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Load Alleviation Control Design Using Harmonic
Decomposition Models, Rotor State Feedback, and
Redundant Control Effectors
Umberto Saetti
PhD Candidate
Department of Aerospace Engineering
The Pennsylvania State University
University Park, PA 16802
Joseph F. Horn
Professor
Department of Aerospace Engineering
The Pennsylvania State University
University Park, PA 16802
Abstract—The present study considers two notional rotorcraft
models: a conventional utility helicopter, representative of an H-
60, and a wing-only compound utility rotorcraft, representative
of an H-60 with with a wing similar to the X-49A wing. An
Explicit Model Following (EMF) control scheme is designed to
achieve stability and desired Rate Command / Attitude Hold
(RCAH) response around the roll, pitch and yaw axes, while
alleviating vibratory loads through both feed-forward and feed-
back compensation. The harmonic decomposition methodology
is extended to enable optimization of primary flight control
laws that mitigate vibratory loads. Specifically, Linear Time
Periodic (LTP) systems representative of the periodic rotorcraft
dynamics are approximated by Linear Time Invariant (LTI)
models, which are then reduced and used in LQR design to
constrain the harmonics of the vibratory loads. The LQR gains
are incorporated in the EMF scheme for feedback compensation.
One innovative approach is the addition of rotor state feedback
to standard rigid body state feedback. A Pseudo Inverse (PI)
strategy is incorporated into the EMF scheme for redundant
control allocation. Finally, simulation results with and without
load alleviation are compared and the impact of PI feed-forward
and rotor state feedback compensation on handling qualities is
assessed in terms of ADS-33E specifications.
I. INTRODUCTION
Many of the critical structural limits on rotorcraft are
associated with vibratory loads and fatigue limits. These loads
are strongly influenced by higher harmonic (greater than
1/rev) dynamics in the rotor systems. These dynamics are not
modelled in the Linear Time Invariant (LTI) dynamic models
normally used for rotorcraft primary flight control design. On
the other hand, Linear Time Periodic (LTP) models are well-
suited for representing vibratory loads on rotorcraft, including
the dominant Nb/rev vibratory forces and moments at the hub
and associated dynamic components, and they can be derived
directly from the physics-based models. Methods have been
developed for approximating LTP systems using high order
LTI models [3], [4]. The harmonic decomposition method
transforms higher frequency harmonics into states of an LTI
state space model.
Not only has the use of the harmonic decomposition
method for Automatic Flight Control Systems (AFCS) and
Higher Harmonic Control (HHC) design been well-studied [3],
[4], it has also been applied towards the design of load allevi-
ation control and cueing methods that act solely through the
primarily flight controls (1st harmonic swashplate control) and
AFCS [9]. Although these studies demonstrated good vibratory
loads reduction in maneuvering flight, the technique can be
further extended and improved. The objective of this research
is to continue the development of load alleviation control
design that act solely through the primarily flight controls by
incorporating rotor state feedback and pseudo inverse control
allocation for compound rotorcraft with redundant control
effectors.
II. LINEAR MO DE LS EXTRACTION
A. Nonlinear Models
This investigation uses two FLIGHTLAB R
models: a
notional conventional utility helicopter representative of an H-
60, as shown in Fig. 1, and a notional wing-only compound
utility rotorcraft representative of an H-60 with a wing similar
to the X-49A wing, as shown in Fig. 2. The models include
flexible blades with representative in-plane, out-of-plane, and
torsional bending modes, in addition to the rigid blade flap and
lag dynamics. A six-state Peters-He inflow model is utilized
and complete non-linear aerodynamic look-up tables are used
for airframe and rotor blade aerodynamic coefficients. The H-
60 model is described in further detail in [1].
B. Linearization Procedure
Starting from the nonlinear model, a first order LTP system
representative of the periodic rotorcraft dynamics is obtained
at 120 kts level flight by following the procedure described in
[9]
˙x =F(ψ)x+G(ψ)u(1a)
y=P(ψ)x+R(ψ)u(1b)
The state and output vectors of the LTP system are given by
xT=xT
BxT
R(2a)
yT=xT
BFT
P L(2b)
where the subscript ( )Bindicates the rigid body states and
( )Rdenotes the states relative to the rotor dynamics. FPL
is the vector of forces acting on a reference pitch link. The
rigid body state vector is given by the body velocities, u,v,
w, the body angular rates, p,q,r, and the Euler angles, φ,θ,
Fig. 1: UH-60 Black Hawk.
ψ. The rotor states include inflow, rigid flap, lag, and torsion
in Multiple Blade Coordinates (MBC), the slowest 11 bending
modes in MBC, and the time derivatives of all the variables
in MBC. The input vector for the conventional model is
uT= [δlat δlong δped](3)
where δlat and δlong are respectively the lateral and longitu-
dinal sticks, and δped is the pedal. The input vector for the
compound model is
uT= [δlat δlong δped δsym δdif f ](4)
where δsym and δdif f are the sticks controlling respectively
the symmetric and differential motion of the flaperons. The
systems have a total of n= 116 states, l= 14 outputs, and
m= 3 or m= 5 inputs, depending on the configuration.
C. Harmonic Decomposition
The state, input, and output of the LTP system obtained
above are decomposed into a finite number of harmonics via
Fourier analysis.
x=x0+
N
X
i=1
xic cos iψ +xis sin iψ (5a)
u=u0(5b)
y=y0+
L
X
j=1
yjc cos jψ +yjs sin jψ (5c)
Note that only the 0th harmonic of the input vector is retained.
This is because the present study considers solely primary
flight control actuation, thus disregarding any sort of higher
harmonic control. In fact, the desired maximum frequency
of the control signal is significantly less than the main rotor
angular speed. As it is shown in [3], the LTP model can be
transformed into an approximate high order LTI model
X=AX +BU (6a)
Y=CX +DU (6b)
where
XT=xT
0xT
1sxT
1c. . . xT
Nc xT
Nc (7a)
U=u0(7b)
YT=yT
0yT
1syT
1c. . . yT
LsyT
Lc(7c)
Fig. 2: X-49A Speed Hawk.
are respectively the augmented state, control, and output vec-
tors.
In the present study the number of harmonics retained for
the state and the output is N=L= 4, leading to a high
order LTI system with n(2N+ 1) = 1044 states, 4inputs, and
l(2L+ 1) = 108 outputs. The state, input and output vectors
of the high order LTI system are given by
XT=xT
B0xT
R0xT
B1sxT
R1s. . . xT
B4cxT
R4c(8a)
U= [δlat0δlong0δped0](8b)
YT=xT
B0FT
P L0xT
B1sFT
P L1s· · · xT
B4cFT
P L4c(8c)
D. Residualization
Low order models make the control design more tractable
and is difficult measuring or estimating the states associated
with rotor dynamics, therefore this research proposes a reduced
order model approach to feedback control design. The prob-
lem is addressed through residualization, a method based on
singular perturbation theory [6] that accurately models low
frequency and steady state but neglects high frequency. It
assumes that the “fast” states reach steady state more quickly
than the “slow” states. The state vector is hence divided into
fast and slow components:
X=Xs
Xf(9)
where, in previous studies, the slow component comprised the
body states and fast component the remaining states [9]. One
innovative approach of the present study is to also retain the
0th harmonic component of the flapping states in the slow
component of the state vector
XT
s=xT
B0β1c0β1s0β01cβ0d1cβ01sβ0d1s(10a)
XT
f=ˆ
xT
R0ˆ
xT
B1sˆ
xT
R1s· · · xT
B4cxT
R4c(10b)
where ˆ
(.)denotes the 0th and 1st harmonic rotor states de-
prived of the longitudinal and lateral flapping states. Note that
β01c, β0d1c, β01sand β0d1sare retained in the slow component
of the state as they are redundant states arising from harmonic
decomposition; failure to retain them all causes problems in the
residualization. The dynamical system can then be re-written
in the following form:
˙
Xs
˙
Xf=AsAsf
Afs AfXs
Xf+Bs
BfU(11)
By applying residualization, a 15-state reduced order model is
thus derived:
˙
Xs=ˆ
AXs+ˆ
BU (12a)
Y=ˆ
CXs+ˆ
DU (12b)
where
ˆ
A=As−Asf Af−1Afs (13a)
ˆ
B=Bs−Asf Af−1Bf(13b)
ˆ
C=Cs−CfAf−1Afs (13c)
ˆ
D=D−CfAf−1Bf(13d)
More details on the derivation of the formulas above are found
in [9].
E. Algebraic Constraints Removal
Temporarily disregarding the notation used so far, consider
the following LTI system
˙
x=Ax +Bu (14a)
y=Cx +Du (14b)
where x∈ <N,u∈ <M, and y∈ <P. Consider the state
vector
xT= [x1. . . xi. . . xj. . . xN](15)
and suppose two states are linearly dependent. Without any
loss of generality pick one of the linearly dependent states to
be the last one
xN=kxi(16)
Then, a reduced state space system can be defined
˙
ˆ
x=ˆ
Aˆ
x+ˆ
Bu (17a)
y=ˆ
Cˆ
x+Du (17b)
where ˆ
x∈ <N−1since the state xNis removed from the new
state vector. The reduced order model state and output matrices
are
ˆ
A=
a1,1. . . a1,i −ka1,j . . . a1,N−1
.
.
.....
.
.....
.
.
aN−1,1. . . aN−1,i −kaN−1,j . . . aN−1,N −1
(18a)
ˆ
C=
c1,1. . . c1,i −kc1,N . . . c1,N−1
.
.
.....
.
.....
.
.
cP,1. . . cP,i −kcP,N . . . cP,N −1
(18b)
where am,n and cp,n are respectively the elements of Aand
C. The input matrix ˆ
Bis obtained by removing the Nth row
from B.
Going back to the standard notation, the above results can
be applied to the 15-state model derived earlier in this section
in light of the following relations
β1c0=β01c=−β0d1c(19a)
β1s0=β01s=−β0d1s(19b)
Fig. 3: EMF block diagram.
An 11-state model can then be derived
˙
ˆ
Xs=ˆ
ˆ
Aˆ
Xs+ˆ
ˆ
BU (20a)
Y=ˆ
ˆ
Csˆ
Xs+ˆ
DU (20b)
where
ˆ
XT
s=xT
B0β1c0β1s0(21)
Since the higher order harmonics of the rigid body and flapping
states are impractical to observe, and, in general, of negligible
amplitude when compared to their respective 0th harmonic,
they are truncated from the output (meaning also from ˆ
Cand
ˆ
D). The output therefore reduces to
YT=xT
B0β1s0β1c0FT
P L0FT
P L1sFT
P L1c· · · FT
P L4c
(22)
where FPL denotes the pitch link load vector. The idea is to
predict the vibratory loads with sufficient accuracy using the
11-state approximation of the high order LTI model. Note that
all of the higher harmonics of the pitch link loads are kept
in the output to capture the dependence of controls and rigid
body states on the vibratory loads. This model is used in the
feedback design to the EMF controller presented in the next
section.
III. CON TRO LL ER DESIGN
A controller is designed to achieve stability and desired
RCAH response around the roll, pitch and yaw axes, while
alleviating unsteady loads during maneuver with only conven-
tional primary control. This restricts the maximum allowable
frequency of the input to be considerably less than the main
rotor angular speed, excluding any higher harmonic control.
Note that the collective stick, which is usually used to mainly
control altitude, remains open-loop.
A. Explicit Model Following
A general EMF scheme for a SISO system is shown in
Fig. 3. The present study does not assume perfect inverse
plant dynamics; in fact, the inverse plant is based on a set
of decoupled 1st order linear models, where the stability and
control derivatives are pulled from the 9-state model derived
in [9]. Specifically, the inverse plant on the roll, pitch, and yaw
axes are approximated, respectively, by the following inverse
transfer functions
δlat
p=s−Lp
Lδlat
(23a)
δlong
q=s−Mq
Mδlong
(23b)
δped
r=s−Nr
Nδped
(23c)
The stability derivatives Lp,Mq, and Nrare obtained from
the following portion of the 9-state model system matrix, as
in 24,
"
p q r
˙pLp. .
˙q. Mq.
˙r. . Nr#(24)
whereas control derivatives Lδlat ,Mδlong and Nδped are ob-
tained from the input matrix of the 9-state model, as shown in
Eq. 25.
δlat δlong δped
˙pLδlat . .
˙q. Mδlong .
˙r. . Nδped
(25)
The equivalent rotor delay time constant τf, used to delay the
ideal response such that it can be physically followed by the
controlled system, is taken as max{τf1c, τf1s}, where τf1cand
τf1sare extracted from the portion of the system matrix of the
11-state model, as shown in Eq. 26.
"
β1cβ1a
˙
β1c−1
τf1c.
˙
β1s.−1
τf1s#(26)
Note that the gains Kp,Kq, and Krrelate each pilot input
to the respective commanded angular rate whereas the time
constants τp,τqand τrdetermine the quickness of the ideal
response on each axis. The turn coordination and turn com-
pensation blocks are discussed in detail in [1], [11].
For the conventional model, a Linear Quadratic regulator
(LQR) is used both for disturbance rejection and feedback
load alleviation. Fig. 4 shows the practical implementation of
LQR in an EMF design. For the compound model, the LQR is
utilized for both for disturbance rejection and for allocation of
the feedback control action to the redundant effectors in such
a way to achieve load alleviation. Fig. 5 shows the practical
implementation of LQR in an EMF design in conjunction with
pseudo inverse control allocation. Note that the body velocity
perturbations, ∆u,∆v, and ∆w, being fed back to the LQR,
are obtained by passing the body velocities, u,v, and w,
through a washout filter. The same applies to the longitudinal
and lateral flapping angles in the rotor state feedback case.
B. Pseudo Inverse
Consider the compound model input vector
U= [δlat δlong δped δsym δdif f ](27)
Allocation of the feed-forward control action to the redundant
control effectors is obtained by a pseudo-inverse strategy, as
described in [10]. The state equation is re-written as
˙
ˆ
Xs=ˆ
ˆ
Aˆ
Xs+ˆ
ˆ
BU
=ˆ
ˆ
Aˆ
Xs+ˆ
ˆ
BGd
=ˆ
ˆ
Aˆ
Xs+˜
Bd
(28)
where
dT= [δlat δlong δped](29)
G=W−1BT
r(BrW−1BT
r)−1(30)
The ganging matrix Gis 5-by-3, Bris a 3-by-5 matrix
of which the rows corresponding to the axes that are being
controlled, ˙p,˙q, and ˙r, are pulled from ˆ
ˆ
B, and Wis a weight-
ing matrix used to place different cost weightings on certain
effectors. Note that Wis taken as a 5-by-5 identity matrix as
all the control effectors are in %units, therefore no scaling is
needed. Further, the feed-forward is now designed pulling the
control derivatives from ˜
B. This makes the inverse plant be
approximated by the following inverse transfer functions
δlat
p=s−Lp(31a)
δlong
q=s−Mq(31b)
δped
r=s−Nr(31c)
C. LQR Design
Since the loop on the right hand side of Fig. 3 is inde-
pendent from the the inverse plant, the LQR gains can be
determined by using the 11-state model previously derived.
The idea is to minimize a cost function that takes into account
the output, such as
J=Zt
0YT
sˆ
QYs+UTˆ
RUdτ (32)
The state cost can be restored by substituting the output
equation of the reduced order system in Eq. 32, obtaining
J=Zt
0XT
s
ˆ
ˆ
CTˆ
Qˆ
ˆ
CXs+UT(ˆ
R+ˆ
DTˆ
Qˆ
D)Udτ (33)
The weight matrices are designed according to [7] and are of
the form of
ˆ
Q=diag hα2
1
(Y1)2
max
α2
2
(Y2)2
max · · · α2
l
(Yl)2
max i(34)
ˆ
R=ρ diag hβ2
1
(U1)2
max
β2
2
(U2)2
max · · · β2
n
(Um)2
max i(35)
where (Yi)2
max and (Uj)2
max are the largest desired response
and input for that particular component of the output/input,
Pl
i=1 α2
i= 1 and Pm
j=1 β2
i= 1 are used to add an
additional relative weighting on the various components of the
output/control input, and ρis used as the relative weighting
between the control and state penalties.
The method described allows to transfer the constraints on
each harmonic of the pitch link load response to the fuselage
and flapping states, effectively providing a load limiting control
action based on the feedback of the 0th harmonic fuselage and
flapping dynamics. Note that what is actually being minimized
Fig. 4: EMF block diagram with LQR disturbance rejection.
Fig. 5: EMF block diagram with LQR disturbance rejection and pseudo inverse.
is the perturbations of pitch link loads from their periodic
equilibrium. It follows that the load alleviating action is
effective only in maneuvering flight.
IV. RES ULT S
A. Linear Models Validation
This section focuses on the validation of the linear mod-
els relative to the conventional model. The results for the
compound model are not shown as they do not provide any
additional insight. The open-loop response of the linear models
is compared to the open-loop response of the nonlinear system
to ensure that both the fuselage dynamics and the vibratory
loads are accurately modeled. The angular rates time histories
from the simulations are shown in Fig. 6. As it appears in
the plots, the LTP model response overlaps one of the LTI
system for each fuselage output. Both the high order LTI and
LTP on-axis responses match the nonlinear response well. The
off-axis responses seem to give a good match as well, being
just a few degrees off in pitch. The 11-state and 9-state LTI
responses are very similar to the high order LTI one. The
only noticeable difference is that the the 11-state model has an
increased phase delay when compared to the 9-state model as
it retains longitudinal and lateral flapping states. The vibratory
response of the z-component of a reference pitch link load is
shown in Fig. 7. Mind that the linear responses in the plots
have the periodic equilibrium added in to ease the comparison
with the nonlinear response. All the linear models seem to
predict the peaks and harmonic content of the vibratory loads
accurately.
Fig. 6: Open loop angular rates response.
The fidelity of the reduced order model is also assessed
by comparing its eigenvalues to the eigenvalues of the high
order LTI system, as shown in Fig. 8. As expected, the rigid
body eigenvalues of the reduced order models match closely
to the eigenvalues of the high order LTI system. Further, the
eigenvalues associated to the regressive flap mode of the 11-
state model are close to the ones of the high order LTI. Fig. 9
illustrates the on-axis frequency response of the angular rates
with respect to the control inputs. It appears that the frequency
response of the 11-state model matches closely the one of the
high order LTI up to about 10 rad/s. This is because the
Fig. 7: Open loop pitch link load response.
11-state model has increased phase delay due to the presence
of the flapping states. It is concluded that both the fuselage
dynamics and the vibratory loads are properly captured by the
11-state model.
Fig. 8: LTI models eigenvalues.
B. Load Alleviation with Rotor State Feedback
In this section the EMF controllers with and without
load alleviation, derived in [9], are respectively referred to as
”Baseline” and ”LA”; the EMF controller with load alleviating
and rotor feedback is referred to as ”LA RFB”. The three
are implemented in nonlinear simulations of the conventional
model and compared. The results of a pullup/pushover maneu-
ver starting at 120 kts level flight are presented to understand
how the load alleviation is effectively performed in terms of
control actuation and rate response. The chosen roll, pitch and
yaw command natural frequencies are respectively 2.5,2.5and
2rad/s. The LQR gains are designed such that, following
ADS-33E-PRF regulations, roll and yaw angular deviations are
maintained within 15◦from the initial steady condition. The
maneuver consists in a 0.2rad/s nose-up/nose-down doublet.
The closed-loop responses are shown in Fig. 10 and 11. The
control input histories are shown in Fig. 12. Noticeably, the
Fig. 9: Open loop frequency response.
Fig. 10: Closed loop nonlinear attitude response following a
pitch rate doublet.
magnitude of LA RFB longitudinal cyclic signal is lower than
LA, leading to a lower achieved pitch rate. Considering the
off-axis response, LA RFB roll response is considerably better
than LA, whereas the yaw response is slightly higher. The pitch
link loads are shown in Fig. 13. For this particular maneuver,
load alleviation between LA and LA RFB appears similar. No
particular improvement is noticed for RFB. Figure 14 shows
the pitch link loads relative to an aggressive 0.5rad/s pitch
doublet. Notice that the load alleviation strategies effectively
achieve load alleviation by limiting the load factor, as shown
in Fig. 15. From a pilot standpoint, limitation of the load factor
is an undesirable effect both in pitch and roll maneuvers. A
possible solution is to add a redundant control surface that
is able to produce a pitching moment comparable with the
one provided by the main rotor, such as a moving horizontal
stabilizer. As the wing of the compound model considered in
this study has its aerodynamic center coincident with the CG,
it is not capable of producing significant pitching moment.
The efficacy of the load alleviation control system with
rotor feedback is quantified for varying pilot input amplitudes
Fig. 11: Closed loop nonlinear attitude response following a
pitch rate doublet.
Fig. 12: Closed loop control signal following a pitch rate
doublet.
Fig. 13: Closed loop nonlinear pitch link load response
following a pitch rate doublet.
Fig. 14: Closed loop nonlinear pitch link load response
following an aggressive pitch rate doublet.
Fig. 15: Closed loop nonlinear load factor response following
a pitch rate doublet.
and command natural frequencies, both on the roll and pitch
axes. The metrics used to evaluate such efficacy are the percent
reduction in Root Mean Squared (RMS) of the pitch link load
and the percent reduction in maximum peak-to-peak pitch
link load, with respect to the baseline control system. The
representative maneuver in roll is chosen to be a roll reversal.
The performance of LA RFB with respect to the baseline
controller, in terms of vibratory load mitigation, is shown in
Fig. 16. LA RFB appears to lead to similar results as LA, as it
appears to give very modest reductions in the vibratory pitch
link loads in roll. LA RFB seems to give a similar performance
to LA also for the pullup/pushover maneuver, although the
percent reduction in both pitch link load RMS and maximum
peak-to-peak is more significant then for roll reversal. Fig.
17 shows an improvement of up to 6% of the peak-to-peak
pitch link load at high command filter natural frequencies.
It is also apparent that lower command natural frequencies
result in higher percent reductions for LA RFB, whereas the
opposite applies for LA. Note that it is possible that there exists
a combination of output costs that leads to a better LA RFB
controller in terms of load alleviation; in the present study
Fig. 16: Peak-to-peak and RMS percent reduction for a roll
reversal maneuver.
Fig. 17: Peak-to-peak and RMS percent reduction for a
pullup/pushover maneuver.
the LQR gains are tuned manually by the authors. Numerical
optimization will possibly be carried on in the future.
C. Load Alleviation with Rotor State Feedback and Pseudo
Inverse Redundant Control Allocation
In this section the baseline controller is compared to the
load alleviation EMF pseudo inverse controllers with and
without rotor feedback, respectively referred to as ”LA PI”
and ”LA PI RFB”. The three are implemented in nonlinear
simulations of the compound model. The results of a roll
rate doublet starting at 120 kts level flight are presented to
understand how the load alleviation is effectively performed in
terms of control actuation and rate response. The chosen roll,
pitch and yaw command natural frequencies are respectively
2.5,2.5and 2rad/s. The LQR gains are designed such that,
following ADS-33E-PRF regulations, roll and yaw angular
deviations are maintained within 15◦from the initial steady
condition. The maneuver consists in a 0.2rad/s roll rate
doublet. The closed-loop responses are shown in Fig. 18 and
19. The control input histories are shown in Fig. 20. Both LA
PI and LA PI RFB nicely re-allocate part of the control signal
from the lateral cyclic to the left and right flaperons to off-load
the rotor and thus decrease the pitch link loads. The flaperons
are thus used to generate part of the roll moment necessary for
the maneuver. Fig. 22 shows how both LA PI and LA PI RFB
give different hub forces and moments when compared to the
baseline controller. The pitch link loads are shown in Fig. 21.
Figure 23 shows the pitch link loads relative to an aggressive
0.5rad/s roll doublet. Noticeably, the aggressive maneuver
results in increased pitch link loads both during the roll-in
and roll-out part of the maneuver. LA PI shows exceptional
load alleviation in the roll-out phase. The introduction of the
pseudo inverse strategy appears to be far more promising than
LA and LA RFB for load alleviation in roll. However, the
addition of rotor feedback does not seem to bring particular
improvements.
Fig. 18: Closed loop nonlinear attitude response following a
roll rate doublet.
The performance of LA PI and LA PI RFB with respect to
the baseline controller, in terms of vibratory load mitigation,
is shown in Fig. 24 and 25. It is evident that the introduction
of the pseudo inverse strategy leads to a substantial increase
Fig. 19: Closed loop nonlinear attitude response following a
roll rate doublet.
Fig. 20: Closed loop control signal following a roll rate
doublet.
Fig. 21: Closed loop nonlinear pitch link load response
following a roll rate doublet.
Fig. 22: Closed loop nonlinear hub forces and moment
response following a roll rate doublet.
Fig. 23: Closed loop nonlinear pitch link load response
following an aggressive roll rate doublet.
Fig. 24: Peak-to-peak and RMS percent reduction for a roll
reversal maneuver.
in load alleviation both in roll and in pitch, with up to 17%
reduction in peak-to-peak PL load in roll and 30% in pitch.
LA PI RFB seems to perform similarly to LA PI, however, its
performance in pitch is about half as LA PI.
D. Handling Qualities Evaluation
An analysis is performed to assess the impact that the
different command filter natural frequencies and the differ-
ent load alleviating controllers have on handling qualities.
Handling qualities are evaluated in terms of ADS-33E-PRF
regulations for Target Acquisition and Tracking in both roll and
pitch [8]. The study is based on closed-loop Simulink models,
representative of the baseline and load alleviating controllers,
that use the high order LTI system as plant. As shown in
Fig. 26 and 27, it appears that performances degrade both
with decreasing natural frequency and standard feedback load
alleviation. However, the use of both pseudo inverse and rotor
feedback tends to increase handling qualities. Although Level
1 is not achieved in roll with LA and LA PI, it is still met
in all other Mission Task Elements (MTEs) specifications; at
least for the higher command filter natural frequencies. It is
Fig. 25: Peak-to-peak and RMS percent reduction for a
pullup/pushover maneuver.
Fig. 26: Conventional model handling qualities in terms of
ADS-33E-PRF regulations for Targer Acquisition and
Tracking.
concluded that the use of both pseudo inverse and rotor feed-
back constitutes an improvement in terms of handling qualities
when compared to the load alleviation strategy developed in
[9].
V. C ONCLUDING REM AR KS A ND FUTURE WORK
A. Conclusion
A 15-state model is derived from a high order LTI system
by retaining the 0th harmonic of the fuselage and flapping
states. The model is further reduced to an 11-state model by
removing the algebraic constraint associated with redundant
flapping states. The higher harmonics of the pitch link loads
are retained in the output. The model appears to predict
both fuselage dynamics and pitch link loads adequately. In
particular, the 11-state model has increased phase delay due
to the presence of the flapping states. This constitutes an
improvement with respect to the 9-state model derived in [9].
Controllers that optimize primary flight control laws to
Fig. 27: Compound model handling qualities in terms of
ADS-33E-PRF regulations for Targer Acquisition and
Tracking.
minimize vibratory loads are developed for both conventional
and compound helicopter configurations. An EMF controller
with rotor state feedback is used on the conventional model,
whereas an EMF controller with both rotor state feedback
and pseudo inverse control allocation strategy is used on the
compound model. Specifically, the pseudo inverse strategy is
used for feed-forward control allocation, whereas LQR deals
with the feedback control allocation. Rotor state feedback does
not show particular improvements in terms of load alleviation.
However, pseudo inverse control allocation proves to be very
effective both in pitch and in roll.
The controllers derived effectively achieve load alleviation
by limiting the load factor, which is undesirable from a pilot
standpoint. One possible solution is to is to add a redundant
control surface that can produce a pitching moment compara-
ble to the main rotor, such as a moving horizontal tail.
An analysis is performed to understand the impact that
rotor state feedback and pseudo inverse have on handling
qualities for different command filter natural frequencies. It
appears that both pseudo inverse and rotor feedback tend to
improve handling qualities, as opposed to the load alleviation
strategy developed in [9].
B. Future Work
The present study concentrates on extracting the linear
models only at one flight condition. A goal for the near future
is to derive the linear models across the entire flight envelope
such that the load alleviating controller can be gain-scheduled
with speed. This is important since the maneuvers studied show
appreciable changes in speed.
The controllers will be tested in piloted simulation studies
to evaluate impact on handling qualities and life extension.
Specifically, the incremental fatigue damage, as measured by
the crack growth rate in the pitch link loads, will be assessed
to understand the actual practical benefit of the method.
More advanced control allocation methods for compound
rotorcraft with redundant controls, such as cascaded pseudo
inverse for position and rate limiting, will also be explored.
The control allocation will adapt to mission requirements and
damage state of the rotorcraft. A moving horizontal stabilizer
will be implemented along with a pusher propeller.
Rubust control techniques such as H2,H∞and Q-design
will be explored to improve the disturbance rejection currently
handled by LQR.
ACKNOWLEDGMENT
This research was partially funded by the Government
under Agreement No. W911W6-17-2-0003. The U.S. Gov-
ernment is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation
thereon.The views and conclusions contained in this document
are those of the authors and should not be interpreted as rep-
resenting the official policies, either expressed or implied, of
the Aviation Development Directorate or the U.S Government.
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