NASA/TM—2014–216616

Virtual Deformation Control of the X-56A 

Model with Simulated Fiber Optic Sensors

Peter M. Suh and Alexander W. Chin

Dryden Flight Research Center, Edwards, California

Dimitri N. Mavris

Georgia Institute of Technology, Atlanta, Georgia

 

Click here: Press F1 key (Windows) or Help key (Mac) for help 

January 2014
NASA STI Program ... in Profile

Since its founding, NASA has been dedicated 

to the advancement of aeronautics and space 

science. The NASA scientific and technical 

information (STI) program plays a key part in 

helping NASA maintain this important role.

The NASA STI program operates under the 

auspices of the Agency Chief Information Officer. 

It collects, organizes, provides for archiving, and 

disseminates NASA’s STI. The NASA STI 

program provides access to the NTRS Registered

and its public interface, the NASA Technical 

Reports Server, thus providing one of the largest 

collections of aeronautical and space science STI 

in the world. Results are published in both nonNASA channels and by NASA in the NASA STI 

Report Series, which includes the following report 

types:

TECHNICAL PUBLICATION. Reports of 

completed research or a major significant 

phase of research that present the results of 

NASA Programs and include extensive data 

or theoretical analysis. Includes compilations of significant scientific and technical 

data and information deemed to be of 

continuing reference value. NASA counterpart of peer-reviewed formal professional 

papers but has less stringent limitations on 

manuscript length and extent of graphic 

presentations.

TECHNICAL MEMORANDUM. 

Scientific and technical findings that are 

preliminary or of specialized interest, 

e.g., quick release reports, working 

papers, and bibliographies that contain 

minimal annotation. Does not contain 

extensive analysis.

CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, seminars, 

or other meetings sponsored or 

co-sponsored by NASA.

SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest.

TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission.

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and 

feeds, providing information desk and personal 

search support, and enabling data exchange 

services.

For more information about the NASA STI 

program, see the following:

Access the NASA STI program home page 

at http://www.sti.nasa.gov

E-mail your question to help@sti.nasa.gov

Phone the NASA STI Information Desk at 

757-864-9658

Write to:

NASA STI Information Desk

Mail Stop 148

NASA Langley Research Center

Hampton, VA 23681-2199

This page is required and contains approved text that cannot be changed.
NASA/TM—2014–216616

Virtual Deformation Control of the X-56A 

Model with Simulated Fiber Optic Sensors

Peter M. Suh and Alexander W. Chin

Dryden Flight Research Center, Edwards, California

Dimitri N. Mavris

Georgia Institute of Technology, Atlanta, Georgia

Insert conference information, if applicable; otherwise delete 

Click here: Press F1 key (Windows) or Help key (Mac) for help 

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, CA 93523-0273

January 2014
Acknowledgments

This research was funded by the NASA Aeronautics Research Mission Directorate Subsonic 

Fixed Wing Project and was approved by Steve Jacobson. Mr. Jacobson, the Dryden Controls 

Branch Chief; John Bosworth, the X-56A Chief Engineer; Jack Ryan, the X-56A Principal 

Investigator; Ryan Stillwater, the X-56A Controls Lead; Chan-gi Pak, the creator of the X-56A 

aeroservoelastic model; and the aeroservoelasticity expert Marty Brenner; have all been 

excellent advisors in the development of these results.

Click here: Press F1 key (Windows) or Help key (Mac) for help 

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
1

Abstract

A robust control law design methodology is presented to stabilize the X-56A model and command its 

wing shape. The X-56A was purposely designed to experience flutter modes in its flight envelope. The 

methodology introduces three phases: the controller design phase, the modal filter design phase, and the 

reference signal design phase. A mu-optimal controller is designed and made robust to speed and 

parameter variations. A conversion technique is presented for generating sensor strain modes from sensor 

deformation mode shapes. The sensor modes are utilized for modal filtering and simulating fiber optic 

sensors for feedback to the controller. To generate appropriate virtual deformation reference signals, 

rigid-body corrections are introduced to the deformation mode shapes. After successful completion of the 

phases, virtual deformation control is demonstrated. The wing is deformed and it is shown that angle-ofattack changes occur which could potentially be used to an advantage. The X-56A program must 

demonstrate active flutter suppression. It is shown that the virtual deformation controller can achieve

active flutter suppression on the X-56A simulation model.

Nomenclature

state matrix

AFS active flutter suppression

AF airframe

 number of airframe states

AWBT antisymmetric wing bending and torsion flutter mode

AW1B antisymmetric wing first bending

AW1T antisymmetric wing first torsion

 rational approximation matrix

 rational approximation matrix

 rational approximation matrix

 number of control surfaces

 additive uncertainty signal

	 control input matrix

BFF body freedom flutter mode

BFL body flap left

BFR body flap right


 sensor output matrix

 center of gravity


 direct feed-through matrix

DOF degrees of freedom


 frequency weighting matrix

 reference deformations

(, , , ) vector of deformations at positions , ,  at time 

 rational approximation matrix of lag terms

FEM finite element model

FOS fiber optic sensors

 objective function for the 

 mode

 number of aerodynamic lag states

freq frequency

 plant

GAF generalized aerodynamic force

! set of plants over a velocity range
2

"# design plant

 damping 

I identity matrix

 input uncertainty signal

$% imaginary number

& generalized damping matrix

' controller

*+ stability margin

- characteristic length

LFT linear fractional transformation

LWLE left-wing leading edge

LWTE left-wing trailing edge

. sensor row index vector for reference deformations

/ generalized mass matrix

/ matrix transfer function

0 number of modes in model

0 modal column index vector for reference modes

1 number of nodes in finite element model

NS nominal stability

NP nominal performance

1 matrix transfer function

2 number of states

2 rigid body sensor noise

2# fiber optic sensor noise

23 input control noise

4# original sensor position before modal deformation

4#

5 original sensor position after modal deformation

467777 line segment distance from point 4 to point 6

4868 777777 line segment distance from point 48 to point 68

9 number of measurements

9 output uncertainty signal

P sensor location point

6: generalized plant

6; sensor position on the fore side of original sensor before modal deformation

6;

5 sensor position on the fore side of original sensor after modal deformation

6< sensor position on the aft side of original sensor before modal deformation

6<

5 sensor position on the aft side of original sensor after modal deformation

6=9>3?@(ABCD ??????????@)4#

5

6; ?????????@5 projection of directed segment 4#

5

6; ?????????@5 on unit vector in the direction of 4#6; ?????????@

6=9>3?@(ABCE ??????????@)4#

5

6< ?????????@5 projection of directed segment 4#

5

6< ?????????@5 on unit vector in the direction of 4#6< ?????????@

6=9>3?@(AC?????@)4868 ????????@ projection of directed segment 4868 ????????@ on unit vector in the direction of 46?????@

F generalized aerodynamic force matrix

FG(H) rational function approximation of the generalized aerodynamic forces

I() vector of modal displacements at time 

IJ() vector of modal velocities at time 

IK() vector of modal accelerations at time 

I() 

 modal displacement defined at time 

I modal reference displacement command

L diagonal matrix of lag roots
3

RP robust performance

RS robust stability

RWLE right-wing leading edge

RWTE right-wing leading edge (M) reference command of the argument  generalized stiffness matrix

SFOS simulated fiber optic sensors

6 short-period mode

SWBT symmetric wing bending and torsion flutter mode

SW1B symmetric wing first bending

SW1T symmetric wing first torsion N input sensitivity matrix A output sensitivity matrix " strain-displacement transformation matrix H Laplace variable

H0 simulation H+(, , , ) strain at measurement locations at time 



O(M) right-hand-rotation matrix ON input complementary sensitivity matrix OA output complementary sensitivity matrix  time P control signal vector

P() control input signal vector at time 

P?@(M) unit vector

V-f frequency versus velocity

V-g damping versus velocity Q"# design speed QR free-stream velocity S error vector from reference input minus measurements T weighting

WF wing flap

WF1R wing flap first right

WF1L wing flap first left

WF2R wing flap second right

WF2L wing flap second left

WF3R wing flap third right

WF3L wing flap third left

WF4R wing flap fourth right

WF4L wing flap fourth left TU matrix of proper additive weights TN matrix of proper input weights TA matrix of proper output weights TV matrix of proper sensitivity weights TW matrix of proper complementary sensitivity weights T3 matrix of proper control output weights X disturbance inputs Y data matrix

() state vector at time 

J() state derivative vector at time 



 Cartesian coordinate in the x-direction
4

UZ vector of airframe states

UZ

 reference airframe states

UZ() vector of airframe states at time 

[\() vector of actuator displacements, velocities and accelerations at time 

\: c.g. location of the aircraft in the  direction

 vector of modal displacements

] uni-axial coordinate in a moving frame always in the direction of the SFOS layout 

][:() vector of aero lag states at time 

 initial state at time 0

 modal displacements

() vector of modal displacements at time 

J

() vector of modal velocities at time 

3 un-deformed location of the aircraft nodes in the  direction

^

 deformed location of aircraft nodes for the 

 mode shape in 

_ estimated modal displacement states

` rational approximation matrix of lag terms

 output sensor vector

() output sensor vector at time 

 Cartesian coordinate in the y-direction

\: c.g. location of the aircraft in the  direction

3 un-deformed location of the aircraft nodes in the  direction 

^

 deformed location of aircraft nodes for the 

 mode shape in 

 number of actuator states

 Cartesian coordinate in the z-direction

aV goal regulated ' states

V goal regulated sensitivity states

W goal regulated complementary sensitivity states

\: c.g. location of the aircraft in the  direction

3 un-deformed location of the aircraft nodes in the  direction

^

 deformed location of aircraft nodes for the 

 mode shape in 

b angle of attack

c angle of sideslip

d structured uncertainty

dN input uncertainty

eQ change of velocity

eb change of angle of attack

ef change in pitch angle

e scalar global displacement in 

e\+ forward change of aircraft’s center of mass from trim

e scalar global displacement in 

e\+ side change of aircraft’s center of mass from trim

e scalar global displacement in 

e\+ heave change of the aircraft’s center of mass from trim

gf scalar global angle about  axis

gh scalar global angle about  axis

gi scalar global angle about  axis

e-T- change in deformation at LWLE

e-TO change in deformation at LWTE
5

eLT- change in deformation at RWLE

eLTO change in deformation at RWTE

gT1O change in SW1T modal displacement

gT1	 change in SW1B modal displacement

eT1L change in WF1R rotation 

eT1- change in WF1L rotation 

eT2L change in WF2R rotation 

eT2- change in WF2L rotation 

eT3L change in WF3R rotation 

eT3- change in WF3L rotation 

eT4L change in WF4R rotation 

eT4- change in WF4L rotation

n measurement error

noo axial strain

noo

; axial strain measured on fore side of a sensor

noo

< axial strain measured on aft side of a sensor

f pitch angle

p structured singular value

qR free-stream density

r7 maximum singular value

s deformation modal matrix, a collection of mode shapes, h+

h bank angle

h 

th natural deformation mode shape 

h(, , ) th natural deformation mode shape defined over , , 

t strain mode matrix, a collection of strain mode shapes

tZAV strain matrix defined at the SFOS locations

i yaw angle

i 

 strain mode

i+ 0 strain mode

u frequency

v3

vow

derivative in axial beam deformation in coordinate direction ]

xR Hardy space norm

Introduction

The NASA Aeronautics Research Mission Directorate (ARMD) is focusing on developing technology 

for dramatically reducing noise and emissions while improving performance for transport aircraft (ref. 1).

New technology must support the decrease of aircraft fuel and energy consumption by 60 percent within a 

timeline of 15 years. One method of achieving this goal is to design new lightweight aircraft. The 

consequence of using a lighter structure with increased flexibility is that the structure is more likely to 

experience adverse aeroelastic effects such as flutter and divergence (ref. 2). Researchers suggest that 

flexible aircraft may be more susceptible than traditional aircraft to gusts and maneuver loading (ref. 3).

Aircraft designed with lightweight flexible structures will require active control technology to be feasible 

alternatives to stiffer aircraft (ref. 4).

Research into active flutter suppression (AFS) technology on flexible aircraft is ongoing. The United 

States Air Force Research Laboratory (AFRL) Multi-utility Aeroelastic Demonstration (MAD) Program 

has developed the Multi Utility Technology Test-bed (MUTT), also known as the X-56A (Lockheed 

Martin, Bethesda, Maryland) (ref. 5). The X-56A is a flutter demonstrator designed for the sole purpose 

of testing various AFS solutions. Lockheed Martin is the prime contractor for design and development of 
6

the vehicle. The X-56A stems from previous work between Lockheed Martin and the AFRL to design and 

develop high-altitude, subsonic, long-endurance autonomous aircraft (refs. 6 and 7).

The X-56A is a remotely-piloted aircraft with a stiff center body and flexible detachable wings. Due to 

the high flexibility of the wings, at certain speeds within the flight envelope the aircraft modes couple 

unfavorably, leading to three flutter instabilities. A control system must be developed with existing 

sensors and actuators to suppress the instabilities. Robust control of these flutter modes will demonstrate 

that lightweight structures can be implemented and flown safely. As part of its sensor array, the X-56A 

aircraft wings will be instrumented with fiber optic sensors (FOS) and fiber Bragg gratings (ref. 8). Fiber 

optic sensors fall into a class of distributed sensors and can measure strain at thousands of locations. 

A research objective in the X-56A program is to use these sensors in a feedback control system. This is 

the problem this report specifically addresses.

Active flutter suppression technology may be extended to not only suppress flutter modes, but to also

control the shape of the aircraft. Research has been published which indicates that drag can be reduced

due to structural deformation changes (ref. 9). The X-53 Active Aeroelastic Wing aircraft (The Boeing 

Company, Chicago, Illinois) warped its wing to achieve higher roll rates, improving its maneuverability

(ref. 10). Research in shape control is ongoing. Smart technologies such as adaptive control surfaces

(ref. 11), adaptive ribs (ref. 12), piezoelectric (ref. 13), and piezoceramic actuators (ref. 14) are under 

investigation. Smart actuation will not be available on the X-56A, so this report investigates the use of 

trailing-edge control surfaces for shape control.

Presented in this report is a robust control law design methodology for stabilizing X-56A aircraft 

models and controlling the wing shapes. The models contain all six rigid-body degrees of freedom (DOF), 

flexible modes, and ten control surfaces and flight actuators. To use the FOS, the methodology requires

implementation of a least-squares estimation (LSE) modal filter. In previous work (ref. 15), the LSE 

modal filter was tested on a clamped aeroelastic plate model, reinforced with spars and ribs and controlled 

by two trailing-edge control surfaces. In that work, simulated shape data at the sensor locations was used 

for sensor inputs to the shape controller. Here, we utilize simulated fiber optic sensor (SFOS) 

measurements for the shape controller. We derive strain modes from the deformation mode shapes and 

use these modes for both modal estimation and strain simulation.

The use of spatial filtering (that is, modal filtering) and distributed sensing enables practical shape 

control. Point sensors such as strain gages or accelerometers could be utilized to support shape control. 

But arrays of point sensors must be frequency-filtered or placed optimally to capture important modal 

coordinates of interest, or both (ref. 16). Distributed sensors with modal filtering do not introduce 

unwanted dynamics into the problem, and the modal displacement estimates will be resistant to the effects 

of residual modes (refs. 15, 17, and 18).

Three objectives are achieved in this report. The first objective is to verify that the modal filter is 

a valid estimator for an aircraft. The second objective is to show that virtual deformation control

(see ref. 15) on an aircraft in flight is achievable. This is accomplished with SFOS feedback (ref. 8) and 

trailing-edge control surface effectors. The third objective is to show that a controller designed to 

incorporate a modal filter can be made to be robust to structured uncertainties and speed variations. We

demonstrate that the popular p-optimal approach (refs. 19 and 20) can be used to design a modal 

controller which achieves multiple-input multiple-output robust stability and performance.

The X-56A Model

The wings of the X-56A model are fitted with four trailing-edge control surfaces. There are two body 

flaps on the rigid center body. All of these control surfaces are available to the controller for both 

AFS and flight control. The medium-sized unmanned aerial vehicle aircraft will be operated in 

experimental studies at subsonic speeds and at low altitude. A top-down view of the X-56A aircraft is 

shown in figure 1.
7

Figure 1. Top-down configuration of the X-56A (L: left; R: right) (figure courtesy Lockheed 

Martin, Bethesda, Maryland).

To simulate fuel in the wings for a passenger transport aircraft, water tanks are included on each wing. 

This generally makes control design more difficult because the controller will either need to adapt or be 

robust to the varying modal properties. This challenge may be compared to the wing stores AFS problem 

of the Northrop YF-17 airplane (ref. 21). For simplicity, the simulation model used here assumes that the 

water tanks are empty. The aircraft has no rudder, so its yaw control will be limited. To yaw, the control 

surfaces must move in opposing directions to generate more drag on one side of the vehicle. To identify 

the flutter characteristics of the vehicle, a normal modes analysis is first completed.

Normal Modes Analysis

The normal modes analysis was completed in MSC NastranTM (ref. 22) (MSC Software Corporation, 

Newport Beach, California) on a detailed finite element model (FEM) developed by Lockheed Martin.

The FEM corresponding to the water-empty case was utilized. The detailed FEM is shown in figure 2.

Figure 2. The finite element model of the X-56A aircraft.
8

The FEM contains over 6,000 nodes. The FEM has not yet been updated through a ground vibration 

test. For model development and flutter analysis, the modes were computed in MSC NastranTM using 

the Lanczos eigensolver. The modes are shown here amplified by a large amount (for visualization) in 

figure 3.

Figure 3. Normal mode shapes: a) SW1B; b) AW1B; c) SW1T; and d) AW1T.

Only four modes are shown here, but 14 flexible modes were included in all of the X-56A models. 

Fourteen were found to be sufficient to capture the salient flexible motion characteristics. Visual 

inspection reveals that the first two mode shapes appear to be strongly coupled with rigid-body motion. 

The coupling is pronounced in the symmetric wing first bending (SW1B) mode, where rigid-body pitch 

and heave are observed. The antisymmetric wing first bending (AW1B) mode has a substantial rigid-body

roll component. The symmetric wing first torsion (SW1T) mode has a slight pitch coupling which is 

difficult to see. The antisymmetric wing first torsion (AW1T) is slightly coupled with roll. The flutter

analysis presented below elucidates how the modes couple in flight.

Flutter Analysis

To determine the theoretical flutter margins of a vehicle the eigensolution of the aeroelastic system 

equation is solved (ref. 2) as shown in equation (1):

/IK() + &IJ() + I() = 1

2 qRQR

FI() (1)

where /{|+×+ is the generalized mass matrix, &{|+×+ is the generalized damping matrix, 

{|+×+ is the generalized stiffness matrix, F{|+×+ is a generalized aerodynamic force (GAF)

matrix, and I() { |+× are the modal displacements of the vehicle. The aerodynamic interaction of the 

modes with the structure is captured by the GAF matrices. For flutter analysis, the GAFs must be 

converted from individual frequencies to Laplace domain. To achieve this, the GAF matrices, F, are 

computed over a range of reduced frequencies and are fit in a rational function approximation (RFA) as

shown in equation (2):
9

FG(H) =  +

-

QR

H +

-

QR

 H + ` }H~ 

QR

-

L

<

H (2)

where , , and  are rational approximation matrices, ` and  are rational approximation matrices of 

lag terms, L is a diagonal matrix of lag roots, - is the characteristic length, and QR is the freestream 

velocity. Roger’s method (ref. 23) with three aerodynamic lag orders was used for the aerodynamic 

approximation in the ZONA Technology Inc. (Scottsdale, Arizona) ZAERO software package (ref. 24).

After substituting equation (2) into equation (1), and performing an inverse Laplace transform, the time 

domain aerodynamics in the equations of motion is achieved. It is important to verify that the frequency 

domain aerodynamics match the time domain aerodynamics for model accuracy. Therefore, in the flutter 

analysis, both frequency and time domain solutions are presented together in figure 4.

Figure 4. Flutter analysis: a) velocity versus frequency, V-f ; and b) velocity versus damping, V-g.

At first glance, figure 4 indicates that the aerodynamic approximation for the state space models 

matches the frequency domain aerodynamic solutions. The primary purpose of figure 4 is to reveal the 
10

flutter characteristics of the X-56A model. Theoretically, flutter appears at a velocity in the model when 

the damping, (), goes to 0. In practice, flutter predicted in this way is conservative, as structural damping 

is unaccounted for.

Figure 4(b) reveals that three flutter modes exist. The first mode that goes unstable (that is, crosses 0)

is the body freedom flutter mode (BFF) at a ratio of design speed, Q"# , of approximately 0.82. The BFF 

frequency is 2.7 Hz (16.9 rad/s). The viscous damping ratio for the BFF is -0.13. Recall that the  value is

equal to the negative of twice the viscous damping (ref. 25). Therefore, positive  corresponds to an 

unstable flight condition. The symmetric wing bending and torsion flutter mode (SWBT) appears at 

approximately 89 percent of design speed at 6.5Hz (41 rad/s). The viscous damping ratio of the SWBT is 

lower than the BFF at -0.06. The velocity versus frequency plot, V-f, indicates that unfavorable coupling 

occurs between the SW1B and SW1T modes, similar to what one might expect when performing a flutter 

analysis on a clamped plate (ref. 25). Typically, the coupled modes’ frequencies shift down but do not 

coalesce. The antisymmetric wing bending and torsion flutter mode (AWBT) appears at 94 percent of 

design speed at 4.9 Hz (30.7 rad/s). The damping ratio of the AWBT is -0.024. This represents another 

unfavorable coupling between the AW1B and AW1T modes.

Figure 4 shows three strong flutter modes in the flight envelope which must be designed for in the 

active control system. The viscous damping of each mode decreases (becoming more unstable) very 

quickly, relative to velocity changes. This is especially true for the BFF. The flutter characteristics 

(damping ratios and frequencies) presented above represent the salient characteristics of the state space 

models over a range of speed. The development of the X-56A state space models is further discussed 

below.

X-56A State Space Modeling

The X-56A aeroservoelastic state space model assumes the matrix time derivative form for simulation 

and control design shown in equation (3) (see ref. 24 or ref. 26):

J() = () + 	P()

() = 
() + 
P() (0) =  (3)

with the initial state (0) = . The 2-dimensional vector () is referred to as a state vector and at any 

discrete time during a simulation can be accessed to give the current “state” of the system. The 

9-dimensional vector  is the system measurements. According to equation (3), only the current state and 

the -dimensional input P are required to know the state in the next time step. The , 	, 
, and 


matrices are real constant matrices with 2×2, 2×, 9×2, and 9× dimensions. States are not unique 

to any given system; however, the X-56A state vector is defined as shown in equation (4):

()  {UZ(), (), J

(), ][:(), [\()}W (4)

where UZ() { |UV× is a vector of airframe states, () { |+× is a vector of modal displacements, 

J

() { |+× is a vector of modal velocities, ][:() { |× is a vector of aerodynamic lag states and 

[\() { |× is formed from a vector of actuator accelerations, velocities, and displacements. For the 

X-56A model, the state space models were calculated at small velocity increments over a range of speed. 

The resulting models are collected into the aggregate model, !, to be used later for analysis.

Figure 4 predicts that the design plant "# will be subject to three flutter modes. To verify that these 

modes are present at the design speed, the X-56A state space model was perturbed with a unit-scaled 

control deflection command to the right-wing control surfaces (see fig. 1), lasting 0.001 s. The pertinent 

state space variations are plotted in figure 5.
11

Figure 5. The X-56A model in open-loop flutter at design speed: a) BFF; b) SWBT; and c) AWBT.

The BFF in figure 5(a) is demonstrated by the interaction of the scaled states, rigid-body heave state,

g\+ , and modal displacement eT	1. The effect of gravity is also modeled. Note that altitude is lost

over time. It is observed that the amplitude of the interacting states increases with time and oscillates at 

the same frequency, out of phase. The SWBT, figure 5(b), also illustrates the unfavorable coupling of

modal displacements: eT1	 and eT1O. The nature of the coupling is difficult to discern because it

appears random in nature. The AWBT, figure 5(c), shows an in-phase interaction of modal displacements:
12

eT1	 and eT1O. The scaled modal amplitudes are small but clearly grow in time, verifying that this 

flutter mode also exists at the design flight condition. Without control, the aircraft is clearly going 

unstable and will require AFS at a minimum.

Methodology

The model of the unstable X-56A aircraft was introduced above. Three flutter modes were identified,

including BFF, SWBT, and AWBT before the design speed. Active flutter suppression is desired to 

suppress the flutter modes; however, a higher objective of this work is to sense and command the shape of 

the aircraft. The available sensors include SFOS and flight sensors. It is shown how the SFOS may be 

used to estimate the modal displacements with the modal filter. This is followed by incorporating the

modal filter into an aircraft shape control design paradigm. The overall methodology is designed to 

support the simulation framework given in figure 6.

Figure 6. The virtual deformation control architecture for the X-56A model.

In this simulation framework, there are three important steps which must be taken. First, the controller,

', must be designed. Second, the modal filter and SFOS must be developed; and third, the modal 

transformation must also be developed, which is used to generate modal displacement reference signals 

for the shape controller. The methodology below is presented, which supports the simulation shown in 

figure 6. The connections and variables are more completely described farther on in this report.

Modal Filtering in the Aircraft State Space Model

The purpose of using the modal filter (see fig. 6) is to estimate the modal displacements of a flexible 

structure. The modal filter was originally developed for the purpose of developing independent modal 

control systems, and was reliably found to remove the possibility of exciting residual modes in the control 

system (ref. 17). In aircraft, frequency-based filters have been traditionally used for this purpose (ref. 4).

The first modal filter was derived using the modal expansion theorem (ref. 27), which states that for 

structures with distinct natural frequencies, the deformation of all nodes of the structure (, , , )
13

can be set equal to a linear combination of the natural mode shapes of the structure

{h(, , ),  = 1 ... 1} as shown in equation (5):

(, , , ) = I()h(, , )





(5)

The mode shapes are made orthogonal to the mass matrix, and are thus orthogonal to each other. 

The orthogonality property makes residual modes invisible to the estimator. Thus, it achieves 

spatially-filtered estimates without corruption from residual modes if enough sensors are available

(ref. 18). The highly-distributed nature of the SFOS is why the SFOS so naturally support modal filtering. 

Strain has a similar relationship, which has been used to define the strain-based modal filter. The

relationship of the strain-based modal filter to the modal displacement state vector () is defined in 

equation (6) (see refs. 28 and 15):

() = (tWt)<tWH+(, , , ) + n (6)

where t = {"h(, , ),  = 1 ... 1} is a 1×1 strain matrix with 1 strain modes, 

H+(, , , ) { |× is the strain and n is the error. This represents the modal filter in its 

pseudo-inverse form (ref. 28). Since the modal filter gives a partial state estimate of the full state vector,

(), the form of the output matrix is simply identity for measured modes.

The sensor output matrix, 
, is a matrix of row vectors relating the sensory information to the state 

vector, (). Previously (see ref. 15) the output matrix was developed for the case of a rectangular wing, 

where rigid-body motion was restrained. Here, the output matrix is adjusted for rigid-body state feedback

concurrent with modal deformation state feedback. It is assumed that only 0 modes are retained for 

measurement. Similar to before, the output matrix is formed assuming all modal displacements,  , and 

rigid-body states, UZ , are measured, as shown in equation (7).


  }

~UV×UV 0UV×+ 0UV×+ 0UV× 0UV×

0+×UV ~+×+ 0+×+ 0+× 0+×

 (7)

Measuring all flexible states (or rigid states) is not required to adequately sense the vehicle state.

Modes which are unstable and contribute strongly to the deformation of the structure are good candidates.

Higher-order modes generally do not significantly contribute strongly to the deformation (refs. 16 and 

17). These modes may be cautiously ignored to reduce the size of the 
 matrix. An excellent method for

choosing which modal displacements to feed back is that of identifying interacting modes from the flutter 

analysis presented in figure 4. Modes which interact strongly and lead to flutter are important to the

feedback control system (ref. 15). One may also look at the modal mass of each mode to determine which 

modal displacements will be significant to the response (ref. 15). A convergence study (ref. 29) may be 

useful for choosing which modes to feed back to control a fluttering structure.

Incorporation of Modal Filtering into Aircraft Control System Design

Described above was how the modal filter may be incorporated mathematically into the state space 

model, which is in turn used for control design. The modal filter implementation was also demonstrated 

in figure 6. The implementation of the modal filter is split into phases. The phases suggested here include:

the control design phase, the modal filter design phase, and the shape reference signal design phase.
14

A methodology for incorporating a modal filter was given in reference 17, however, it was not specific to

an aircraft or for shape control. The phases are presented as follows:

1. Control Design: Phase I

1) Define the robustness and performance requirements.

2) Identify which modes are significant to the response of the model, through the flutter analysis 

(see fig. 4).

3) Update the output matrix to measure the identified modes in the state space model as in 

equation (7).

4) Scale and reduce the order of the plant.

5) Design the controller, K , with the chosen control synthesis technique.

6) Reduce the controller order.

7) Verify that the performance and robustness requirements were met, while simulating the 

controller on a range of plants around the design plant.

8) Iterate steps 2 through 7.

2. Modal Filter Design: Phase II

1) After selecting the modes for feedback, place the sensors either using an optimal sensor 

placement (ref. 30) technique or intuition of which modes are being measured.

2) Form the sensor strain modal matrix, tZAV .

3) Select an operator to calculate the modal displacements with the modal matrix.

3. Shape Reference Signal Design: Phase III

1) Identify the reference points on the vehicle for shape tracking, using the optimal sensor placement 

techniques or intuition.

2) Correct the elastic mode shapes to remove the rigid-body translations or rotations, if any.

3) Form a modal set-point transformation matrix with the corrected mode shapes.

4) Verify that the flexible modes included in the controller lead to acceptable deformation control.

If they do not, include more modes in the controller for integral tracking within the actuator 

bandwidth and repeat the control design phase.

Depending on the platform in which the modal filter is implemented, some of the steps may not need

to be taken. For example, if a plant is small enough, order reduction may not be required. During 

reference signal construction for a less-flexible structure, flexible modes may not be coupled with 

rigid-body translation and rotation. These modes will not need correction.

Control Design

The methodology defines the way the modal filter fits into an aircraft shape control design paradigm.

It was shown that only simple changes were required for the state space matrices to utilize the modal 

filter; see equations (6) and (7). To design the controller as stated in Phase 1, the actual fiber optic sensor 

locations need not be determined yet; only the modal displacements which are desired to be controlled 

should be known. The requirements of the vehicle controller are now derived, and a μ-optimal controller 

is designed, which can track both rigid-body states and the first two modal displacements.
15

Control Design Overview

It is desired to suppress the flutter modes (see figs. 4 and 5) and track the rigid-body commands of the 

X-56A aircraft. In addition, one of the primary objectives is to demonstrate virtual deformation control on 

a flight vehicle. To this end, the contributing structural modal displacements will be tracked. In this 

regard, the shape of the aircraft will be controlled. The following discussion will be primarily concerned 

with how Phase I was accomplished. The first step of Phase I is to define the robustness and performance 

requirements.

Since flutter is a potentially destructive phenomenon, the modal controller must be robust to 

uncertainty. This is especially important for a modal controller, since during a ground vibration test, mode 

shapes are typically not predicted exactly (ref. 16). In fact, cross-coupling is a primary argument against 

using a modal filter for control. Cross-coupling can occur when any one mode shape in the modal matrix 

is not accurate and propagates into other modal displacement estimates through projection (ref. 31). To 

reduce these uncertainties, the following uncertainty requirements were defined: 10 percent multiplicative 

uncertainty on the inputs and outputs, and 10 percent additive uncertainty on the scaled plant. Since flutter 

speeds are hard to predict precisely and structural damping is not modeled, the controller requirements 

must also be satisfied at off-design conditions. A notional requirement is that the controller must be 

robust to a 3-percent speed variation. Since the flutter mode damping decreases so rapidly with speed 

variations (see fig. 4), this requirement is seen as practical.

Robustness to parameter uncertainty is highly desirable, but the controller must also meet performance

specifications. The tracked measurements must respond to doublet inputs with low rise times and small 

overshoots. The performance must also be achievable under the aforementioned uncertainty conditions. 

The uncertain plant and required control system is summarized in figure 7.

Figure 7. The uncertain plant and the required controller.

With the uncertainties in figure 7 in mind an p-optimal control approach was taken (ref. 19). The 

design approach uses a hybrid of performance weights and uncertainty weights. Robust stability (RS) is 

achieved if and only if a system is stable for all perturbed plants about the nominal model up to the 

worst-case uncertainty (ref. 19). The robust stability condition which must be met for the p-optimal 

controller is shown in equation (8).

L  p/($u% ) < 1, u (8)

The p is calculated over the frequency range with the relation shown in equation (9) (see ref. 19):

p/($u% ) = 1

02{*+|(~*+/($u% )d) = 0 9= H=PP= d, r7(d)  1} (9)
16

where *+ is the stability margin, r7 is the maximum singular value, and d is the structured uncertainty.

The transfer function matrix from the input of the uncertainty blocks to the outputs of them as shown in

figure 7 is presented in equation (10):

/ = 

TNON TN'A TN'A

TUN TU'A TU'A

TAN TAA TAOA

, ! (10)

where TN is a matrix of proper input weights, TU is a matrix of proper additive weights, TA is a matrix of 

proper output weights, and ' is the controller. From this matrix, the salient sensitivity and 

complementary sensitivities from the / structure are identified which correspond to the requirements.

The magnitude of the singular values predict the performance of the control system. These closed-loop 

transfer functions are defined in equation (11) (see ref. 19):

ON = '(~ + ')<

'A = '(~ + ')<

A = (~ + ')<

N = (~ + ')<

OA = '(~ + ')<

(11)

where  is the input complementary sensitivity,  is the output sensitivity,  is the input sensitivity,

and  is the output complementary sensivitiy. To improve rejection of control input uncertainty, the 

controller was designed around the plant as shown in figure 8.

Figure 8. The control design framework.

The design plant, "# , at the design speed, Q"# , was of order 130. To improve controller synthesis,

"# was scaled by the full range of actuator movement and expected sensor output changes (see ref. 19).

The translational states, e\+, e\+, and e\+, and the velocity state were removed from the model. 

With the resulting 126th-order model, balanced reduction (ref. 32) was performed to bring the model 

order to 90 states. The selected states to be tracked were: T1	 and T1O modal displacements; pitch

angle, f; and bank angle, h. The angle of attack, b; angle of sideslip, c; yaw angle, i ; and a body 

accelerometer were also sensed but were chosen to be suppressed.
17

Traditional proper weights from a mixed R synthesis were utilized, along with multiplicative 

uncertainty at the plant inputs. The input uncertainty weight,  , was adjusted to achieve maximum 

amplitude near the actuator break frequency. Sensitivity weight, , was adjusted for integral tracking on 

tracking states and for suppression on suppression states. It was found that the break frequencies of the 

modal displacement performance weights had to be increased 10 rad/s relative to the airframe state weight 

break frequencies of 1 rad/s. The break frequency of the control weight,  , was adjusted to 5 rad/s for 

reduced control surface movement. The break frequency of the complementary sensitivity weight, ,

was set to 30 rad/s to improve high-frequency noise rejection. The uncertainty transfer function, , was 

calculated (see fig. 8) to be as shown in equation (12):

1 = 

TNON TN'A

TVN TVA

T3ON T3'A

TWN TWOA

  (12)

One may verify that all of the pertinent closed-loop transfer functions corresponding to those shown in 

equation (11) are present in equation (12). Thus, by reducing the xR norm of 1, the robustness will 

increase and it will be easier to meet the specified requirements (see fig. 7). For this control design 

architecture, the p-optimal controller was computed using the MATLAB® (The MathWorks, Inc., Natick, 

Massachusetts) Robust Control Tool Box. To find the controller, DK-iteration (see ref. 19) was 

performed, which solves the iterative optimization problem shown in equation (13):

min

a ¤min ¥¦

§
1(')
<§R ̈ (13)

The DK-iteration resulted in a 162th-order controller after some trial and error with the weights in 

figure 8. The controller was then internally balanced (ref. 32) and truncated to 44 states without a 

substantial loss of robustness or performance. This resulted in an xR norm of 3.29. It was difficult to 

meet both performance and robustness requirements with the scaled plant. Therefore, the desired

xR norm of 1 was not achieved. Rescaling may improve the controller.

Nominal stability (NS) was verified for the reduced-order controller by verifying that eigenvalues of 

the closed-loop system had real negative parts. An analysis of the singular values of the open-loop plant 

and controller as well as the closed-loop sensitivity functions given in equation (11) are presented in

figure 9.
18

Figure 9. Maximum singular values of open-loop (GK) and closed-loop sensitivity functions.

At first glance, figure 9 reveals that most of the closed-loop transfer functions stay at or beneath 

10 dB, corresponding to a magnification of the inputs to these transfer functions of approximately 3.2.

The controller bandwidth is 25 rad/s, which is below the actuator break frequency. After this frequency, 

the complementary sensitivities fall sharply at 60 dB/decade. Therefore, noise will be attenuated well past 

25 rad/s.

Recall that it is desired that the controller be robust to uncertainty and, of course, disturbances.

Robustness to input multiplicative uncertainty is strongly impacted by the maximum singular values of 

ON; see equation (10). The maximum singular value of ON within the bandwidth of the controller has a

magnitude of 10 dB. Therefore the controller will amplify input uncertainties at frequencies where flutter 

is most likely to occur.

Robustness to additive uncertainty or gust responses is strongly affected by the peak of the closed-loop 

transfer function, 'A. Over the entire bandwidth, the singular values of 'A are below 0 dB and, thus,

gust-like disturbances will be attenuated. The magnitude of the singular values of 'A falls off quickly

after 6 rad/s. This predicts that higher-frequency turbulence will be rejected.

Robustness to sensor uncertainty is predominantly dependent on the peaks of A and OA. From 

0.1 rad/s to 1 rad/s the singular values of A show that plant uncertainty is neither amplified nor reduced;

however, near flutter frequencies the plant uncertainty is amplified. The singular values of OA indicate 

that sensor noise or output uncertainty will not be strongly amplified past 25 rad/s.

Some aspects related to tracking may also be identified from figure 9. For best tracking accuracy, OA

must be equal to 0 dB over most of the controller bandwidth. Figure 9 shows that there are several peaks

above 0 dB, which may lead to overshoots or poor performance. The tracking history can be investigated

by inputting doublet reference inputs to the controller. The set of charts presented in figures 10-12

expands on the performance and robustness characteristics of the controller.
19

Figure 10. Performance chart for doublet inputs to tracked variables: a) control surfaces; b) pitch 

angle tracking; c) bank angle tracking; d) SW1B modal displacement tracking; and e) SW1T modal 

displacement tracking.
20

Figure 11. Mu analysis over speed range: a) © chart; and b) corresponding uncertainty weightings.
21

Figure 12. Performance over 3.3% variation of velocity and uncertainty range: a) pitch angle

tracking; b) bank angle tracking; c) SW1B modal displacement tracking; and d) SW1T modal 

displacement tracking.

It was desired that the responses of the system to a doublet have low rise time and small overshoots.

This goal was for the most part achieved (see figs. 10 and 12). All of the signals had low rise times of 

0.25 s to 1.5 s. The change in pitch angle, ef, had excellent tracking characteristics. The change in bank 

angle, eh, showed a reasonable response. The tracked modal displacements were non-minimum phase.
22

The modal displacement, eT1O, experienced a 25-percent overshoot, which was predicted by the 

singular values of OA.

There was very little rigid-body longitudinal and lateral coupling. It was interesting that the 

commanded rigid-body signals moved off their reference point when modes were commanded. The 

converse also happened; pitch angle commands tended to disturb the modal displacements. This indicates

that a strong coupling between rigid-body motion and flexible motion in the models exists, which cannot 

be easily avoided. 

The normalized control movements for ef and eh tracking were reasonably small. In fact, the 

movement of the control surfaces for eh tracking was almost unnoticeable. During modal displacement 

commands, the normalized control movements moved to a maximum amplitude of 0.4 on the body flaps.

Since the outputs are scaled to half of the actuator limits, this is a relatively large control movement, but 

is not unreasonable. In figure 10(a) it can be seen that the normalized control movement was significantly 

larger when commanding the modal displacements, than when commanding the flight variables.

Significant control authority is required to perturb the X-56A structure from equilibrium; however, 

improper scaling may be to blame. Based on this analysis, the nominal performance (NP) appears to be 

adequate, although it might be improved with better weights and plant scaling.

Figure 9 indicates that some disturbances and uncertainties are amplified; however, the actual impact 

of these uncertainties on the stability must be checked with a p analysis as shown in figure 11. The 

p analysis calculates robustness of the modal controller across dimensions of speed and uncertainty. For a 

range of plants ! additive, multiplicative input and output uncertainties with 10 percent weightings was

added; see figure 11(b). The chosen shapes of the weightings were meant to replicate worst-case 

scenarios. Additive uncertainties may be more likely to occur in the 0.1 rad/s to 10 rad/s range due to gust 

disturbances. Multiplicative uncertainties may be more likely to occur at higher frequencies due to 

actuator dynamics and sensor uncertainties and noise. For the generalized plant with uncertainties, p were 

then calculated across the operational frequency range with one controller, as shown in figure 11(b).

Recall that RS is guaranteed if the condition given in equation (8) is met. Therefore, it is desirable that 

p be less than one for all structured uncertainties. The maximum p for all perturbed plants was 0.92. 

Therefore, RS is achieved. The variation of p across the frequency range shows that it tended to increase 

at closed-loop flutter mode frequencies. The p tended to increase for speeds past the design speed.

Conversely, the p decreased for plants below the design speed. This happens because as speed increases 

the damping of the flutter modes decreases; see figure 4(b). The reverse happens when decreasing in 

speed. Figure 11 indicates that the design requirement to be RS to a 3-percent variation in speed was met.

Hence, it is shown here that the modal controller can be designed to be a robust to modeled uncertainties.

Performance at off-design conditions must also be analyzed.

Robust performance (RP) is achieved if and only if the performance objectives are achieved for all 

possible plants about the nominal plant up to the worst-case uncertainty (ref. 19). There were no hard 

performance requirements defined other than that the tracked signals have low rise times and small 

oscillations. Figure 12 indicates that the responses from the perturbed plants were tight around the 

acceptable nominal plant responses shown in figure 10. The overshoots on the modal displacements for 

some perturbed plants did increase to 50 percent, but this overshoot increase is acceptable for the current 

study. Therefore, RP is achieved for these lenient performance requirements. Since NP, RP, NS, and RS

are adequately achieved, the modal controller is acceptable. It meets basic criteria necessary for utilization 

in an aircraft. This completes Phase I. The controller designed here is used in the below discussion for

virtual deformation control simulation.

Virtual Deformation Control Simulation

The modal controller derived above adequately satisfies the robustness and performance objectives for 

the X-56A model. The simulation study presented here demonstrates the use of the modal filter for virtual 

deformation control of the aircraft. To complete Phase II, the simulated fiber optic sensors are placed on 
23

the X-56A FEM. The sensor strain modes are calculated and the sensor strain mode matrix is formed. It is 

then shown that in order to create a proper deformation signal for Phase III, rigid-body translations and 

rotations must be removed from the deformation mode shapes. An objective function is given, which is

minimized to achieve this goal. For demonstration, the controller designed in Phase I is put into a 

simulation with the modal filter designed in Phase II. The wing tips are commanded to a particular 

deformation using the reference signal derived in Phase III.

Fiber Optic Sensor Placement

Traditionally, strain sensors or accelerometers used for active structural control are placed in an 

optimal sense. The FOS can measure strain at locations that are less than a one-half-inch apart, and can 

contain several thousand measurement stations along a single fiber. One of the benefits of having so much 

spatial sensor resolution is that optimal sensor placement (OSP) routines (refs. 30 and 32) lose 

significance. The first few structural modes in bending and torsion are most significant to flutter, so the 

sensors are placed from root to wing tip to capture bending information. The sensors are laid from trailing 

edge to leading edge to capture torsional motion. The resulting sensor configuration looks like three claw 

marks across each wing. The resulting sensor configuration is given in figure 13.

Figure 13. The fiber optic sensor layout on the X-56A model.

With the six fibers placed, points on the aircraft were also chosen where deformation could be

commanded with the reference signal derived in Phase III. These points can be anywhere on the aircraft 

(see ref. 15). A multitude of points could have been chosen, but for simplicity four points were selected.

While not required, an OSP technique, known as Kammer’s effective independence (EI) procedure

(ref. 33), was utilized to locate the four points. This procedure starts by forming a sensor projection 

matrix from a selection of mode shapes. In this case, the sensor deformation mode shapes were utilized. 

The rows of the modal matrix were iteratively reduced based on the smallest diagonal of the projection 

matrix. After implementing this procedure, the resulting four deformation command locations were 

determined and are shown in figure 13.

These points are right-wing trailing edge (RWTE), right-wing leading edge (RWLE), left-wing trailing 

edge (LWTE), and left-wing leading edge (LWLE). Their index locations in the modal matrix are referred 
24

to as a«. As in previous work (ref. 15) results indicate that the wing tips carry the most modal information. 

Note that the selected points do not have to be on sensor locations for the virtual deformation concept to 

be employed.

X-56A Sensor Strain Modes

The strain and deformation modes defined at the SFOS locations are necessary to develop an 

operational modal filter. To model sensor locations on the model, a spline was made on existing nodes on 

the FEM in MSC NastranTM from root to tip on the wing. The distance between each node on the spline 

was set at one-half-inch, which was the spacing between each measurement grating on the fiber (ref. 8).

At each node of the spline or sensor location a minute mass was added. The sensor nodes on the spline

were then interpolated with RBE3 MSC NastranTM cards. The modal analysis conducted in MSC 

NastranTM gives sensor deformation mode shapes corresponding to each flexible mode.

The close proximity of the sensors (of approximately one-half-inch) along the SFOS gives a unique 

opportunity to calculate axial strain modes directly from the modal deformation at the sensor locations. 

To our knowledge, this is not a common practice. It is also possible to calculate the strain-displacement 

relation using the finite element method (ref. 34) which may be more reliable for complex structures. One 

might also perform a static strain analysis for each deformation mode shape to obtain a strain mode shape. 

For simplicity, the following technique is utilized to convert each sensor deformation mode shape from 

MSC NastranTM to a strain mode shape. Recall the axial strain in an element (ref. 35), where ] is in the 

direction of the strain measure, shown in equation (14):

noo = ¬P

¬]

(14)

The FOS measure axial strain, so it is assumed that ] is in the local coordinate direction of the SFOS

layout. Figure 14 is presented as a visual aid for the breakdown of equation (14) into a usable format.

Figure 14. The fiber optic sensor locations deformed for notional mode shape.

Figure 14 represents sensors from the SFOS along the aircraft wing, which have been labeled for 

clarity. Each sensor has a line drawn between it, where a fictitious isotropic beam is assumed to exist. The

deformed sensors are connected with blue lines. The original sensors are connected with black lines.

Consider the deformed fictitious beam, to be in blue. Linear algebra may be used to show that the strain in 

any fictitious beam element of the SFOS shown in figure 14 is calculated as shown in equation (15):
25

noo = 6=9>3?@(AC?????@)4868 ????????@  467777

467777 (15)

where 46?????@ is a directed line segment from the un-deformed sensor location 4­|® to the un-deformed 

sensor location 6­|®, and 4868 ????????@ is a directed line segment from the deformed sensor location 48­|® to the 

deformed sensor location 68­|® and P?@ is the unit vector. The projection in figure 14 calculates the final 

length of the beam section with respect to its original beam orientation. The difference between the 

original length and the deformed length, divided by the original beam length, gives the strain in the beam 

element. The strain in one beam may differ from the strain in the beam next to it. The strain from modal 

deformation on the = sensor (or sensor location 4) may be defined to be the average of the strain on 

either side (see fig. 14) as shown in equation (16):

n(H) 

1

2 [noo

; + noo< ] = 1

2 ±

6=9>3?@(ABCD ??????????@)4#

5

6; ?????????@5  4#6; 777777

4#6; 777777 +

6=9>3?@(ABCE ??????????@)4#

5

6< ?????????@5  4#6< 777777

4#6< 777777 2 , Hn (16)

where noo

; is the axial strain measured on the fore side of the sensor and noo< is the axial strain measured 

on the aft side of the sensor. This operation can be carried out for each mode and at each sensor node, and 

strain modes i+ may be formed by repeating equation (16) for each sensor. The sensor strain modes that

result are collected into the desired strain mode matrix defined at SFOS sensor locations, as shown in 

equation (17):

tZAV  [i, i ... i ... i+] (17)

To compute the strain at point 4 with good precision, candidate sensor locations 6;

5 and 6<5 must be 

selected close to the original sensor. Since the change in strain was so small between two adjacent 

sensors, two sensors were selected a few inches away from the desired sensor location. Even with this 

technique, the resulting strain modes from equation (16) appeared noisy. Thus, the strain was fit to a

polynomial surface for each wing and an M-estimator (ref. 36) was utilized to trim outliers. The first four 

sensor strain modes used for the X-56A simulation studies are given in figure 15.
26

Figure 15. Sensor strain mode shapes: a) SW1B; b) AW1B; c) SW1T; and d) AW1T.

The sensor strain modes were computed for all 14 flexible modes of the X-56A, however, only the 

first four modes are shown here. The SW1B mode has a strain distribution which starts out very high near 

the root of the wing (see fig. 15) and goes to nearly zero at the wing tips. It is typical that in bending, 

strain is highest near the root of the beam (ref. 16). The AW1B mode shows an antisymmetric strain 

distribution in bending. The SW1T mode shows a distribution of strain which has an inflection near zero 

close to the mid-span of the wing, indicating that for torsion modes where the wing is twisting, very little 

strain exists in the mid-span of the wing. The AW1T mode, like the AW1B, mode has an antisymmetric 

strain distribution in torsion. The sensor modes are collected into the modal matrix defined in 

equation (17) and, together with equation (6), the modal states of the aircraft can be measured. With the 

strain mode matrix prepared, attention is turned to the reference signal, which is needed for virtual 

deformation control. For reference signal creation, the deformation mode shapes shown in figure 3 must 

be transformed.

Reference Signal Creation

Phase III from the modal filter control design methodology is to design a modal reference command.

From equation (5) a least-squares approximation may be used and the modal reference command is 

shown in equation (18) (see ref. 15):

I = sW(., 0)s(., 0)

<sW(., 0) (18)

where  is a vector of deformations from the un-deformed aircraft corresponding to the index vector,

., in the pure elastic deformation modal matrix, s. Recall that each row of the modal matrix corresponds 

to a physical location on the aircraft, (, , ). The index of modes, 0 , corresponds to the index of

modes within the modal matrix, which the controller is designed to track. In this case, 0 corresponds to 

{1,3}, since the SW1B and SW1T modal displacements are tracked.
27

Performing virtual deformation control on the X-56A model is not straightforward because the flexible 

modes of the vehicle are not purely elastic. The normal mode shapes (see fig. 3) are corrupted by strong 

rigid-body motion. The use of these modes to recreate the modal reference signal will lead to large 

displacement errors. The mode shapes required in equation (18) are calculated by an optimization 

procedure on the original mode shapes. The optimization procedure rotates and translates every node in 

the mode shape about the aircraft center of gravity (cg).

Recall that a mode shape corresponds to deformations at 1 grid points. At each grid point, the mode 

shape is defined for six DOF, so each mode shape technically has 61 DOF. The number of DOF 

is increased with global DOF parameters so the total DOF in each mode equate to 61 + 6 DOF. The 

global DOF are the only variables in the objective function defined for the 

 mode shape given by

equation (19):

 =  ́

 ́

μ

¶

·

¶

 ̧



\: × 1W

\: × 1W

\: × 1W

  + O(gf)O(gh)O(gi)

1

o

o

o

»3 + ^

  \: × 1

W

3 + ^

  \: × 1

W

3 + ^

  \: × 1

W

1⁄4

1⁄2

1⁄2

1⁄2

3⁄4

+ ±

e × 1W

e × 1W

e × 1W

2

¿

¶

À

¶

Á

 ±

3

W

3

W

3

W

2 ́

 ́



 (19)

where \:, \: and \: are the scalar coordinates of the cg of the aircraft, O(M) is a right-hand-rotation 

matrix operator, 3, 3 and 3 are 1×1 coordinate vectors of the un-deformed aircraft, ^

 , ^ , and ^

are 1×1 coordinate vectors of the 

 modal deformation vector, h. The six scalar variables in the 

optimization function include translational coordinates: e, e, and e , and rotations about all three 

axes: ef, eh, and ei.

By minimizing the objective function, equation (19), for the 

 mode, with respect to these six scalar

variables the relative distance between the un-deformed aircraft and the deformed aircraft is minimized. 

Since the squared Euclidean norm is a strictly convex function of its respective arguments, the objective 

function is also convex (ref. 37); therefore, there are numerous discrete optimization methods available in 

the literature which can be used to solve it. For each mode , the objective function is minimized with a 

locally convergent random search technique known as the compass method (ref. 37). Four of the resulting 

corrected mode shapes are given in figure 16.
28

Figure 16. Corrected mode shapes a) SW1B; b) AW1B; c) SW1T; and d) AW1T.

The mode shapes with rigid-body components removed (see fig. 16) may be used with equation (18)

to develop appropriate reference signals. Note that the flexible movement within each mode appears 

amplified when compared to figure 3. Deformation commands can now be adequately transformed to 

achieve virtual deformation control.

Simulation

The simulation architecture introduced in figure 6 represents the inner-loop design for the X-56A

simulation model. The inputs to the controller are references, provided by a notional outer-loop control 

system. The references in figure 6 are separated into rigid-body commands, UZ

, from the flight 

computer and vertical deformation commands, , at selected locations (see fig. 13), namely, LWLE, 

LWTE, RWLE, and RWTE. The reference vertical deformations were chosen to be 0.6 percent of the 

span of the vehicle, which corresponds to modal displacement commands of approximately twice the 

scaled value. This value was thought to be reasonably small, since the control movement for scaled modal 

displacement commands was less than 50 percent of the actuator range as shown in figure 10(a).

Since the reference deformations were equal, a bending-type motion for the aircraft wing shape 

results. Different deformation choices could lead to torsion-type motion instead. The deformation 

references can be verified to generate a bending motion when transformed with equation (18) into a 

modal reference. For the following study, the airframe state references are set to 0.

The controller that was designed above (see figs. 9-12) takes the difference of the references and the 

feedback signals to produce a control input signal. Multiplicative noise is added to the controller inputs,

with a mean of 0 and standard deviation of 0.1. The plant takes the inputs via equation (3) and produces 

the state vector, Â , in the output, which includes the measurements SW1B, SW1T, and the residual

modes. The plant also outputs the rigid-body measurement vector, UZ.

The SFOS measurements are simulated by multiplying the sensor strain matrix, tZAV , with the vector,

. Normal noise, 2# , with mean 0 and a standard deviation of 3 p# is then added to this strain vector. 

Since the rigid measurements are scaled, a normal distribution with mean 0 and standard deviation of .01 
29

is added to the measurements to simulate noise. The results of the simulation to a reference doublet 

command are shown in figure 17.

Figure 17. Virtual deformation tracking time history: a) deformation tracking at wing tips; b) 

tracked modal displacements; and c) airframe states.

The change in deformation near the wing tips, figure 17(a), indicates that the actual deformations 

moved very close to the deformation set-points. The error is due to the modal displacements from residual 
30

modes (see ref. 15). This is expected, since the trailing-edge actuators are not collocated with the 

structure’s sensors. Due to this fact and since the modes are coupled aerodynamically, independent modal 

control cannot be achieved (ref. 16).

The commanded modal displacements are shown in figure 17(b), and they are tracked well. Since the 

command is primarily due to bending, the SW1T modal reference is very small. The scaled SW1B modal 

displacement amplitude moves the greatest amount. Both modes had some overshoot and oscillations

during the initial part of the command, which was predicted and is seen in in figure 10. The steady-state

response has oscillations primarily due to the rather large multiplicative input noise modeled in the 

system.

During the shape-deformation maneuver, the aircraft experienced no enduring change of velocity or 

pitch angle. The angle of attack, however, achieved a steady-state change; the change in angle of attack 

should have also effected an enduring change in velocity. An angle-of-attack change generally leads to a 

drag increase or decrease; however, this was not observed, likely due to improper drag modeling. If the 

results were taken as truth, it could be concluded from this analytical model that the effective lift over 

drag (L/D) ratio has changed. This observation could be a result of control surface movements, the 

structural deformation itself, or both. The caveat here is that the results are achieved on a linearized plant 

model defined for small perturbations around the trim point. In flight, the aircraft achieves a new trim 

point from the angle-of-attack change, which may result in a lower or higher L/D ratio.

Outer-loop control requirements are also observed from figure 17. The results indicate that the 

flight-path angle has changed due to shape deformation commands. Therefore, the aircraft will require a

heading-angle-hold outer-loop controller to make virtual deformation control feasible. The outer-loop 

controller could command the pitch angle to control the flight-path angle; or it could command engine 

throttle settings, which are not currently modeled. If the throttle is changed to achieve trim, fuel efficiency 

degradation or improvement may be realizable, depending on the command to the structure.

Previously, it was shown that the X-56A model experiences strong flutter at the design speed 

(see fig. 5). Technically, the shape controller demonstrated above is capable of suppressing flutter. From 

the previous plots, this has not been shown explicitly. To demonstrate this in a traditional sense, the 

aircraft is perturbed as before and the control system is turned on after a period of time of 1.6 s. To 

reduce control-induced oscillations, it was ramped in over a 1-s period, with all references set to 0. In 2 s, 

the shape tracking controller achieves AFS, as shown in figure 18.
31

Figure 18 Active flutter suppression time history: a) BFF suppression; b) SWBT suppression; and

c) AWBT suppression.

During AFS there was an altitude change, as can be seen in figure 18(a); however, this change 

occurred because an outer-loop guidance system is not currently modeled. The achieved flutter 

suppression was not unexpected, since control was implemented on a linear plant model. Nonlinear 

effects have not been included. Only the poles must be considered in such a model. The poles of the 
32

closed-loop system have real negative parts and the closed-loop system is, thus, stable. The open-loop 

poles and the closed-loop poles are plotted together in figure 19.

Figure 19. Open-loop versus closed-loop poles: a) open-loop poles; and b) closed-loop poles.

The poles of the open-loop system indicate the presence of the three flutter modes identified above

(see fig. 4). There are also poles from lightly-damped structural modes near the vertical axis starting at 

54 rad/s. When the loop is closed with the controller, the flutter poles migrate to the stable region in the

left half-plane. The lightly-damping structural mode poles remain unchanged, as they are outside the 

controller bandwidth. The controller introduces poles seen tightly grouped near the lower right-hand

corner of figure 19(b). Integrator and estimator poles are present. The lowest damping of the closed-loop 

system poles within the actuator bandwidth is 0.08. This represents a significant improvement over the 

open loop BFF damping of -0.13.

The results of all simulations must be accepted as notional. Recall that all of the simulations are 

completed on a linear plant model with simulated strain measurements. Further work with time-marching

nonlinear simulations will be necessary to validate the controller architecture for AFS and active shape 

control; however, these preliminary simulation results are promising. Time histories indicate that the 

modal filter is a suitable estimator for both AFS and virtual deformation control. Simulation results 

similar to this were published previously in reference 15, but those results were for a clamped-wing 

model only. 

Conclusions and Recommendations

There were three primary objectives set out to be accomplished in this report. The first objective was 

to demonstrate that the least-squares modal filter is a candidate aircraft modal estimator and that it can be

adequately used for control feedback. For practical implementation, three phases for incorporating the 

modal filter were presented and followed throughout the report. The modal filter was demonstrated with 

simulated fiber optic sensors. The estimated modal displacements were used for both shape control and 

active flutter suppression of the X-56A simulation model. It was found that significant angle-of-attack
33

changes occurred during shape control. The angle-of-attack change may lead to improved lift to drag 

ratios on the aircraft. Therefore, structural shape control may someday enhance performance in fuel 

efficiency or maneuvers. The caveat is that it is not known what lift to drag ratio will result at the new 

trim condition. Nonlinear simulations and experimental studies are required to explore this further.

The second objective was to demonstrate that the modal filter and distributed sensing can be used to 

control the deformation of the aircraft wing structure in flight. By designing a controller for the modal 

filter and simulated fiber optic sensors, and commanding the wing-tip deflections, this objective was 

satisfied. Only bending commands were presented here, however. The deteriorating effects of residual 

modes may be more prominent with other command shapes. Modeling errors could also appear 

prominently in a realistic bench case study. In-flight experiments are also required to validate its use for 

aircraft state estimation and control.

The third objective was to show that the modal controller can be designed to have good performance 

and be robust to parameter variations. This objective was satisfied by employing the powerful mu

synthesis technique. It was shown that the controller had robust performance and was robustly stable to 

structured uncertainty and model speed variations. The requirements for robust performance were lenient, 

however, and further improvements can certainly be made here. Future endeavors will extend the 

controller to be robust to fiber optic sensor failures.

Acknowledgments

This research was funded by the NASA Aeronautics Research Mission Directorate Subsonic Fixed 

Wing Project and was approved by Steve Jacobson. Mr. Jacobson, the Dryden Controls Branch Chief;

John Bosworth, the X-56A Chief Engineer; Jack Ryan, the X-56A Principal Investigator; Ryan Stillwater,

the X-56A Controls Lead; Chan-gi Pak, the creator of the X-56A aeroservoelastic model; and the

aeroservoelasticity expert Marty Brenner; have all been excellent advisors in the development of these

results.

References

1. Taminger, Karen, “Technical Challenges to Reducing Subsonic Transport Weight,” AIAA Aerospace 

Sciences Meeting, January 2012.

2. Karpel, Mordechay, “Design for Active Flutter Suppression and Gust Alleviation Using State-Space 

Aeroelastic Modeling,” Journal of Aircraft, Vol. 19, No. 3, 1982, pp. 221-227.

3. O'Connel, R. F., and A. F. Messina, “Development of an Active Flutter Margin Augmentation System 

for a Commercial Transport,” Journal of Guidance and Control, Vol. 3, No. 4, 1979, pp.352-360.

4. Zeng, Jie, Jiang Wang, Raymond de Callafon, and Martin Brenner, “Suppression of the 

Aeroelastic/Aeroservoelastic Interaction Using Adaptive Feedback Control Instead of Notching 

Filters,” AIAA-2011-6459, August 2011.

5. Norris, Guy, “Breaking the Flutter Barrier,” Aviation Week & Space Technology, Vol. 174, Issue 5,

2012, pp. 28-29.

6. Beranek, Jeff, Lee Nicolai, Mike Buonanno, Edward Burnett, Christopher Atkinson, Brian 

Holm-Hansen, and Pete Flick, “Conceptual Design of a Multi-utility Aeroelastic Demonstrator,” 

AIAA-2010-9350, September 2010.
34

7. Nicolai, Leland, Keith Hunten, Scott Zink, and Pete Flick, “System Benefits of Active Flutter 

Suppression for a SensorCraft-Type Vehicle,” AIAA-2010-9349, September 2010.

8. Richards, Lance, Allen R. Parker, William L. Ko, and Anthony Piazza, “Real-time In-Flight Strain and 

Deflection Monitoring with Fiber Optic Sensors,” Space Sensors and Measurements Techniques 

Workshop, Nashville, Tennessee, August 2008.

9. Kusmina, S., F. Ishmuratov, and V. Kuzmin, V., “Minimization of induced drag of elastic airplane,” 

AIAA-2004-4611, August 2004.

10. Pendleton, Edmund W., Denis Bessette, Peter B. Field, Gerald D. Miller, and Kenneth E. Griffin,

“Active Aeroelastic Wing Flight Research Program: Technical Program and Model Analytical 

Development,” Journal of Aircraft, Vol. 37, No. 4, 2000, pp. 554-561.

11. Weisshaar, Terrence A., and David K. Duke, “Induced Drag Reduction Using Aeroelastic Tailoring 

with Adaptive Control Surfaces,” Journal of Aircraft, Vol. 43, No. 1, 2006, pp. 157-164.

12. Austin, Fred, Michael J. Rossi, William Van Nostrand, Gareth Knowles, and Antony Jameson, “Static 

Shape Control for Adaptive Wings,” AIAA Journal, Vol. 32, No. 9, 1994, pp. 1895-1901.

13. Ehlers, S. M., and T. A. Weisshaar, “Static Aeroelastic Control of an Adaptive Lifting Surface,”

Journal of Aircraft, Vol. 30, No. 4, 1993, pp. 534-540.

14. Hu, Yan-Ru, and George Vukovich, "Active robust shape control of flexible structures," 

Mechatronics, Vol. 15, No. 7, 2005, pp. 807-820.

15. Suh, Peter M., and Dimitri N. Mavris, “Modal Filtering for Control of Flexible Aircraft,” 

AIAA-2013-1741, April 2013.

16. Preumont, Andre, Vibration Control of Active Structures An Introduction, 2nd ed.; Kluwer Academic 

Publishers, Dodrecht, The Netherlands, 2002.

17. Tzafestas, S. G., ed., Applied Control: Current Trends and Modern Methodologies, Marcel Dekker, 

Inc., New York, 1993.

18. Preumont, A., Francois, A., De Man, P., and Piefort, V., "Spatial filters in structural control," Journal 

of Sound and Vibration, Vol. 265, No. 1, 2003, pp. 61-79.

19. Skogestad, Sigurd, and Ian Postlethwaite, Multivariable Feedback Control Analysis and Design, 2nd 

ed., John Wiley & Sons, Ltd, West Sussex, 2005.

20. Zhou, Kemin, John C. Doyle, and Keith Glover, Robust and Optimal Control, Prentice-Hall, Inc., 

Upper Saddle River, NewJersey, 1996.

21. Nissim, E., and I. Lottati, “Active External Store Flutter Suppression in the YF-17 Flutter Model,” J. 

Guidance and Control, Vol. 2, No. 5, 1979, pp. 395-401.

22. The NASTRAN Theoretical Manual, Level 17.5, NASA SP-221(05), 1978.

23. Roger, K. L. “Airplane Math Modeling Methods for Active Control Design,” AGARD-CP-228, 1977.
35

24. ZONA Technology Inc., ZAERO Version 8.5 Theoretical Manual, ZONA Technology Inc., 

Scottsdale, Arizona, 2011.

25. Wright, Jan R., and Jonathan E. Cooper, Introduction to Aircraft Aeroelasticity and Loads, John 

Wiley & Sons, Ltd, West Sussex, England, 2007.

26. Gupta, K. K., M. J. Brenner, and L. S. Voelker, “Development of an Integrated Aeroservoelastic 

Analysis Program and Correlation with Test Data,” NASA TP 3120; NASA Technical Paper 3120, 

1991.

27. Meirovitch, Leonard, Analytical Methods in Vibrations, The Macmillan Company, New York, 1967.

28. Kang, Lae-Hyong, Dae-Kwan Kim, and Jae-Hung Han, “Estimation of dynamic structural 

displacements using fiber bragg grating strains sensors,” Journal of Sound and Vibration, Vol. 305, 

No. 3, 2007, pp. 534-542.

29. Pak, Chan-gi, “Unsteady Aerodynamic Model Tuning for Precise Flutter Prediction,”

NASA/TM-2011-215963, 2011.

30. Kammer, Daniel C., “Sensor Placement for On-Orbit Modal Identification and Correlation of Large 

Space Structures,” American Control Conference, 1990, pp. 2984-2990.

31. Matsumura, Yuichi, “Spatial Filter Design for Observation Spillover Suppression,” 2005 

IMAC-XXIII: Conference & Exposition on Structural Dynamics, Orlando, Forida, 2005.

32. Gawronski, Wodek, Advanced Structural Dynamics and Active Control of Structures,

Springer-Verlag, New York, 2004.

33. Kammer, D. C., “A Hybrid Approach to Test-Analysis-Model Development for Large Space 

Structures,” Journal of Vibration and Acoustics, Vol. 113, No. 3, 1991, pp. 325-332.

34. Lee, Jongho, Hunsang Jung, and Youngjin Park, “Feedback FE model updating using strain mode 

shapes,” ICCAS2002, Jeonbuk, Korea, 2002.

35. Levy, Robert, and William R. Spillers, Analysis of Geometrically Nonlinear Structures, 2nd ed.; 

Kluwer Academic Publishers, Dordrecht, Netherlands, 2003.

36. Huber, Peter J., “Robust Estimation of a Location Parameter,” The Annals of Mathematical Statistics, 

Vol. 35, No. 1, 1964, pp. 73-101.

37. Vanderplaats, Garret N., Numerical Optimization Techniques for Engineering Design, 4th ed.; 

Vanderplaats Research & Development Inc., Colorado Springs, Colorado, 2005.