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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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