Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011, pp.52–62

Available online at www.jseepub.com

Multiobjective fault detection and isolation forflexible air-breathing hypersonic vehicle

Xuejing Cai and Fen Wu∗

Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA

Abstract: An application of the multiobjective fault detection andisolation (FDI) approach to an air-breathing hypersonic vehicle(HSV) longitudinal dynamics subject to disturbances is presented.Maintaining sustainable and safe flight of HSV is a challengingtask due to its strong coupling effects, variable operating condi-tions and possible failures of system components. A common typeof system faults for aircraft including HSV is the loss of effective-ness of its actuators and sensors. To detect and isolate multipleactuator/sensor failures, a faulty linear parameter-varying (LPV)model of HSV is derived by converting actuator/system compo-nent faults into equivalent sensor faults. Then a bank of LPV FDIobservers is designed to track individual fault with minimum errorand suppress the effects of disturbances and other fault signals.The simulation results based on the nonlinear flexible HSV modeland a nominal LPV controller demonstrate the effectiveness of thefault estimation technique for HSV.

Keywords: fault detection and isolation (FDI), hypersonic vehicle(HSV), actuator and sensor faults, multiobjective optimization.

DOI: 10.3969/j.issn.1004-4132.2011.01.006

1. Introduction

Air-breathing hypersonic vehicles (HSVs) may representa more cost-effective way to access outer space and makeintercontinental travel routinely. Research on air-breathinghypersonics started during 1960s and continued through1990 with the National Aerospace Plane. The most re-cent progress in hypersonic research is NASA’s flight testsof the scramjet-powered X-43A [1]. Hypersonic flightis defined as at least 5 times the speed of sound. Toachieve sustainable hypersonic flight in the atmosphere, anair-breathing HSV requires a scramjet propulsion systemwhich is highly integrated with the airframe. Moreover, thevehicle tends to have lightweight, then its flexible structurewill introduce the elastic effect at low natural frequenciesduring the flight.

Similar to all other airplanes and space vehicles,

Manuscript received November 18, 2010.*Corresponding author.This work was supported by NASA (NNX07AC40A).

possible sensor/actuator failures are not avoidable in theHSV. Fault detection and isolation (FDI) have become in-creasingly important for HSVs, due to the safety concerns.To improve reliability of HSV, it is critical to detect andidentify possible failures and faults in the system as earlyas possible to prevent significant performance degradationsand disasters. The identified fault information is also criti-cal for reconfigurable control to mitigate detrimental ef-fects of these faults.

The longitudinal motion dynamics of HSV were studi-ed in several papers [2–5] and the modeling and controlproblems of such hypersonic aircrafts were discussed. Themain challenges in hypersonic modeling include the stronginteractions between the aerodynamics and propulsion sys-tems, as well as the coupling between the rigid body accel-erations and flexible body dynamics. The newtonian im-pact theory was first used to determine the pressure dis-tributions over the vehicle in the earliest studies of air-breathing HSV modeling [2]. The pressures were calcu-lated from expressions dependent on vehicle Mach num-ber, free-stream pressure, angle of attack and vehicle ge-ometry. Since this method was more suitable to deter-mine the pressures on blunt shapes, the oblique shock andPradtl-Meyer theory was used in [3] by assuming the air-flow was quasi-steady. The longitudinal nonlinear modelderived in [3] could describe the complex flow propertieswith the flexible coupling effects in a scramjet-powered ve-hicle. It was further modified in several studies to make itmore accurate to capture the flow effect in propulsive sys-tem [5] and the governing flexible dynamics [4]. Never-theless, the quasi-unsteady assumption could be violateddue to the vehicle vibrations. To accurately capture theunsteady components of the flow field, the piston the-ory was also applied in the later aerodynamics modelingeffort [5, 6].

Model-based FDI schemes exploiting analytical redun-dancy have recently received increasing attention. Vari-ous FDI techniques based on observer schemes, parame-ter estimation methods, or statistical approaches have been

Xuejing Cai et al.: Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle 53

developed (for detailed surveys of different FDI ap-proaches, see [7, 8]). Among which, the observer-basedFDI approach is deemed most promising, because of itscapability in exploiting the known relation between sys-tem inputs and outputs and its flexibility in selecting ob-server gains to match performance requirements. To ac-complish the task of fault detection and identification inthe presence of disturbances and other possible faults, theeffect of a specific fault in the residual signals has to be de-coupled from those of other faults and disturbances. Earlyapproaches for disturbance decoupling for robust FDI de-signs include eigenvalue assignment [9] and unknown in-put observer [10].

However, in most perfect decoupling and isolationcases, solvability conditions are generally restrictive andhard to be satisfied. To alleviate such a difficulty, almostdisturbance decoupling has been widely investigated usingH∞ techniques [11, 12]. Optimization-based FDI schemeshave been proposed where a suitable performance index(such as H−/H∞) is optimized to provide robust FDI inthe presence of disturbances. H−/H∞ filter minimizes thesensitivity of the residual signal to disturbance while main-taining a minimum level of sensitivity to faults. However,the objectives of increasing the sensitivity of the residual todisturbances and simultaneously reducing the sensitivity todisturbances and plant/model mismatch are often conflict-ing with each other. In [13], an optimal fault detectionfilter in H−/H∞ setting was proposed to detect faults inthe presence of disturbances, and the solution is charac-terized by a Riccati equation. FDI filter designs based onH−/H∞ criterion were also addressed in [14, 15] and thesolvability conditions were formulated as non-convex op-timization problem and solved through some iterative pro-cedures. Recently, we have developed an alternative FDIapproach for linear parameter-varying (LPV) systems byextending the work in [16]. The study of LPV systems aremotivated by the gain-scheduling control design methodo-logy [17] for nonlinear systems. Through a multiobjec-tive formulation of the FDI problem, a bank of FDI filterswill be constructed such that the effect of disturbance tothe residual is minimized. In addition, each residual sig-nal will track one of the faults while decoupling it fromother faults. This approach not only provides a computa-tional efficient solution to detect and isolate faults, but alsoprovides the estimation of fault signals.

The main contribution of this paper is on the applicationof the multiobjective FDI scheme to a flexible air-breathingHSV model developed by Bolender and Doman and theirco-workers [3 – 5]. In this model, the oblique-shock the-ory and Prandtl-Meyer expansion are used to determine thepressure distributions on any surface of the vehicle and theflow properties inside the scramjet. The flexible body ef-

fects are derived by applying the assume mode method ona free-free beam structure. This nonlinear model could belinearized into an LPV model at multiple trim points whichrepresents the HSV dynamics over a large flight envelop.Then we apply the multiobjective FDI algorithm developedin [18] to detect and isolate possible actuator and sensorfaults of HSV. The performance of synthesized FDI ob-servers is examined under different operating conditions tojustify their usefulness.

In this paper, given P ⊂ Rs as a compact set and Vas a convex polytope in Rs that contains the origin, i.e.V = {v : νi � v � νi, i ∈ I[1, s]}, we can define a pa-rameter ν-variation set as

FνP :=

{ρ∈C1(R+, Rs) : ρ(t)∈P , ρi(t)∈V , ∀ t∈R+

}For x ∈ Rn, its Euclidean norm is ‖x‖ := (xTx)1/2, L2

is the space of square-integrable signals with norm ‖ · ‖2.The L2 gain of an LPV system G is defined as

‖G‖i2 = supu∈L2,‖u‖2 �=0

supρ(·)∈Fν

P

‖y‖2

‖u‖2

If G is independent of parameter ρ, it becomes a lineartime-invariant (LTI) system and its L2 gain degenerates tothe familiar H∞ norm, i.e.

‖G‖∞ = supω

σ[G(jω)]

2. Multiobjective FDI scheme: a review

Now we consider a faulty LPV system G0 with faults anddisturbance inputs as

x = A(ρ)x + Bd(ρ)d + Bu(ρ)u + Bf1(ρ)f1 + Bf2(ρ)f2

(1)y = C(ρ)x + Dd(ρ)d + Df1(ρ)f1 (2)

where ρ(·) ∈ FνP , x ∈ Rn and y ∈ Rny are states and out-

puts of the system, d ∈ Rnd is external disturbances, f1 ∈Rnf1 and f2 ∈ Rnf2 are two different groups of possi-ble faults, A(ρ), C(ρ), Bd(ρ), Bf1(ρ), Bf2 (ρ), Dd(ρ) andDf1(ρ) are parameter-dependent matrices of appropriatedimensions and not necessarily affine in ρ. We assume thatthe system has at least as many outputs as potential faults,i.e. ny � nf1 + nf2 . As a typical requirement for FDIproblem, it is also assumed that the pair (A(ρ), C(ρ)) isquadratically detectable. Therefore, there exist a positivedefinite matrix P > 0 and a matrix function L(ρ, ρ) ofappropriate dimensions such that

[A(ρ) + L(ρ, ρ)C(ρ)]TP + P [A(ρ) + L(ρ, ρ)C(ρ)] < 0

for all ρ(·) ∈ FνP . Moreover, we assume that Df1 has full

column rank, which means every fault in f1 group could af-fect both system dynamics and measurements as well. So

54 Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011

f1 corresponds to “sensor faults”. For f2 group, Df2 = 0in G0 and the faults will not affect the measurement di-rectly. This type of faults will be used to represent “actua-tor and system component faults”. Many previous researchon FDI mainly considered actuator faults but convertingsensor faults to their equivalent actuator faults. Since it ismore difficult to deal with the case of actuator faults com-pared with sensor faults, we will transform G0 to a generalfaulty system by recasting actuator and system componentfaults to sensor faults.

Lemma 1 For Gdy : Lnd2 → Lny

2 given by (1) and(2), there always exist an LTI filter T0 and an LPV systemGf2y such that Gf2y = Gf2yT0, where x1 ∈ Rnf2 and

T0 :{

x1 =AT x1 + BT f2

f2 =x1

Gf2y :

{x2 =A(ρ)x2 + Bf2(ρ, ρ)f2

y =C(ρ)x2 + Df2(ρ)f2

withBf2(ρ, ρ) = A(ρ)Bf2(ρ)B−1

T −Bf2(ρ)B−1

T AT − Bf2(ρ)B−1T

Df2(ρ) = C(ρ)Bf2(ρ)B−1T

Applying Lemma 1, the original faulty LPV system canbe transformed to its equivalent faulty system G that is af-fected by generalized sensor fault f = [fT

1 fT2 ]T only

x = A(ρ)x + Bd(ρ)d + Bu(ρ)u + Bf (ρ, ρ)f (3)

y = C(ρ)x + Dd(ρ)d + Df (ρ)f (4)

where f ∈ Rnf , nf = nf1 + nf2 and

Bf (ρ, ρ) = [Bf1(ρ) Bf2(ρ, ρ)]

Df (ρ) = [Df1(ρ) Df2(ρ)]In the transformed system G, note that theDf matrix has

full column rank andBf(ρ, ρ)is affinely dependent on ρ.Based on the transformed LPV system (3) and (4), we

will introduce a bank of FDI observers to detect and isolatemultiple actuator/sensor faults. Specifically, the ith LPVobserver Fi (i ∈ I[1, nf ]) is defined as

˙xi = A(ρ)xi + Bu(ρ)u − Li(ρ, ρ)[y − C(ρ)xi] (5)

ri = Hi(ρ)[y − C(ρ)xi] (6)

where x ∈ Rn is the observer states and ri ∈ R is theresidual signal, Li : Rs × Rs → Rn×ny and Hi : Rs →R1×ny are unknown parameter-dependent observer gainsto be designed. To achieve robust FDI, it is required thatresidual ri is only sensitive to the ith fault signal fi in f .Moreover, the effect of other faults and disturbance on ri

should be minimal to prevent false alarm.Let ei = xi − x be the state estimation error and

ri = ri − fi as the fault signal tracking error, then theerror dynamics of FDI observer Fi will be

ei = [A(ρ) + Li(ρ, ρ)C(ρ)]ei + [Bd(ρ)+

Li(ρ, ρ)Dd(ρ)]d + [Bf (ρ) + Li(ρ, ρ)Df (ρ)]fri = Hi(ρ)C(ρ)ei + Hi(ρ)Dd(ρ)d +

[Hi(ρ)Df (ρ) − Ei]fwhere Ei is the ith row of a unity matrix of dimension nf .

Different from typical H∞/H− formulation used inpast FDI research, we will enforce the residual signal ri

to track the fault fi instead of amplifying fault effect in ri.The proposed FDI approach would transform the difficultH∞/H− min-max problem into a tractable optimizationproblem which is amenable to convex solutions. To thisend, the LPV FDI observer design problem will be solvedas a multiobjective optimization

minLi,Hi

w1γi + w2εi

s. t. ‖Gdri‖i2 < γi

‖Gfri‖i2 < εi

(7)

where w1, w2 > 0 are pre-specified weighting factors totradeoff between fault isolation and disturbance rejectionperformances. If the tracking error ri is small enough, onecould recover fault fi from ri to achieve robust FDI objec-tive.

Using real bounded lemma [19] and the mechanismfrom [20], we have derived the following FDI synthesisconditions in terms of linear matrix inequalities (LMIs) tosolve the multiobjective optimization problem.

Theorem 1 Given compact sets P and V , and a faultyLPV system G in (3) and (4), there is a bank of FDI ob-servers Fi (i ∈ I[1, nf ]) to solve the multiobjective FDIproblem (7) if there exist positive definite matrix func-tions Pi(ρ) > 0, rectangular matrix functions PLi(ρ, v)and Hi(ρ) such that for any (ρ, v) ∈ P × V⎡

⎣Mi(ρ, v) Nid(ρ, v) CT(ρ)HTi (ρ)

∗ −γiI DTd (ρ)HT

i (ρ)∗ ∗ −γiI

⎤⎦ < 0 (8)

⎡⎣Mi(ρ, v) Nif (ρ, v) CT(ρ)HT

i (ρ)∗ −εiI DT

f (ρ)HTi (ρ) − ET

i

∗ ∗ −εiI

⎤⎦ < 0

(9)where

Mi(ρ, v) =s∑

j=1

{vj , vj

} ∂Pi

∂ρj+ AT(ρ)Pi(ρ) +

Pi(ρ)A(ρ) + CT(ρ)PTLi(ρ, v) + PLi(ρ, v)C(ρ)

Nid(ρ, v) = Pi(ρ)Bd(ρ) + PLi(ρ, v)Dd(ρ)Nif (ρ, v) = Pi(ρ)Bf (ρ) + PLi(ρ, v)Df (ρ)

Consequently, the observer gains are Li(ρ, v) = P−1i (ρ)·

PLi(ρ, v) and Hi(ρ).Note that the FDI observer synthesis conditions (8) and

(9) are infinite-dimensional LMIs. In order to solve these

Xuejing Cai et al.: Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle 55

conditions, it is necessary to parameterize function spacesof Pi and Li, Hi. By choosing a set of scalar basis func-tions {rs(ρ)}nr

s=1 and {gj(ρ, v)}ng

j=1 , {hk(ρ)}nh

k=1, one canspecify Li and Hi as

P (ρ) =nr∑

s=1

rs(ρ)Pis

Li(ρ, v) =ng∑j=1

gj(ρ, v)Lij

Hi(ρ) =nh∑

k=1

hk(ρ)Hik

and the new optimization variables will be {Pis}ns

s=1 and{Lij}ng

j=1 , {Hik}nh

k=1. Moreover, it is numerically impos-sible to solve the above LMIs over infinite numbers ofρ ∈ P . A practical way is to grid the parameter set P intofinite points. When the gridding points are dense enough,we will be able to obtain a suboptimal solution over theentire set P by continuity. On the other hand, parameter v

enters LMIs (8) and (9) in affine form, therefore it is onlynecessary to check the vertices of set V . If the number ofgridding points of P is np, then it will result in a total of2s+1np + 1 number of LMIs in (8) and (9).

As εi, γi → 0, it can be seen that ‖Gdri‖i2 → 0 and

Hi(ρ)Df (ρ) − Ei → 0

Bf (ρ, v) + Li(ρ, v)Df (ρ) → 0 or Hi(ρ)C(ρ) → 0Therefore we will achieve asymptotic fault recovery and

disturbance decoupling as εi, γi approach zero.

3. Flexible air-breathing HSV model

The basic geometry of an HSV is shown in Fig. 1. It isassumed that the air around vehicle behaves as a perfectgas.

Fig. 1 Hypersonic air-breathing vehicle diagram

By assuming the structure of vehicle body as a free-freebeam, it decouples the rigid body dynamics from the elas-tic modes and the equation of motion (EOM) consist of aset of equations for the traditional rigid body motion anda set for the flexible mode vibrations [3]. As a result, thelongitudinal equations of motion of the flexible HSV aregiven by

Vt =T cosα − D

m− g sin(θ − α) (10)

α =L + T sin α

m+

g

Vtsin(θ − α) + Q (11)

Q =Mp

Iyy(12)

h = Vt sin(θ − α) (13)

θ = Q (14)

ηi = −2ζiωni ηi − ω2ni

ηi + Ni, i = 1, . . . , n (15)

where Vt, α, Q, h, θ are true airspeed, angle-of-attack,pitch rate, altitude and pitch attitude of the vehicle. T, D, L

are thrust, drag, and lift forces. Mp is pitching moment andIyy is the moment of inertia. The elasticity of the vehicle isdescribed by the flexible modes in (15), where ηi, ωi and ζi

are the generalized flexible coordinate, natural frequenciesand damping coefficients of the ith elastic mode. The vi-brational mode of fuselage is derived using assumed modesmethod [4], which determines the evolution of natural fre-quencies and their associated mode shapes. This techniquechooses the mode shapes of a simple structure (e.g., a can-tilever beam) as a set of basis functions to generate approx-imated mode shapes for the actual structure and accountsfor the actual mass and stiffness distribution of the truestructure. Typically, as the number of basis functions in-creases, the first few approximated nature frequencies andmode shapes will converge to their true values quickly.

The state variable xn = [ xTr xT

f ]T of HSV longitudi-nal dynamics includes 5 rigid body states xr and 6 flexiblestates xf describing its first three elastic modes

xr = [ Vt α Q h θ ]T

xf = [ η1 η1 η2 η2 η3 η3 ]T

In the latest HSV configuration, a canard is added asa redundant pitch control effector at the forebody of ve-hicle to enlarge the angle and velocity control bandwidth[5]. Thus the control inputs are un = [ δe δc φ Ad ]T,where δe, δc, φ, Ad are elevator angle, canard angle, throt-tle ratio and diffuser area ratio. These control inputs donot appear directly in the equations of motion (10)–(15),but enter them through the forces and moment T, L, D, Ni

and Mp. Since the modeling of HSV is not the main focusof this research, the derivation of aerodynamics, propul-sion system and flexible modes governing equations willbe omitted here. Interested readers may refer to [3 – 5] formore details.

Note that it is often difficult to obtain the flexible shapesof the fuselage in real-time operation of the HSV. More-over, angle-of-attack α is also hard to measure accurately.On the other hand, the values of velocity Vt, altitude h, andpitch attitude θ are measurable, and the pitch rate Q can beobtained by taking derivative of θ. Therefore, we will usethese four states as the measurement for FDI observer de-sign purpose, i.e. yn = [ Vt Q h θ ]T.

56 Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011

To derive LPV models of HSV, we choose its veloc-ity and altitude to be the scheduling parameters, i.e. ρ =[ Vt h ]T, and specify a flight envelop with Mach numberM ∈ [ 8.5, 9 ] and altitude h ∈ [ 85 000, 90 000 ]ft. Notethat the Mach number is defined as a ratio of the local flowvelocity Vt to local speed of sound a by

M =Vt

a=

Vt√γλTp

where γ is the ratio of specific heats, λ is the specificgas constant, Tp is the atmosphere temperature as a func-tion of altitude h. For the given Mach number and al-titude envelop, one could determine its equivalent speedrange Vt ∈ [ 8 336, 8 857 ] ft/s and the changing rateof velocity Vt ∈ [−500, 500 ] ft/s2 and that of altitudeh ∈ [−200, 200 ] ft/s. This defines the compact sets Pand V .

For better FDI performance, it is critical to have LPVmodels as close to the nonlinear HSV dynamics as possi-ble. Basically, the more gridding is chosen, the better ap-proximation that the LPV plant will be. Nevertheless, thecomputational cost of solving LPV observer synthesis con-dition will increase exponentially for more gridding points.To overcome the computational difficulty, two griddingschemes will be used in this study. The coarse griddingwith 5 × 5 points for the design of FDI observer and nom-inal controller, and a finer one with 5 × 17 points for theimplementation of the resulting LPV observer/controller.In particular, the 5 × 17 gridding points are derived fromthe coarse gridding set by adding 3 more points evenly be-tween each gridded altitude region.

Based on the nonlinear HSV model (10)–(15), the trimconditions (equilibrium points) can be calculated. At eachtrim point, Vt and h will be fixed at their gridding value.The pitch rate Q and all elastic mode derivatives ηi arealso set to 0. Moreover, all other states and inputs are con-strained within desired states and actuator ranges as listedin Table 1.

Table 1 State and control input bounds in solving trim points

States Control inputsα ∈ [0, π/60] rad δe ∈ [π/30, π/12] rad

Q = 0 rad/s δc ∈ [−π/9, π/9] radθ ∈ [0, π/60] rad φ ∈ [0.15, 0.5]

ηi ∈ [0, 1] Ad ∈ [0.75, 0.95]ηi = 0

Through a nonlinear optimization code, we obtain theset of coarse trim points X 5×5

e and another set of refinedtrim pointsX 5×17

e . As an example, a solved trim conditionat M = 8.875 0 and h = 86 250 ft is provided in Table 2.

Subsequently, the local linear models could be obtainedby linearizing the nonlinear equations of motion at these

trim points through Jacobian linearization with linearizedstate x, input u and output y as

x = xn − xe, u = un − ue, y = yn − ye

Then the LPV system describing healthy dynamic rela-tion from input u to output y will be

x = A(ρ)x + Bu(ρ)u (16)y = Cx (17)

where A(ρ) and B(ρ) matrices are scheduled by parameterρ, C is a constant matrix for any ρ in this LPV model. Bychecking all linearized models point by point, it was foundthat all LPV models are unstable. Moreover, {A(ρ), B(ρ)}pair is indeed controllable and {A(ρ), C(ρ)} pair is ob-servable. In addition, the noise on the measurements ofV and h will be introduced and treated as external distur-bances, that is, nd = 2.

Table 2 Trim condition of the HSV model at M = 8.875 0 and h =86 500 ft

Rigid body states Flexible body states Control inputsVt = 8 711.9 ft/s η1 = 0.629 15 δe = 0.105 5 radα = 0.031 854 rad η1 = 0 δc = −0.178 5 rad

Q = 0 rad/s η2 = −0.139 07 φ = 0.255 5

h = 86 250 ft η2 = 0 Ad = 0.950 0

θ = 0.031 854 rad η3 = −0.040 575η3 = 0

Using the LPV model (16) and (17) with added dis-turbances, one can design a nominal LPV controller overthe coarse gridding set X 5×5

e to track altitude trajectoryover the specified flight envelop [21, 22]. To eliminatethe steady-state tracking error, we have introduced integralcontrol action into the feedback control. The integrationof tracking errors will add one additional state to the HSVLPV model. For nominal HSV control, an output feedbackcontrol law is designed using the measurements of 4 rigidbody states, i.e. Vt, Q, h and θ. Other than minimizingtracking errors, more requirements on control magnitudeshould be taken into consideration to maintain reasonablecontrol efforts. Usually, the control actions are expectedto be as small as possible to satisfy the saturation limits.Moreover, the vibration of the flexible HSV is dominatedby its first flexible mode. So this mode has been selectedas one of the controlled variables to minimize the elasticeffects during the flight [23]. After LPV control synthesis,the controller gains on the refined gridding set X 5×17

e willthen be generated through interpolation.

4. FDI observer design for HSV

In this research, we would like to detect and isolate HSVactuator and sensor effectiveness losses from disturbancesand noises. Actuator/sensor effectiveness loss is a com-mon type of system component fault in the aircraft and canbe quantified by the degradation level of individual faultyactuator/sensor, i.e. 0 � fi � 1 (i ∈ I[1, nf ]).

Xuejing Cai et al.: Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle 57

Suppose the faults happened on jth sensor and kth ac-tuator. For sensor fault, the faulty jth measurement will beyj = (1 − f1)Cjx with Cj as the jth row of matrix C. Bydefining the sensor absolute fault signal as f1 = −f1Cjx,we have the following faulty output equation

y = Cx + Df1f1 (18)

where Df1 = Eyj , and Eyj denotes the jth column of anidentity matrix with dimension ny . Moreover, the corre-sponding Bf1 = 0n×1. On the other hand, the kth inputfault will render the input uk to be (1− f2)uk. By definingthe actuator absolute fault signal as f2 = −f2uk, we haveBf2(ρ) to be the kth column of Bu(ρ). This will lead tofaulty state equation as

x = A(ρ)x + Bu(ρ)u + Bf1f1 + Bf2(ρ)f2 (19)

For multiple sensor or actuator faults in f1 or f2, thefaulty system matrices can be derived similarly.

Combining (19) and (18), the faulty HSV LPV modelcould be written in the forms of (1) and (2). By applyingLemma 1, and choosing AT = −BT = −20Inf2 for T0, itwill be transformed to the generalized faulty LPV plant G

in (3) and (4) with nf = nf1 + nf2 .For each fault signal, we will design an LPV FDI ob-

server in (5) and (6). The functional spaces of Li and Hi

will be parameterized as

Pi(ρ) = Pi0 + Pi1ρ1 + Pi2ρ2

Li(ρ, v) = Li0 + Li1ρ1 + Li2ρ2

Hi(ρ) = Hi0 + Hi1ρ1 + Hi2ρ2

It is found that the FDI observer performance does notchange significantly by letting Li depending on v. Ac-cording to the fault decoupling requirement, the proposedFDI algorithm could handle up to 4 faulty signals whenny = 4. To examine the FDI performances under variousfault cases, we solve FDI synthesis conditions (8) and (9)on the trim points ofX 5×5

e with different faulty componentcombinations as shown in Table 3. Upon solving the FDIsynthesis problem, we will construct observer gains for re-fined gridding points by interpolating those of neighboringcoarse gridding points.

Table 3 FDI performance comparison of different fault cases

Fault cases Q θ δe δc φ

1 (nf = 2) γ 0.014 1 0.208 2ε 0.010 2 0.348 8

2 (nf = 2) γ 0.002 3 0.156 4ε 0.001 9 0.093 0

3 (nf = 2) γ 0.000 1 0.660 2ε 0.000 1 0.220 3

4 (nf = 3) γ 0.054 3 0.084 9 0.044 1ε 0.118 6 0.774 0 0.956 8

5 (nf = 3) γ 0.020 7 0.277 7 0.552 4ε 0.013 1 0.398 7 0.794 8

6 (nf = 3) γ 0.015 7 0.136 2 0.019 9ε 0.979 0 0.161 4 0.981 9

In the comparison of FDI performances from fault cases1 to 6, it can be seen that the disturbance rejection andfault decoupling performances become worse for the caseswith more faulty components. Moreover, the fault decou-pling performance not only depends on nf but also closelyrelates to the characteristics of the HSV dynamics. For ex-ample, in all cases with nf = 3, the fault decoupling anddisturbance rejection performances become worse for dif-ferent sensor and actuator channels due to different faultycomponent combinations.

In the sequel, we will focus on the first fault case in Ta-ble 3 with faulty pitch attitude θ and faulty canard δc. Thefrequency responses Tdri , Tf1ri and Tf2ri of these FDIobservers at two gridding points of ρ are plotted in Fig. 2.

Fig. 2 Frequency responses of FDI observers at two trim conditions

58 Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011

It is observed that each fault fi has dominated its resid-ual ri over other faults and disturbance. Nevertheless, thefault recovery and disturbance rejection for canard fault isexpected to be worse than that for pitch fault. Moreover,Tf1r1 for sensor fault maintains constant value 1 over allfrequencies; while the amplitude of Tf2r2 for actuator faultdrops at high frequencies due to limited bandwidth of T0.This would render residual r2 less sensitive to high fre-quency canard fault.

5. Simulation results

The FDI observer designed for the first fault case has beenapplied to the nonlinear HSV model (10)–(15). A nomi-nal LPV controller (discussed at the end of Section 3) isalso used in the simulation to maintain closed-loop controlof HSV. In order to examine FDI observer performanceunder different tracking commands, we will consider twooperating scenarios of HSV:

(i) multiple step altitude tracking with the step size of100 ft starting from 86 250 ft,

(ii) ramp altitude tracking with a slope of 20 ft/s startingat 86 250 ft.

The first operating scenario is to check the performanceof FDI observers under different trim conditions; while thesecond one is aimed to show FDI observers’ capability inslow-varying operating environment. Since the nominalcontroller is designed based on the healthy HSV dynamicsand could only tolerate small amount of actuator/sensorfaults, a relatively small degradation level of 0.1 (that is,maximum 10% efficiency loss) is introduced when faultshappening. To simulate FDI observers’ performance inrealistic situation, we also assume that all measurementshave been contaminated by different levels of white noiseswith zero mean and standard variation of 5 × 10−4.

Operation 1 Multiple step altitude tracking

In this operation, the nonlinear HSV will track a se-quence of step altitude commands. The altitude commandand its tracking trajectory are shown as dash and solid linesin Fig. 3. The first step starts at t = 20 s and after every20 s another step will be applied. For most of the steps,The altitude could converge to the reference in about 8 s.However, the controller switching at 82.5 s introduces ad-ditional tracking error and causes longer convergent time(about 10 s). The altitude of HSV tracks multiple step com-mands with zero steady-state tracking error. To avoid falsealarm during transient settlement at new operating condi-tions, the FDI algorithm will be shut down temporarily for3 s after each step input.

For simulation purpose, we assume sensor and actua-tor faults with degradation level f1(t), f2(t) in periodictriangle shapes as the upper plots of Fig. 4. To reduce

Fig. 3 Step response: altitude tracking under actuator/sensor faults

Fig. 4 Actuator/sensor faults and estimates in step altitude tracking

Xuejing Cai et al.: Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle 59

high frequency noise effects on the fault estimation, twolow pass filters

W1(s) = W2(s) =5

s + 5are applied to residues. The corresponding absolute faultsignals and filtered residues from FDI observers Fi are alsoplotted in Fig. 4. It is observed that each residual signalrecovers the absolute fault quite well and has very littlecoupling effect from each other.

As a statistical measure of the difference between faultestimation ri and absolute fault fi, we calculate the meanvalue ei and standard deviation σi of the estimation errorusing

ei =1N

N∑j=1

(ri,j − fi,j) (20)

σi =

√√√√ 1N

N∑j=1

(ri,j − fi,j)2 (21)

where N is the number of sampling points, ri,j and fi,j

are the ri and fi values at the jth sampling point, respec-tively. The statistic results are provided in Table 4. As canbe seen that the mean value and the standard deviation arequite small compared with the fault magnitude. Moreover,the sensor fault estimation is slightly better than the actua-tor fault estimation.

Table 4 Mean and standard deviation of fault estimation errors inOperation 1

Statistics F1 : θ F2 : δc

ei 1.808 8 × 10−4 −3.121 5 × 10−4

σi 7.787 1 × 10−4 4.507 5 × 10−3

The control inputs and closed-loop responses are pro-vided in Fig. 5 and Fig. 6. It is observed that control

Fig. 5 Step response: control inputs and rigid body states

60 Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011

Fig. 6 Step response: flexible body states

inputs and rigid body states respond quickly to each stepcommandwith minimal fault effect. The elastic mode exci-tation caused by controller switching is slightly larger thanthat of tracking command effect. There is no obvious inter-action between flexible body states and slow varying faultsignals.

Operation 2 Ramp altitude trackingThe tracking command for this operation has a slope of

20 ft/s and start from 86 250 ft. In Fig. 7, the dash andsolid lines represent altitude command and actual altitudeoutput, respectively. The controlled HSV undergo 5 timescontroller switching during 90 s simulation. The controllerswitchings introduce some initial tracking errors right af-ter the switching instant. Nevertheless, the tracking erroris always less than 90 ft.

Fig. 7 Ramp response: altitude tracking under actuator/sensorfaults

In the simulation, the degradation levels f1(t) and f2(t)are chosen as periodic square signals as shown in the up-per plots of Fig. 8. The noisy residual signals are then

filtered by the low pass filters W1(s), W2(s). Their corre-sponding absolute fault signals and filtered residues fromFDI observers Fi are also plotted in Fig. 8. The residuesfrom FDI observers could recover the absolute fault sig-nals under the slow varying operating conditions. Recallthat observer F2 could be less sensitive to fast changingfault signals due to its limited bandwidth. It is found thatresidue r2 is able to catch quick jump of fault f2 but withsacrificed accuracy in fault estimation.

Fig. 8 Actuator/sensor faults and estimates in ramp altitude track-ing

Applying (20) and (21), we obtain the statistical mea-sure of estimation errors in Table 5. Again the mean valueand the standard deviation are relatively small. Neverthe-less, compared with the slow varying fault signal in the firstoperation, the estimation errors have increased for suddenchanging actuator/sensor faults.

Xuejing Cai et al.: Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle 61

Table 5 Mean and standard deviation of fault estimation errors inOperation 2

Statistics F1 : θ F2 : δc

ei −1.917 1 × 10−4 −1.640 6 × 10−3

σi 6.389 2 × 10−4 4.657 7 × 10−3

From the system responses in Fig. 9 and Fig. 10, itis observed that both sudden fault change and controllerswitching have shown more effect on control inputs andrigid/flexible body states. Moreover, the elastic mode ex-citation is dominated by η1 and its magnitude is within anacceptable range.

Fig. 9 Ramp response: control inputs and rigid body statesFig. 10 Ramp response: flexible body states

62 Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011

6. Conclusion

We present the multiobjective FDI filter design for aflexible air-breathing HSV longitudinal dynamics. TheHSV has complicated dynamics behavior including inter-actions among fuselage, propulsion and elastic modes andpresents great challenge for FDI task. From the non-linear HSV model, the faulty LPV model is derived byassuming the actuator and sensor faults of partial effec-tiveness loss. The FDI observers are then designed us-ing LMI technique by minimizing the fault estimation er-ror and rejecting other faults and disturbance simultane-ously. This approach is effective in achieving the bal-anced requirements of fault identification and disturbancerejection. Furthermore, based on altitude tracking controlof HSV, nonlinear simulation is conducted for pitch atti-tude sensor and canard actuator faults. In the followedresearch, we have compared the proposed FDI algorithmwith another FDI approach for HSV in [23] to demonstrateits advantages. The simulation results indicate that theLPV FDI observers could achieve satisfactory fault esti-mation and isolation from disturbances for the faulty HSVunder multiple trim points and slowly varying operatingconditions.

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Biographies

Xuejing Cai received her B.S. degree in au-tomatic control and M.S. degree in mechani-cal engineering from Tsinghua University in2003 and 2006, respectively. She enrolledin graduate school of North Carolina StateUniversity (NCSU) in 2006 and received herPh.D. degree in mechanical engineering ofNCSU in 2010. Her research interests includegain-scheduling control, fault detection and

isolation, fault tolerant control and switching control.

Fen Wu is a professor with mechanical andaerospace engineering, in North Carolina StateUniversity. He received the B.S. and the M.S.degrees in automatic control from Beihang Uni-versity, Beijing, in 1985 and 1988, respectively,and the Ph.D. degree in mechanical engineer-ing from University of California at Berkeley,CA, in 1995. Subsequently he worked eighteenmonths in the Centre for Process Systems Engi-

neering, Imperial College as a research associate. His research in-terests include robust analysis and control, linear parameter-varyingcontrol of nonlinear systems, fault detection and fault tolerant con-trol, and the application of advanced control and optimization tech-niques to aerospace, mechanical and chemical engineering problems.E-mail:fwu@eos.ncsu.edu