JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
Vol. 22, No. 4, July
–
August 1999
Gain-Scheduled Linear Fractional Control
for Active Flutter Suppression
Jeffrey M. Barker
¤
and Gary J. Balas
†
University of Minnesota, Minneap olis, Minnesota 5 5455
and
Paul A. Blue
‡
U.S. Air Force Research Laboratory, Wright
–
Patterson Air Force Base, Ohio 45433
A gain-scheduled controller for active utter suppression of the NASA Langley Research Center’s Benchmark
Active Contro ls Technology wing section is presented. The wing section changes signi cantly as a function of Mach
and dynamic pressure and is modeled as a linear system whose parameters depend in a linear fractional manner on
Mach and dynamic pressure. T he resulting gain-scheduled controller also depends in a linear fractional manner
on Mach and dynamic pressure. Stability of the closed-loop system over a wide range o f Mach and dynamic
pressure is dem onstrated. Closed-loop stability is demonstrated via time simulations in which both Mach and
dynamic pressure are allowed to vary in the presence of input disturbances. The linear fractional gain-scheduled
controller and an optimized linear controller
(
designed for comparison
)
both achieve closed-loop stability, but the
gain-scheduled controller outperforms the linear controller throughout the operating region.
Nomenclature
A; B; C; D = state-space representation of a system
D
= stable, rational polynomial for ¹ design
e
= generic element of a state-space matrix
F
u
;
F
l
= upper and lower linear fractional transformations
I
= identity matrix
K
= controller
M
= Mach number
N
= constant 2
£
2 block matrix
P = parameter dependent plant
Nq
= dynamic pressure, kPa
u
= control signal
y
= measured plant variables
1 = uncertainty model
±
M
= normalized variation in Mach
±
Nq
= normalized variation in dyna mic pressure
¹ = structured singular value
6 = summation
Subscripts
act = actuator
d
= disturbance
in = input
(
to plant
)
multiplicative
LFT = linear fractional transformation
n
= noise
p
= performance
I. Introduction
F
LUTTER is a dynamic instability, characterized in detail in
1935
(
Ref. 1
)
and written of as early as 1916
(
Ref. 2
)
, that can
result in catastrophicmechanicalfailure of an aircraftwing. Because
of the severity of the potential problem, aircraft today typically
operate at conditions well b elow the utter boundary. However, as
aircraft design moves toward lighter weight materials in efforts to
Received 8 June 1998; revision received 21 December 1998; accepted
for publication 22 December 1998. Copyright
c°
1999 by the authors. Pub-
lished by the American Institute of Aeronautics and Astronautics, Inc., with
permission.
¤
Graduate Research Assistant, Department of Aerospace Engineeringand
Mechanics, 110 Union Street SE. Student Member AIAA.
†
Associate Professor,Department of Aerospace Engineeringand Mechan-
ics, 110 Union Street SE. Member AIAA.
‡
First Lieutenant, U.S. Air Force, Flight Control Div ision.
improve fuel ef ciencyand aircraftagility, active utter suppression
will likely become increasingly important.
The problem of active utter suppressionthus has receivedmuch
attention in the form of the active exible wing
(
AFW
)
program
3; 4
and the Benchmark Active Controls Technology
(
BACT
)
wing,
developed at NASA Langley Research Center speci cally to bet-
ter understand utter and its suppression.
5¡11
The AFW has been
used for testing various single-input/single-output and multi-input/
multi-output contr ollers.
12
The BACT model has been used to de-
velop adaptive neural control schemes
13
and generalized predictive
control.
14
Other control approachesto active utter suppressionthat
have been investigated include optimal control using acoustics,
15
multirate control,
16
nonlinear control,
17
and
H
2
=
H
1
control.
18; 19
This paper f ocuses on the design of an active utter suppres-
sion gain-scheduled controller using linear fractional transforma-
tions
(
LFT
)
. Gain-scheduled LFT control is a natural extension of
H
1
control for systems that vary smoothly as a function of measu r-
able parameters. Gain-scheduledLFT controller synthesis has been
successfully applied to a wide variety of problems with parameter
dependent plants, including lateral-directional control of the F-14
aircraft during powered approach.
20
This control methodology is
applied here to the BACT model. Sc heduling the controller as a
function of Mach and dynamic pressure allows the smooth han-
dling of transitions in the dynamic model throughout the operating
envelope. The gain-scheduled controller presented has a designed
operating range o f Mach from 0.5
–
0.82 an d a dynamic pressure
of 6.5
–
10.77 kPa. The range of operating conditions includes most
of the unstable operating region and demonstrates how a single
gain-scheduledLFT controller can stabilize the BACT model for a
wide range of Mach and dynamic pressure. In addition to the gain-
scheduled controller, a single linear controller is designed using a
D
-
K
iterationtechnique,which attemptsto be robustto the changing
utter dynamics. Comparison of the two controllers demonstrates
the improvedperformanceobtainablethrough use of gain-scheduled
control techniques.
The paper is presented in the following six sections. Section II
describes the BACT facility, the wing section test bed, and the lin-
ear, time-invariant
(
LTI
)
models derived at speci c Mach numbers
and dynamic pressures. In Sec. III an LFT model of the BACT
wing section, which i s a function of
M
and
Nq
, is derived from the
LTI models. This LFT model is used for control design. Section IV
presentsthe controltheory assoc iatedwith gain-scheduledLFT con-
trol. The gain-scheduled control problem requires the solution of
linear matrix inequalities that can be solved ef ciently
21; 22
using
convex optimization te chniques. Control design and synthesis of
507
508 BARKER, BALAS, AND BLUE
Table 1 Flight conditions of LTI models
Mach Dynamic pressure, kPa
0.50 3.59 4.79 5:84
a
6:32
a
7:18
a
8:38
a
9:58
a
10:77
a
0.70 3.59 4.79 5.99 6:51
a
6:99
a
8:38
a
9:58
a
10:77
a
0.78 3.59 4.79 5.99 6:75
a
7:22
a
8:38
a
9:58
a
10:77
a
0.82 3.59 4.79 5.99 6:84
a
7:33
a
8:38
a
9:58
a
10:77
a
a
Open-loop unstable.
the gain-scheduled LFT controller and a linear ¹ controller for the
BACT model are presented in Sec. V. The controllers designed in
Sec. V are analyzed and simulated in Sec. VI. The stability and per-
formance of both the gain-scheduleddesign and the ¹ controller are
evaluated over varying
M
and
Nq
. These results of the simulations
are then compared and co ntrasted. The nal section summarizes the
results and presents conclusions.
II. BACT Model
The BACT model is an elementof NASA Langley Research Cen-
ter’s Benchmark Models Program
(
BMP
)
.
5
The BMP consists of
several models used in the investigation of aeroelastic effects and
to acquire experimental data for the validation of the computational
uid dynamics
(
CFD
)
code. The BACT model has been used to ob-
tain experimental data over a wide range of operating conditions,
which researchers use t o develop active utter suppression design
tools and to calibrate unsteady CFD code.
The BACT model is a rigid, rectangularwing with an NACA 0012
airfoil.
8
The airfoil has three control surfaces driven by hydraulic
actuators: a trailing-edge ap with a
§
15-deg operating range and
upper and lower surface spoilers,each with a range from 0 to 45 deg.
The primary sensors used for control are four linear accelerometers,
placed at the four corners of the wing section. The wing model is
mounted on a exible device called the Pitch and Plunge Apparatus
(
PAPA
)
. PAPA is designed to allow rotation
(
pitch
)
and vertical
translation
(
plunge
)
.
7; 8
Using Interaction of Structures, Aerodynamics, and Controls
(
ISAC
)
,
23; 24
a computer pro gram for calculating the interactive ef-
fects of exible structures, unsteady aerodynamics, and active con-
trols, NASA researchersdeveloped LTI mo dels for the BACT wind-
tunnel model at a wide range of operating conditions. The program
uses a doublet-lattice method to calculate the three- dimensional
aerodynamicforces on the wing for a speci ed operating condition.
LTI models have been generated to represent the wing at various
wind-tunnel operating conditions. The LTI models made available
(
Table 1
)
are at four different Mach numbers and range in dynamic
pressure from 3.59 to 10.77 kPa. These LTI models have only one
control input
(
the trailing-edge ap
)
and leading- and trailing-edge
acceleration measurements. The LTI models have 14 states: four
states correspond to the pitch and plunge d ynamics, six states char-
acterize unsteady aerodynamics, two states characterize actuator
dynamics, and two states are a second-order Dryden turbulence
model.
Examination of Bode plots of the LTI models reveals that they
are primarily functions of the pitch, plunge, and actuator states,
with the unsteady aerodynamics playing a small role and the gust
model resulting in a small disturbance. Figure 1 shows re presenta-
tive magnitudeplots for the full
(
14th
)
order and reduced
(
6th
)
order
LTI systems. Clearly, the salient features of the utter problem for
the BACT win g are captured in the reduced-ord er systems. Essen-
tially, the BACT wing appears to be well represented by a model
similar to a wing model with quasisteadyaerodynamics.This is not
surprising because the wing was designed so that the aerodynamic
instabilities would be relatively benign, which made building in
safety mechanisms ea sier.
14
III. LFT Model Synthesis
The implementation of gain-scheduled LFT control requires the
construction of an LFT plant from the BACT LTI model data. The
32 LTI models of Table 1 are the basis fo r the LFT model of the
system used for control design.
(
The operating range of the LFT
model used for control design includes the 19 models at dynamic
Fig. 1 Full- and reduced-order transfer function models from trailing-
edge actuator to trailing- and leading-edge accelerometers at Åq =
8.38 kPa and Mach = 0.7.
pressures of greater than 6.5 kPa.
)
The model, previously described
in detail,
25
is presented here in a much abbreviated format.
First, a model depending on Mach and dynamic pressure was
developed in which each element of the state matrices
(
A; B; C; D
)
is a function of
M
and
Nq
. The two sta te equations correspondingto
the actuator model are constant across the 32 LTI models, and the
pitch and plungestate equationsare just integrationsof thepitch-rate
and plunge-ratestates.Each element
(
e
)
of the remaining state-space
equations could be accurately modeled over the range of operating
conditions in Table 1, as
e
.
Nq
;
M
/
D e
o
C e
Nq
¢ Nq C e
M
¢ M C e
Nq M
¢ Nq ¢ M
(
1
)
This relationship allows t he 32 LTI models to be written as one
parameter-varying model of the form
Px D
A.
Nq
;
M
/
¢ x C
B.
Nq
;
M
/
¢ u
(
2
)
y D
C.
Nq
;
M
/
¢ x C
D.
Nq
;
M
/
¢ u
(
3
)
where A.
Nq
;
M
/, B.
Nq
;
M
/, etc.,takethe form of Eq.
(
1
)
. Furtherstudy
of the models reveals that C.
Nq
;
M
/ and D.
Nq
;
M
/ can be constructed
from A.
Nq
;
M
/ and B.
Nq
;
M
/ because the outputs are accelerations
(
linear combinations of t he state rates
)
.
To ascertain the accuracyof th e parameter-varyingrepresentation
[Eqs.
(
2
)
and
(
3
)
], its poles were compared wi th the poles of the in-
dividual LTI models at the 32 ight conditionsa nd were found to be
very similar.
25
Indeed,the parameter-varyingmodel was constructed
to match the LTI models exactly at Mach 0.5 and 0.82 and is a go od
representation at intermediate Ma ch. Thus, the parameter-varying
model accurately describes the LTI utter models. This parameter-
varying model is used as the basis for the reduced-orderLFT model
of the system used in control design.
Using the information from Fig. 1, one can see that a sixth-order
model captures the primary effects of wing u tter for the BACT
wing. Using a reduced-ordermodel of the plant so that the resulting
controllerwill also be of low orderis desirable;thus,thesix unsteady
aerodynamic states and the second-orderDryden wind-gust model
are truncated from the model because they have little effect on the
dynamics of the system. For modeling purposes the wind gust is
replaced with an input disturbanceinjected on the actuator position.
Although this does not accurately repla ce th e effects of the gust
model, the added distu rbance path provid es for added robustness.
Thus, the single-input,two-output design model contains six states:
the four states associated with the pitch and plunge modes and the
two-state actuator model.
Before introducing the LFT model, some backgroundon LFTs is
appropriate. The upper and lower linear fractional transformations
of a block-partitionedmatrix
N D
N
11
N
12
N
21
N
22
BARKER, BALAS, AND BLUE 509
Fig. 2 Upper and lower linear fractional transformations.
Fig. 3 LFT plant as a function of
Åq and M.
with a matrix 1 are de ned as
F
u
.
N
; 1/
D N
22
C N
21
1.
I ¡ N
11
1/
¡1
N
12
F
l
.
N
; 1/
D N
11
C N
12
1.
I ¡ N
22
1/
¡1
N
21
shown in a block-diagram representationin Fig. 2.
Equation
(
1
)
can be usedto formulatethe parameter-varyingprob-
lem in LFT form
25
with four copiesof the parameter
Nq
and two copies
of
M
. Figure 3 shows the resulting LFT plant P as a function of
Nq
and
M
, which vary from 3.59
–
10.77 kPa and from Mach 0.5
–
0.82,
respectively.Thus the parameterizationincludes both the open-loop
stable and unstable models.
(
Recall that although
H
1
control the-
ory speci es stable perturbations, a stable 1 can either stabilize or
destabilizea given plant.
)
For control design purposes scaling
Nq
and
M
so that they vary between
¡
1 and 1 is necessary. To do this, a
constant LFT
N
is employed:
F
u
N
;
±
Nq
.
t
/ 0
0 ±
M
.
t
/
D
Nq
.
t
/
I
4
0
0
M
.
t
/
I
2
The same process could be applied to the 14 state LTI models
with additional copies of parameters
Nq
and
M
. The choice to use a
six-state LFT model is thus one of engineering judgement, and the
resultingcontrollerswill of coursebe validatedagainstthe full-order
LTI models.
The gain-scheduled LFT control design methodology is a direct
extensionof ¹-synthesistheory.Thus ¹ synthesis is a naturalchoice
for the lin ear control design technique.Because t he control problem
in the ¹ framework is posed a s maximizing robust performance,this
is equivalent to minimizing the structured singular value ¹. Thus
nding a controller
K
to minimize ¹ is a reasonable objective. An
approximationto the ¹-synthesisproblem is given by the
D
-
K
itera-
tion, which is a two-step iterative solutionmethod. First a controller
K
is found through standard
H
1
methods. Next, the scaling matrix
D
.
s
/ that mi nimizes
kD
.
s
/
F
l
.
P
;
K
/.
s
/
D
¡1
.
s
/
k
1
is determined.
These steps are iterated until an acceptable controller is obtained.
Further details of the structured si ngular value theory and
D
-
K
it-
eration may be found els ewhere.
26¡28
IV. Gai n-Scheduled LFT Control
There are two main performance objectives for any utter sup-
pressionsystem. The rs t is to extendthe utter boundary,i.e., to use
feedback control to stabilize the wing over a larger region of operat-
ing conditions. Second, utter control is used to suppressvibrations
in the operating region where the wing is open-loop stable. Extend-
ing the utter boundary implies that the boundary is a function of
measurable parameters
(
e.g.,
M
and
Nq
)
. The variation of the plant
as a function of one or more parameters is an essential feature of the
problem. By allowing the cont roller to depend explicitly on these
Fig. 4a Parameter-dependent plant.
Fig. 4b Parameter-dependent controller.
parameters, attainingimprovedclosed-loopperfo rmanceand stabil-
ity should be possible. Because the essential dynamics of the utter
problem are well described by the LFT plant, using gain-scheduled
LFT controller synthesis techniques for active utter suppressionof
the BACT mo del is appropriate.
The central idea of gain-scheduled LFT control is that a plant
often can be represented as a linear fractional transformation of a
nominal plant an d physical parameters that vary within a known
range. If we can measure these physical parameters in real time,
then the controller can use this knowledge to schedu le as a function
of these parameters.
Consider a parameter-dependent plant modeled as an LFT of a
time-varying block diagonal matrix 1.
t
/ and a three-input, three-
output LTI plant P
(
Fig. 4a
)
. The parameter dependenceof the sys-
tem is becauseof the time-varying 1 matrix. For the utter problem
the 1.
t
/ block is
1.
t
/
D
±
Nq
.
t
/
I
4
0
0 ±
M
.
t
/
I
2
The assumption is made that 1.
t
/ takes values in a known set
D
and that 1.
t
/ can be measured in real time.
The parameter-dependentcontrolleris restrictedto havinga struc-
ture similar to that of the plant
(
Fig. 4b
)
. By interconnecting the
parameter-dependent plant and controller the closed-loop system
appears as a nite dimensional LTI system subjected to the time-
varying perturbation1.
t
/. The perturbationhas a structure consist-
ing of two parts: the physical parametersthat affect the plant and the
measured parameters that are used by the controller. The actual and
measured parameters may be different, bu t for notationalsimplicity
1.
t
/ will be used to repr esent both.
The control objective is to design the controller
K
LFT
such that
for all allowable perturbations 1.
t
/
2 D
the parameter d ependent
closed-loop system is internally exponentially stable with small in-
duced L
2
norm from disturbancesto erro rs
(
includingmeasurement
noise and bounds on control authority
)
. The small-gain theorem can
be employed to bound
(
conservatively
)
the stability of the system
and the induced L
2
norm o f the disturbance to error channels of the
parameter-dependent closed-loop system. Scaling matrices are re-
stricted to be constant diagonal matrices so that they will commute
with the repeated structu re of the perturbation. The main result of
the theory that allows the determination of such a contro ller is that
existence of a controller satisfying the scaled small-gain bound can
be expressed exactly as the feasibility of a nite dimensional af ne
matrix inequality
21
(
AMI
)
. Because of the convexity of the AMI,
this problem can be computed numerically. The details of gain-
scheduled LFT control theory are covered in detail by Packard.
29
V. Control Design and Synthesis
The control design methodology is very similar for both the gain-
scheduled LFT controller and the ¹ controller, and so will be pre-
sented only o nce in Sec. V.A. Controller synthesis is discussed for
each method in Sec. V.B.
A. Control Design
The reduced six-state LFT plant
(
4 aerodynamicstates, 2 actuator
states
)
is used to design both utter suppression controllers. This
reduced plant has seven inputs and eight outputs. The rst six inputs
510 BARKER, BALAS, AND BLUE
Fig. 5 LFT control design block diagram.
and outputs connect to the parameter block 1.
t
/ a s discussed in
Sec. IV. The other input is the control signal, and the two outputs,
trailing-edge and leading-edgeacceleration,are used fo r feedback.
The block diagram
(
Fig. 5
)
is used in the synthesisof both the LFT
gain-scheduledcontroller and the ¹ controller. This diagram corre-
sponds to the integration of performance objectives and robust sta-
bility objectivesinto a single controldesignframework.The stabi lity
objectives are to stabilize the wing throughout the oper ating region
and to be robust to unc ertaintyin the modeling processand to errors
in model reductio n.These objectives are incorporatedt hrough both
input multiplicativeuncertainty and the LFT model
(
which entersas
parametric uncertainty in the ¹ framework
)
. Performance require-
ments are formulated through the choice of the weighting functions
applied to the input and output signals of the open-loop system.
The output leading- and trailing-edgeaccelerations are the primary
performance signals. The ap command signal from th e controller
is also restricted.A disturbance si gnal on the input to the open-loop
plant allows fo r unknown exogenous disturbances to the system,
representingwind gusts. A noise signal is also added to the leading-
and trailing-edge signals to corrupt the measurements.
Multiplicative uncertainty, represented in the block diagram by
the multiplicativeuncertaintyweight
W
in
and the uncertaintyset 1
in
,
is used to capture modeli ng error at high frequency, differences be-
tween the LFT model and the individual LTI models, and the uncer-
tainty introducedby model reduction.For this system the multiplica-
tive uncertaintyweight used is
W
in
D
[0:1.
s
=2/
C
1]=[.
s
=200/
C
1],
representing10% uncertainty in the LFT model at low frequencies,
100% uncertainty at 20 rad/s, increasing to 1000% at high frequen-
cies. The level of uncertainty at high frequency ensures that the
controllers will not be amplifying the system dynamics in this fre-
quency range.
The LFT model, whi ch is a function of
Nq
and
M
, is treated as para-
metric uncertaintyin the ¹-synthesis control design. Thus the nom-
inal plant for the linear control design is simply the plant that corre-
spondsto ±
Nq
D
0; ±
M
D
0
(
correspondingto 7.18 kPa and Mach 0 .66,
an unstable open-loop plant
)
, whereas in the LFT gain-scheduled
design the variation in
Nq
and
M
is part of the model.
The primary performance objective for active utter suppression
is to decrease the peak responseof the wing at the utter frequencies
and the pitch and plunge modes. This objective is captured via a
constant diagonal performance weighting
W
p
, which restricts the
maximum magnitude of the transfer functions. The constants are
rst chosen to normalize the output channel
(
in cm/s
2
/rad
)
to have a
peak value of approximately one
(
approximately, because the peak
values of the transfer functions from ap command to leading- and
trailing-edge ac celeration vary as functions of
Nq
and M
)
. These
constants are th en multipliedby two, asking that the peak magnitude
be reduced to half of its initial value. Thus,
W
p
D
2
1
27;500
0
0
1
25;000
is chosen. These constant weights, applied to the trailing- and
leading-edge acceleration output channels, request a reduction of
the maximum singular values from all inputs
(
disturbance on the
actuator signal, uncertainty, and noise
)
to these outputs. Thu s th ese
weights c orrespond to asking for a decrease to 50% of the open-
loop peak response at the natural frequencies for the stable plants
and to similar magnitudes at the utter frequency for the open-loop
unstable plants. Because the basic performance problem is one of
vibration attenuation, the constant performance weight is all that is
needed and is selected to suppress the peak singular values. Addi-
tionally, by choosing a constant performance weight, the order of
the gain-scheduledLFT controller is kept low.
(
A rst-order pe rfor-
mance weight on each output channel would add t wo states to the
controller, for example.
)
The trailing-edge ap used as the actuator h as limits of
§
15 d eg
or ¼=12 rad. As in the performance weight, the actuator weight
W
act
D
12=¼ is chosen to scale the largest allowable actuator com-
mand to
§
1. No ra te limits are imposed on the actuator in this
control design formulation although the high frequency gain of the
multiplicative uncertainty effectively limits the bandwidth of the
controller.
Sensor noise is added to the feedback signals to corrup t the mea-
surements and to satisfy the
H
1
control algorithm used for design.
The weight
W
n
D
diag[250; 250] was chosen so that the maximum
noise to signalratio is about 10% in the frequencyra nge 10
–
50 rad/s.
Disturbances are introduced through a weighted dis turbance in-
put on the a ctuator command to the plant. The disturbance weight
is chosen as
W
d
D
¼=36 rad, which represents maximum actuator
positioning error on the order of
1
3
the size of the maximum allow-
able actuator c ommand. This constant disturbance weight was used
for control design rather than using the Dryden gust model, which
would add two states to the plant and introduce additional copies of
the LFT parameters to the problem. The added disturban ce model
helps ensure the resulting controller is not overly aggressive in its
attempt to damp out vibrations. The resulting controllers will be
tested against the full-order system with the gust model included to
validate the designs.
B. Controller Synthesis
The linear ¹ contoller
(
K
¹
)
was synthesized via the
D
-
K
it-
erative control design technique using the MATLAB
TM
¹-Analysis
and Synthesis To olbox.
26
The weighted open-loopsystem has seven
states
(
the reduced six-statesystem plus one state for the multiplica-
tive uncertainty weight
)
. The structured uncertaintyis introducedas
a complex parameter variation. To keep the total order of the ¹-
controller low, the four copies of the parameter
Nq
are allowed to
vary independently
(
adding additional conservatism to the control
design
)
whereas the two copies of
M
vary together. Three iterations
resulted in a 43rd-order controller, and further iterations decreased
¹ by less than 1%. A balanced realization of the controller is ob-
tained, and the 29 states with the smallest Hankel singular values are
truncated from the system. The resulting 14-state controller differs
from the full-o rder controller in H
1
norm by less than 0.1%.
(
H
2
norms cannot be compared because the high frequency gain of the
reduced-ordercontrolleris small but constant, resultingin an in nite
H
2
norm.
)
The resulting reduced-ordercontroller has ¹ < 2:7.
A gain-scheduled LFT c ontroller
(
K
LFT
)
was synthesized using
an algorithm that allows the parameters ±
Nq
and ±
M
to be complex,
which is conservative.
K
LFT
stabilizes the open-loop LFT plant over
the full range of Mach
(
0.5
–
0.82
)
and a reduced range of dynamic
pressures
(
6.5
–
10.77 kPa
)
. The gain-scheduledcontrolleris synthe-
sized using a formulation of the
H
1
problem as a linear objective
minimization.
29
The resulting controllers have the same number of
states as the weighted open-loop system. Thus the resulting con-
troller is seventh order. A fourth-order truncated balanced realiza-
tion of the controller results in a r educed-order controller differing
from the original controller in H
1
norm by less than 0.1% in the
frequency range of interest.
Because of the structure of the problem, using the frequency-
varying
D
scalings on the LFT parameters in the manner used in the
D
-
K
iteration
26
to determine the robust stability of the system is not
possible.However, we are able to usea constant
D
scalingcalculated
just below the utter frequency at 20 rad/s and get an approximation
BARKER, BALAS, AND BLUE 511
of robust stability information. Performing the preceding scaling,
then calculating frequency-dependent scalings for only the multi-
plicative uncertainty channel, a structured singula r value of under
14 is obtained. The high value of ¹ is consistent with the ¹ value
satis ed by the ¹-synthesis controller after the rst iteration.
VI. Results
The primary objectives were to improve the disturbance rejec-
tion characteristics of the wing a nd increase the range of operating
conditions at which the wing is stable. In examining the success of
the gain-scheduled controller at meeting these goals, several per-
formance characteristics are considered. First, the stability of the
closed-loopsystem is examinedusing the full-orderLTI single-point
models and the point controllersobtainedby specifyingconstantdy-
namic pressureand Mach. For example, the full-order
(
14 state
)
LTI
model at Mach 0.7 and 10.77 kPa was closed with
K
LFT
operating
at the same Mach and
Nq
. Second, the maximum singular value pl ots
of the open- and closed-loop systems are compared. Finally, ti me
simulations of the response of the wing to var ious inputs are used
to ex amine the disturbance rejection characteristics, as well as to
demonstrate reasonable actuator usage.
The gain-scheduled controller stabilizes the LFT plant over the
range indicated in Sec. V.B. It remains to be shown that the con-
troller also stabilizesall of the full-orderLTI plantswithin this range
(
Nq
> 6:5 and 0:5 <
M
< 0:82
)
. For
K
LFT
, there are 19 LTI plants to
be examined
(
Table 1
)
. All 16 of the unstableLTI plants in this range
are stablewhen closedwith the LTI controllerobtainedfrom the LFT
controller
K
LFT
at the appropriate Mach and dynamic pressure. The
three open-loop stable plants in the region remain stable. Addition-
ally, although
K
LFT
was designed over only part of the operating
range, it actually stabilizes and improves disturbancerejection over
all 32 full-orderLTI models. For
K
LFT
, this means that the point con-
troller at a xed Mach
M
0
and 6.5 kPa stabilizesthe LTI model at
M
0
and 3.59 kPa and rejects disturbancesbetter th an the open-loopplant
at Mach 0.5 and 3.59 kPa. The gain-scheduledcontrollerthus shows
a greater than 50% increase in the utter boundary as a function of
dynamic pressure, which may indicate that a single gain-scheduled
LFT controller operating over the entire range of LTI plants c ould
be synthesized, perhaps by using real-valued
(
instead of complex
)
parameters in the LFT synthesis problem.
The linear ¹ controller
K
¹
also stabilizes the closed-loopsystem
over all of the operatingconditions.To accomplishthis, performance
is traded off for stability, as will be shown next.
In evaluating the success of controllers in attenuating vibrations
of exible structures,examining the magnitude of the transfer func-
tions or the maximum singular value plots of the system is t ypical.
These generally give a good indication of how well the closed-loop
systems will reject disturbances, especially in comparison to the
open-loop plant.
Figure 6 shows the full-order LTI open-loop plant and the plant
closed with
K
LFT
and
K
¹
at the two unstableextremes of the rangeof
operating conditions
(
at high dynamic pressure
)
. The two left-hand
Fig. 6 Open- and closed-loop maximum singular value plots from
trailing-edge ap to trailing- and leading-edge acceleration for both
controllers.
plots show maximum singular value plots of the open-loop plant
and the plant closed with
K
LFT
, whereas the right-hand plots show
the open-loop plant at the same operating points closed with
K
¹
.
The upper plots are maximum singular value plots for the system at
Mach 0.5 and 10.77 kPa for each of the controllers, and the lower
plots give the same at Mach 0.82 and 10.77 kPa. The rst peak, at
approximately 20 rad/s, correspondsto the pitch mode in the stable
operating range whereas the second, at ap proximately26 rad/s, cor-
respondstotheplungemode in theo pen-loopstablerange.The mode
that becomes unstable at the utter boundary
(
at 26 rad/s
)
exhibits
characteristicsof both p itch and plunge, indicatingthat the pitch and
plunge modes are coupled. This instab ility occurs at 6.08 kPa for
Mach 0.5 and at 7.09 kPa for Mach 0.82. The signi cant reduction
in th e peak singular values with
K
LFT
at both Mach 0.5 and 0.82
and 10.77 kPa is representativeof the reductions seen at other oper-
ating points, indicating that good vibration attenuation/disturbance
rejection throughoutt he operating range is likely. Time simulations
will show that these disturbancerejection conclusions are valid.
K
¹
also attenuates the peaks at both operating conditions, but by sig-
ni cantly less than
K
LFT
; this suggests that while the ¹-synthesis
controller stabilizes the plant throughout the operating range the
disturbanceattenuationwill be signi cantly smaller at the extremes
of the operating range than that achieved by
K
LFT
.
Time simulations of the closed-loop systems are used to investi-
gate both the response of the system to input disturbances and the
stability of the closed-loopsystems as
Nq
and Mach vary as f unctions
of time. Figure 7 demonstrates that the closed-loop systems remain
stable in the presence of a windgust as
Nq
and
M
vary. Here,
Nq
varies
linearly from 3.59 to 10.77 kPa over 12 s, and Mach varies from 0.5
to 0.82 as t he function .0:25
C
0:0352
t
/
1
2
. This function was cho-
sen taking into account the p hysical relationship between dynamic
pressure and Mach. In other words air density was held nearly con-
stant while still allowing Mach to var y only over the range for which
linear models were available. The disturbance input is bandwidth-
limited
(
50 Hz
)
white noise input to the gust model. Note that the
variation in
Nq
and
M
takes the plant through the region over which
K
LFT
was not desig ned
(
below 6.5 kPa
)
. While the simulated system
was in this region,
K
LFT
operated at 6.5 kPa and at the simulated
Mach. Thus the gain-scheduled controller remained on the edge of
its opera ting range until
Nq
increased to 6.5 kPa.
Figure 7 demonstratesthe greaterdisturbanceattenuationachiev-
ed by
K
LFT
in comparison to
K
¹
, particularly for
t
> 4:8 s where
the operating point has moved into the range for which
K
LFT
was
designed. The peak leading- and trailing-edge accelerations for the
simulation are 35:5 and 31.4 cm/s
2
for
K
¹
and 32:0 and 28 .5 cm/s
2
for
K
LFT
. For
t
> 4:8 s the rms accelerations of the leading- and
trailing-edge aps are 12:3 and 11.7 cm/s
2
for
K
¹
and 12:2 and
11.2 cm/s
2
for
K
LFT
. Throughout
K
LFT
’s operating range it displ ays
similarperformanceimprovementin comparisonto
K
¹
. Simulations
of both controllersat the LTI models in
K
LFT
’s operatingrange show
an averageof a 4% reductionin peak accelerationsandno signi cant
improvement in rms accelerations.
K
¹
, however, displays slightly
Fig. 7 Leading- and trailing-edge acceleration for K
LFT
(
left
)
and K
¹
(
right
)
as
Å
q and M vary with time.
