NASA Technical Reports Server (NTRS) 20010029664: Aeroelastic Stability of A Soft-Inplane Gimballed Tiltrotor Model In Hover

14 Pages·0.96 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA 2001 - 1533
Aeroelastic Stability Of A Soft-lnplane
Gimballed Tiltrotor Model In Hover
Mark W. Nixon, Chester W. Langston, Jeffrey D. Singleton
U.S. Army Research Laboratory
Hampton, VA
David J. Piatak, Raymond G. Kvaternik
NASA Langley Research Center
Hampton, VA
Lawrence M. Corso, Ross Brown
Bell Helicopter Textron, Inc.
Fort Worth, TX
AIAA/ASM E/ASCE/AHS/ASC Structures,
Structural Dynamics and Materials Conference
April 16-19, 2001 / Seattle, WA
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500 Reston, VA 20191

AIAA-2001-1533
AEROELASTIC STABILITY OF A SOFT-INPLANE
GIMBALLED TILTROTOR MODEL IN HOVER*
Mark W. Nixon Chester W. Langston Jeffrey D. Singleton
Aerospace Engineer Aerospace Engineer Aerospace Engineer
U.S. Army Vehicle Technology Directorate
Langley Research Center
Hampton, VA
David J. Piatak Raymond G. Kvaternik Lawrence M. Corso Ross Brown
Aerospace Technologist Senior Research Engineer Engineer Engineer
NASA Aeroelasticity Branch Rotor Dynanfics Research Projects
Langley Research Center Bell Helicopter Textron, Inc.
Hampton, VA Fort Worth, TX
ABSTRACT INTRODUCTION
Soft-inplane rotor systems can significantly reduce While three-bladed, gimballed, stiff-inplane rotor
the inplane rotor loads generated during the maneu- systems are employed for the current generation of
vers of large tiltrotors, thereby reducing the strength tiltrotors (XV-15, V-22, BA609), the weight and per-
requirements and the associated structural weight, of fornmnce penalties of stiff-inplane rotors may become
the hub. Soft-inplane rotor systems, however, arc too significant for use on future systems larger than
subject to instabilities associated with ground reso- the V-22. In airplane mode, rotor loads are usually
nance, and for tiltrotors this instability has increased small compared to those in helicopter mode, due to
complexity a,s compared to a conventional helicopter. reduced disk loading and axisymmetric inflow con-
Researchers at Langley Research Center and Bell ditions. However, during airplane-mode maneuvers
Helicopter-Textron, Inc. have completed an initial such as a rapid pull-up, high oscillatory inplane ro-
study of a sofl-inplane gimballed tiltrotor model sub- tor loads are developed. Tiltrotors employ large
ject to ground resonance conditions in hover. Para- highly-twisted blades, creating a significant aerody-
metric variations of the rotor collective pitch and nanfie forcing in the inplane direction where the cen-
blade root damping, and their associated effects on trifugal restoring forces are much lower than those
the model stability were examined. Also considered associated with the flapping direction. Stiff-inplane
in the study was the effectiveness of an active swash- hubs must be designed for these inplane loads us-
plate and a generalized predictive control (GPC) al- ing additional structure which increases weight and
gorithm for stability augmentation of the ground res- can make the inplane stiffness higher still, leading to
onance conditions. Results of this study show that more increases in the inplane loads. For larger tiltro-
the ground resonance behavior of a gimballed soft- tors, this cyclic design challenge can lead to very high
inplane tiltrotor can be significantly different, from structural weights or even to an infeasible design al-
that of a classical soff-inplane helicopter rotor. The together. As a design trade-off, the maximum aero-
GPC-based active swashplate was successfully imple- dynamic load capability of a large tiltrotor may be
mented, and served to significantly augment damping limited to reduce the structural weight of the hub.
of the critical modes to an acceptable value. Such is the case for the V-22 which currently em-
ploys a controller to linfit the body pitch-rate motion
of the aircraft and thereby curtail rotor inplane loads
which could otherwise rise above their design allow-
Thi,_ paper isdeclared a work of the U.S. Clovermm_nt and ables (Ref. 1).
is not subject to copyrighl protection in the Uniled State.s.
Soft-inplane rotor systems can significantly reduce
Presented at the 42nd AIAA/ASME/ASCE/AHS/ASC
the inplane rotor loads generated during maneuvers
Sturctures, Structural Dynamics, and Materials Conference.
Seattle, Washington, April 16-19, 2001, of larger tiltrotors, thereby reducing strength re-
search presented in Refs. 2-10 explain well the phe-
nomenon and characteristics of the onset of ground
resonance instabilities. However, there are several
differences between helicopters and tiltrotors which
suggest that ground resonance behavior of tiltrotors
may be significantly different than ground resonance
of a helicopter. There is increased complexity for
stiff-inplane
a tiltrotor associated with the participation of elastic
¢.) design point
-& wing modes, in addition to the body and rotor modes,
E which couple through the changing rotor speed during
<
wind-up (start-up). The ground resonance condition
)lane design point is improved with an increase in damping in both the
rotor system lag mode and the body or wing mode
with which it participates. While the damping of the
0 1 2 3 body modes may be supplemented through mechan-
ical design of the landing gear, there is little that
Inplane Natural Frequency/_l
can be adjusted in the elastic modes of a tiltrotor
wing without difficulty and significant wing design
changes.
Figur, .: Effect of lag frequency on hub loads.
The type of hub used for a tiltrotor is another
quirements on the hub, leading to reduced structural important consideration influencing the characteris-
weight and improved aircraft agility. The plot of tics of ground resonance. Previous studies, as is
Fig. 1 illustrates the benefit with respect to loads discussed in the review paper of Ref. 10, have con-
for moving from a stiff-inplane design point to a soft- sidered the important differences between the fully-
inplane design point. For a soft-inplane rotor system, articulated, hingeless, and bearingless rotor systems
the lower the inplane natural fre, luency becomes, the with respect to ground resonance on helicopter sys-
lower the associated hub loads become. The plot tems. However, these past efforts may not be ap-
also shows that there is a theoretical asymptotic in- plicable to the present effort because none of these
crease in the load factors at 1P due to the resonance conventional hub systems are currently employed on
of the inplane mode (lead-lag mode) with rotor speed tiltrotors. Standard bearingless rotor systems do not
("lag mode" will be used throughout the remainder of provide enough pitch control for tiltrotor applicati,,ns
the paper as a description of this mode, however, the while hingeles s rotors do not provide enough flap_ ";
terms "inplane" and "lag" will be used interchange- control during differential cyclic maneuvers. Arti. _-
ably to be consistent with conimon usage of the in- lated rotor systems are heavier and produce higher
dustry). For a soft-inplane rotor system to be ef- drag than the gimballed rotor systems that have be-
fective, it should have a lag frequency below about come the standard for tiltrotors to date. The current
0.9/rev, otherwise the loads are about the same as study will focus on the ground resonance !)ehavior
those of the stiff inplane rotor system. A desirable of a soft-inplane gimballed rotor system, _ ich may
design point for development of a soft-inplane tiltro- display results significantly different from those asso-
tot is with a lag frequency of about 0.7/rev (as in- ciated with the hub systems considered in past stud-
dicated oil Fig. 1). This lag frequency provides a ies. However, it should be noted that this type of
good reduction in design loads and it is about the rotor system may not be feasible for implementation
lowest lag frequency that is practical for the types of on tiltrotors due to other stability considerations such
soft-inplane hubs currently under consideration for as high-speed whirlflutter. Although fully-articulated
full-scale development. Soft-inplane rotor systems, rotors may produce a greater drag penalty in airplane
however, are subject to instabilities associated with mode, some modification of this hub type may need to
ground resonance. Before so_-inplane rotor systems be considered to make soft-inplane tiltrotors feasible
become viable for application to tiltrotors, a com- for all aeroelastic considerations. Additional analy-
prehensive understanding of the means for avoiding sis and testing is required to identify the ideal hub
ground resonance instabilities must be established, configuration for soft-inplane tiltrotor applications.
but there have been few investigations related to this To date, the most significant studies of tiltro-
subject to date. tor systems subject to ground resonance conditions
The ground resonance phenomenon is well- were made by Boeing Helicopter with the develop-
understood for conventional helicopters, and the re- ment of their Model 222 tiltrotor concept, a hinge-
lesssoft-inplanreotorsystem,whichwastheiren-
try intoaNASA/Army-sponsorteildtrotorresearch
aircraftprogram(eventuallynamedtheXV-15and
wonby theBellHelicopterstiff-inplanerotorsys-
tem). Groundandair resonancbeehaviorof the
Boeingsoft-inplaneconfigurationwasaddresseidn
severaelxperimentaanl danalyticasltudiesusingdif-
ferentsizerotortestapparatusebs,eginningwitha
1/10-scawleind-tunneml odeal sdescribeidnRef.11,
andendingwithafull-scale26-ft.diametesremispan
modetlestedintheNASAAmes40-x 80-ft. tun-
nelasdescribedin Refs.12and13. TheBoeing
soft-inplandeesignhadarelativelyhighinplanenat-
uralfrequencys,uchthatthedesignrotorspeedin
hovermodedidnotcreateagroundresonancperob
lem. Theonlyexperimentarelsultsassociatewdith
aeromechanicinasltabilityofthisconfiguratiownas
obtainedwiththesystemin airplanemodesubject
toairresonancceonditions.Thisconfiguratiownas
Figure 2: Hover-cell home of the Wing and Rotor
alsonotgimballedi,t wasastandardhingelesssoft-
Aeroelastic Testing System at the TDT.
inplanedesign. Thus,the tiltrotor configuration
ofthecurrentstudyisuniqueintwoimportantde-
also to assess the feasibility of using a generalized pre-
signparameters1:)theuseofagimbaal ndconstant
dictive controller (GPC) with an actiw_ swashplate
velocityjoint whichhasa significanetffectonthe
control system to augment dmnping and eliminate
Coriolis forces and therefore the flap-lag blade cou- the ground resonance instahilities. Parametric vari-
pling, and 2) the use of a "low" lag frequency (about ations of the rotor collective pitch and blade root
0.5/rev) which creates a resonance condition between damping, and their associated effects on the model
the low-fl'equency lag mode and the critical tiltrotor stability were also examined in the. study.
pylon/wing mode that is welt within the design ro-
tor speed envelope. While this latter parameter is
APPARATUS
not considered a desirable design goal, it does pro-
vide some benefits for the current study ms shall be
The experimental study was performed in a 30' x
discussed.
30' hover cell (Fig. 2) located in a high-bay building
Researchers at NASA Langley Research Center adjacent to the Transonic Dynamics Tmmel (TDT)
(LaRC) and Bell Helicopter-Textron, Inc. (BHTI) at the NASA Langley Research Center in Hamp-
have completed an initial study of a soft-inplane gim- ton, Virginia. Notable features of the facility are
balled tiltrotor model subject to ground resonance a backstop mount which centers the tiltrotor model
conditions in hover. This effort was planned as part in the hover cell at the same height above the floor as
of a Memorandum of Agreement between the NASA when mounted in the TDT wind-tuime] test section
LaRC and BHTI to "Perform Experimental Aeroelas- (8 ft.); a snubber stand to halt pyhm motion in the
tic Studies of a Tiltrotor Model," and represent,_ the advent of air instability; a 3000 psi hydraulic sys-
first experiment in a series of tests to 1)e conducted tem; 100 psi shop air; 440 volt electric motor power;
with this soft-inplane hubs on the Wing mM Rotor and a closed-circuit chilled-water system for motor
Aeroelastic Testing System (WRATS). The phrase cooling. The model mount was designed to provide
"initial study", as used in the first sentence of this stiffness similar to that of the TDT test section side-
paragraph, indicates that the current hub design is wM1 mount so that system frequencies are identical
not scaled from a fifll-scale design that is deemed in either the hover or wind-tmmel facilities. Instru-
practical for large scale application. Rather, tire cur- mentation wiring runs from the model into an un-
rent design was developed as a low-cost soft-inplane derground cable tray, out of the hover cage, and into
modification to the existing stiff-inplane gimballed a block house control room. Testing is monitored in
hub representative of the V-22. The objective of the control rooln using a closed-circuit camera system
the current study was to evaluate the stability char- and standard televisions. Signal conditioning, data
acteristics of the gimballed soft-inplane rotor system aeqnisiti(m equipment and the model pilot control
subject to ground resonance conditions in hover, and console are located side-t)y-side in the control room.
iigure3:WRATSgimballedsoft-inplanpearametri- Figure 4: WRATS soff-inplane system in hover.
callyvariablehub.
with the normal stiff-inplane hub modified with the
AremotecontrolunitisusedtooperatetheMGset
addition of lag hinges to provide a soft-inplane ro-
whichcontrolsmoderlotorspeed.It shouldbenoted
tor configuration. As illustrated in Fig. 3, the soft-
thatwhileseveral
comparisons between the hover cell inplane hub is parametrically variable with a set of re-
and TDT test section were mentioned in this para- placeable coil springs (located behind the lag spring
graph, the current stud_" was performed entirely in catches) used to tune the lag stiffness, and an ad-
the hover test cell. N_ wind-tunnel test results have justable hydraulic damper used to tune the lag damp-
yet been conducted on the soft-inplane version of the ing. The dampers are adjusted by turning a screw
tiltrotor model. which controls the flow of fluid within the damper
casing. With the screw open to about 5 turns the
The hover test facility described in the previous dmnper is ineffective, and only the structural and
paragraph was developed in 1995 msa dedicated area mechanical damping of the system is present (about
for the Wing and Rotor Aeroelastic Testing System 3%). With the screw shut, the damper is almost
(WRATS), a semi-span 1/5-size acroelastic tiltrotor immovable and the lag motion is critically damped.
model based on the V-22. This tiltrotor model has
Another notable feature of the WRATS tiltro-
been used in several aeroelastic experimental efforts
tor model is the hydraulically-controlled swashplate
beginning in 1984 as part of the Navy's JVX pro-
which has high bandwidth controller capability.
gram, and more recently has been on loan to NASA
Three oil cylinder actuators are used to position the
LaRC. Since 1994, BHTI and NASA LaRC have had
swashplate, each controlled by a Moog servo valve
an ongoing cooperative research agreement in place
using an attached linear variable displacement trans-
to perform experimental aeroelastic investigations in-
ducer (LVDT) for position feedback. The pilot con-
volving the WRATS model with several associated
trol console has three inputs for AC/DC signals (ac-
modifications. Some of the more notable investi-
tive control commands) which may be summed with
gations include stability augmentation using a com-
the pilot DC input commands and sent to the swash-
posite tailored wing (Ref. 14), vibration reduction
plate control system. The model is shown in Fig. 4
using an active flap (Ref. 15), and vibration reduc-
configured for hover testing; and surrounded by the
tion using an active swashplate (Ref. 16). Important
snubber system, but loosely attached to it so that the
general features of the model are listed as follows:
system properties are not effected.
an aeroelastically-scaled wing with renmvable airfoil
panels, a dynamically-scaled pylon with a downstop For testing ground resonance instability, the snub-
spring tuned to provide elastic mode shapes and fre- ber system provided two benefits. In addition to pro-
quencies close to those associated with the full-scale viding a means for preventing damage to the model
conversion actuator (different springs are used for dif- with the onset of an instability, it also provided
ferent conversion actuator positions), a gimballed 3- a mechanism for performing isolated rotor testing.
bladed tmb with a constant-velocity joint, and a set While the snubber was activated to restrain the up-
of aeroelastically-scaled rotor blades. The current per portion of the pylon, the lower portion of the
study was performed using the WRATS model, but pylon could be clamped to the support stand. This
lon with a cyclic stick-stir at the natural frequency of
the mode of interest. After removing the excitation,
data was processed for approximately 5 seconds, and
the frequency and damping were determined using
a moving block method (Ref. 17) from the digitized
time histories. Obtaining tile lag response required
slightly more work at some rotor speeds because the
aubber
response was highly damped and thus did not pro-
vide enough cycles outside of the "noise" level to
accurately calculate a frequency or damping of the
mode. To determine the frequency of the lead lag
mode in these instances, the amplitude of tile lag re-
st)onse was measured during the excitation produced
by tile cyclic stick stir. This measurenmnt was re-
peated for several perturbations of the excitation fie-
quency. When the excitation matched up to the
Figure 5: WRATS tiltrotor model setup for isolated natural frequency of the lag mode then tile response
rotor testing. amplitude of this mode reached a peak.
The fixed-system mode of interest for the tiltro-
combination would raise the lowest fixed-system fre-
tor model in hover is best described as a wing tor-
quencies to about 20 Hz. The isolated rotor test
sion/chord coupled elastic mode at approximately 4.9
setup was used to determine frequencies and damp- Hz. For hover, the pylon is oriented vertically above
ing for the lag mode in the uncoupled system, which
the wing elastic axis creating a significant mass-offset
are beneficial to know before initiating the part of the
• which creates the large coupling between the chord
experiment where instabilities are likely to occur. A
and torsion modes. A second wing mode, beam (ver-
photo of the model setup for isolated rotor testing, tical) bending, occurs at 5.1 Hz, but since the beam
with snubber activated and lower pylon clamped to
motion is not highly-coupled with the lag rotor mo-
the support stand, is shown in Fig. 5.
tion (the dominant associated t)ylon motion is per-
pendicular to the rotor plane), there is a negligit)le
PARAMETRIC STUDY IN HOVER participation of this mode in the ground resonance
phenomenon of interest. Also, it should be noted
The stability characteristics of a gimballed soft- that while in airplane mode the wing beam mode is
inplane tiltrotor model were determined experimen- highly coupled with the wing torsion mode, this is
tally as a function of collective pitch and lag damper not the case for the hover mode, because of the po-
settings for a lag frequency setting of 0.5/rev. The sition of tile pylon and the location of its associated
choice of the lag frequency used for the study is mass offset. A third mode at 11.5 Hz is best de-
a crucial one because it defines the rotor speed at scribed as a wing chord/torsion mode, but because
which ground resonance will occur. In general, the of the relatively high frequency, this mode is also in-
lower the lag frequency, the greater the problem with significant with respect to the ground resonance con-
ground resonance instability. While a practical dition. The 4.9 Hz wing torsion/chord mode (here-
tiltrotor configuration may most likely have a lag fre- after referred to msWTC) and rotor lag frequencies
quency of 0.7/rev to 0.8/rev, the current test was de- are shown as a function of rotor st)eed in Fig. 6 for a
veloped using a nmch lower value as indicated. The condition of 8° collective pitch and damper setting of
reason for using 0.5/rev was two-fold: 1) there is a 3% critical (nonrotating). The plot shows that there
limited number of stock coil springs that could be is only a slight drop in the wing frequency with rotor
modified to fit into the spring cage which would pro- speed. Tile regressive h)w-frequency lag mode be-
duce reasonable lag frequencies, and 2) this value comes a progressive mode at at)out 490 RPM. As the
of lag frequency provided instabilities at low rotor rotor speed continues to increase the h)w-frequency
speeds so that the ground resonance related charac- lag mode and tile WTC mode t)ecome more coupled
teristics of the system could be fully examined be- and the associated frequencies begin to converge.
fore, during, and beyond the rotor speed at which The damping of these modes is illustrated in tile
resonance occurred.
plot of Fig. 7. The plot shows that the lag mode is
The system frequencies and associated damping highly damped, greater than 3_ critical except over
were determined experimentally by exciting the py- the rotor speed range where an instability is develop-
6.0 3.0
IPCrossoIv/ er
4.0
LagMode
II La_Mode
2.0
t
RotorSpeed,RPM RotorSpeed,RPM
Figure 6: Coupled frequencies of the wing and rotor Figure 7: Damping of the wing and rotor modes of
modes of interest for ground resonance (8° collective interest for ground resonance (8° collective and 3%
and 3% critical lag damper setting). critical lag damper setting).
ing. Damping at very low rotor speeds could not be damping and the associated ground resonance insta-
measured because there is not sufficient aerodynamic bility, even with significant changes in the lag damper
forcing to produce excitations using the stick stirs. setting. In fact, these results are not shown using a
The WTC mode damping is shown to decrease at a plot, as all the curves essentially are overlaid with
nearly constant rate with rotor speed, and the mode those shown in Fig. 8. While there is suspicion that
eventually becomes unstable at about 720 RPM. these results may be a consequence of the particular
The variation of WTC mode damping with rotor design, and that some structural nonlinearities as-
speed is shown in Fig. 8 for three collective pitch set- sociated with the lag mechanics may play a role in
this behavior, this remains a significant and surpris-
tings. The results show that the system behavior is
extremely sensitive to collective pitch, and this trend ing result of the study. It can also be seen that the
is more sensitive than that associated with a typi- damper does have an effect on the lag motion itself.
cal helicopter system. With an increase in collective The plot of Fig. 9 shows the 1P response of the lag
motion as a function of rotor speed, in the vicinity
pitch, the aerodynamic forces at higher rotor speeds
of the !ag mode 1P cross-over. The amplitude of
may produce significant hub forces which couple the
WTC and blade lag modes, leading to an aeroelastic the response is shown to be greatly influenced by the
instability. The WTC mode damping shown in Fig. 8 damper setting, but this effect does not seem to in-
as a function of rotor speed at the 5° and 12° collec- fluence stability of the system.
tive pitch settings also illustrates a trend which is un-
like that of classical helicopter ground resonance, but
more like a helicopter pylon instability. In classical GENERALIZED PREDICTIVE CONTROL
ground resonance conditions with insufficient rotor
lag damping, a sudden and steep reduction followed Control Algorithm
by a recovery in the lag mode damping is often ob-
served in the vicinity of the resonant frequency. This This section describes the theory of GPC in gen-
plot shows that the 5° and 12° collective cases have eral terms. Implementation of GPC to the specific
a relatively slow, but constant, reduction in WTC model used in the study is described in a subsequent
fixed-system damping as rotor speed increases. Also, section. The essential features of the GPC adaptive
following the instability, the WTC mode damping control process are depicted in Fig. 10. The system
does not recover at higher rotor speeds. (plant) has r number of control inputs u, m number
The parametric effect of the damper setting was of measured outputs y, and is subject to unknown
also considered in this study. Surprisingly, there external disturbances d. Measurement noise is also
was little difference in the results of the fixed-system present. There are two fundamental steps involved:
0.6
+ 12deg.
+ 5deg. Damper,
Collective
2.0 + 0deg.
%Critical
0.4 • 10% ]t_
A6% / \
e_
_, <
0.2
N
._. 1.0
I I
0 q'
400 500 600
I I I RotorSpeed. RPM
0 400 600 800
RotorSpeed,RPM
Figure 9: Effect of lag dalnper on 1P lead-lag re-
sponse (8° collective).
Figure 8: Effect of collective pitch oil ground reso-
difference model:
nance stability with lag damper set to 3% critical.
(1) identification of the system; and (2) use of the y(a-)=
identified model to design a controller. A finite-
(_ly(k - 1) + a2y(k - 2) + ... + opy(k - p)
difference model in the form of an auto-regressive
moving average with exogenous input (ARX) model 4-/_()'lt(/x') -_- /_lll(_" -- 1) + ... + Jvu(k - p) (1)
is used here. This model is used for both system
This equation states that the current output y(k) at
identification (SID) and controller design. System
time step k may be estimated by using p sets of tile
identification is done on-line in the presence of any
previous output and input measurenmnts, y(k- 1),
disturbances acting on the system, as indicated in tile
..., y(k - p) and u(k - 1)..... u(k - p); and the cur-
center box of the diagram. In this way, an estimate
rent input me_urement u(k). The integer p is called
of the disturbance model is reflected in the identi-
the order of the ARX model. The coefficient matri-
fied system model, and does not have to be modeled
ces (_i and _3iappearing in Eqn. 1 are referred to as
separately. This approach represents a case of feed-
observer Markov parameters (OMP), or ARX para-
back with embedded feedforward. Because the dis-
meters, and are the quantities to be determined by
turbance information is embedded in tile feedforward
the identification algorithm.
control parameter, there is no need for measurement
Closed-loop robustness is enhanced by performing
of the disturbance signal (Ref. 18). The parameters
the system identification in the presence of the exter-
of the identified model are used to compute the pre-
nal disturbances acting on the system, thereby ensur-
dictive control law. A random excitation uid (some-
ing that disturbance information will be incorporated
times called dither) is applied initially with the vector
into the system model. The goal of SID is to deter-
of closed-loop control inputs ,_,.equal to zero to iden-
mine the OMP based on input and output data. The
ti_' the open-loop system. Dither is added to u_ if it
OMP may be determined by any SID techniques that
is necessary to re-identify tile system while operating
return an ARX lnodel of the system.
in the closed-loop mode.
The ARX model is used to design the controller and
leads to a control law that in the case of a regulator
problem has the general form given by:
_,,,(k)=
The relationship between the input and output
(_gJ(_-: 1)+ (_'_;V(-_"2) + ...+ (_;;._(k- p)
time histories of a MIMO system are described by the
time-dolnain auto-regressive exogenous (ARX) finite- + ;3'fu(t,')+ d_u(L'- 1)+ ...+ ;_;;u(/,-" p) (2)
This equation indicates that the current control in- Disturbance (d)
)
put uc(k) may be computed using p sets of the previ-
ous input and output measurements. The coefficient
; : Plant
matrices ct_ and/3_ appearing in Eqn. 2 are the con- Uid
)
trol gain matrices.
System identification in the presence of the oper-
ation disturbances acting on the system is the first & Disturbance Estimation I
U{.
of the two major computational steps. The external 1
disturbances acting on the system are assumed to be
t System Identification
unknown (unmeasurable). The number of control in-
--- (FeePdrebdaicckt/iFveeed Confotrrowlard) __
puts is r and the number of measured outputs is m.
The system is excited with band-limited white noise
for SID. These random excitations are input to all r
Figure 10: The GPC adaptive control process block
control inputs simultaneously and the corresponding
diagram.
m responses are measured. The input and output
time histories (u and y) are digitized according to
complete discussion of these aspects of the develop-
the 1 immber of time points, an(l these parameters
ment may be found in Ref. 19.
are then used to form the data matrices y and V as
_7 is the nmtrix of observer Markov parameters
which are to be identified and has the form
v= Fv (3)
where = {./30 _10_1 _20l2 _3 (:_3 -.. /_p OQ, } (7)
By solving Eqn. 3 for Y, the solution mav be written
(4)
y = [y(0) y(1) y(2).., y(p).., y(l - 1)]
as
and
= yV -1 = yVT[VVT] -1 (8)
If the product VV r is well-conditioned, the ordinary
v(0) v(1) v(p- 1) v(l- 2)
inverse may be taken• Otherwise, a pseudo-inverse
V-- v(0) v(p 2) v(/- 3) must be used. It should be noted that because the
_,(0) u(1) u(2).., u(p) ... u(l-1) size of VV r is nmch smaller then V, a pseudo-inverse
v(0) ...v(l-p-1) may be appropriate even if the product is well condi-
(5) tioned.
The order of the ARX model (p) and the number of At this point in the derivation the output predic-
time steps (1) must be specified by the user. The size tion equation has been established for one time step.
of y is m x l and the size of V is [r+(r+m)p] × 1. The control law is enhanced by developing the output
Equations 4 and 5 follow from writing the discrete- prediction equation several time steps ahead (multi-
time state-space equations for a linear time-invariant step approach)• With the o:i and L_idetermined by
system at a sequence of time steps k -- 0, l, ..., (l- 1) the system identification process, the multi-step out-
and gro.uping them into matrix form. The vector v put prediction equation may be derived at time step
is defined as k + j. Equation 1 may be written at time step k + j
as
v(k)= [ } (6)
y(k + j) =
which has size (r + m) × 1. In forming the matrices
of Eqns. 4 and 5, it has been assumed that the state a_J)y(k - 1) .-(r(j)y(k - 2) + ... + c_(vJ)y(k - p)
matrix A is asymptotically stable so that for some
+ L_ott(k + j) + 1301u(k+ j - 1) + ... + 3_J)u(k)
sufficiently large p, Ak _-.0 for all time steps k > p,
and that an observer has been added to the system. + - + - 2)+. + - ;)
It is through these expedients that the matrix V is (9)
reduced to a size amenable for practical numerical
computation of its pseudo-inverse. The SID process This equation shows that the output y(k +j) at time
yields OMP rather than system Markov parameters step k + j may be estimated by using p sets of the
(SMP) because of the inclusion of an observer. A previous output and input measurements, y(k-1) .....