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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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