotor1 is µ = 1.

The states sl,D refer to the occurrence of concurrent fault sl and a large disturbance. Finally, the statessF and sF,D correspond to the occurrence of an incipient fault and the concurrent occurrence of an incipientfault and a large disturbance, respectively. Consequently, the cardinality of the state set of the quad-rotor1 is 23. The input set of the diagnoser is I = (R1

1, ..., R17, R

21, ..., R

26) ∈ B13 and the output set is equal to

SH . The transition function λH can be found by following the results in Section IV.B.Figure 5 shows the residual evaluation functions corresponding to a concurrent fault scenario considered

in Section III and Figure 6 depicts the state of the hybrid fault diagnoser. As shown in Figure 6, thediagnoser state first changes to sD at t = 58 seconds after the occurrence of the wind gust with no falsealarm generated. Later on when a fault is injected in the first actuator of quad-rotor 2, the diagnoser stateswitches to s3 at 123.2 seconds. Finally, after the occurrence of the fault in the second actuator of quad-rotor1 at t = 150 seconds, the diagnoser switches to the state s2,3 at t = 163 seconds (note that τ0 is selected as10 seconds. Refer to Table 2.)

Consequently, we can conclude that the hybrid fault diagnoser can perfectly detect and isolate concurrentfaults despite the presence of large concurrent disturbances. However, as shown in Section III, if one only

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uses the first six residuals r1, ..., r6 with the threshold values J2thk

, although no false alarms will be generateddue to large wind gust but the faults at quad-rotors 1 and 2 cannot be detected and isolated. However, byusing our proposed hybrid FDI methodology, we are able to distinguish the occurrence of concurrent faultsas well as large disturbances by designing only one additional residual.

Figure 7 depicts the residual evaluation functions associated with occurrence of a fault in the first actuatorof quad-rotor 3 with m5 = m31 = 0.01 at t = 70 seconds while the quad-rotors are passing through the areabetween x = 50 and x = 100 meters and are subject to wind gust. Figure 8 shows the states correspondingto the hybrid fault diagnoser. As seen from this figure, the diagnoser first detects the occurrence of a largedisturbance between t = 58.6 seconds and t = 71.8 seconds and its state is changed to sD, then it detectsthe occurrence of a fault at t = 71.8 and changes its state to sF,D. At t = 88.2 seconds it detects theremoval of the wind gust and changes its state to sF and finally at t = 126.3 seconds it can isolate thefault and its state is changed to s5. It should be noted that the occurrence of wind gust is also detectedby the diagnoser while its state is at sF when the diagnoser switches between sF and sF,D states duringt = 88.2 and t = 126.3 second. The above fault scenario shows the capability of our proposed FDI algorithmfor detecting the occurrence of a fault while the quad-rotor is subjected to large disturbances and no falsealarms are generated due to the presence of disturbances. Moreover, it is demonstrated that the if one usesconventional approaches in the literature that are based on only the first six residuals, faults cannot bedetected and isolated in the network of unmanned vehicles consisting of multiple quad-rotor helicopters.

Figure 11 depicts the residual evaluation functions associated with occurrence of a fault in the firstactuator of quad-rotor 1 with m1 = m11 = 0.01 at t = 30 seconds and the quad-rotors are passing throughthe area between x = 50 and x = 100 meters and are subject to wind gust. Figure 12 shows the statescorresponding to the hybrid fault diagnoser. As seen from this figure, the diagnoser first detects and isolatesthe occurrence of fault in m1 at t = 33.6 and its state is changed to s1. Later, it also detects the occurrenceof a large disturbance between t = 58.6 seconds and t = 89 seconds and its state is changed to s1,D and aftert = 89 seconds its state remain at s1 given that the quad-rotors have passed through the wind gust area.

Finally, Figure 9 depicts the residual evaluation functions associated with occurrence of a incipient faultin the second actuator of quad-rotor 3 with m6 = m32 = 0.001 at t = 25 seconds and the quad-rotors arepassing through the area between x = 50 and x = 100 meters and are subject to wind gust. Figure 10 showsthe states corresponding to the hybrid fault diagnoser. As seen from this figure, the diagnoser first detectsthe occurrence of fault in m6 at t = 32.6 and its state is changed to sF . Later, it also detects the occurrenceof a large disturbance between t = 58.6 seconds and t = 115 seconds and its state is changed between sF,D

and sF and after t = 115 seconds its state remain at sF given that the quad-rotors have passed through thewind gust area. This fault scenario shows another advantage of our proposed FDI algorithm in detectionlow-severity faults.

To summarize, the above fault scenarios highlight the capability of our proposed FDI algorithm in thefollowing cases:

1. Detection and isolation of concurrent faults in two components in the network in the presence of largeenvironmental disturbance (first fault scenario).

2. Detection and isolation of single fault when the quad-rotors are subjected to large disturbance (secondand third fault scenarios).

3. Detection of occurrence of concurrent incipient fault and large disturbance (fourth fault scenario).

VI. Conclusions

A novel hybrid fault detection and isolation scheme is proposed for a network of unmanned systems thatis subjected to large external disturbances. Our proposed scheme consists of two modules, namely, a bank ofresidual generators and a discrete-event system (DES)-based fault diagnoser. A new set of complementaryresiduals is proposed and constructed that is developed based on a linear geometric fault diagnosis approach.The hybrid fault diagnoser uses the residuals and their temporal behavior to robustly detect and isolate thefaulty channels. Our proposed hybrid FDI methodology is applied to the problem of actuator fault detectionand isolation for a network of unmanned systems consisting of multiple quad-rotors operating under severeexternal disturbances such as wind gust.

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0 50 100 150 2000

0.02

0.04

tJ r 1

0 50 100 150 2000

0.02

0.04

t

J r 2

0 50 100 150 2000

0.02

0.04

t

J r 3

0 50 100 150 2000

0.02

0.04

t

J r 4

0 50 100 150 2000

0.02

0.04

t

J r 5

0 50 100 150 2000

0.02

0.04

t

J r 6

0 50 100 150 2000

0.01

0.02

t

J r 7

Figure 5. Residual evaluation functions corresponding to concurrent faults in the first actuator of quad-rotor 1 andthe second actuator of quad-rotor 2.

References

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0 20 40 60 80 100 120 140 160 180 200t

Faul

t Dia

gnos

er S

tate

s

0

sD

s3

s2,3

Figure 6. Fault diagnoser state corresponding to concurrent faults in the first actuator of quad-rotor 1 and the secondactuator of quad-rotor 2.

0 50 100 150 2000

0.02

0.04

t

J r 1

0 50 100 150 2000

0.02

0.04

t

J r 2

0 50 100 150 2000

0.02

0.04

t

J r 3

0 50 100 150 2000

0.02

0.04

t

J r 4

0 50 100 150 2000

0.02

0.04

t

J r 5

0 50 100 150 2000

0.02

0.04

t

J r 6

0 50 100 150 2000

0.005

0.01

t

J r 7

Figure 7. Residual evaluation functions corresponding to a fault in the first actuator of quad-rotor 3.

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0 20 40 60 80 100 120 140 160 180 200t

Faul

t Dia

gnos

er S

tate

s0

sD

sF,D

sF

s5

Figure 8. Hybrid fault diagnoser state corresponding to a fault in the first actuator of quad-rotor 3.

0 50 100 150 2000

0.02

0.04

t

J r 1

0 50 100 150 2000

0.02

0.04

t

J r 2

0 50 100 150 2000

0.02

0.04

t

J r 3

0 50 100 150 2000

0.02

0.04

t

J r 4

0 50 100 150 2000

0.02

0.04

t

J r 5

0 50 100 150 2000

0.02

0.04

t

J r 6

0 50 100 150 2000

0.01

0.02

t

J r 7

Figure 9. Residual evaluation functions corresponding to a fault in the first actuator of quad-rotor 1.

0 20 40 60 80 100 120 140 160 180 200

t

Faul

t Dia

gnos

er S

tate

s0

s1

s1,D

Figure 10. Hybrid fault diagnoser state corresponding to a fault in the first actuator of quad-rotor 1.

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0 50 100 150 2000

0.02

0.04

t

J r 1

0 50 100 150 2000

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t

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0 50 100 150 2000

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t

J r 3

0 50 100 150 2000

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t

J r 4

0 50 100 150 2000

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t

J r 5

0 50 100 150 2000

0.02

0.04

tJ r 6

0 50 100 150 2000

0.5

1x 10

−3

t

J r 7

Figure 11. Residual evaluation functions corresponding to a fault in the second actuator of quad-rotor 3.

0 20 40 60 80 100 120 140 160 180 200

t

Faul

t Dia

gnos

er S

tate

s0

sF

sF,D

Figure 12. Hybrid fault diagnoser state corresponding to a fault in the second actuator of quad-rotor 3.

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