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Use of Harmonic Decomposition Models in
Rotorcraft Flight Control Design with Alleviation of
Vibratory Loads
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—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 feedback compensation. First, the effect of command model
tailoring is explored to understand the effect of feed-forward
compensation on vibratory loads, with a focus on the main rotor
pitch links. Secondly, 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 gains derived
are incorporated in the EMF scheme for feedback compensation.
Finally, simulation results with and without load alleviation
are compared and the impact of feed-forward and feedback
compensation on handling qualities is assessed in terms of ADS-
33E specifications.
I. INTRODUCTION
The benefits of load alleviation control and envelope cueing
have been demonstrated in numerous simulation studies [1],
[2], [3], [4], [5], [6]. The use of Automatic Flight Control
System (AFCS) or active control sticks to help the pilot
observe structural constraints can extend the life of critical
dynamic components and reduce cost of operation. These
technologies can also improve handling qualities by alleviating
pilot workload associated with monitoring envelope limits.
Load alleviating controls have been implemented on the V-22
tilt-rotor aircraft, using cyclic pitch control to reduce in-plane
loads during forward flight maneuvers [7].
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. Past
work in the design of load limiting control laws has used proxy
models of the vibratory loading. An example is the Equivalent
Retreating Indicated Tip Speed (ERITS) parameter, which has
been correlated with vibratory pitch link loads that occur
with retreating blade stall onset [8]. Vibratory load limiting
has also been demonstrated using basic correlations, curve
fits, or neural network approximations of vibratory loads as
a function of aircraft states (angular rates, accelerations, load
factor, airspeed) [1], [2], [3].
Reliance on non-physics-based models and curve fits to
approximate vibratory loads is a limitation of past work. 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. Recently, methods
have been developed for approximating LTP systems using
high order LTI models [9], [10]. The harmonic decomposition
method transforms higher frequency harmonics into states of
an LTI state space model. Harmonic decomposition methods
have been used to model interactions between Higher Har-
monic Control (HHC) systems and the primary Automatic
Flight Control System (AFCS) and to optimize the design of
these systems to reduce vibration and provide stability and
handling qualities augmentation, where the HHC is primary
responsible for vibratory load reduction.
While the use of harmonic decomposition method for
HHC/AFCS design has been well-studied, the method has not
yet been applied towards design of load alleviation control and
cueing methods that act solely through the primarily flight con-
trols (1st harmonic swashplate control) and AFCS. Previous
studies have shown that tailoring of response characteristics
through limiting or modification of response bandwidth can
significantly reduce vibratory loads, and could be significantly
cheaper to implement on existing rotorcraft. The objective of
this research is to extend the harmonic decomposition method-
ology to enable optimization of primary flight control laws
that alleviate vibratory loads while meeting desired handling
qualities. The use of high order LTI models is used to derive
correlations of the controlled aircraft states to main rotor
vibratory loads directly from the linearized physics model. In
particular, the high order LTI models are reduced and used to
design an EMF controller with LQR feedback optimized to
reduce changes in vibratory loads.
II. HIGH OR DE R LTI MO DE L EXTRACTION
A. Nonlinear Model
This investigation uses a FLIGHTLAB R
model of a no-
tional conventional utility helicopter representative of a UH-60,
as shown in Fig. 1. The model includes 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 model is described
in further detail in [1].
Fig. 1: UH-60 helicopter.
B. Linearization Procedure
The following procedure is similar to the one found in [11]
but focuses on a first order formulation of a LTP. Consider a
nonlinear system of the form
˙
x=f(x,˙
x,u)(1a)
y=g(x,˙
x,u)(1b)
where xis the state vector of dimension n,uis the control
vector of dimension m, and yis the output vector of dimension
l. Considering now consider the case of small disturbances
x=xe(ψ) + ∆x (2a)
u=ue(ψ) + ∆u (2b)
y=ye(ψ) + ∆y (2c)
where xe(ψ),ye(ψ), and ue(ψ)define a periodic equilibrium
condition
˙
xe=f(xe,˙
xe,ue)(3a)
ye=g(xe,˙
xe,ue)(3b)
A Taylor series expansion can now be performed on the
state vector time derivative. After neglecting the terms of
second order and higher the following equation is derived
f(xe+∆x,˙
xe+∆ ˙x,ue+∆u) = f(xe,˙
xe,ue)
+F(ψ)∆x +K(ψ)∆ ˙x +G(ψ)∆u (4)
where
F(ψ) = ∂f(x,˙
x,u)
∂uxe,˙
xe,ue
(5a)
K(ψ) = ∂f(x,˙
x,u)
∂˙
xxe,˙
xe,ue
(5b)
G(ψ) = ∂f(x,˙
x,u)
∂uxe,˙
xe,ue
(5c)
With a few steps of algebraic manipulation, one can derive
∆ ˙x =ˆ
F(ψ)∆x +ˆ
G(ψ)∆u (6)
where
ˆ
F(ψ) = (I−K)−1F(7a)
ˆ
G(ψ)=(I−K)−1G(7b)
A Taylor series expansion can be performed also on the output.
After neglecting the terms of second order and higher the
following equation is derived
g(xe+∆x,˙
xe+∆ ˙x,ue+∆u) = g(xe,˙
xe,ue)
+P(ψ)∆x +Q(ψ)∆ ˙x +R(ψ)∆u (8)
where
P(ψ) = ∂g(x,˙
x,u)
∂uxe,˙
xe,ue
(9a)
Q(ψ) = ∂g(x,˙
x,u)
∂˙
xxe,˙
xe,ue
(9b)
R(ψ) = ∂g(x,˙
x,u)
∂uxe,˙
xe,ue
(9c)
After substituting 6 in 8 and carrying on the calculations, one
can derive
∆y =ˆ
P(ψ)∆x +ˆ
R(ψ)∆u (10)
where
ˆ
P(ψ) = P+Qˆ
F(11a)
ˆ
R(ψ) = R+Qˆ
G(11b)
It is thus obtained a first order formulation of a LTP system
representative of the periodic rotorcraft dynamics.
In practice, the nonlinear FLIGHTLAB R
model is first
trimmed at a desired flight condition. Then a nonlinear simu-
lation is run until the azimuthal position of a reference blade
reaches ψ= 0◦. Finally, the model is linearized at incremental
azimuth positions over one rotor revolution. In the present
study the flight condition is chosen to be level flight at 120
knots and the time step to be ∆ψ= 0.5◦, which gives a total
of 720 azimuthal positions.
The notation is simplified by dropping the ∆in front
of the linearized variables, remembering that they are indeed
perturbations from a periodic equilibrium. The state, input, and
output vectors of the LTP system are given by
xT=xT
BxT
R(12a)
uT= [δlat δlong δped](12b)
yT=xT
BFT
P L(12c)
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, in order, the body velocities
u,v,w, the body angular rates p,q,r, and the Euler angles
φ,θ,ψ. The rotor states include inflow, rigid flap, lag and
torsion in Multiple Blade Coordinates (MBC), the slowest 11
bending modes also in MBC, and the time derivatives of all
the variables in MBC. The system has a total of n= 116
states, m= 3 inputs, and l= 12 outputs.
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 +xic sin iψ (13a)
u=u0(13b)
y=y0+
L
X
j=1
yjc cos jψ +yjc sin jψ (13c)
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 [9], the LTP model can be
transformed into an approximate high order LTI model
X=AX +BU (14a)
Y=CX +DU (14b)
where
XT=xT
0xT
1sxT
1c. . . xT
Nc xT
Nc (15a)
U=u0(15b)
YT=yT
0yT
1syT
1c. . . yT
LsyT
Lc(15c)
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(16a)
U= [δlat0δlong0δped0](16b)
YT=xT
B0FT
P L0xT
B1sFT
P L1s· · · xT
B4cFT
P L4c(16c)
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, is left open-loop.
A. Reduced-Order Models
Due to the fact that low order models make the con-
trol design more tractable, and because of the difficulty of
measuring or estimating the states associated with flapping
and inflow dynamics, this research proposes a reduced order
model approach to feedback control design. The problem is
addressed through residualization, a method based on singular
perturbation theory [12] 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(17)
where
Xs=xB0(18)
XT
f=xT
R0xT
B1sxT
R1s· · · xT
B4cxT
R4c(19)
The dynamical system can then be re-written in the following
form ˙
Xs
˙
Xf=AsAsf
Afs AfXs
Xf+Bs
BfU(20)
By assuming that the fast states reach steady state quickly, the
algebraic constraint ˙
xf= 0 can be imposed. It follows that
AfsXs+Af sXf+BfU= 0 (21)
Solving for the fast states leads to
Xf=Af−1(−AfsXs−BfU)(22)
By substituting the latter result into 20, a new expression for
the slow states can be found
˙
Xs=ˆ
AXs+ˆ
BU (23)
where
ˆ
A=As−Asf Af−1Afs (24a)
ˆ
B=Bs−Asf Af−1Bf(24b)
Considering now the output equation
Y= [CsCf]Xs
Xf+Du (25)
and by substituting 22 in it, one can derive
Y=ˆ
CXs+ˆ
DU (26)
where
ˆ
C=Cs−CfAf−1Afs (27a)
ˆ
D=D−CfAf−1Bf(27b)
Now, since the higher order harmonics of the rigid body
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
B0FT
P L0FT
P L1sFT
P L1c· · · FT
P L4c(28)
where FPL denotes the pitch link load vector. The idea is
to predict the vibratory loads with sufficient accuracy using
Fig. 2: Explicit model following block diagram.
the 9th order approximation of the high order LTI model just
derived. 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 of the EMF controller
presented in the next section.
An 11th order model is also derived in a similar fashion.
The 0th harmonic of the longitudinal and lateral flapping is
retained in the slow state, along with the rigid body states.
Since this model is only used to extract time delay parameters
used in the feed-forward design of the EMF controller, there
is no need for it to predict the vibratory loads. Hence, the
harmonics higher than the 0th are truncated and the slow
and fast components of the state vector are selected from the
averaged model as follows
XT
s=xT
B0β1s0β1c0(29a)
Xf=ˆ
xR0(29b)
where ˆ
xR0denotes the vector of the 0th harmonic rotor states
deprived of longitudinal and lateral flapping.
B. Explicit Model Following
A general EMF scheme for a SISO system is shown in
Fig. 2. 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 9th order model.
Specifically, the inverse plant on the roll, pitch, and yaw axes
is approximated respectively by the following inverse transfer
functions
δlat
p=s−Lp
Lδlat
(30a)
δlong
q=s−Mq
Mδlong
(30b)
δped
r=s−Nr
Nδped
(30c)
The stability derivatives Lp,Mq, and Nrare obtained from
the following portion of the 9th order model system matrix, as
in 31,
"
p q r
˙pLp. .
˙q. Mq.
˙r. . Nr#(31)
whereas control derivatives Lδlat ,Mδlong and Nδped are ob-
tained from the input matrix of the 9th order model, as shown
in 32.
δlat δlong δped
˙pLδlat . .
˙q. Mδlong .
˙r. . Nδped
(32)
The present study utilizes a Linear Quadratic regulator (LQR)
both for disturbance rejection and to pursue load alleviation.
Fig. 3 shows the practical implementation of LQR in the EMF
design. The turn coordination block is explained in detail in
[1]. 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 τf1c
and τf1sare extracted from the portion of the system matrix
of the 11th order model, as shown in 33.
"
β1cβ1a
˙
β1c−1
τf1c.
˙
β1s.−1
τf1s#(33)
Note that the gains Kp,Kqand 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.
C. LQR Design
Since the loop on the right hand side of Fig. 2 is inde-
pendent from the the inverse plant, the LQR gains can be
determined by using the 9th order model previously derived.
The idea is to minimize a cost function that takes into account
the output, such as
J=Zt
0YTˆ
QY +UTˆ
RUdτ (34)
The state cost can be restored by substituting the output
equation of the reduced order system in Eq. 34, obtaining
J=Zt
0XT
sQXs+UTRUdτ (35)
where the state and control cost matrices are defined as
Q=ˆ
CTˆ
Qˆ
C(36a)
R=ˆ
R+ˆ
DTˆ
Qˆ
D(36b)
The weight matrices are designed according to [13] and are of
the form of
ˆ
Q=diag hα2
1
(Y1)2
max
α2
2
(Y2)2
max · · · α2
n
(Yl)2
max i(37)
ˆ
R=ρ diag hβ2
1
(U1)2
max
β2
2
(U2)2
max · · · β2
n
(Um)2
max i(38)
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
Fig. 3: Explicit model following block diagram with LQR disturbance rejection.
states, effectively providing a load limiting control action based
on the feedback of the 0th harmonic fuselage dynamics. Note
that what is actually being minimized is the perturbations of
pitch link loads from their periodic equilibrium. This means
that the controller tends to bring the pitch link loads to their
periodic equilibrium. It follows that the load alleviating action
is effective only in maneuvering flight.
IV. RES ULT S
A. Correlation of Vibratory Loads to Response Bandwidth
In preliminary studies the effect of command model tai-
loring is explored to understand its impact on vibratory loads,
with focus on the main rotor pitch links. Initial simulation
studies illustrate the correlation of vibratory loads to response
bandwidth. The selected flight condition is level flight at 120
kts. To study the effects of roll rate and roll acceleration on
the vibratory loads independently, the roll controller has been
set up as a rate command (RCAH) system. This way, roll
acceleration due to a step input can be modified by acting
on the time constant of the first order command filter while
keeping the magnitude of the roll rate perturbation constant.
Note that the roll acceleration is equivalent to the natural
frequency of the command filter, which is the inverse of the
time constant: ωn=1
τ. The LQR gains are found by imposing
constraints only on the fuselage states.
Fig. 4 shows a roll rate doublet response for varying natural
frequencies. Fig. 5 shows the correlation between pitch link
loads and roll rates for different roll accelerations, given by
the different natural frequencies of the rate command filter. It
appears that increasing roll accelerations has a greater effect
on pitch link loads for higher roll rates. It is concluded
that command model tailoring does indeed have an effect
on vibratory loads and is a viable way to load alleviation.
Nonetheless, it comes at the cost of performance since the
agility of the rotorcraft decreases with decreasing command
filter natural frequencies.
Fig. 4: Roll attitude and roll rate time response in RCAH
mode with varying natural frequencies.
B. Validation of the Linear Models
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 rate command time history of the
maneuver in consideration is the same as Fig. 4. The attitude
and angular rates from the simulations are shown respectively
in Fig. 6 and 7. As it appears in the plots, the LTP model
response overlaps the one of the LTI system for each fuselage
output. Both the high order LTI and LTP on-axis responses
Fig. 5: Maximum peak-to-peak pitch link load with varying
roll rate command filter natural frequencies.
match the nonlinear response well. The off-axis responses
seem to give a good match in in yaw, less in pitch. The reduced
order LTI response is very similar to the high order LTI one.
The only noticeable difference is that the high order LTI system
contains a model of the actuators acting on the swashplate,
of which the states are lost during reduction. This causes the
roll response not to have a delay, as it can be appreciated
in the angular rates plot. The vibratory response of the z-
component of a reference pitch link load and its perturbation
from the periodic equilibrium are shown respectively in Fig.
8 and Fig. 9. Mind that the linear responses in the plots have
the periodic equilibrium added in to facilitate 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: Attitude time history of the reduced order, high order
LTI, LTP and nonlinear models.
The fidelity of the reduced order model is also assessed
by comparing its eigenvalues to the ones of the high order
LTI system, as shown in Fig. 10. As expected, the rigid body
eigenvalues of the reduced order model match closely to the
Fig. 7: Angular rates time history of the reduced order, high
order LTI, LTP and nonlinear models.
Fig. 8: Pitch link load z-component time history of the
reduced order, high order LTI, LTP and nonlinear models.
Fig. 9: Pitch link load perturbation z-component time history
of the reduced order, high order LTI, LTP and nonlinear
models.
ones of the high order LTI system. Fig. 11 illustrates the on-
axis frequency response of the angular rates with respect to
the control inputs. It appears that up to about 4rad/s the
frequency response of the reduced order model matches closely
the one of the high order LTI on each axis. It is concluded that
both the fuselage dynamics and the vibratory loads are properly
captured by linear systems.
Fig. 10: Comparison between the eigenvalues of the reduced
order model and high order LTI system.
Fig. 11: Comparison between the on-axis frequency responses
of the reduced order model and high order LTI system.
C. Load Alleviating Control System
The load alleviating control system is compared to EMF
with LQR disturbance rejection but no load alleviating action,
which will be referred to as the baseline control system. The
results of a pullup/pushover maneuver 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 fre-
quencies 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 unaccelerated condition. Also, the output harmonics
included in the cost function are 0th,1st and 4th ; these appear
to give the best compromise between load alleviation and flight
performance. The maneuver consists in a 0.2rad/s nose-up /
nose-down doublet, similar to the one in Fig. 4 but on the pitch
axis. The closed-loop responses are shown in Fig. 12 and 13.
The control input histories is shown in Fig. 14. Noticeably, the
pitch rate response with the load alleviating control system is
less aggressive and possibly slightly slower when compared to
the response of the baseline controller. Furthermore, the off-
axis response, specifically around the yaw axis, looks worse
for the load alleviation case. Degradation in flight performance
is reasonable since load alleviation is achieved solely through
the 0th harmonic of the fuselage dynamics. To put it simply, the
load alleviating control system makes the helicopter fly more
“gently”. The pitch link loads and their perturbation from trim
can be appreciated in Fig. 15 and Fig. 16. It is clear that the
vibratory load perturbation is minimized. But the question is:
how much?
Fig. 12: Comparison between attitude time histories for
baseline and load alleviating control systems.
Fig. 13: Comparison between the angular rates time histories
for baseline and load alleviating control systems.
The efficacy of the load alleviation control system is
quantified for varying pilot input amplitudes and command
Fig. 14: Comparison between the control signals for baseline
and load alleviating control systems.
Fig. 15: Comparison between the pitch link loads for
baseline and load alleviating control systems.
Fig. 16: Comparison between the pitch link load perturbation
for baseline and load alleviating control systems.
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 perturbations from
trim 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 the load alleviating controller with respect
to the baseline one in terns of vibratory load mitigation is
shown in Fig. 17. Although giving some improvement, the
load alleviating system provides only very modest reductions
in the vibratory pitch link loads in roll. However, for the
pullup/pushover maneuver the percent reduction in both pitch
link load perturbation RMS and maximum peak-to-peak pitch
link load is more significant. Fig. 18 shows an improvement
of up to 29 % of the RMS of the pitch link load perturbation
at low command filter natural frequencies. It also appears that
lower command natural frequencies result in higher maximum
peak-to-peak percent reduction, up to around a 20 %. This is
due to the fact that the part of the response where the maximum
peak-to-peak occurs is dominated by the feed-forward signal
and thus by command model tailoring. Decreasing the quick-
ness of the response leaves more space for the load alleviating
controller to act. Mind that it is possible that there exists a
combination of output costs that leads to a better controller
in terms of load alleviation; in the present study the LQR
gains have been optimized manually by the authors. Numerical
optimization will possibly be carried on in the future.
Fig. 17: Percent reduction in Root Mean Squared (RMS) of
the pitch link load perturbations from trim and percent
reduction in maximum peak-to-peak pitch link load for a roll
reversal maneuver with load alleviation with respect to the
baseline controller.
D. Handling Qualities Evaluation
An analysis is performed to assess the impact that the
different command filter natural frequencies and the load
alleviating controller 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
[14]. 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.
19, it appears that performances degrade both with decreasing
natural frequency and feedback load alleviation. Although
Level 1 is not achieved in roll with the load alleviation mode
Fig. 18: Percent reduction in Root Mean Squared (RMS) of
the pitch link load perturbations from trim and percent
reduction in maximum peak-to-peak pitch link load for a
pullup/pushover maneuver with load alleviation with respect
to the baseline controller.
on, it still is met in all other Mission Task Elements (MTEs)
specifications; at least for the higher command filter natural
frequencies. It is concluded that load alleviation through both
feed-forward and feedback compensation comes at the cost of
a degradation in handling qualities.
Fig. 19: Handling qualities in roll and pitch in terms of
ADS-33E-PRF regulations for Targer Acquisition and
Tracking, for varying command filter natural frequencies and
control systems.
V. C ONCLUDING REM AR KS A ND FUTURE WO RK
A. Conclusion
The effect of command model tailoring is assessed to
understand impact on vibratory loads. It appears that increasing
roll accelerations have a greater effect on pitch link loads for
higher roll rates. It is concluded that command model tailoring
does indeed have an effect on vibratory loads and is a viable
way to load alleviation.
A9th order model is derived from a high order LTI system
by retaining the 0th harmonic of the fuselage states and the
higher harmonics of the pitch link loads in the output. The
model appears to predict fuselage dynamics and vibratory
loads adequately.
A controller that optimizes primary flight control laws to
minimize vibratory loads is developed by incorporating LQR
gains for disturbance rejection in an Explicit Model Following
scheme. The gains are derived by using a reduced order model
to impose constraints on the harmonics of the pitch link loads.
The resulting load alleviating control action translates into a
degradation of flight performance. The controller gives a good
load alleviation in pullup/pushover maneuvers but appears to
be less effective in a roll reversal maneuver.
Previous work limitations, such as the reliance on non-
physics-based models and curve fits to approximate vibratory
loads, is lifted. Since both command model tailoring and the
load alleviating controller act solely through 1st harmonics
swashplate control, a combination of the two could be readily
integrated with existing or future AFCS on military rotorcraft.
An analysis is performed to understand the impact that
the different command filter natural frequencies and the load
alleviating controller have on handling qualities. It appears
that load alleviation through both feed-forward and feedback
compensation comes at the cost of a degradation in handling
qualities.
B. Future Work
The present study concentrates on extracting the LTP, high
order LTI and reduced order systems only at one flight condi-
tion. A goal for the near future is to derive the aforementioned
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.
Command model and effectiveness of load alleviating ac-
tion might be dependent on operating conditions, mission task,
and current damage of the rotorcraft, and thus the controller
will be integrated with a notional regime recognition algorithm.
The controller will be tested in piloted simulations 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.
Control allocation methods for compound rotorcraft with
redundant controls will also be explored. The harmonic de-
composition LTI models will be used to develop a weighted
pseudo-inverse in the control allocation that minimizes critical
vibratory loads. The control allocation will adapt to mission
requirements and damage state of the rotorcraft.
Rotor state feedback will be studied. The idea is to retain
the 0th and possibly higher harmonics of the flapping states
during reduction. The output would include all the harmonics.
The resulting model would be used in the LQR gain design.
An approach where the command model and LQR gains
are simultaneously tuned to meet Level 1 handling qualities
while maximizing load alleviation will be considered.
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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