NASA Technical Reports Server (NTRS) 20010059025: An Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic Stability Augmentation in Airplane Mode of Flight

15 Pages·1.2 MB·English

by  NASA Technical Reports Server (NTRS)

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An Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic
Stability Augmentation in Airplane Mode of Flight
Raymond G. Kvaternik and David J. Piatak
NASA Langley Research Center
Hampton, Virginia
Mark W. Nixon, Chester W. Langston, and Jeffrey D. Singleton
Army Research Laboratory
Vehicle Technology Directorate
NASA Langley Research Center
Hampton, Virginia
Richard L. Bennett and Ross K. Brown
Bell Helicopter Textron
Fort Worth, Texas
The results of ajoint NASA/Army/Bell Helicopter Textron wind-tunnel test to assess the potential of Gener-
alized Predictive Control (GPC) for actively controlling the swashplate of tiltrotor aircraft to enhance aeroelastic
stability in the airplane mode of flight are presented. GPC is an adaptive time-domain predictive control method
that uses a linear difference equation to describe the input-output relationship of the system and to design the con-
troller. The test was conducted in the Langley Transonic Dynamics Tunnel using an unpowered I/5-scale semispan
aeroelastic model of the V-22 that was modified to incorporate a GPC-based multi-input multi-output control algo-
rithm to individually control each of the three swashplate actuators. Wing responses were used for feedback. The
GPC-based control system was highly effective in increasing the stability of the critical wing mode for all of the
conditions tested, without measurable degradation of the damping in the other modes. The algorithm was also ro-
bust with respect to its performance in adjusting torapid changes in both the rotor speed and the tunnel airspeed.
Notation y(k) vector of digitized response time histories
o__'if control law gain matrices
A, 13,T matrices (formed t?om aiand 13i)in multi- _i, fli coefficient matrices (OMP) appearing in
ARX equation and determined by SID
step output prediction equation
control horizon blade kinematic pitch-flap coupling angle
h# error function (difference between target
hp prediction horizon
J objective function to be minimized and predicted responses)
k time index Abbreviations:
I number of data time steps used for SID ARX autoregressive with exogenous input
Ell number of feedback outputs to GPC MIMO multi-input/multi-output
P order of ARX equation describing system OMP observer Markov parameters
Q,R weighting matrices for inputs and outputs SID system identification
r number of control inputs from GPC
u(k) vector of digitized input time histories
INTRODUCTION
vector of computed control inputs
IIc
V matrix of input and output measurements
weighting factors tbr inputs and outputs Tiltrotor aircraft operating at high speeds in the
W_., W r
airplane mode of flight are susceptible to a propro-
matrix of OMP determined by SID
tor/pylon instability akin to propeller whirl flutter. Such
an instability was first encountered during full-scale
testing of the Bell XV-3 tiltrotor in the NASA-Ames
Presented at the American Helicopter Society 57th Annual
40- by 80-foot Wind Tunnel in 1962. Following this
Forum, Washington, DC, May 9-11, 2001. Copyright ©2001
incident, extensive analytical and experimental studies
by the American Helicopter Society International, Inc. No
of small dynamically-scaled models were initiated by
COl_vrightisasserted in the United States under Title 17, U.S.
Code. The U.S. Government has a royalty-free license under Bell with the threefold objective of providing a physical
the copyright claimed herein for Government Purposes. All understanding of the phenomenon, developing an ana-
other rights are reserved by the copyright owner lytical method to predict such instabilities, and identify-
ingcorrectivdeesignchangesB.y1965b,othanexpla-
nationforandameanfsoreliminatintgheinstability
werefound(refs.1-2). TheXV-3aircraftwasthen
modifiedtoincorporactehangeinsdicatebdyanalyses
andretestesduccessfuilnlytheNASA-Ame4s0-by80-
footWindTunneiln 1966.Theproprotor/pyldoyn-
namicstudieisnitiatedbyBellinsupporotftheXV-3
investigatiownerefollowedby severaoltherstudies
conductebdygovernmeanntdindustroyverthenext
decad(eseef,orexamplere,['.,3;.-6).Takenasawhole,
thesestudieshelpedtoestablishthetechnologbyase
needetdolatersuccessfualldydresthseissueofpropro-
tor/pylon/winageroelastsictabilityinthedesignofthe
XV-15tiltrotorresearcahircraftinthemid-1970asnd
theV-22Ospreiynthemid-1980s.
Figure1.- Perturbatiornotor-induceaderodynamic
Themechanisomftheproprotor/pylionnstability forcesactingonatiltrotoraircrafdturingpitchingand
experiencbeydtheXV-3wasidentifiedbyBellduring yawingoscillation(rsotorsinterconnected).
theinvestigatioonfthatincidenatndwasreportebdy
Hall(ref. 1). Theprimarydestabilizinfagctorwas Proprotor/pylaoenroelastsictabilitycontinuetos
foundtobehubinplanesheafrorcesresultinfgromro- beaprimaryconsideratiionnthedesignoftiltrotorair-
torprecessiownhenoperatinagthighforwardspeedins craft. Becausweingtorsionasltiffnessis themajor
theairplanmeodeofflight.Thehubshearidsentifieads structuradlesignparameteinrfluencintghisphenome-
beingresponsibfloerthesubjecint stabilitayreactually non,thestabilityrequiremenotfscurrenttiltrotorair-
asubseotfalargersetofhubforcesandmomentthsat craft(XV-15,V-22,BA-609h)avebeenattainedby
actonatiltrotoraircraf(tref.4).Disturbancoecscurringusingthick,torsionallsytiffwingshavinga23%thick-
in flightcanexciteeithertheelasticor rigid-body ness-to-chorardtio. Suchwingsprovidethetorsional
modesofanaircraftinanoscillatormyanner.Fora stiffnesrsequirefdorstabilityinthoseaircrafta,lbeiatt
tiltrotoraircraft,anymotionsof thistypeeffectively theexpensoefcruiseefficiencyandmaximumspeed.
represeonstcillatortyranslationaanldrotationamlotions Theuseofthinnerwingswouldpermithighercruise
oftheproprotosrhaftinspaceT.hisleadstoproprotor-speedsin, creaserdange,andimprovedproductivity.
generateaderodynamfoicrcesandmomenttshatarea Howevert,heattendanrteductionin wingstiffness
functionoftheseoscillatormyotions.Figure1,from wouldbringwithittheproblemofproprotor/pylionn-
referenc4e,showtsheperturbatiorontor-induceadero- stability.Thus,aeroelastiincstabilityof thepropro-
dynamifcorcesactingatthehubsofatiltrotoraircraft tor/pylon/winsgystemstandassamajorbarriertoin-
executinsgmalpl itchingandyawingmotionwshenop- creasintghemaximumspeecdapabilitoyftiltrotorair-
eratinginanairplanmeodeofflight.Fromtheposition craft. Bothpassivaendactivemethodfsorextending
oftheseforcesontheaircrafti,tisclearthattheforces proprotor/pylosntabilityboundariefosr conventional
showncaninfluenceaircraftlongitudinaalndlateral- tiltrotoraircrafhtavebeeninvestigatedR.eference7s-
directionastlabilityH. owevetrh,esheafrorcesHandY 12,whichdescribsetudiesintotheuseofwingand
can,quiteindependenotlfyanyaircrafrtigid-bodmyo- bladeaeroelastticailoringtechniquefsor improving
tions,alsodestabilizteheproprotor-pylon-wsinysgtem stabilitya,reillustrativoefpassivaepproaches.
aeroelasticalTlyh.esearethesheafrorcesidentifiebdy
Whiletherehavebeenanumbeorfstudiedseal-
Hall(ref.1)asthedriversforproprotor-pylionnstabil-
ity. Theyareadirectconsequenocfeairloadmomentsingwiththeuseofactivecontrolsforimprovingthe
thataregeneratetodprecestsherotorinspaceinre- aeroelastbicehavioorftiltrotoraircraftm, ostofthese
sponsteoshaftpitchingandyawingmotionsI.naddi- haveaddressethdeproblemofgustandmaneuvelorad
tiontoatruewhirlinstabilityinvolvingbothpitching alleviationandtherehasbeenonlylimitedattention
andyawingmotionosfthepylonp,roprotor/pylionnsta- giventotheuseofactivecontrolfsorstabilityaugmen-
bilitycanoccurinasingleplane(eithepritchoryaw), tation(refs.13-16).Referenc1e3investigatethdeap-
dependinognthepylonsupporsttiffnessesP.ropel- plicationofswashplafteeedbacfkoraugmentatioofn
ler/pylonwhirlflutter,ontheotherhandi,sdrivenby aeroelastsictabilityaspartofabroadesrtudyoffeed-
aerodynamcicross-stiffnemssomentasndcanoccur backcontrolforimprovingtheaeroelastaicndrigid-
onlyifthereispylonflexibilityinbothpitchandyaw. bodyflightcharacteristoicfstiltrotoraircraft.Thebase-
Adiscussioonftheseandotherimportandtifferenceins lineconfiguratioonfthosestudiewsasaBoeingsoft-
theacromechanibceahlavioorfpropellerasndpropro- inplanteiltrotorknownastheMode2l22.Bodeanaly-
torsisgiveninreferenc4e. seswereusedtodefinetheappropriagteainsandphases
tobeappliedtothewingresponstehsatweretobefed
A 1/5-scale, semispan aeroelastic model of the
backtotheswashplactyeclicinputs.Reference1s4-t5
V-22 designed and built by Bell in 1981 is being used in
wereanalyticasltudieosffeedbacckontroflorincreas-
the cxperimental evaluation of GPC. That model, on
ingproprotor/pylsotnabilityonanXV-15sizctiltrotor
loan to the AB from the Navy, has been refurbished to
aircraft.Afeedbacgkainmatrixwasintroduceindtothe
form a tiltrotor research testbed called the Wing and
formulatiobnyaddingafeedbaclokoptotheequations
Rotor Aeroelastic Test System (WRATS) tor use in the
ofmotionlinearizeadbouatflightconditioonfinterest.
Langley Transonic Dynamics Tunnel (TDT) (fig. 2).
Wing tip vertical velocities and accelerations were used
tor feedback. The gains needed to stabilize the system
were determined by simply varying the terms in the gain
matrix until an eigenvalue analysis of the closed-loop
system indicated a stable system. Reference 16 was an
analytical study into the use of linear quadratic regulator
(LQR) techniques for determining the wing feedback
gains needed to stabilize the whirl modes of a tiltrotor
aircraft using an active swashplate. The method was
studied using a mathematical model that had been de-
veloped earlier for a full-size semi-span configuration of
the XV-15. Control design was done in modal space.
The discrete state-space equations were transformed to
modal form to allow for separation of stable and unsta-
ble modes during the design of the controller. A Kal-
man-Bucy filter was employed as the state estimator to
account for disturbances and noise.
The Aeroelasticity Branch (AB) and the Army Figure 2.- WRATS tiltrotor testbed installed inTDT.
Research Laboratory's Vehicle Technology Directorate
at NASA Langley Research Center, in collaboration Three exploratory experimental studies to investigate
with Bell Helicopter Textron Inc (BHTI), is evaluating the use of a multi-input/multi-output (MIMO) GPC-
based method tbr active control have been conducted on
an adaptive control technique known as Generalized
the WRATS model to-date (ref. 18). These include a
Predictive Control (GPC) (ref. 17) to assess its potential
lk_ractively controlling the swashplate of tiltrotor air- hover test with the baseline stiff-inplane gimbaled rotor
craft to enhance aeroelastic stability in both hclicopter (May 1998), a ground resonance test employing a soft-
and airplane modes of flight. The term adaptive as used inplane variant of the gimbaled rotor (October 1999),
and a wind-tunnel test with the baseline stiffness rotor
here refers to a control system in which the parameters
describing the model are identified under operational (April 2000).
conditions using measured input/output data, and the
The purpose of this paper is to summarize the re-
updated system parameters are then used to compute a
new set of control gain matrices and the next set of sults of the joint NASA/Army/BHTI investigation that
commands to be sent to the swashplate actuators, with was conducted in the TDT in April 2000 in the initial
assessment of the potential of GPC for augmenting sta-
all computations being done on-line. GPC is a time-
bility in tittrotor aircraft operating in the airplane mode
domain method that uses a linear difference equation to
of flight. The WRATS model was modified to incorpo-
describe the input-output relationship of the system and
rate a GPC system for independently controlling the
to design the controller. An ARX-type difference equa-
actuators of the fixed-system swashplate. Active con-
tion is being employed in the present study. The ARX
trol was introduced into the fixed-system swashplate
model is used for both system identification and control
using three high-frequency servo-controlled hydraulic
design. The coefficient matrices of the ARX equation
actuators mounted aft of the swashplate inside the pylon
are the quantities determined by the identification algo-
fairing. The actuator commands included the steady
rithm. Closed-loop control is enhanced by performing
pilot commands as well as the actuator motions called
the system identification in the presence of the external
for by the active control algorithm. Wing bending and
disturbances acting on the system. The coefficients of
torsion strain gages located near the wing root were the
the ARX model are assembled into a multi-step (finite-
sensors used to provide the response measurements
horizon) output prediction equation, the desired (target)
needed by the active control algorithm. The GPC-
response is specified, and the resulting expression is
computed commands were summed with the pilot trim
used to form an objective function. Minimization of the
commands to produce the desired swashplate actuation
objective function leads to an expression for the control
about the trim state. The effectiveness of the rotor
to be applied to the system.
swashplate in increasing stability (damping) was inves-
tigatedoverarangeoftunnealirspeedfosrasinglerotor The essential features of the adaptive control
rotationaslpeed.Therobustnesosftheactivecontrol
process used in the present GPC investigation are de-
systemtorapidvariationisntunnealirspeeadndrotor picted in figure 3. The system has r control inputs u, m
speedb,othsinglyandincombinatiown,asstudied. measured outputs y, and is subject to unknown external
TheeffectofGPConbladeloadsandpitchlinkloads
wasnotassessed.
Thepertinenetquationusnderlyintghemethod Disturbances (d)_
arepresenteadnddiscusseadndthestrategoyfcom-
puterimplementatioisndescribedin,cludingsystem Uicl_._+ ,)Inputs (u),.._lIv Plant IOutputs (y)
identificatiocna,lculatioonfcontrollawmatricesa,nd
calculatioonfthecontroclommandssenttotheswash-
plateactuatorsC.onsideratiorneslatetdoimplementa- uC _l SDyissttuermbanceIdentifEicsatitmioantion & _.
tionforon-linecomputationasrediscussedE.xperi- ,
mentarlesultasrethenpresenteilldustratintghestability
augmentatitohnatwasobtainedwiththeGPC-based
activecontroslystem. _[ (FeePdraecdki/cFtieveedforwCaorndt)rol _
Figure 3. - Block diagram of identification and control
GENERALIZED PREDICTIVE CONTROL procedure.
Introductory Remarks
disturbances d. Measurement noise is also present.
There are two fundamental steps involved: (1) identifi-
Predictive controllers were initially introduced in
cation of the system, and (2) use of the identified model
the chemical industries for controlling chemical proc-
to design a controller. As mentioned earlier, a linear
esses and have found applications in a wide variety of
difference equation or model is used to describe the
industrial processes (e.g., ref. 19). Predictive control
relationship between the input and the output of the sys-
refers to a strategy wherein the decision for the current
tem. A linear input-output model gives the current out-
control action is based on minimization of a quadratic
puts as a linear combination of past input and output
objective function that involves a prediction of the sys-
measurements. The input-output model used in the pre-
tem response at some number of time steps into the fu-
sent work has the form of what is referred to as an ARX
ture. A variety of predictive controllers have been pro-
(autoregressive with exogenous input) model. The
posed (e.g., ref. 20). Among these, Generalized Predic-
ARX model is used for both system identification and
tive Control (GPC), which was introduced in 1987 (rel\
controller design. System identification is done on-line,
17), has received notable attention by researchers. GPC
but not at every time point, in the presence of any dis-
is a time-domain multi-input-multi-output (MIMO) pre-
dictive control method that uses a linear difference turbances acting on the system. In this way, an estimate
of the disturbance model is reflected in the identified
equation to describe the input-output relationship of the
system model and the disturbance does not have to be
system. The input-output equation is used to form a
modeled separately. This approach represents a case of
multi-step output prediction equation over a finite pre-
feedback with embedded feedforward. Because the
diction horizon while subject to controls imposed over a
disturbance information is embedded in the feedforward
finite control horizon. The control to be imposed at the
next time step is determined by minimizing the devia- control parameters, there is no need for measurement of
tion of the predicted controlled plant outputs from the the disturbance signal (ref. 28). The parameters of the
desired (or target) outputs, subject to a penalty on con- identified model are used to compute the predictive con-
trol effort. trol law. A random excitation Uia(sometimes called
dither) is applied initially with u,.equal to zero to iden-
A novel version of the GPC procedure was de- tify the open-loop system. Dither is added to the
veloped at NASA Langley Research Center in 1997 fl)r closed-loop control input u, if it is necessary to re-
efficient computation and unknown disturbance rejec- identify the system while operating in the closed-loop
tion by Dr. Jet-Nan Juang and his co-workers. Their mode.
work has resulted in a suite of MATLAB m-files that
have been collected into a Predictive Toolbox that can Form of Model and Control Law Equations
be used by researchers for GPC studies. A summary of
the SID and control theory underlying their develop- The relationship between the input and output
ment is found in references 21-29, among others. digitized time histories of a MIMO system are described
byanARXmodetlhat
has the form where
y(k) = a, y(k-1) + _2y(k-2) + ... +o_, y(k-p)
(1)
y=[y(O) y(I) y(2)...y(p)...y(l-1)] (4)
+ fl0u(k) +fl_u(k-1) + ... +fl,,u(k-p)
m x l
This equation states that the current output y(k) at time and
step k may he estimated by using p sets of the previous
output and input measurements, y(k-l) ..... y(k-p) and
1
u(k-l) ..... u(k-p); and the current input measurement
v(0) v(1) .-. v(p-l) .-- v(/-2)
u(k). The integer p is called the order of the ARX
model. The coefficient matrices ai and fit appearing in (5)
V= v(0) iiI v(p.-2) ... v(/-3)
this equation are referred to as observer Markov pa- u(0) u(l) u(2).., u(p) ... u(l-1) /
rameters (OMP) or ARX parameters and are the quanti- v(O) *'" v(l- ip -1)
ties to be determined by the identification algorithm.
[r +(r +m)p] x l
Closed-loop robustness is enhanced by performing the
system identification in the presence of the external Equations 4 and 5 follow from writing the discrete-time
disturbances acting on the system, thereby ensuring that state-space equations for a linear time-invariant system
disturbance information will be incorporated into the at a sequence of time steps k = 0, I..... l-1and grouping
system model. The goal of SID is to determine the
them into matrix form. The vector v(k) appearing in the
OMP based on input and output data. The OMP may be
data matrix V is formed from the vectors u(k) and y(k)
determined by any SID technique that returns an ARX
according to
model of the system.
The ARX model is used to design the controller
(6)
and leads to a control law that in the case of a regulator
problem has the general form given by (r+m} xI
The order of the ARX model (p) and the number of time
u_(k)=6r[y(k-I) + a';y(k-2) + --- +@ y(k-p)
(2) steps (/) must be specified by the user. Some guidelines
+,8;,(k-D +¢_;y(k-2) + ... +¢_i;,(k-p) for their selection are given later. The sizes of the vec-
tors and arrays are noted.
Equation 2 indicates that the current control input u,(k)
may be computed using p sets of the previous input and In forming the matrices given in equations 4 and
output measurements. The coefficient matrices c( and 5, it has been assumed that the state matrix A is asymp-
if appearing in this equation are the control gain matri- totically stable so that for some sufficiently large p, Ak=
ces. 0 for all time steps k > p, and that an observer has been
added to the system. It is through these expedients that
System Identification the matrix V is reduced to a size amenable for practical
numerical computation of its pseudo-inverse. The SID
System identification in the presence of the op- process yields OMP rather than system Markov parame-
erational disturbances acting on the system is the first of ters (SMP) because of the inclusion of an observer. A
the two major computational steps. The external distur- complete discussion of these aspects of the development
bances acting on the system are assumed to be unknown may be found inreference 21.
(unmeasurable). The number of control inputs is r and
the number of measured outputs is m. The system is
Y is the matrix of observer Markov parameters
excited with band-limited white noise for SID, These
that are to be identified and has the form
random excitations are input to all r control inputs si-
multaneously and the m responses are measured. The
digitized input and output time histories (u and 3') at l
time points are then used to form the data matrices 3'and
V in the equation t?lxr tHxr 71lxIH?Hxr m xIH mxr 71lxm tHx r nl xt?l (7)
mx[r +{r +m}p]
y= Y V (3)
ThesolutionfbrY is obtained by solving equation 3 and fig. Equation 9 shows that the output y(k+j) at time
step k+j may be estimated by using p sets of the previ-
for Y according to ous output and input measurements, y(k-l) ..... y(k-p)
and u(k-l) ..... u(k-p), and the (unknown) current and
-y = y Vt= y vr[v (8) future inputs, u(k), u(k+l) ..... u(k+j). The GPC algo-
vT_ -1
rithm is based on system output predictions over a finite
where 4idenotes the pseudo-inverse. If the product V Vr horizon hi, known as the prediction horizon. To predict
is a well-conditioned matrix of reasonable size, the or- future plant outputs, some assumption needs to be made
dinary inverse can be taken as shown. Otherwise, a about future control inputs. In determining the future
pseudo-inverse must be used• It should be noted that control inputs for GPC, it is assumed that control is ap-
because the size of V Vr is much smaller than V, a plied over a finite horizon h,.known as the control hori-
pseudo-inverse might be appropriate even if the product zon. Beyond the control horizon the control input is
is well conditioned. assumed to be zero. In GPC, the control horizon is al-
ways equal to or less than the prediction horizon. Let-
Multi-Step Output Prediction Equation ting j in equation 9 range over the set of values j = I, 2,
.... hi,-1, the resulting equations can be assembled into a
The one-step ahead output prediction equation given multi-step output prediction equation having the form
in equation I is the starting point for deriving the multi-
step output prediction equation that is needed for de- ' (k)= 7- uj,(k)+ B uv(k-p)+ .,4 yp(k-p)
)hp h_tlxh "c hl,nlXl., hl,mXl_t I"
signing a GPC controller• Using equation l, the output hpm x[ ' hrxI pr xI pinxI
at time step k+j may be written in the form (10)
The coefficient matrices T, B, and ..4are formed from
.v...(k +j)=o'li_v(k-I) + o'_j_v(k-2) + "'" +o:tj!)(,k-p)
combinations of the observer Markov parameters ai and
+fl,,u(k +j)+fl',"u(k +j - I) +..-+ fl,,Ijl u(k) (9) fl,. The quantity y;,;,(k) is the vector containing the fu-
+fl,"u(k-l)+fl_i'u(k - 2) + ... +fll/'u(k-l,) ture outputs, whereas Uh,(k) is the vector containing the
future control inputs yet to be determined. The quanti-
where the coefficient matrices are given by recursive ties ul,(k-p) and yp(k-p) are vectors containing the previ-
ous p sets of control inputs and outputs, respectively.
expressions involving the matrices ai and fli appearing
The expanded form of this multi-step output prediction
in the ARX equation (ref. 25). The system identifica-
equation is shown in equation 11.
tion process described earlier determines the matrices as
= "+a("'%1
y(k) A_
A:" A,
y(k +1)
u(k) ]
y(k + q- 1) _{/i-I) /_f(jq-2) .. AJ hc< hp
AI" u(k.+ 1) /
y(k +q) A(//) j_f I/-I } .,
u(k +/t - l)J
: - -.
y(k + h_,- 1) _(,h,-l) t_,{/,,,-2) .. [/_e-h, )
(11)
O'I O_2 •' • O'p_ I 19//, _2 "" 1_I,I /8/,
0_/I) 0(_ I) ... _il)l _'/)) y(k -l) ]_i( I) 2 .. R_pI_-I BrpII_ u(k - 1)
: : ".. : : y(k -2) u(k - 2)
+ OYlqM) _2/T(q-I ) """ _pgY(q-[)l _l'(q-II •q- ]_l('l-l) 2(q-I) .• _/p_(q-l) _rp(q-I)
al" a_" "" Gp(q)_, a'p(q) y(k-p+l) t,_lIq) j_(q2) ... Rt"(q)p-I /__plql u(k-p+l)
: : ".. - : y(k - p) ".. : : u(k - p)
•.. P2,-"E"-"
t_( hr-I ) _/_Ihe-I)
I 2
=Gr + GI'I t_Gv-Il t"p'_-I(q) ---- _I(q-Ip) "l-o(lq-l)j_/,_l ]
The OMP ai and fli determined in SID form the first
ul_(k)=- (7-rRT+ Q )+7-rR (_3y(k)+/3t_,(k- p)+.Ayp(k- p))
block rows of the coefficient matrices T,, ,4, and /3 in
: _/ (-3y(k)+13t_,(k-p)+A)),(k-p))
equation 11. The terms in the remaining rows are com-
puted using the recursive relations indicated in the I_.rxl
boxes (ref. 25). All terms in equation 11 are known, (14)
except for the h, sets of future commands and the he,sets
of predicted responses. The goal of the GPC control as the control sequence to be applied to the system over
the next h,. time steps. However, only the first r values
algorithm is to determine the set of future commands
u(k), u(k+l) ..... u(k+h,.-l) that are required to achieve (corresponding to the first future time step)
adesired predicted response y(k), y(k+ 1)..... y(k+hl,- I).
u, (k) = -7_ vT(k) + fl'u,, (k - p) a v,,(k - p)
It should be remarked that the system Markov rxl
parameters (SMP), which are commonly used as the (15)
basis for identifying discrete-time models lbr linear dy-
arc applied to the r control inputs, the remainder are
namical systems, form the first block column in the ma-
discarded, and a new control sequence is calculated at
trix U, the remaining block columns are formed from
the next time step.
subsets of the SMP. The Markov parameters are the
pulse response of a system and are unique for a given The target response is zero for a regulator prob-
system. The discrete-time state-space matrices A, B, C, lem and non-zero for a tracking problem. The matrix Q
and D are embedded in the SMP. must be tuned to ensure a stable closed-loop system.
Typically, h,.is chosen equal to hi,. However, h,.may be
Derivation of Control Law
chosen less than hpresulting in a more stable, but slug-
gish, regulator.
The predictive control law is obtained by mini-
mizing the deviation of the predicted controlled re- An expression for estimating the order of the
sponse (as computed from the multi-step output predic- ARX model that is to be used for SID is given by
tion equation) from a specified target response over a
prediction horizon hi,. To this end, one first defines an
number qf + number of
error function that is the difference between the desired
._3,stemstates disturbance states
p>ceil (16)
(target) response YT (k) and the predicted response
in
yhj,(k):
where ceil denotes rounding up the value of the quantity
e = YT(k) - Yhp (k)
in parentheses to the next higher integer. The number
of system states is typically chosen to be twice the num-
= Yv(k)-'Tut_ (k)-]3tlp(k - p)-Ayl,(k- p)
ber of significant structural modes; the number of dis-
(12)
turbance states is set to twice the number of frequencies
in the disturbance; m is the number of output measure-
An objective function J quadratic in the error and the ments. If measurement noise is of concern, the order of
unknown future controls is then formed:
p so computed should be increased to allow for compu-
tational poles and zeros to improve system identification
J = E'TR _"+ u_ Q uj_. (13) in the presence of noise. In practice, simply choosing p
to be 5-6 times the number of significant modes in the
system is often adequate. The prediction and control
Two weighting matrices are included in the objective
horizons are set according to the relations
function: Q (symmetric and positive definite) is used to
weight the control effort and stabilize the closed-loop
system; R (symmetric and positive semi-definite) is hp ->p hc _<hp (17)
used to weight the relative importance of the differences
Although hi, can be set equal to p, h_, is typically set
between the target and predicted responses. Typically,
greater than p to weight the control effort so as not to
Q and R are assumed to be diagonal and for Q to have
saturate the control actuators. If the control horizon is
the same value w,. along its diagonal and R to have the
greater than the system order a minimum energy (mini-
same value w, along its diagonal. Minimizing J with
mum norm) solution is obtained wherein the com-
respect to uh,(k) and then solving for uh,.(k) gives
manded output is shared so that the control actuators
don't fight each other. If hl,is set equal to p one obtains
aso-calleddeadbecaotntrolle(rref.25).Byextending SID should be done with the external distur-
thepredictionandcontrohl orizon(sandhence
p, the bances acting on the system so that information about
order of the ARX model) to very large values, the GPC the disturbances is embedded in the OMP. However,
solution approaches that of the linear quadratic regulator depending on the nature of the external disturbances, it
(LQR). Thus, GPC approximates an optimal controller may be possible to perform a SID on a system without
[or large p (ref. 17). the external disturbances and still determine a control
law that results in satisfactory closed-loop performance.
Weighting matrices Q and R are used to weight
the control effort and to weight the relative importance The computation of pseudo-inverses should be
of the differences between the target and predicted re- performed using Singular Value Decomposition (SVD)
sponses, respectively. As mentioned earlier, Q and R because of the latter's ability to deal with matrices that
are usually assumed to be diagonal and for Q to have are numerically ill conditioned. The use of pseudo-
the same value w,. along its diagonal and R to have the inverses (via SVD) is recommended even in cases
same value w, along its diagonal. The control weight where the ordinary inverse may seem appropriate (such
must be tuned to produce an acceptable solution without as in the operation (VVr) nindicated in equation 8).
going unstable. Reducing w, increases control authority
A number of decisions also need to be made.
and performance but eventually drives the system un-
stable. For example, should the system be re-identified and the
control law matrices recalculated in every sampling
Key Features of Present Method period, every specified number of time steps (block up-
dating), or only if some event requiring a re-ID occurs'?
An ARX model is employed to represent the sys- Should all the calculations be done in real time, or can
tem and is used for both system identification and con- some be done in near real time'? Should updating of the
troller design. The SID process used makes recourse to OMP be done in batch mode or recursively?
an observer to enable numerical computation of the
pseudo-inverse needed for calculation of the OMP that Microprocessor speeds are such that it is now of-
comprise the coefficients of the ARX equation. The ten possible to complete the full cycle of GPC computa-
controller is thus inherently observer-based but no ex- tions and apply the commands to the actuators within
plicit consideration of the observer needs to be taken one sampling period. However, the need for efficient
into account in the implementation. In practice, the computational algorithms and attendant coding is not
disturbances acting on the system are unknown or un- expected to diminish.
measurable. However, as discussed in reference 28, by
performing the SID in the presence of the external dis- Implementation Considerations
turbances acting on a system, a disturbance model is
Several considerations must be taken into ac-
implicitly incorporated into the identified observer
Markov parameters. However, the identified model count when actually implementing GPC algorithms in
unust be larger than the true system model to accommo- hardware for active controls work.
date the unknown disturbances. Thus, the effects of the
unknown disturbances acting on the system are embed- The measured response time histories must be
ded in the matrices .,4and/3, and hence the control law passed through a low-pass filter with a cut-off frequency
set equal to the Nyquist frequency)'u. The latter is cho-
matrices a" and if. If only the disturbances acting on
sen so that the maximum frequency of interest is about
the system change, there is no need to recalculate
75% offN. The sampling frequency f_should be at least
7" (and hence y') because 7" is formed solely from the twice fN to prevent aliasing. However, iffs is made too
SMP, which are unique for a given system (ref. 28). large the low frequency modes will be poorly identified
The solution for Y indicated in equation 8 involves due to a loss of frequency resolution. A sampling rate
between 2to 3timesfN is generally sufficient. Once the
forming the matrix products yVTand VVr. Here, these
sampling frequency has been selected, the minimum
products are obtained using the computationally effi-
number of data points that should be used for SID fol-
cient procedure described in reference 23.
lows from the requirement of having 5-10 cycles of the
lowest frequency mode in the measured response time
Computational Considerations
histories. Normalization of the input and output data
that is used for SID on the maximum actual or expected
Several considerations dealing with computa-
values of the data is often helpful numerically. The
tions should be kept in mind when developing algo-
procedure will depend on whether the computations are
rithms for GPC applications.
being done in batch mode or recursively.
Thecomputintgaskscanbedistributeadmong ness is provided by means of a hollow, rectangular cross
computeorsrdifferenCtPUsonasinglecomputeTr.he section, composite spar having chordwise flanges. The
choicewill influencetheextentofuserinvolvement.
4.6-foot spar, which lies along the calculated elastic axis
Thevaluesoftheinputandoutpudtatathatarebeing of the wing, has segmented, nonstructural, aluminum
usedduringclosed-looopperationmsustbecarefully aerodynamic fairings that provide the spanwise distribu-
monitoretdo ensurethattheyfall withinacceptabletion of airloil contour. To provide surface continuity
boundsT.hisiseasilydoneusingIF/THEN-typcehecks over the lifting surface of the wing, the space between
inthecode.
the segments is filled with strips of foam rubber. The
wing-tip-mounted pylon contains the transmission and
gearbox components for the rotor drive system, the
EXPERIMENTALSETUP
lower part of the mast, and the swashplate control sys-
tem. Because these internal components can be treated
Wind Tunnel
as rigid, the pylon is scaled dynamically so as to pre-
The experiment was conducted in the Langley serve its overall mass and inertia properties. The pylon
Transonic Dynamics Tunnel, which is a continuous is attached to the wing tip by means of a "racetrack"
flow, single return, variable pressure tunnel having a spring assembly that simulates the combined stiffness of
test section 16-feet square with cropped corners. The the lull-scale conversion actuator and downstop lock
control room and test section walls are provided with mechanism.
large windows for close viewing of the model. The
tunnel is capable of operation at stagnation pressures The 3-bladed, 7.6-foot diameter stiff-inplane ro-
from near vacuum to slightly above atmospheric and at tor has a gimbaled hub that is connected to the mast by a
Mach numbers from near zero up to about 1.2. Either constant speed joint (2 coincident universal joints). The
air or a heavy gas (R-134a) can be used as the test me- rotor yoke has 2.5 degrees of built-in precone and is
dium. Both the density and test-section Math number flexible to allow the blade coning angle to adjust under
are continuously controllable. The present investigation centrifugal loading. The blades have a nonlinear distri-
was conducted in air near atmospheric conditions and at bution of built-in twist with an overall root-to-tip twist
free-stream Math numbers less that 0.30. of 47.5 degrees (leading edge down).
Model A large plywood panel through the vertical plane
of symmetry of the model and attached to a support
The model used in the investigation is a modified
structure consisting of a spider-like arrangement of steel
version of a Froude-scaled semispan aeroelastic model
tubes was employed to seal the open back side of the
of the V-22 tiltrotor that was used by Bell/Boeing to semi-fuselage and to serve as a reflection plane. The
support the preliminary and full-scale design phases of structure also provided the offset needed to position the
the aircraft (Ref. 30). Upon completion of that series of rotor axis near the centerline of the wind tunnel.
tests, the Navy transferred the model to the Aeroelastic-
ity Branch (AB) at NASA Langley under a loan agree- The fundamental natural frequencies of the
ment to be used as the experimental testbed of a tiltrotor model (as measured on the test stand) with the pylon
aeroelastic research program that was initiated by AB in locked to the downstop in an airplane mode configura-
1994. The tiltrotor testbed has been designated the tion are summarized in Table I. Typical wind-on natural
Wing and Rotor Aeroelastic Testing System (WRATS).
General Characteristics
Table I- System Natural Frequencies*
The WRATS tiltrotor testbed as installed in the
Langley Transonic Dynamics Tunnel lor this study is Mode Frequency, Hz
shown in figure 2. The model has alength scale factor Wing Beam Bending 5.83
of 1/5 and was designed to maintain full-scale values of Wing Chord Bending 8.67
the Froude, Lock, and Strouhal numbers when operated Wing Torsion 12.0
in air (Ref. 30). The wing and rotor are dynamically Pylon Yaw (2nd wing chord) 19.4
and aeroelastically scaled; the pylon is only dynamically Blade 1st Elastic Flap 7.20
scaled. The fuselage is rigid and only maintains the Blade 1st Elastic Lag 12.5
scaled external aerodynamic shape. The model is at- Blade Rigid Body Torsion 95.6
tached to a support structure that is effectively rigid, its Blade 1stElastic Torsion llO.
lowest frequency being well above any important elastic
Rotor on, flapping locked out, zero rpm.
mode frequency of the model. Simulation of the dis-
tributed wing beamwise, chordwise, and torsional stiff-
frequencioefstherotorsystem(e.g.a,tV=100 knots and eluding gimbal flapping angles and downstop spring
load. Pitch-link and blade loads were not measured.
f_ = 742 rpm) are: gimbal flapping at 0.85P, first cyclic
lag at 1.2P, and collective flapping (coning mode) at
about 1.7 P. The present test was run with the drive Control System Architecture
system disconnected so the collective lag mode had zero
frequency. A schematic diagram of the control system for
active stability augmentation is shown in figure 5. The
Active Swashplate Hardware flapping response of the rotor is displayed
Three high-frequency electro-hydraulic actuators
having a bandwidth of 50 Hertz were used in the control Pilot I _ 1P
system to drive the swashplate (fig. 4). The actuators ,Conclrol _ ofnlapping .
are positioned azimuthally at the 12, 4, and 8 o'clock
positions around and slightly aft of the nonrotating oo OPIrim SyNem responses ]_
swashplate. These actuators were used to input the
steady pilot commands as well as the swashplate inputs
called for by the GPC active control algorithm. Three
J/AcI u_o r -
3000-psi servo-valves were used to meter hydraulic
fluid to the actuators. These servo-valves were located
inside the pylon near their corresponding actuators (see
GPC _ IFeedback
fig. 4). Conlrol ] _ I responses
I sy._,. I
Servo-valve system
High-frequency Figure 5. -Schematic of active stability control system.
to the pilot as 1P lateral and longitudinal flapping of the
rotor tip-path-plane. The pilot trims the windmilling
rotor by setting both the collective pitch (to maintain the
desired rotor speed) and the cyclic pitch (to maintain
zero flapping with respect to the shaft). A mixer in the
pilot control console calculates the commands to be sent
to the individual actuators to trim the rotor based on the
pilot stick commands. The GPC control system calcu-
lates the actuator commands needed for stability aug-
mentation. The steady 0P (DC) swashplate actuator
commands required for trim are summed with the oscil-
Active swashplate latory actuator commands calculated by the GPC system
(after those commands are converted from digital to
Figure 4. - Major hardware components of the GPC analog signals).
active control system.
Data Acquisition and Processing
Feedback Signals and Instrumentation The data acquisition system (DAS) that is part of
the WRATS testbed is a 64-channel, PC-based system
Feedback signals for the active control system consisting of a National Instruments data acquisition
were developed from the measured responses of the board housed in a 750 MHz Pentium processor com-
pylon/wing system. The suite of candidate responses puter with a per-channel sampling rate of 1000 Hz. The
included wing strain gages (beam, chord, and torsion) WRATS DAS is controlled using a LabVIEW front
and six pylon accelerations (normal, lateral, and axial). panel that has been customized to support WRATS
The beam and chord strain gages are located 17.0 inches model testing. LabVIEW applications have been devel-
out from the root and the torsion strain gages are located oped for on-line FFT analyses, harmonic analyses, ex-
23.0 inches out from the root. The wing bending and citing the model via stick-stir inputs to the swashplate,
torsion gages were the sensors used for feedback. A damping estimations using moving block and logarith-
number of other measurements were monitored and mic decrement analyses, and plotting. The digitized
recorded either to trim the rotor or for flight safety, in- data records are written in a binary file format to the
10