-
ADAPTIVE OUTPUT-FEEDBACK GAIN SCHEDULINGAPPLIED TO FLEXIBLE
AIRCRAFT
Rafael M. Bertolin∗ , Antônio B. Guimarães Neto∗ , Guilherme C.
Barbosa∗ , Flávio J. Silvestre∗∗Instituto Tecnológico de
Aeronáutica, São José dos Campos, SP, 12228-900, Brazil
Keywords: Adaptive control, flexible aircraft, MRAC, X-HALE
Abstract
The advantages of full-state feedback adaptivecontrol in dealing
with uncertainties of a flexibleaircraft model are demonstrated
with the use ofmodel reference adaptive control. Design of
anoutput-feedback system with an observer, basedon the separation
principle, is attempted. Differ-ences in both stability and
performance charac-teristics of the full-state feedback and the
output-feedback closed-loop systems demonstrate thatfurther
investigation is needed to design adaptiveoutput-feedback
controllers.
1 Introduction
Flexible aircraft (FA) are caracterized by low orvery low
frequencies of their aeroelastic modesand, as a consequence,
strong, dangerous and un-desirable coupling between the structural
dynam-ics and the rigid-body flight dynamics may oc-cur. For
example, the short-period mode of a veryFA can become unstable as
the dihedral angle in-creases [1].
A special class of FA that has motivated thescientific community
in recent decades is knownas High-Altitude Long-Endurance (HALE)
air-craft. The mission profile of a HALE aircraftinvolves cruising
at very high altitudes (above20 km) and flying for weeks, months
and evenyears [2]. It turns out that, due to the
missionrequirements, these aircraft may undergo actua-tor anomalies
such as power surge in motors orstructural damage in control
surfaces.
The described adversities give rise to a series
of challenges in the flight control law design pro-cess [3] and
may sometimes exceed the stabilitymargins of the system. In such
cases, traditionallinear control techniques are no longer
adequate.On the other hand, adaptive control is an appro-priate
solution because it should be able to over-come all these
adversities [4].
In this paper, the advantages of adaptive con-trol in dealing
with uncertainties of a control sys-tem applied to flexible
aircraft will be demon-strated for an experimental HALE aircraft,
the X-HALE [5].
At first and assuming that all the system statesare measurable,
a linear baseline control systemfor velocity, altitude, sideslip
and roll angle track-ing will be designed and afterwards
augmentedby a model reference adaptive control (MRAC)law [6]. A
comparison between the two systems(with and without the MRAC
augmentation) willbe made by which the usefulness of the
adaptivecontroller will become apparent.
At a second moment and to address the statefeedback problem, an
attempt to design an ob-server based on the separation principle
will beperformed.
2 Problem Statement
The FA flight dynamics under small disturbancesaround an
equilibrium flight condition can be rep-resented by a class of
multi-input multi-output(MIMO) linear time-invariant (LTI)
uncertain
1
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BERTOLIN, R. M. , GUIMARÃES NETO. A. B. , BARBOSA, G. C. ,
SILVESTRE, F. J.
systems in the following form:
ẋp = Apxp +BpΛ[u+ΘT Φ(xp)
](1)
yp = Cpxp +DpΛ[u+ΘT Φ(xp)
](2)
where xp ∈ Rnx corresponds to the state vec-tor, u ∈ Rnu are the
control inputs, yp ∈ Rny isthe system output vector, composed of
measure-ment outputs (ym ∈ Rnm) and tracking outputs(yt ∈ Rnt )
that are also measured, according to:[
ymyt
]︸ ︷︷ ︸
yp
=
[CmCt
]︸ ︷︷ ︸
Cp
xp+[
DmDt
]︸ ︷︷ ︸
Dp
Λ[u+ΘT Φ(xp)
](3)
The matrices Ap ∈ Rnx×nx , Bp ∈ Rnx×nu , Cm ∈Rnm×nx , Ct ∈
Rnt×nx , Dm ∈ Rnm×nu and Dt ∈Rnt×nu are assumed to be constant and
known.This assumption corresponds to an ideal case. Inreality,
several types of uncertainties exist in thedynamic model and such
matrices are unknownand even time-variant.
Aiming at inserting more realism into theideal system, two kinds
of parametric uncertain-ties are considered in Eqs. (1) and (2): an
un-known constant multiplicative diagonal matrixΛ ∈ Rnu×nu with
strictly positive diagonal ele-ments, to represent control actuator
uncertainties,control effectiveness reduction and other
controlfailures, damage or anomalies; and an additiveterm f(xp) =
ΘT Φ(xp) that represents uncertain-ties present in Ap through the
input channels,where f(·) : Rnx →Rnu , Θ ∈Rnx×nu is a matrix
ofunknown constant parameters and Φ(xp)∈Rnx isa known regressor
vector.
The control problem is to design u such thatyt tracks a bounded
time-variant reference signalycmd in the presence of the
aforementioned con-stant parametric uncertainties, whereas the rest
ofthe signals in the closed-loop system as well astracking errors
remain bounded.
3 Control Design
In face of the problem described in section 2, thecontrol signal
u is selected as:
u = ubl +uad (4)
where ubl corresponds to a baseline linear con-troller and uad
is an MRAC.
The reason for using this augmentation ap-proach (baseline +
adaptive) stems from the factthat in most realistic applications a
system al-ready has a baseline controller designed to op-erate
under (or very close to) nominal condi-tions. When subjected to
excessive disturbancesthis controller has its performance degraded.
Insuch situations, the adaptive term acts to recoverthe desired
performance (and ensure stability) bymeans of an online adjustment
and in a real-timefashion [6].
This section presents the methodologies usedto design ubl and
uad: section 3.1 shows the ar-chitecture and the design procedure
of the base-line linear control system, which will also serveas
reference model in the design of the adaptivecontrol law described
in section 3.2.
In the following section ubl will be designedin the form of a
full state feedback. This cor-responds to assume that all states
are availablefor feedback. Evidently, this assumption consti-tutes
a practical limitation of the designed controllaw because the
states of a system usually cannotbe completely measured. However,
the design ofoutput-feedback based controllers for
nonlinearuncertain MIMO systems represents a challeng-ing problem
[6]. Regarding adaptive controllers,these challenges represent
several restrictive as-sumptions that the plant has to fulfill.
Recent re-search [7, 8] relaxed these assumptions, but
thecomplexity of the problem remains. To addressthis issue, an
attempt to apply the separation prin-ciple [9] will be investigated
in more detail in sec-tion 5.
Moreover, the pair (Ap,Bp) is assumed con-trollable and (Ap,Cm)
observable. Controllabil-ity is necessary to ensure model matching
condi-tions of the adaptive law, which will be explainedin section
3.2. Observality is necessary for theanalysis of section 5.
3.1 Baseline Control Design
For the purpose of tracking with null steady stateerror, the
baseline control system corresponds to
2
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ADAPTIVE OUTPUT-FEEDBACK GAIN SCHEDULING APPLIED TO FLEXIBLE
AIRCRAFT
FA 𝒆𝒚 ⅆ𝒕
𝐊𝒙
--
-
+
𝐊𝑰𝒆𝒚𝑰(𝒕)𝒆𝒚(𝒕)𝒚𝒄𝒎ⅆ(𝒕) 𝒚𝒕(𝒕)𝒖𝒄(𝒕)
𝒖𝒔𝒂𝒔(𝒕)
𝒖𝒃𝒍(𝒕)
𝒙𝒑(𝒕)
Fig. 1 Baseline linear control system block diagram.
a linear quadratic regulator (LQR) with propor-tional and
integral feedback connections (Figure1) [10].
Let ycmd ∈ Rnt be a bounded command thatyt must track and ey =
yt− ycmd be the outputtracking error whose integral is denoted by
eyI:
ėyI = ey = yt−ycmd (5)
From Eqs. (1), (3) and (5), the extendedopen-loop dynamics can
be written as:[
ėyIẋp
]︸ ︷︷ ︸
ẋ
=
[0 Ct0 Ap
]︸ ︷︷ ︸
A
[eyIxp
]︸ ︷︷ ︸
x
+
[−I0
]︸ ︷︷ ︸
Bref
ycmd
+
[DtBp
]︸ ︷︷ ︸
B
Λ [u+ f(xp)] (6)
In terms of the tracking outputs:
yt︸︷︷︸y
=[
0 Ct]︸ ︷︷ ︸
C
[eyIxp
]+ Dt︸︷︷︸
D
Λ [u+ f(xp)]
(7)Equations (6) and (7) can be written com-
pactly as:
ẋ = Ax+BΛ [u+ f(xp)]+Brefycmd (8)y = Cx+DΛ [u+ f(xp)] (9)
The baseline control law is designed assum-ing that the system
operates in the nominal con-ditions. It corresponds to set Λ = I
and Θ = 0in the previous equations, resulting in the linearbaseline
open-loop system:
ẋ = Ax+Bubl +Brefycmd (10)y = Cx+Dubl (11)
where:
ubl =−[
KI Kx]︸ ︷︷ ︸
KT
x =−KTx (12)
It is well-known [11] that the optimal LQRsolution is given
by:
KT = R−1BTP (13)
with P being the unique symmetric positive-definite solution of
the algebraic Riccati equation(ARE):
ATP+PA+Q−PBR−1BTP = 0 (14)
which is solved using the symmetric positive-denite design
parameters Q and R.
Therefore, the baseline closed-loop system isgiven by:
ẋ =(
A−BKT)
x+Brefycmd (15)
y =(
C−DKT)
x (16)
3.2 MRAC Design
The adaptive control law that composes the to-tal control input
(Eq. 4) is based on the MRACapproach [6, 12].
The baseline closed-loop dynamic given byEq. (15) corresponds to
the desired behavior forthe actual closed-loop system. Therefore,
the ref-erence model is assumed to be:
ẋref = Arefxref +Brefycmd (17)yref = Crefxref (18)
3
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BERTOLIN, R. M. , GUIMARÃES NETO. A. B. , BARBOSA, G. C. ,
SILVESTRE, F. J.
where:
Aref = A−BKT (19)Cref = C−DKT (20)
correspond to the model matching conditions andmust be
ensured.
As previously stated, the adaptive term inEq. (4) acts to
recover the desired behavior forthe actual closed-loop system when
operating inthe presence of uncertainties or disturbances, andmust
be designed in such a way that the dynamicsin Eq. (8) when under
the effect of the input givenin Eq. (4) matches Eq. (17). So,
substituting Eq.(4) into (8):
ẋ = Ax+BΛ[ubl +uad +ΘT Φ(xp)
]+Brefycmd
(21)From Eqs. (12), (19) and (21):
ẋ=Arefx+BΛ[uad+
KTu︷ ︸︸ ︷(I−Λ−1
)ubl+ΘT Φ(xp)
]+Brefycmd (22)
which can be rewritten as:
ẋ = Arefx+BΛ[uad + Θ̄T Φ̄(ubl,xp)
]+Brefycmd
(23)with:
Θ̄T =[
KTu ΘT]
(24)
Φ̄(ubl,xp) =[
ublΦ(xp)
](25)
Equivalently:
y = Crefx+DΛ[uad + Θ̄T Φ̄(ubl,xp)
](26)
Comparing Eqs. (23) and (26) with (17) and(18), respectively, it
is evident that if the adaptivelaw uad is chosen to dominate the
system uncer-tainties Θ̄T Φ̄(ubl,xp), x→ xref and consequentlyy→
yref. Therefore, proposing:
uad =− ˆ̄ΘT Φ̄(ubl,xp) (27)
where ˆ̄Θ ∈ R(nu+nx)×nu corresponds to the matrixof adaptive
parameters, defining the matrix of pa-rameter estimation error
as:
∆Θ̄ = ˆ̄Θ− Θ̄ (28)
and substituting Eq. (27) into Eqs. (23) and (26)results in:
ẋ = Arefx−BΛ∆Θ̄T Φ̄(ubl,xp)+Brefycmd (29)y = Crefx−DΛ∆Θ̄T
Φ̄(ubl,xp) (30)
Introducing the state tracking error as:
e = x−xref (31)
the state tracking error dynamics can now be cal-culated by
subtracting the reference model dy-namics in Eq. (17) from the
actual closed-loopextended system in Eq. (29):
ė = Arefe−BΛ∆Θ̄T Φ̄(ubl,xp) (32)
It is possible to demonstrate using Lya-punov’s direct method
(Ref. [6], chapter 10) that,if the adaptive law is chosen in the
form:
˙̄̂Θ = ΓΘ̄Φ̄(ubl,xp)eT PrefB (33)
then the closed-loop state tracking error dynam-ics in Eq. (32)
is globally asymptotically sta-ble. In other words, the closed-loop
system fromEq. (29) globally asymptotically tracks the ref-erence
model from Eq. (17), as t → ∞ and forany bounded command ycmd. At
the same time,y (Eq. (30)) also track ycmd with bounded errors.
In Eq. (33), ΓΘ̄ = ΓTΘ̄ > 0 represents ratesof adaptation and
Pref = PrefT > 0 is the uniquesymmetric positive-definite
solution of the alge-braic Lyapunov equation:
ArefT Pref +PrefAref =−Qref (34)
where Qref = QrefT > 0 is a matrix of design pa-rameters.
4 Numerical Application
This section presents a numerical application ofthe formulation
developed in the previous sec-tion. Section 4.1 introduces the
flexible aircraftconsidered here, the X-HALE. Section 4.2
ad-dresses the model order reduction for control pur-pose. Lastly,
section 4.3 describes the design pro-cedure and the simulation
cases analyzed.
4
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ADAPTIVE OUTPUT-FEEDBACK GAIN SCHEDULING APPLIED TO FLEXIBLE
AIRCRAFT
4.1 The X-HALE Aircraft
The X-HALE is a radio-controlled unmanned ex-perimental airplane
developed to be a test plat-form to collect aeroelastic data
coupled with theaircraft rigid-body motion, in order to
validatemathematical formulations of flexible-aircraftflight
dynamics as well as control system designtechniques [5].
It can be configured to fly as a four-, six- oreight-meter-span
configuration. In all of them,the outer panels of the wing have 10
degree di-hedral angle. A central stabilizer is used witha flipping
mode (horizontal or vertical position)to increase or decrease
lateral-directional stabil-ity of the aircraft. In this work, the
four-meter-span configuration is considered, only. Figure
2illustrates the aircraft.
elevators
motors
aileron
Fig. 2 Four-meter-span, vertical-central-tail X-HALE
configuration.
The mathematical formulation employed tomodel the X-HALE flight
dynamics was devel-oped by Guimarães Neto [13].
The state variables of the full nonlinear modelare given by:
xfull =[V α q θ H x β φ p r ψ y · · ·
λrbT ηT η̇T ληT]T (35)
The model includes the kinematic equationsin the inertial
reference frame, for all six degreesof freedom: displacements in
the x and y direc-tions, altitude H and roll, pitch and yaw
angles(φ, θ and ψ, respectively). Furthermore, the ve-locity V ,
the angle of attack α, the sideslip angle
β, as well as the angular rates p, q and r also havetheir
corresponding equations of motion. A par-ticular feature of the
model is the modeling of thestructural dynamics using modal
amplitudes andtheir time-derivatives (η and η̇). Modes of
vi-bration with frequencies up to 25 Hz are retainedin the model.
Aerodynamic lag states arise dueto rigid-body and control-surface
dynamics (λrb)and due to the aeroelastic dynamics (λη). There-fore,
the full model is composed of 210 states: 12from rigid body motion
plus 63 from rigid-bodyand control-surface aerodynamic lag states
plus30 from aeroelastic states plus 105 from aeroe-lastic
aerodynamic lag states.
The four-meter-span aircraft is composed oftwo boom-mounted
elevators, two ailerons andthree motors. All these actuators can be
inde-pendently controlled. However, to accomplishaircraft control
in a more conventional way, thelongitudinal attitude is controlled
by the eleva-tors (δe), the rolling motion is controlled by
theailerons (δa), whereas the yawing motion is con-trolled using
differential thrust of the externalmotors (δr). The global thrust
level of the threemotors responds to the throttle command
(δt).Then, the input vector can be rewritten as:
u =[
δt δe δa δr]T (36)
The output vector yp ∈ R90×1 comprisesmodel outputs such as
displacements and atti-tudes, linear and angular velocities, load
factors,at different points of the wing and close to thecenter of
gravity (CG) of the aircraft. The mea-surements made possible by
the aircraft sensorsare a subset of the model outputs.
The full nonlinear model may be linearizedaround different
equilibrium conditions. In thispaper, all subsequent development is
performedconsidering linearized models around the straightand level
flight condition with velocity of 14 m/sand altitude of 650 meters,
ISA+10. Moreover,the full linear model (around such flight
condi-tion) of the vertical-central-tail configuration air-craft is
assumed as the nominal open-loop model.
5
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BERTOLIN, R. M. , GUIMARÃES NETO. A. B. , BARBOSA, G. C. ,
SILVESTRE, F. J.
10-4 10-3 10-2 10-1 100 101 102
Frequency [Hz]
-20
-15
-10
-5
0
5
10
15
20
25
30
Singu
larValue[dB]
σmin
10-4 10-3 10-2 10-1 100 101 102
Frequency [Hz]
30
40
50
60
70
80
90
Singu
larValue[dB]
σmax
FullReduced
Fig. 3 Comparison between the maximum and minimum singular
values of the transfer function matricesfor both full and reduced
models.
4.2 Model Reduction
The high order of the X-HALE model is a chal-lenge to most of
the control techniques and there-fore a state-space reduced-order
model is appro-priate. For this purpose, a residualization
tech-nique is applied to all the aerodynamic lag statesof the
nominal model [13]. The resulting reducedlinear model (Ap, Bp, Cp,
Dp) comprises ninerigid-body states of the full model (discarding
ig-norable variables x, y and ψ) as well as the aeroe-lastic
ones:
xp =[
V α q θ H β φ p r ηT η̇T]T (37)
totalizing thirty-nine states. Fig. 3 examinesthe maximum and
minimum singular values ofthe MIMO transfer function matrix for
both fulland reduced models. It is notorious that the re-duced
model preserves sufficient characteristicsof the full one. This
makes sense, since the aero-dynamic lag states have a greater
impact on thephase of the system.
4.3 Simulation Results
In order to illustrate the advantages of the MRACaugmentation of
a baseline linear controller, acontrol system for velocity,
altitude, sideslip androll angle tracking is considered.
From the reduced linear model, the first stepis to obtain the
augmented open-loop dynamics,according to Eq. (10), including the
integral of
the following output tracking error:
ey = yt−ycmd =
VHφβ
−
VcmdHcmdφcmdβcmd
(38)The baseline linear controller is then de-
signed from Eqs. (12), (13) and (14), with theappropriate
choices for the ARE parameters:
Q =[
I4×4 04×39039×4 10−3I39×39
](39)
R = diag(25, 0.1, 0.1, 25) (40)
where diag(•) is a diagonal matrix for which themain diagonal
elements are given by •.
The next step is to design the MRAC system(from Eqs. (34) and
(33)) in order to recover thedesired closed-loop performance given
by Eqs.(17) and (18). After some iterations focusing ona fast
tracking with reduced transient oscillations,the following
parameters were selected:
Qref = 10−3 I4×4 04×9 04×3009×4 10−2I9×9 09×30
030×4 030×9 10−1I30×30
(41)
ΓΘ̄ = 5∗10−3[
I4×4 04×39039×4 I39×39
](42)
with Φ(xp) = xp being the choice for the regres-sion vector of
Eq. (25) [6].
6
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ADAPTIVE OUTPUT-FEEDBACK GAIN SCHEDULING APPLIED TO FLEXIBLE
AIRCRAFT
0 10 20 30 40 50 60 70
Time [s]
-1.5
-1
-0.5
0
0.5
1
1.5
∆V
[m/s]
CommandReferenceBaselineAdaptive
0 10 20 30 40 50 60 70
Time [s]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
∆H
[m]
0 10 20 30 40 50 60 70
Time [s]
-20
-15
-10
-5
0
5
10
15
∆φ[◦]
0 10 20 30 40 50 60 70
Time [s]
-5
-4
-3
-2
-1
0
1
2
3
4
5
∆β[◦]
0 10 20 30 40 50 60 70
Time [s]
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
∆δt[-]
0 10 20 30 40 50 60 70
Time [s]
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆δe[◦]
0 10 20 30 40 50 60 70
Time [s]
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆δa[◦]
0 10 20 30 40 50 60 70
Time [s]
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
∆δr[-]
0 10 20 30 40 50 60 70
Time [s]
0
2
4
6
8
10
12
14
16
18
‖e‖
Fig. 4 Simulation results of the case (i) (vertical-central-tail
aircraft in nominal condition).
To validate the designed controller as well asto demonstrate the
potentiality of the adaptivecontrol, three cases are analyzed: (i)
consider-ing the vertical-central-tail aircraft operating inthe
nominal condition, that is, straight and levelflight at 14 m/s and
650 meters; (ii) consideringthe horizontal-central-tail aircraft
also flying at14 m/s and 650 meters, however after a damageof the
right motor; (iii) the same flight conditionas in (ii) but with the
residual effectiveness of theright motor degraded.
Some aspects of the previous cases deserveattention: (1) in all
of them the simulations areperformed using the respective full
linear model.As a consequence, the effects of the aerodynamiclag
states are treated as model parametric un-certainties; (2) the
simulations consider the dy-namics as well as the saturation of the
actuators,
whereas the design was carried out without both.All actuator
dynamics are first-order functions,with time constant of 75 ms for
the control sur-faces and 150 ms for the motors. For the
controlsurfaces, the saturation magnitude is ±8 deg, forthe rudder
±30% of the throttle command andfor the throtlle [30%,−70%]. These
limits cor-respond to the maximum possible perturbationsaround the
equilibrium condition; (3) about thefault tolerance test of cases
(ii) and (iii), it wasassumed that, to represent some sort of
damageto the right motor, its maximum throttle com-mand was limited
to 50%, and therefore the air-craft assumed a new equilibrium
condition. Thisrepresents a new set of matrices Ap, Bp, Cp andDp;
(4) another uncertainty present in this new setof matrices
corresponds to the horizontal-central-tail configuration of the
aircraft; (5) lastly, in (iii),
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BERTOLIN, R. M. , GUIMARÃES NETO. A. B. , BARBOSA, G. C. ,
SILVESTRE, F. J.
0 10 20 30 40 50 60 70
Time [s]
-1.5
-1
-0.5
0
0.5
1
1.5
∆V
[m/s]
CommandReferenceBaselineAdaptive
0 10 20 30 40 50 60 70
Time [s]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
∆H
[m]
0 10 20 30 40 50 60 70
Time [s]
-20
-15
-10
-5
0
5
10
15
∆φ[◦]
0 10 20 30 40 50 60 70
Time [s]
-5
-4
-3
-2
-1
0
1
2
3
4
5
∆β[◦]
0 10 20 30 40 50 60 70
Time [s]
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
∆δt[-]
0 10 20 30 40 50 60 70
Time [s]
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆δe[◦]
0 10 20 30 40 50 60 70
Time [s]
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆δa[◦]
0 10 20 30 40 50 60 70
Time [s]
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
∆δr[-]
0 10 20 30 40 50 60 70
Time [s]
0
2
4
6
8
10
12
14
16
18
‖e‖
Fig. 5 Simulation results of the case (ii)
(horizontal-central-tail aircraft after a right motor damage).
the residual effectiveness of the right motor is de-graded by a
factor Λ = 0.85.
The simulation results are presented in Fig-ures 4 to 7, where
the curves called Referencecorrespond to the reference model
response, theBaseline curves are the response of the closed-loop
system considering u = ubl (just the base-line control law) and the
curves called Adaptiveare the response of the closed-loop system
con-sidering u = ubl +uad (baseline + adaptive con-trol laws).
It is desired to track velocity (Vcmd) and rollangle (φcmd)
commands, while the commandedaltitude (Hcmd) and sideslip (βcmd)
are kept con-stant and equal to zero. An initial condition ofβ(0) =
3◦ is considered.
Figure 4 shows the simulation results of case(i). In an ideal
case, the reference, baseline and
adaptive curves must be identical, once with-out uncertainties
the reference model is exactlythe baseline linear control system,
and thereforethe term uad must remain null. It turns outthat the
simulation model for this case containssome parametric
uncertainties, and this is prob-ably the reason for the mismatch
between thecurves. Even so, both (baseline and adaptive)perform
very similarly to the reference model.Regarding the control
signals, it is observed thatall of them are feasible, that is, they
operate withmagnitudes smaller than the stipulated limits
andpresenting feasible rates. Lastly, is is possibleto see that the
2-norm of the state tracking errortends asymptotically to zero, as
expected.
Figure 5 shows the simulation results of case(ii). For this
case, it is observed that the uncer-tainties of the model have a
significant influence.
8
-
ADAPTIVE OUTPUT-FEEDBACK GAIN SCHEDULING APPLIED TO FLEXIBLE
AIRCRAFT
0 10 20 30 40 50 60 70
Time [s]
-1.5
-1
-0.5
0
0.5
1
1.5
∆V
[m/s]
CommandReferenceBaselineAdaptive
0 10 20 30 40 50 60 70
Time [s]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
∆H
[m]
0 10 20 30 40 50 60 70
Time [s]
-20
-15
-10
-5
0
5
10
15
∆φ[◦]
0 10 20 30 40 50 60 70
Time [s]
-5
-4
-3
-2
-1
0
1
2
3
4
5
∆β[◦]
0 10 20 30 40 50 60 70
Time [s]
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
∆δt[-]
0 10 20 30 40 50 60 70
Time [s]
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆δe[◦]
0 10 20 30 40 50 60 70
Time [s]
-10
-8
-6
-4
-2
0
2
4
6
8
10
∆δa[◦]
0 10 20 30 40 50 60 70
Time [s]
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
∆δr[-]
0 10 20 30 40 50 60 70
Time [s]
0
2
4
6
8
10
12
14
16
18
‖e‖
Fig. 6 Simulation results of the case (iii)
(horizontal-central-tail aircraft after a right motor damage
andresidual effectiveness degradation).
Although the baseline system still performs sat-isfactorily, its
response is clearly compromised.Among all the uncertainties
considered, the fail-ure of the right motor is the most critical,
and itsimpacts are evident, as clearly seen in the evo-lution of β
and δr. More differential thrust isneeded to compensate the
asymmetric failure. Adirect consequence of the right motor damage
isthe worsening of the β regulation. On the otherhand, the adaptive
controller performs much bet-ter and indicates its usefulness in
dealing withproblems of this nature. Once more the 2-normof the
state tracking error tends asymptotically tozero.
Case (iii) is certainly the most interesting ofall, and the
simulation results are presented inFigure 6. In addition to
preserving all the para-
metric uncertainties of case (ii), the right motorhas its
residual effectiveness degraded by a mul-tiplicative factor of
0.85. In this case the baselinecontrol system diverges. However,
the MRACnot only ensures stability of the closed-loop sys-tem, but
also very good performance, with feasi-ble control signals. Only β
presents some oscil-lations in the initial instants. The 2-norm of
thestate tracking error remains converging to zero.Figure 7
presents the temporal evolution of the 2-norm of the adaptive gain
vectors with respect toeach one of the control inputs. Note that
with theasymptotic convergence to zero of the state track-ing error
2-norm, the 2-norm of the adaptive gainvectors tends to a steady
state value.
9
-
BERTOLIN, R. M. , GUIMARÃES NETO. A. B. , BARBOSA, G. C. ,
SILVESTRE, F. J.
0 20 40 60 80
Time [s]
0
2
4‖K
∆δt‖
×10-3
0 20 40 60 80
Time [s]
0
2
4
6
‖K∆δe‖
×10-4
0 20 40 60 80
Time [s]
0
1
2
3
‖K∆δa‖
×10-3
0 20 40 60 80
Time [s]
0
0.02
0.04
‖K∆δr‖
Fig. 7 Temporal evolution of the 2-norm of theadaptive gain
vectors with respect to each of thecontrol channels. Results of
case (iii).
5 Observer-Based Adaptive Controllers
In linear control theory, a very well-establishedapproach to
deal with the practical problem of de-signing an output-feedback
controller is the sep-aration principle, according to which a
full-state-variable feedback can be coupled to an observerthat
provides state estimates based on a reducednumber of measurements
[9]. There are evenworks where such an approach was applied to
thecontrol of flexible aircraft [14].
The question that arises is whether the appli-cation of such
principle together with adaptivecontrol technique remains valid. In
general, theseparation principle does not exist for
nonlinearcontrol systems [15]. Some works on the sepa-ration
principle in adaptive control can be cited[16, 17]. However no
global stability results arereported. In Ref. [15] an alternative
and muchmore promising approach is proposed in whichthe author
makes use of a closed-loop referencemodel as an observer and
ensures global stabilityunder certain assumptions.
In this section, an attempt is made to employthe separation
principle together with the MRAC.The observer here proposed is
designed accord-ing to Ref. [14] and the detailed theory about
thisapproach can be found in Ref. [9].
0 5 10 15 20 25 30
Time [s]
-0.04
-0.02
0
0.02
0.04
0.06
∆η1[-] Full
Observed
0 5 10 15 20 25 30
Time [s]
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
∆η5[-]
0 5 10 15 20 25 30
Time [s]
-6
-4
-2
0
2
4
6
∆η6[-]
×10-3
Fig. 8 Observer response versus full-order modelfor coupled
sinusoidal elevator (0.1 to 15 Hz) andaileron (0.1 to 10 Hz)
commands.
To observe the states of the reduced model(section 4.2),
acceleration measurements in thethree axes as well as angular rates
(p, q andr) are taken from three different points of theaircraft:
left and right wing tips and close toCG. In addition, strain gauges
located in boththe right and left wing are also considered.
Ob-servability of the system considering such mea-surements was
checked. Choosing weightingmatrices 10−3I39×39 for the observer
states and102I27×27 for the measurements, the observergain matrix L
is calculated.
Figure 8 shows the comparison betweenthe observer response and
the full-order linearmodel, for coupled sinusoidal elevator (0.1 to
15Hz) and aileron (0.1 to 10 Hz) commands. Thefirst, fifth and
sixth aeroelastic states are pre-sented. They correspond to the
first torsion andbending modes. It is noticed that the
observergenerates good estimates of the states. The case(ii)
simulation results considering now the ob-server in the loop are
presented in Figure 9.
Regarding the MRAC law, the closed-loopsystem remains stable but
has its performancemuch degraded. On the other hand, the base-line
controller that, in case (ii), assuming full
10
-
ADAPTIVE OUTPUT-FEEDBACK GAIN SCHEDULING APPLIED TO FLEXIBLE
AIRCRAFT
state feedback, is stable and presents good per-formance, is
unstable for the case in which theobserver is considered.
0 50 100 150
Time [s]
-1
0
1
∆V
[m/s]
0 50 100 150
Time [s]
-2
-1
0
1
2
∆H
[m]
0 50 100 150
Time [s]
-20
-10
0
10
∆φ[◦]
CommandReferenceBaselineAdaptive
0 50 100 150
Time [s]
-5
0
5
∆β[◦]
Fig. 9 Simulation results of the case (ii), but con-sidering the
observer in the loop.
6 Conclusions
This paper aimed at demonstrating the advan-tages of the MRAC in
dealing with uncertaintiesof a control system applied to the X-HALE
air-craft. A comparison between the linear baselinecontrol system
and the MRAC augmentation wasperformed in which the usefulness of
the adaptivecontroller is evident.
A discussion on the use of the separationprinciple with the MRAC
was presented. It hasbeen verified that the observer inclusion is
able todestabilize the baseline control system subject
touncertainties, whereas the adaptive one ensuresstability but with
degraded performance.
It is clear that further investigation is neces-sary to develop
an adaptive output-feedback con-trol system that is both stable and
of performancecomparable to that of full-state feedback.
Contact Author Email Address
[email protected]
Acknowledgement
This work has been funded by FINEP andEMBRAER under the research
project Ad-vanced Studies in Flight Physics, contract num-ber
01.14.0185.00.
Copyright Statement
The authors confirm that they, and/or their companyor
organization, hold copyright on all of the origi-nal material
included in this paper. The authors alsoconfirm that they have
obtained permission, from thecopyright holder of any third party
material includedin this paper, to publish it as part of their
paper. Theauthors confirm that they give permission, or have
ob-tained permission from the copyright holder of thispaper, for
the publication and distribution of this pa-per as part of the ICAS
proceedings or as individualoff-prints from the proceedings.
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12
IntroductionProblem StatementControl DesignBaseline Control
DesignMRAC Design
Numerical ApplicationThe X-HALE AircraftModel
ReductionSimulation Results
Observer-Based Adaptive ControllersConclusions
