NASA Technical Reports Server (NTRS) 20060022131: Adaptive Failure Compensation for Aircraft Tracking Control Using Engine Differential Based Model

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Adaptive Failure Compensation for Aircraft Tracking Control Using
Engine Differential Based Model
Yu Liu, Xidong Tang and Gang Tao Suresh M. Joshi
Department of Electrical and Computer Engineering Mail Stop 308
University of Virginia NASA Langley Research Center
Charlottesville, VA 22903 Hampton, VA 23681
Abstract—Anaircraftmodelthatincorporatesindependently patterns. An algorithm based on multiple model adaptive re-
adjustableenginethrottlesandaileronsisemployedtodevelop configuration control approach was presented and illustrated
anadaptivecontrolschemeinthepresenceofactuatorfailures.
by simulation results of the F/A-18 aircraft during carrier
Thismodelcapturesthekeyfeaturesofaircraftflightdynamics
landing.In[12],anF-16fighteraircraftsubjecttoasymmet-
when in the engine differential mode. Based on this model an
adaptivefeedbackcontrolschemeforasymptoticstatetracking ricactuatorfailurewasdiscussed,includingsystemmodeling
is developed and applied to a transport aircraft model in the and control system design. The problem was formulated as
presence of two types of failures during operation, rudder a nonlinear disturbance rejection problem in the presence
failure and aileron failure. Simulation results are presented to
of actuator failures and simulation results using an F-16
demonstrate the adaptive failure compensation scheme.
aircraft model were discussed. In [6], fault-tolerant control
Keywords:Actuatorfailures,adaptivecompensation,aircraft system design against stuck actuators was investigated using
flight control, engine differentials, tracking. aniterativelearningobserverthatprovidesinformationofthe
systemstateestimatesandfaultcompensationtransients.The
performance of the controller design was evaluated using an
I. INTRODUCTION
F-8 aircraft model.
Effective compensation of control component failures is In this paper, we present a failure compensation scheme
crucial for aircraft flight safety. Considerable research has based on an adaptive control approach that can utilize the
focused on the design of control systems that can provide remaining (functioning) controls to achieve desired perfor-
safe performance when failures occur. In [5], an emergency mance in the presence of uncertain system failures. To com-
flight control system that can utilize engine thrusts to ma- pensate for aircraft failures such as rudder failure or engine
neuver an aircraft was developed and tested on an MD- malfunction, asymmetric engine thrusts may be inevitably
11 airplane. In [7], a propulsion controlled aircraft design needed [10]. For the design of such control schemes, an
by H-infinity model matching was introduced. In [1], an aircraft model with independently adjustable engine thrusts
indirect adaptive LQ controller was developed for aircraft is necessary. In [8], we derived such an aircraft model and
control, which is able to implicitly reconfigure the control used it to develop an adaptive failure compensation control
lawusingon-lineestimatesofthechangedaircraftdynamics, schemeusingenginedifferentialsforstateregulation.Inthis
so that the failures in the pitch control channel or the paper,weshallusethisaircraftmodeltodevelopanadaptive
horizontal stabilizer can be accommodated. In [2], several failure compensation control scheme for state tracking, and
multivariable adaptive control algorithms for flight control apply the scheme to a transport aircraft model. We shall
reconfiguration were presented with a failure characterized considertwosimulationcasesrepresentingrealisticscenarios
byalockedlefthorizontaltailsurface.Anadaptivecontroller in which the rudder or an aileron are stuck at unknown
wasusedtocompensatethisfailure.In[13],adirectadaptive constant values at unknown time instants.
reconfigurableflightcontrolalgorithmwaspresented.Anon- The paper is organized as follows. In Section 2, we
lineadaptiveneuralnetworkwasappliedtoregulatetheerror describe an engine differential based aircraft flight dynamic
between the plant model and the actual aircraft, and appli- model. In Section 3, we develop an adaptive compensation
cation of this control approach to a tailless advanced fighter scheme that is able to handle uncertain actuator failures and
aircraft was demonstrated. In [9], an algorithm for aircraft guarantee asymptotic state tracking. In Section 4, we apply
failure detection and compensation was presented, which this compensation scheme to a transport aircraft model and
incorporated multiple model adaptive estimation methods. presentsimulationresultstoillustratetheeffectivenessofthe
In this approach failures are detected by a bank of parallel scheme.
Kalman filters and a reconfiguration algorithm is used to
redistribute control commands to the non-failed surfaces.
II. ENGINEDIFFERENTIALBASEDMODEL
In [3], a new parametrization for the modeling of control
effector failures in flight control applications was proposed, As described in [8], a nonlinear aircraft dynamic model
including lock in place, hard over and loss of effectiveness in body-axis coordinate system which incorporates engine
1
differentials can be described by the force equations whereA(2) andB(2) arezeromatrices,A(1),A(4),B(1) and
B(4) are of the same forms as in the literature [4], and the
m(u˙ +qw−rv)=X−mgsinθ+(T +T )cosǫ (II.1)
L R matrices
m(v˙ +ru−pw)=Y +mgcosθsinφ (II.2) 0 0 0 0 0 0 0
m(w˙+pv−qu)=Z+mgcosθcosφ−(TL+TR)sinǫ (II.3) A(3)=TT¯¯u′ TT¯¯w′ 00 00,B(3) =00 TT¯¯δ′′t′′′l −−TT¯¯δ′′t′′′r 
and moment equations  00u 00w 00 00 00 00δtl 00δtr 
Ixp˙+Ixzr˙+(Iz−Iy)qr+Ixzqp=L+l(TL−TR)sinǫ (II.4)    (II.13)
I q˙+(I −I )pr+I (r2−p2)=M (II.5) represent the effect of engine thrust differentials, that is, if
y x z xz
the left and right engine thrusts are equal, these matrices are
Izr˙+Ixzp˙+(Iy−Ix)qp−Ixzqr =N +l(TL−TR)cosǫ zero. See [8] for details of this model.
(II.6) We note that this aircraft model is different from standard
wheremisthemassoftheaircraft.u,v andw arethebody- models used in most of the literature [4] that assume equal
axis components of the velocity of the center of mass. p, q engine thrusts and aileron angles. This engine differential
and r are the body-axis components of the angular velocity basedmodelinwhichthetwoenginethrustsandtheailerons
of the aircraft. X, Y and Z are the body-axis aerodynamic are taken into account separately captures the essential
forces about the center of mass. L, M and N are the body- dynamics of the aircraft in the engine differential mode,
axis aerodynamic torques about the center of mass. θ and φ and is capable of coping with some actuator failures such as
aretheEulerpitchandrollanglesoftheaircraft.ǫrepresents rudder failures or engine failure, which cannot be achieved
theanglebetweenthrustandbodyx-axis.Iiarethemoments without using engine differentials. Therefore it is desirable
(or products) of inertia in body axes. g is the gravitational to develop an adaptive control scheme for aircraft actuator
force per unit mass. TL and TR are the left and right engine failure compensation using engine differentials.
thrusts,andl isthedistancebetweenenginesandx–z plane.
By applying the linearization procedure around the equi- III. ADAPTIVEFAILURECOMPENSATION
libriumpointofinterest,wecanobtainthelinearizedaircraft
In this section, we shall first formulate an actuator failure
model with engine differentials. For this purpose, the state
compensation problem for linear systems, and then develop
and control vectors of the linearized model are
an adaptive failure compensation scheme for closed-loop
x = [u w q θ v r p φ ψ]T (II.7) stabilityandasymptotictrackingofthesystemstatevariables
U = [δ δ δ δ δ δ ]T (II.8) in the presence of certain actuator failures.
e tl tr al ar r
where the notation “δ” has been dropped from δx and δU A. Problem Formulation
for simplicity of presentation. Thus (u, v, w) represent the
Consider the linear time-invariant system
velocity perturbations along each axis and (p, q, r) are the
angular velocity perturbations about each axis. (θ, φ, ψ) are x˙ =Ax+Bu, x∈Rn, u∈Rm, (III.1)
the pitch, roll and yaw angle perturbations, and δ , δ , δ ,
e al ar
δr are the deflection perturbations of the elevator, the left whose actuators u = [u1,u2,...,um]T may fail during
and right ailerons and the rudder.δtl and δtr are the left and system operation. A typical failure model is
right throttle perturbations.
In our study, we consider a steady-state rectilinear wings- ui(t)=u¯i, t≥ti, i∈{1,2,...,m}, (III.2)
level flight condition as the equilibrium point. For this
where t is the unknown failure time instant and u¯ is the
steady-state flight condition, the derivatives of all states, the i i
unknown failure constant [11]. An example of such actuator
angular velocity components (p, q, r) and the roll angle φ at
failuresiswhenanaircraftcontrolsurface(suchastherudder
the equilibrium point are all zero, that is,
or an aileron) is stuck at some unknown fixed position at an
[u˙ w˙ q˙ θ˙ v˙ r˙ p˙ φ˙ ψ˙] =0 (II.9) unknown time instant.
xo,Uo
p =q =r =φ =ψ =v =0, (II.10) The control objective is to design an adaptive state feed-
o o o o o o
back control signal to be applied to the actuators in u,
where xo and Uo are determined as to ensure closed-loop signal boundedness, and asymptotic
x =[u w 0 θ 0 0 0 0 0],T tracking:limt→∞(x(t)−xd(t))=0,wherexd(t)isadesired
o o o o
statetrajectory,inthepresenceofunknownactuatorfailures.
U =[δ δ δ δ δ δ ].T (II.11)
o eo tlo tro alo aro ro
B. Adaptive Compensator Designs
By applying the linearization around this equilibrium point,
we can obtain the linearized aircraft model as In the presence of actuator failures,u(t) can be expressed
A(1) A(2) B(1) B(2) as
x˙ = 4×4 4×5 x+ 4×3 4×3 U, (II.12)
(cid:20)A(3) A(4) (cid:21) (cid:20)B(3) B(4) (cid:21) u(t)=v(t)+σ(u¯−v(t)), (III.3)
5×4 5×5 5×3 5×3
2
where v(t) ∈ Rm is the applied control input vector, u¯ = Rm×1, are the parameters updated from the adaptive laws
[u¯ ,u¯ ,...,u¯ ]T is the failure vector, and σ represents the
fai1lure2patternmand is defined as Kˆ˙ = −Γ xeTPb , i=1,2,...,m (III.10)
i i i
κˆ˙ = −γ r eTPb , i=1,2,...,m (III.11)
σ =diag{σ ,σ ,...,σ } (III.4) i i d i
1 2 m θˆ˙ = −λ eTPb , i=1,2,...,m, (III.12)
i i i
with σ = 1 if the ith actuator has failed, that is, u = u¯ ,
i i i
and σi = 0 otherwise. The failures are assumed to occur where Γi = ΓTi > 0, γi = γiT > 0, λi > 0, bi is the ith
instantaneously, i.e., σi are piecewise constant functions of column of B, i = 1,2,...,m, and P = PT > 0 satisfying
time. There are 2m possible combinations of actuator states (III.7). Γ , γ , and λ denote the design parameters for the
i i i
(eachactuatoriseithernormalorfailed),andtherefore2m−1 adaptive laws. This adaptive actuator failure compensation
possible failure patterns that constitute a set denoted by Σ¯. scheme has the following desired properties:
The system (III.1) can then be rewritten as Theorem 3.1: The control law (III.9), updated from
(III.10)–(III.12) and applied to the system (III.1) subject
x˙(t)=Ax(t)+B(I−σ)v(t)+Bσu¯. (III.5)
to the actuator failures (III.2) under Assumption 3.1, en-
For our adaptive control design for actuator failure compen- sures that all closed-loop system signals are bounded and
sation, the following assumption is needed: limt→∞(x(t)−xd(t)) = 0, for any failure pattern σ ∈ Σ
Assumption 3.1: (A,B) is known and stabilizable, and with uncertain parameters.
thereexistsanon-emptysetΣof“recoverable”failuressuch Proof: For Q¯ =Q+PBR−1BTP, using (III.7), we obtain
that rank[B(I−σ)]=rank[B] ∀σ ∈Σ. Σ is a subset of Σ¯.
P(A+BK)+(A+BK)TP =−Q¯ <0. (III.13)
Remark 3.1: This assumption characterizes the built-in
redundancyneededforfailurecompensationaswellasallthe
Suppose that at time t there are p < m actuator failures,
failure patterns that can be accommodated. This condition is
that is, u (t) = u¯ , i = i ,i ,...,i , {i ,i ,..., i } ⊂
needed for the existence of a (fixed) failure compensation i i 1 2 p 1 2 p
{1,2,...,m}, and that actuator failures happen at time
controller that can achieve the desired performance when
instants t , with t <t , k =1,2,...,N.
the system and failure parameters are known. The adaptive k k k+1
From the condition of Assumption 3.1: rank[B(I−σ)]=
control design for unknown failure parameters is developed
rank[B], ∀σ ∈Σ, it follows that for each σ ∈Σ, there exist
based on the same condition. For instance, when the aircraft
rudder fails during flight, this condition can still be satisfied constant matrices Kσ ∈Rm×n and κσ ∈Rm×mr such that
so that the aircraft can be controlled by the remaining
B(I−σ)K =BK, B(I−σ)κ =Bκ. (III.14)
σ σ
actuators and the failure can be accommodated (which is
demonstrated in the simulation in Section 4.2). 2 Therefore, for each σ, there are constant K satisfying
σ
Forasymptotictracking,wefirstpresentadesirednominal
design for the system (III.1) without any actuator failures. P[A+B(I−σ)K ]+[AT+(I−σ)K TBT]P =−Q¯ <0,
σ σ
The nonadaptive nominal controller is (III.15)
and κ satisfying B(I −σ)κ r (t) = Bκr (t). In addition
σ σ d d
u(t)=Kx(t)+κrd(t), (III.6) there exists constant θ =[θ ,θ ,...,θ ]T ∈Rm, where θ ,
1 2 m i
where K = −R−1BTP ∈ Rm×n is an optimal LQ gain i6=i1,i2,...,ip, are solutions of the following equation
with P satisfying the Riccati equation
b θ =− b u¯ , (III.16)
i i j j
ATP +PA−PBR−1BTP +Q=0 (III.7) i6=i1X,i2,...,ip j=i1X,i2,...,ip
forsomechosenn×nmatrixQ=QT >0andm×mmatrix and θi = 0, for i = i1,i2,...,ip, such that B(I −σ)θ =
R = RT > 0, and the reference input rd(t) ∈ Rmr and Bσu¯.
κ∈Rm×mr are chosen for some desired system trajectory. From (III.5), (III.8), and III.9, we have
With this nominal controller, the closed-loop system is
x˙(t)=(A+BK)x(t)+Bκrd(t),basedonwhich,wedefine e˙(t) = Ax(t)+B(I−σ)(Kˆx(t)+κˆrd(t)+θˆ)+Bσu¯
the desired state trajectory x (t) from the reference system −(A+BK)x (t)−Bκr (t)
d d d
=Ax(t)+B(I−σ)(K x(t)+κ r (t)+θ)+Bσu¯
x˙ (t)=(A+BK)x (t)+Bκr (t). (III.8) σ σ d
d d d
−(A+BK)x (t)−Bκr (t)+B(I−σ)[(Kˆ−K )x(t)
d d σ
Define the tracking error e(t) = x(t)−xd(t). As the new +(κˆ−κ )r (t)+(θˆ−θ)] (III.17)
adaptive failure compensation scheme for asymptotic state σ d
tracking, the feedback control law is
With equations (III.14) and (III.16), the dynamic equation
v(t)=Kˆx(t)+κˆr (t)+θˆ, (III.9) for tracking error can be simplified as
d
where Kˆ = [Kˆ1,Kˆ2,...,Kˆm]T ∈ Rm×n, κˆ = e˙(t) = (A+BK)e(t)+B(I−σ)[(Kˆ −Kσ)x(t)
[κˆ1,κˆ2,...,κˆm]T ∈ Rm×mr, and θˆ = [θˆ1,θˆ2,...,θˆm]T ∈ +(κˆ−κσ)rd(t)+(θˆ−θ)]. (III.18)
3
Withtheadaptive laws(III.10)–(III.12),aLyapunov func- we can conclude that e ∈ L2 ∩L∞, Kˆ ∈ L∞, κˆ ∈ L∞,
tion candidate can be chosen as and θˆ ∈ L∞. For the nominal design (III.6), A+BK is
asymptotically stable such that x (t)∈L∞ with a bounded
V= eTPe+ (Kˆ −K )TΓ−1(Kˆ −K ) d
i i i i i reference input r (t). Hence we conclude that x(t) ∈ L∞,
i6=i1X,i2,...,ip so does v(t). Furdthermore, since x˙(t) ∈ L∞ and x˙ ∈ L∞,
d
+ (κˆi−κi)Tγi−1(κˆi−κi)+ λ−i 1(θˆi−θi)2 given that e(t)∈L2, we also have limt→∞e(t)=0. 2
i6=i1X,i2,...,ip i6=i1X,i2,...,ip The physical meaning of state tracking for the aircraft
foreachtimeinterval(tk,tk+1), k =0,1,...,N,witht0 = dynamicmodellinearizedatanequilibriumpoint(xo,Uo)is
0 and t =∞, where K is the ith row of K and κ is that the operation of the aircraft follows a desired trajectory
N+1 i σ i
theithrowofκσ.Thetime-derivativeofV ineach(tk,tk+1) in a neighborhood of the equilibrium point. When we apply
is
this adaptive failure compensation scheme to aircraft flight
V˙ = eT[P(A+BK)+(AT +KTBT)P]e control, we want the aircraft to maintain the desired trajec-
+2eTPB(I−σ)[(Kˆ−Kσ)x(t)+(κˆ−κσ)rd(t)+(θˆ−θ)] torythatwasoriginallysetforthenominalcaseofnofailure,
evenifunknownactuatorfailuresoccur.Theorem3.1givesa
+2 (Kˆi−Ki)TΓ−i 1Kˆ˙i+2 (κˆi−κi)Tγi−1κˆ˙i solution to the problem of state tracking. The stabilizability
i6=i1X,i2,...,ip i6=i1X,i2,...,ip and rank condition in Assumption 3.1 characterizes the
+2 λ−i 1(θˆi−θi)θˆ˙i system redundancy condition needed for actuator failure
i6=i1X,i2,...,ip compensation. As shown in next section, it is satisfied for
= eT[P(A+BK)+(AT +KTBT)P]e the rudder or aileron failure case and the system state is
+2eTPB(I−σ)[(Kˆ−Kσ)x(t)+(κˆ−κσ)rd(t)+(θˆ−θ)] able to track the desired trajectory asymptotically, which
impliesthattheaircraftcanmaintainthedesiredperformance
−2 (Kˆi−Ki)TxeTPbi−2 (κˆi−κi)TrdeTPbi under normal as well as failure conditions. Assumption
i6=i1X,i2,...,ip i6=i1X,i2,...,ip 3.1 also requires the knowledge of A and B. However, it
−2 (θˆi−θi)eTPbi, may be noted that, if the actual system matrix given by
i6=i1X,i2,...,ip Ap =A+δA (where δA is the parameter error) is such that
Fortheconsideredactuatorfailurepattern,thatis,ui(t)= (Ap+BK)TP+P(Ap+BK)<0,theasymptotictracking
u¯ , σ = 1, i = i ,i ,...,i , using the fact that B(I − andsignalboundedness ofTheorem3.1willstillhold.(This
i i 1 2 p
σ)(Kˆ−K )= b (Kˆ −K )T andthecommu- condition basically implies that the nominal LQ regulator
σ i6=i1,i2,...,ip i i i used for generating the desired trajectory is designed to be
tativitypropertyPofthematrixtraceoperator,i.e.,Tr(XY)=
robusttoparameteruncertainties).Furtherresearchisneeded
Tr(YX), the following equalities hold:
in order to investigate robustness of the adaptive scheme to
eTPB(I−σ)(Kˆ−Kσ)x(t) = (Kˆi−Ki)TxeTPbi, model errors, and to relax the requirement of knowledge of
i6=i1X,i2,...,ip A and B.
eTPB(I−σ)(κˆ−κ )r (t) = (κˆ −κ )Tr eTPb ,
σ d i i d i IV. APPLICATIONTOFLIGHTCONTROL
i6=i1X,i2,...,ip Inthissection,wedemonstrateapplicationoftheadaptive
eTPB(I−σ)(θˆ−θ) = (θˆi−θi)eTPbi. failure compensation technique to a transport aircraft by
i6=i1X,i2,...,ip presenting some simulation results for trajectory tracking in
So the time-derivative of V in each (t ,t ) is simplified the presence of unknown rudder and aileron failures. We
k k+1
as shallfirstdescribetheaircraftmodelusedinsimulation,and
then present the simulation results.
V˙ =−eTQ¯e≤0, (III.19)
A. Aircraft Model for Simulation Study
where(III.13)isusedforthelastequality.Itfollowsthate∈
L2∩L∞, and Kˆ ∈L∞ and θˆ ∈L∞ for i6=i ,i ,...,i , Foroursimulationstudy,weuseatransportaircraftmodel.
i i 1 2 p
The airplane flies at a velocity of 774 ft/sec and an altitude
where i ,i ,...,i are the indexes of failed actuators.
1 2 p
From (III.10)–(III.12), we have of 40 kft. The linearized dynamic model is
hΓ−11Kˆ˙ 1,Γ−21Kˆ˙ 2,...,Γ−m1Kˆ˙ mi=−xeTPB, x˙(t)=(cid:20)AA(4(13×))4 AA(4(24×))5(cid:21)x(t)+(cid:20)BB4((13×))3 BB4((24×))3(cid:21)U(t)(IV.1)
5×4 5×5 5×3 5×3
γ−1κˆ˙ ,γ−1κˆ˙ ,...,γ−1κˆ˙ =−r eTPB,
1 1 2 2 m m d whereA(2) andB(2) arezeromatrices,andA(1),A(4),B(1)
h i
λ−1θˆ˙ ,λ−1θˆ˙ ,...,λ−1θˆ˙ =−eTPB, (III.20) and B(4) are of the same forms as in [8], and
1 1 2 2 m m
h i x=[u w q θ v r p φ ψ]T,
which implies that Kˆi ∈ L∞, κˆi ∈ L∞ and θˆi ∈ L∞ for U=[δe δtl δtr δal δar δr]T,
i=i1,i2,...,ip, because B can be represented by a linear 0 0 0 0 0 0 0
combination of b , i6=i ,i ,...,i . 0.001 0.001 0 0 0 0.8 −0.7
The function Vi is not1co2ntinuoups at t , k =0,1,...,N, A(3) =−0.001 −0.001 0 0,B(3)= 0 −0.5 0.6 
k 0 0 0 0 0 0 0
and only has finite value jumps at those time instants. So    
 0 0 0 0  0 0 0 
4
The non-zero terms in A(3) and B(3) represent the engine 1.15
0.5
thrust differential effect. The basic units used in this model 0
−0.5
are ft, sec and crad (0.01 radian). 0 20 40 60 80 100 120 140 160 180 200
y−axis velocity x=v (solid) and desired x (dashed) (ft/sec) vs. time (sec)
Weconsidertwotypesofconstantactuatorfailures:rudder 5 d5
0.5
failure and aileron failure. The rudder failure is denoted as 0
−0.5
−1
U (t)=U (t ) t≥t , (IV.2) 0 20 40 60 80 100 120 140 160 180 200
6 6 f f roll angle x=φ (solid) and desired x (dashed) (deg) vs. time (sec)
8 d8
where t is the failure time instant. It represents the rudder 0
f
stuck in its position at instant t , and cannot be moved. The −0.5
f
aileron failure we consider is −1
0 20 40 60 80 100 120 140 160 180 200
yaw angle x=ψ (solid) and desired x (dashed) (deg) vs. time (sec)
U (t)=0 t≥t , (IV.3) 9 d9
5 f
which indicates that at failure time instant t , the right Fig.1. Systemstatesx5=v,x8=φ,x9=ψ (CaseI).
f
aileron angle drops to zero and is stuck from then on. These
1
failure patterns satisfy Assumption 3.1. 0.5
0
−0.5
B. Simulation Results 0 20 40 60 80 100 120 140 160 180 200
left engine throttle δ (solid) and right engine throttle δ (dashed) (ft/sec2) vs. time (sec)
tl tr
In this subsection, we present the simulation results for 1
the asymptotic tracking of x (t) by x(t) to demonstrate 0
d −1
the performance of the system with the adaptive failure −2
0 20 40 60 80 100 120 140 160 180 200
compensation scheme described in Section 3.2. For our left aileron δ (solid) and right aileron δ (dashed) (deg) vs. time (sec)
al ar
simulation, the values of κ and r were chosen as r = 1,
d d 0.4
and 0.2
0
−0.2
κ=[0.7344 0.6842 1.5974 −0.5817 −0.7853 1]T −0.40 20 40 60 80 100 120 140 160 180 200
rudder angle δ (deg) vs. time (sec)
r
that is, r = r0 ∈ R is a scalar, which leads to the final
d d
values of the desired states: Fig.2. Controlsignals:δtl,δtr,δal,δar,andδr (CaseI).
x∞ =[4 −1.06 0 1 0 0 0 0 −1.8]T
d
immediately after its occurrence by the adaptive controller.
This trajectory represents a steady state flight condition (The other states can also converge to the desired trajectory
in which the aircraft is climbing with a 4 ft/sec velocity after rudder failure, but are not shown in the figures due to
perturbation along the x-axis, a -1.06 ft/sec velocity per- the limitation of space). The initial transients (before the
turbation along the z-axis, a pitch angle of 1 crads (0.57 failure occurs) are because of the adaptive system response
degrees), and a yaw angle of -1.8 crads (-1.0 degree). The whenthesystemisfirstturnedonwithsomearbitraryinitial
initial value of the state vector is zero, i.e., the airplane values of the adaptive control gains.
is in steady wings-level flight. The physical meaning of
Case (II). System performances with adaptive compensa-
state tracking is that the aircraft flies from one steady state
tion scheme and aileron failure (IV.3). The failure instant is
flight condition toanother steadystateflight conditionwhile
t = 20 seconds and the parameter setting is the same as
closely following a reference state-trajectory. The choice f
Case (I). The results are shown in Figures 3 and 4.
of the reference trajectory (in particular, κ and r ) in this
d
paper was arbitrary, the main purpose being demonstration
1.5
oftheadaptivescheme.Forthecontrollerdesign,wechoose 1
0.5
Q=I and R=diag{2,6,6,2,2,2}. 0
9 −0.5
0 50 100 150 200 250 300
For our simulation study, we examine two cases: (I) y−axis velocity x=v (solid) and desired x (dashed) (ft/sec) vs. time (sec)
5 d5
system responses with adaptive failure compensation with 0.5
0
failure (IV.2) and (II) system responses with adaptive failure
−0.5
compensation with aileron failure (IV.3). −10 50 100 150 200 250 300
roll angle x=φ (solid) and desired x (dashed) (deg) vs. time (sec)
8 d8
Case (I). System performances with adaptive 0
compensation scheme and rudder failure (IV.2). The failure
−0.5
instant is t = 5 seconds. Γ (i = 1,...,6) are chosen
f i −1
as [0.01 0.01 0.01 0.06 0.01 0.01 0.01 0.04 0.08]. 0 20 40 60 80 100 120 140 160 180 200
yaw angle x=ψ (solid) and desired x (dashed) (deg) vs. time (sec)
γ and λ (i = 1,...,6) are chosen as 9 d9
i i
[0.01 0.05 0.05 0.02 0.02 0.01]. These design Fig.3. Systemstatesx5=v,x8=φ,x9=ψ (CaseII).
parameters were chosen by trial and error. Some selected
states and control signals are shown in Figures 1 and 2, In summary, in this study we simulated some typical
which demonstrate how the rudder failure is accommodated aircraft motions for realistic rudder and aileron failure con-
5
0.15 ACKNOWLEDGMENTS
0
−0.5 This research was partially supported by NASA Langley
0 20 40 60 80 100 120 140 160 180 200
left engine throttle δ (solid) and right engine throttle δ (dashed) (ft/sec2) vs. time (sec) Research Center under grant NCC-1-02006. The authors
tl tr
1 wouldliketothankthereviewersfortheirhelpfulcomments.
0
−1 REFERENCES
−2
0 20 40 60 80 100 120 140 160 180 200
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which guarantees closed-loop signal boundedness as well
as asymptotic state tracking in the presence of unknown
actuator failures occurring at unknown time instants. Sim-
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performance in the presence of rudder or aileron failures,
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take over for the failed ones.
Several important and challenging issues need to be ad-
dressed in future research. First, robustness of the adaptive
schemetomodelerrors,andrelaxationoftherequirementof
knowledge of the system matrices, need to be investigated.
In addition, the effects of actuator nonlinearities including
outputandratesaturation,aswellasactuatordynamics,need
to be addressed. Also, extensions of the adaptive scheme
to model reference state and output tracking, as well as to
nonlinear aircraft models, should be addressed.
6