NASA/TM—2013–216597

Modal Filtering for Control of 

Flexible Aircraft

Peter M. Suh

Dryden Flight Research Center, Edwards, California

Dimitri N. Mavris 

Georgia Institute of Technology, Atlanta, Georgia

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NASA/TM—2013–216597

Modal Filtering for Control of 

Flexible Aircraft

Peter M. Suh

Dryden Flight Research Center, Edwards, California

Dimitri N. Mavris

Georgia Institute of Technology, Atlanta, Georgia

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National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, CA 93523-0273

September 2013
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1

Abstract

Modal regulators and deformation trackers are designed for an open-loop fluttering wing 

model. The regulators are designed with modal coordinate and accelerometer inputs 

respectively. The modal coordinates are estimated with simulated fiber optics. The robust 

stability of the closed-loop systems is compared in a structured singular-value vector analysis.

Performance is evaluated and compared in a gust alleviation and flutter suppression simulation.

For the same wing and flight condition two wing-shape-tracking control architectures are 

presented, which achieve deformation control at any point on the wing.

Nomenclature

number of accelerometers

 state matrix

Accel accelerometer

AFS active flutter suppression

 balanced and reduced-order plant state matrix

 balanced

 control input matrix

BWB blended wing body

 balanced and reduced-order input matrix

 disturbance input weighting matrix for balanced and reduced-order plant

 control input matrix for balanced and reduced-order plant

	 number of control surfaces 

	
 mean aerodynamic chord

 state space output matrix

 balanced and reduced-order plant output matrix

 output matrix transformation for accelerometers

 state-regulated goal matrix for balanced and reduced-order plant

 output matrix for balanced and reduced-order plant


 vector of deformations at fiber optic measurement stations


 vector of virtual measurement deformations 


 vector of reference deformations at virtual measurement locations


(, , , ) vector of deflections and rotations defined over , ,  at time 


(, , , ) vector of accelerations and angular accelerations defined over , ,  at time 

 direct feedthrough matrix

DFRC Dryden Flight Research Center

 control-regulated goal matrix for balanced and reduced-order plant

 output noise weighting matrix for balanced and reduced-order plant

 number of goal-regulated states and control surface movements

EI Effective Independence

EOM equations of motion

 lower linear fractional transformation

 structural damping coefficient

 plant

GLA gust load alleviation

() class of all generalized plants with multiplicative input uncertainty

 () class of all generalized plants with multiplicative output uncertainty

!"#$ gust model
2

%&(') matrix transfer function from disturbance to regulated output over all controllers '

%&() matrix transfer function from disturbance to regulated output

*++ scalar elements of the -

$. column and row of /

/ sensor projection matrix (also referred to as hat matrix)

/0 h infinity control type

12 imaginary number

3 identity matrix

34(5) imaginary part

6 matrix defined as 78

9 reduced frequency or non-dimensional frequency

9 system stability margin

' controller

' control gain matrix

' filter gain matrix

: row vector index

: row vector index for virtual deformation references

; global stiffness matrix

;< leading edge

LFT linear fractional transformation

LQG linear quadratic Gaussian

LSE least-squares estimation

LTI linear time-invariant

;< reference leading edge deflection at wing tip

; elemental stiffness matrix

;= modal stiffness matrix

4 column vector index

4> total mass of the wing

4 index of reference modes

? global mass matrix

? elemental mass matrix

?+@AB indexing of modal coordinates

?" multiplicative noise input matrix

?C modal mass matrix

D number of states

DA noise vector added to plant outputs

D number of states in reduced order model

D" noise vector added to plant inputs

N number of degrees of freedom in finite element model

NASA National Aeronautics and Space Administration

E number of measurements (or outputs)

F lower linear fractional transformation of generalized plant and '

G generalized plant

G44+ percent modal mass of the total mass 4> for the -

$. mode shape

G generalized plant with multiplicative input uncertainty

G generalized plant with multiplicative output uncertainty

H vector of modal coordinates

H() vector of modal coordinates at time 

H#$ estimated modal coordinates from modal filter in simulation

H+() -

th modal coordinate scalar at time 

H indexed modal coordinates for input to controller
3

H vector of reference modal coordinates

H0 free-stream dynamic pressure

8 left singular matrix IJ

8(9) generalized aerodynamic force matrix

K number of reference deformations

K vector of reference inputs to controller

KJ relative uncertainty magnitude at steady state

K0 relative uncertainty magnitude at high frequency

L similarity transformation matrix

L(5) real part

RS robust stability

(, , , ) strain measurement vector at time 

M strain-displacement matrix

SMI structural modal interaction

SVD singular-value decomposition

M0 solution of control algebraic Ricatti equation

 simulation time

N output complementary sensitivity function

N input complementary function

TE trailing edge

TEref reference trailing edge deflection at wing tip

O vector of control signals

O() vector of control signals at time 

OA reference unit vector of deformations and rotations

O@ control signal noise

OP output to uncertainty block

Q right singular matrix of 6

R input signal to controller

S left singular matrix of 6

Vel velocity

V-g velocity versus structural damping coefficient

V-velocity versus frequency

T vector of disturbances

T() vector of disturbances at time 

T!() gust velocity at time 

TU

!() gust acceleration at time 

T+(s) weight functions on diagonals of I and I 

I diagonal input frequency weighted matrix

I diagonal output frequency weighted matrix

I controllability Gramian matrix

IJ observability Gramian matrix

 state vector of plant model

 Cartesian coordinate

() state vector of a plant model at time 

V$() state vector of actuator states belonging to ()

V#() state vector of modal aero lags belonging to ()

() balanced and reduced-state vector of plant model at time 

W() state vector of modal coordinate displacements belonging to ()

WU

() state vector of modal coordinate velocities belonging to ()
4

J initial state at time 0

U() time derivative of ()

U

() time derivative of ()

X() estimated states of balanced and reduced-order plant

 output vector of plant model

 Cartesian coordinate 

() output vector at time 

P input to uncertainty block

 goal-regulated outputs

 Cartesian coordinate 

() goal regulated outputs at time 

7 left singular matrix of I

Y air density

Z diagonal matrix of singular values of 6

[ structured uncertainty block

[ structured input uncertainty block

[ structured output uncertainty block

\+ -

$. uncertainty

] normal error

 vector of eigenvalues

^+ -

$. system eigenvalue

_ diagonal matrix of eigenvalues

` structured singular-value vector

a spectral radius

b+ -

$. Hankel singular value of the system triple (, , )

bc(5) maximum singular values

d time constant

e+(, , ) -th natural mode shape vector defined over , , 

f matrix of natural mode shapes, e+(, , )

h+(, , ) -th natural strain mode shape vector defined over , , 

i matrix of strain mode shapes, h+(, , )

j natural frequency

k viscous damping ratio

: total column vector index (that is, all columns included)

Introduction

Active flutter suppression (AFS) and gust load alleviation (GLA) may be necessary additions 

to the control systems of future flexible aircraft. One of the ways to reduce the weight and 

improve the fuel efficiency of a vehicle is to remove structure, reducing the structural stiffness of 

the wing. Reducing wing stiffness may lower the natural frequencies of the aircraft. As elastic 

frequencies shift lower, they may coalesce with rigid-body and gust frequencies. Aircraft with 

wing stiffness reduction modifications may require active structural control to maintain stability, 

improve ride quality, and alleviate gust and maneuver loads (ref. 1). These aircraft may also be 

more susceptible to dangerous phenomena such as buffeting or flutter.

Flutter is an aeroelastic phenomenon which is manifested when energy from the air excites 

the structure, and the structure moves, adding energy back to the air. This transfer of energy 

back and forth tends to grow, leading to an increase in structural amplitudes until structural 
5

failure occurs (ref. 2). All aircraft are subject to flutter at some particular flight condition; aircraft 

do not generally operate within this flutter envelope due to its destructive nature.

For this reason, AFS is generally not required in aircraft which operate today. Nevertheless,

some aircraft still require suppression of marginally-stable modes and suppression of control 

system interactions with modes. Modal suppression is often achieved through adding mass 

ballast or putting notch filters in the controller (refs. 3 and 4). These methods, however, tend to 

add weight or reduce controller bandwidth, respectively. Some researchers are seeking AFS

solutions as an alternative to passive methods (ref. 5). A recent study (ref. 6) performed by 

Lockheed Martin (Bethesda, Maryland) in 2010 on the SensorCraft vehicle showed that a 

25-percent total weight savings could be achieved by incorporating AFS into the controller

(ref. 7).

Increasing flexibility in the structure of the wing may come with other consequences. In 

simulation studies, drag has been shown to increase for a flexible aircraft, relative to a stiffer 

one, especially during maneuvers (ref. 8). As a potential solution, some researchers have 

suggested that the flexibility of an aircraft could be shaped to reduce this effect (ref. 9). This will 

naturally require adequate structural control effectors. Research has been conducted with

structural-shaping technologies such as adaptive control surfaces (ref. 10), adaptive ribs

(ref. 11), and piezoelectrics (ref. 12).

Research into of flexible aircraft has recently become of interest to the National Aeronautics 

and Space Administration (NASA). The blended wing body (BWB) aircraft X-56A (Lockheed 

Martin) was selected as a platform in November of 2011 (ref. 13) for experimental applications 

and innovation. As a first application, the X-56A aircraft control engineers at the NASA Dryden 

Flight Research Center (DFRC) (Edwards, California) must demonstrate AFS and GLA on the 

aircraft at subsonic speeds. These requirements must be met because the X-56A aircraft has 

several body freedom flutter modes within its flight envelope and its wings are highly flexible.

Body freedom flutter is characterized by an interaction of rigid-body and low-frequency elastic 

modes. Body freedom flutter occurs more often in aircraft with low tail volume or high fineness 

ratios (ref. 14). By learning to actively control these unstable structural modes, similar methods 

may be applied to full-scale aircraft in the future to realize weight savings.

The X-56A aircraft is also a distributed control and sensing platform, having discrete arrays 

of accelerometers placed along the wing. There are also plans for the layout of distributed strain

sensors along the wings, known as fiber optic sensors with fiber Bragg gratings (ref. 15). With 

its many control effectors and distributed sensing the X-56A aircraft will likely make an excellent 

platform for wing-shape optimization research. Research has already been conducted at NASA 

DFRC with fiber optic sensors and fiber Bragg gratings. These sensors were implemented on 

the wing of the NASA Ikhana Predator B unmanned aircraft (General Atomics Aeronautical 

Systems, San Diego, California); the sensors feed back strain and shape data at high sample 

rates (ref. 15).

In this paper, it is demonstrated that distributed sensors such as the fiber optic sensors with 

fiber Bragg gratings can be used for the control of flexible aircraft. The strain or shape data may 

be passed to a modal filter which can estimate key states of the structure (ref. 16). These states 

may then be passed as a signal to an active controller with modal coordinate inputs (refs. 17 

and 18). A controller with accelerometer inputs is also developed. The two controllers are

juxtaposed and qualitatively rated on the basis of order reduction, robustness, flutter 

suppression, and disturbance rejection. The purpose of the comparison is to discover whether a

controller using a modal filter has at least comparable performance to a controller using 
6

point-sensor feedback. Point sensors, such as accelerometers, could be considered to be the 

state-of-the-art sensors for experimental AFS studies (refs. 19 and 20).

After comparison of the controllers, two virtual deformation-tracking design case studies are 

demonstrated on the open-loop fluttering wing model. A subsonic flight condition was chosen in 

which the model is open-loop unstable in order to simulate what it may be like to perform the 

same type of deformation control on the aeroelastically-unstable X-56A aircraft. The modal filter 

is shown to either predict deformations at unmeasured points (virtual deformation estimation) or 

to estimate modes directly related to deformations, as in prior research (refs. 21 and 22).

In the first virtual deformation-tracking design case study, a reference virtual deformation 

signal is tracked indirectly from modal coordinate references. In the second study, a large 

number of modes are estimated by the modal filter, and the deflections are predicted and 

tracked directly. The work described in this paper is potentially of interest because to our 

knowledge, few if any researchers have attempted to track deformations (especially virtual 

deformations) on a fluttering three-dimensional wing.

Background

Since the modal filter has not yet been implemented on a flexible aircraft for control system 

feedback, solving the problem of integrating of the modal filter into the aeroservoelastic design 

may be the primary contribution of this paper. Various flutter control architectures have been 

developed for AFS, such as the aerodynamic energy concept (ref. 23), classical (ref. 24), linear 

quadratic Gaussian (LQG) (refs. 7 and 25), /0 (ref. 26), eigenspace (ref. 27), minimax (ref. 28),

predictive (ref. 29), neuro-adaptive (refs. 2 and 30), and many others. Most of these designs are 

supported by a few point sensors for feedback. Control designs (refs. 31-34) for AFS and GLA 

have traditionally relied on discrete arrays of single-point sensors, such as accelerometers,

placed at selected locations on the aircraft wing or body. To our knowledge, there is currently no 

ongoing flutter research that utilizes a modal filter in the controller. Numerous control-related 

studies have been performed using the modal filter as an estimator in air-off structural systems

(refs. 16-18, 35, and 36).

The modal filter has quite an extensive history in the structures discipline; a brief review is 

offered below to illustrate the significance of this filter in this field. The modal filter has also seen 

recent advances with regard to distributed sensing; this topic is also discussed below.

Introduction to the Modal Filter

The modal filter was initially developed to solve a problem in active structural control related 

to the structural modal interaction (SMI) problem in aircraft (ref. 4). Aerospace engineers

encounter SMI in aircraft when flexible modes are excited by control surface feedback that is 

meant to excite only rigid-body motion. The parallel problem in structures is known as 

observation spillover, which exhibits many of the same mechanisms (ref. 16). Observation 

spillover was observed by structural control experts utilizing observers to estimate the states for 

model-based controllers. They found that Luenberger (ref. 37) and Kalman (ref. 38) -type 

observers had the potential to feed residual modes back to the control system when reliance 

was placed on arrays of single-point sensors. Because single-point sensors pick up residual 

mode information, the controller tended to generate a control signal in response to residual 

modes, increasing the amplitude of the residual modes. As the amplitudes of residual modes

increased, sensors picked up more residual information, and a positive feedback system 

resembling a flutter interaction developed, leading to instability.
7

Research into spillover reduction moved in several directions, including incorporating more 

constraints into the estimator design. One such constraint was to make the control effectors 

move orthogonally to the residual modes, thus, not exciting the residual modes (ref. 39).

Another direction taken to reduce spillover was to develop estimators for control which were 

naturally resistant to observation spillover using the concept of mutual modal orthogonality.

These estimators are known as modal filters. The first relied on the modal expansion theorem

(ref. 40), which states that for structures with distinct natural frequencies, the deformation of all 

nodes of the structure 
(, , , ) can be set equal to a linear combination of the structure’s

natural mode shapes {e+(, , ), - = 1 ... o} [see eq. (1)]:


(, , , ) = pH+()e+(, , )

q

+

(1)

where the time-varying weights qs(t) of each mode shape may be found as shown in 

equation (2)

H+() = e+(, , )>?
(, , , ) (2)

where ?uvq×q is the global mass matrix, e+(, , )uvq× is the -

$. mode shape, and 


(, , , ) u vq× is a vector of all deflections and rotations from every node in the structure. 

In 1985, Meirovitch and Baruh (ref. 17) presented the first modal filter based on the modal 

expansion theorem presented above. The finite-element method was used and interpolation 

was assumed to measure all degrees of freedom with a small set of sensors. Meirovitch also 

showed analytically that the filter operated well for input to a controller in a noisy environment.

Contemporary researchers thought the modal filter was impractical due to the requirement of a 

highly accurate analytical mass matrix (ref. 16).

In 1991 two types of modal filters were published by Shelley et al. (ref. 36): the 

pseudo-inverse modal filter and the modal filter with reciprocal vectors. The pseudo-inverse 

modal filter relies on a least-squares estimation (LSE) formulation. By forming the error equation 

and solving for H(), equation (1) can be transformed into the LSE problem shown in 

equation (3) (ref. 36),

H() = (f>f)wf>
(, , , ) + ] (3)

where modal matrix f = {e+(, , ), - = 1 ... o} is calculated from the air-off eigenvalue 

problem. The modal matrix is typically column-truncated to lower-frequency mode shapes which 

span the space of the deformation. The Shelley et al. reciprocal modal filter approach utilized

experimentally-derived vectors. Given the measured actuator forces over a time history, the 

reciprocal vectors could be experimentally derived through an optimization procedure. The 

reciprocal vectors can be shown to be orthogonal to all modes except the desired mode. By 

forming a dot product of the measurements with any reciprocal vector, the corresponding modal 

coordinate could be calculated.

Both the pseudo-inverse and reciprocal modal filters were experimentally tested on a 

five-meter truss, and were shown to estimate modal states well, although the pseudo-inverse 

approach underperformed compared to the experiment-based reciprocal vector approach.

Rigid-body modes and very-high-frequency modes were also estimated poorly by the 
8

pseudo-inverse modal filter, possibly due to the use of a small number of accelerometers. The 

reciprocal approach was also utilized by Shelley et al. for structural health monitoring (ref. 41) in 

1993. The approach was also shown to work in an adaptive control framework for control of the 

Big Darby Creek bridge structure (ref. 35) in Ohio in 1995.

Modal Filtering and Smart Sensors

For good modal estimation, the structure should be fitted with many sensors (ref. 16). Since 

most modal filters tend to rely on an LSE calculation, and every sensor produces data 

containing a certain amount of error, more sensors tend to give a better estimate of the state. 

Recently, researchers have favored smart sensors because they are typically lightweight and 

non-intrusive, facilitating distribution on the structure. Modal sensors with shaped polyvinylidene 

fluoride film were implemented by Lee and Moon (ref. 42) in 1990. The linear combiner was 

developed by Sumali et al. (ref. 43) in 1997 which relied on piezoelectric patches to estimate 

modal coordinates and to adaptively predict mode shapes.

Distributed strain-based modal filters with fiber optics were implemented by Kang et al.

(ref. 22) in 2007. The modal coordinates were used to estimate the deflections at any point of a 

cantilever beam. The strain mode filter looks very much like the pseudo-inverse modal filter 

given in equation (3) except the regressor matrix is formed by first multiplying the mode shapes 

by the strain-displacement matrix M (ref. 44). Since the strain-to-displacement relationship is 

linear, and equation (1) holds generally for structures with distinct natural frequencies (ref. 39),

the least-squares solution of the strain mode matrix can be used to estimate the same modal 

coordinates calculated in equation (3), as shown in equation (4) (ref. 22),

H() = (i>i)wi>(, , , ) + ] (4)

where i = {Me+(, , ), - = 1 ... o} is a o×o matrix with o strain modes, and

(, , , ) u vq× is the measured strain.

The strain-mode filter clearly allows so-called virtual deflections (deflections at points which 

are not measured directly) to be measured. The equation can be represented in equation (5)

(ref. 22).


(, , , ) = f(i>i)wi>(, , , ) (5)

Equation (5) demonstrates that once modal coordinates are estimated, they may be used to 

estimate deflections, known as virtual deformations, at any point on the structure. Others have 

used this concept to estimate deflections with just a few strain gauges (ref. 21). Kang et al. 

found that the modal coordinates were not precisely estimated using the technique given in

equation (5) due to small model errors. Treiber et al. (ref. 45) in 2008 attempted to improve 

upon the Kang et al. modal filtering approach with a Kalman filter. Their work showed that the 

filtering approaches are sensitive to structural uncertainty. Related approaches to deformation 

estimation such as Tessler and Spangler’s (ref. 46) strain-based least-squares variational 

method and Ko and Richards’s (ref. 47) Bernoulli-Euler beam bending method have also been 

developed. Photogrammetry (ref. 48) on the F/A-18 airplane (McDonnell Douglas, now the 

Boeing Company, Chicago, Illinois) Active Aeroelastic Wing has been shown to adequately 

capture deflections.
9

Wing Model

The development of the modal filter and deformation estimation was reviewed to illustrate the 

rich background of the modal filter and its relationship to distributed sensing. The modal filter 

may provide the most benefit when it is applied to a finite-element-type structure. To 

demonstrate the modal filtering technique, a three-dimensional wing model with 199 nodes was

developed. The finite-element wing model structural characteristics were tailored to have 

characteristics similar to a high-aspect-ratio wing, and the model was fitted with several control 

surfaces and realistic actuators which could be used for AFS and GLA. A modal analysis was

completed for the wing model to show some of the relevant modes which were estimated by the 

modal filter. The aeroservoelastic analysis is also presented to familiarize the reader with flutter 

and how the flight condition of the model is selected.

Geometry

The wing model half-span was modeled as a 3.354- by 0.838-m aluminum rectangular plate 

clamped at the wing root, and made from the 6061-T6 aluminum metal alloy traditionally utilized 

in aircraft flying today. The dimensioned model is presented in figure 1.

Figure 1. Specifications and layout of wing test model.

Aluminum ribs and spars were added to reinforce the structure, giving it structural properties 

similar to a realistic aircraft wing. The leading-edge spar was made thicker than the trailing-edge 

spar, as it would likely carry more of the lift load. To simulate more load-bearing potential at the 

wing root, the spars and rib dimensions were linearly tapered toward the wing tips. The control 

surface panels shown outlined in red dots in figure 1 were also stiffened with leading-edge and 

trailing-edge spars, as well as cross-wise ribs along the dotted lines. This stiffening was done to 

remove the possibility of a control surface flutter interaction. Each control surface was 

connected to the wing structure by two six-degree-of-freedom springs. The connection joined 

the wing and control surface at 1/3 and 2/3 of the control surface span from the leading edge
10

corner of a control surface. A Cartesian coordinate system was defined for the model, with 

origin at the lower left corner of the wing. The coordinate  is defined as increasing from the 

trailing edge to the leading edge of the wing. The coordinate  increases from the wing root to 

the wing tip. Coordinate  increases out of the page. The natural modes of the structure,

presented below, were all assumed to move only in the direction of .

Modal Analysis

The natural modes of a structure are shapes, which the structure will tend to take when 

excited at the structure’s natural frequencies (ref. 39). They are found through the eigensolution 

of the matrix structural equations of motion (EOM). The formulation of the EOM requires

knowledge of the global mass matrix ? and stiffness matrix ;. The global mass and stiffness 

matrices may be built up from elemental mass ? and stiffness ; matrices using the 

finite-element method (ref. 49). The matrix equation is presented as equation (6).

?
(, , , ) + ;
(, , , ) = 0 (6)

By assuming 
=
z|2~$ the eigensolution of the EOM may be calculated, resulting in

eigenvalues _ = 
-(^+, - = 1 ... o) and eigenvectors f = {e+(, , ), - = 1 ... o}. The 

eigenvectors can be shown to be linearly independent since they are mutually orthogonal to 

each other for structures with distinct natural frequencies (ref. 39). The eigenvectors are also 

referred to as mode shapes.

Since the o×1 mode shapes are of same dimension as the deformation and are orthogonal 

to each other for structures with distinct natural frequencies, the eigenvectors also span the 

space of all possible deformations. Thus, they form a basis for the deformation, and 

the structural deformation may be expressed as a linear combination of mode shapes, as 

discussed above. Aircraft may have higher-order modes with frequencies near to each other 

due to the complexity of the structure, but orthogonality is generally still a good assumption 

even for structures with close frequencies or repeated frequencies (see ref. 39).

The wing model presented in figure 1 was modeled with the finite-element method. The 

model was designed to be a structure with distinct natural frequencies. An eigenvalue solution 

of equation (6) was performed to give the mode shapes and natural frequencies. The modes 

were then normalized with respect to the global mass matrix. When normalizing in this way, 

the mode shapes can be said to be orthogonal to the mass matrix. Ten modes are used in the 

simulations; the mode shapes and natural frequencies for the first four modes are presented in 

figures 2(a) through 2(d).
11

Figure 2. Modal representation of wing model: a) first wing bending; b) first wing torsion; 

c) second wing bending; and d) second wing torsion.

From visual inspection the first four modes are: first wing bending, figure 2(a); first wing torsion, 

figure 2(b); second wing bending, figure 2(c); and second wing torsion, figure 2(d). The 

maximum deformation for each mode tends to stay below 0.2 m. Each mode shape reveals that 

the control surfaces moved together with the rest of the structure in the static mode shape; this 

action is made possible by setting the stiffness in the actuators to extremely large values, as 

displayed in figure 1, effectively clamping the control surface movement.

The control surfaces also have mode shapes that are referred to as control modes

(not shown), which are formed by enforcing a static 1-deg rotation on the actuators (see ref. 50)

with actuator stiffness set to very low values. After the mode shapes (including the natural and 

control modes) are calculated, it is important to predict the behavior of the modes as the air 

interacts with them. This interaction will characterize the nature of the model during operations.

Flight Condition

The aeroelastic behavior of the wing during flight is important for many reasons, including

determining the flutter speed boundary due to the structural interaction with airflow. To identify 

the flutter boundary (the speed at which the aircraft begins to flutter) a velocity (V) versus 

damping (g) (V-g) and 	



V-) analysis is computed by solving the

generalized eigenvalue problem shown in equation (7) and assuming harmonic motion, (that is,

H=Hz|2~$) of the wing modes (ref. 51):

?C + Y	




29 8(9) 

1+12

j ;= H=0 (7)

where ?C is the modal mass matrix, ;= is the modal stiffness matrix,  is the structural damping 

coefficient, j is the natural frequency, Y is the air density, 	
is the mean aerodynamic chord, 9 is 

the reduced frequency or non-dimensional frequency defined by 9 = ~


 , and 8(9) is the 

generalized aerodynamic force matrix. Equation (7) is a generalized eigenvalue problem solely
12

in terms of reduced frequency, 9. The eigenvalues are defined as ^ = |2!

~ and equation (8) 

shows the relationship of the variables in equation (7) to the eigenvalues (ref. 52).

j =

1

L(^) ,  = 2k = 34(^)

L(^) , S=

j	


29 (8)

where k is the viscous damping ratio. By specifying a range of reduced frequencies, 9, the 

eigenvalues of equation (7) may be calculated repeatedly and variations of frequency and

structural damping coefficients with velocity may be found using equation (8). These values are 

then plotted against each other.

Correct interpretation of the plots is important. In the structures discipline, negative damping 

is defined to be good; whereas within the controls discipline, positive damping is defined to be 

good. This difference can cause confusion when interpreting the structural damping coefficient

over the speed regime. In the case at hand, positive structural damping (k>0) corresponds to 

velocity of an unstable flight condition. The V-g plot is presented in figure 3.

Figure 3. Velocity versus damping plot of wing model.

One of the purposes of the V-g plot is to identify the nature of flutter in the aeroelastic model.

The characteristic of a typical two-mode flutter interaction on a V-g plot is that one mode will 

become more and more unstable and the mode it interacts with will become more and more 

stable (ref. 52). Physically, this represents a situation in which energy from the air is transferred 

to the structure moving it, and the movement of the structure adds energy back to the air,

forming a positive feedback system as described above. As the airspeed increases past the 

flutter speed, the strength of the flutter tends to become stronger and more destructive, and 

more system damping is usually required (notice that  grows past the flutter boundary in 

figure 3).
13

For simulation studies it is desired that the flight condition be selected substantially past the 

flutter speed to demonstrate the use of modal filtering in flutter suppression. The V-g plot 

indicates that the first wing bending (mode # 1) passes the stability boundary at 72 m/s. It 

interacts with the first wing torsion (mode # 2) pushing it into the highly-stabilizing region, which 

is typical of a clamped plate bending/torsion flutter mode. Because only these two modes 

interact, only their frequencies are plotted in the V-	 in figure 4. The variation of frequency 

over the speed range is also typical for rectangular plate aeroelastic analysis (ref. 52). For the 

studies to be conducted on this wing model, the flight condition is selected to be at a speed of

80 m/s at an altitude of 305 m, where at least 2.8 percent structural damping is required and 

flutter frequency (see fig. 4) is approximately 4.42 Hz, which is the frequency shown for the first 

mode at 80 m/s. The damping ratio predicted with this frequency matching method tends to be 

incorrect at conditions above and below flutter because aerodynamic behavior is typically not 

harmonic.

Figure 4. Velocity versus frequency plot of wing model.

Methodology

The methodology for incorporating the modal filter into the control design is brief, as the 

primary requirement is to update the state space model so that it reflects the use of the modal 

filter. Fulfilling this requirement is actually as simple as choosing which modal coordinates one 

wishes to measure and updating the output matrix to reflect this choice. The update of 

the output matrix when accelerometers are utilized is given for comparison and because the

accelerometers are used later for a benchmark of the controller with modal filtering. A guideline 

for the incorporation of the modal filter into the controller is also given with a bias toward control 

of flexible aircraft. This guideline also serves as a roadmap for the work described in this paper.
14

State Space Model – Modal Filter or Accelerometers

The aeroservoelastic wing model described above may be represented by a linear 

time-invariant (LTI) system of finite dimensions. The LTI system may be cast into linear constant 

coefficient differential equations which are valid only at the flight condition selected in figure 3.

The Mach number and free-stream dynamic pressure H0 are required to be known to formulate 

the state space equations. Assuming known quantities, the aeroservoelastic state space model 

can assume the matrix time derivative form for simulation and control design shown in 

equation (9) (ref. 50):

U() = () + O()

() = () + O() (0) = J (9)

with the initial state (0) = J. The D-dimensional vector () is referred to as a state vector and 

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

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

state and the 	-dimensional input O is required to know the state in the next time step. The 

, ,  and  matrices are real constant matrices with D×D, D×	, E×D and E×	 dimensions. 

The exact formulation of these matrices is not the focus of this study. The matrices are derived

from the finite-element nodal model, linear aero-paneling and frequency-based methods,

rational function approximations, and modal transformations. An excellent development of the

state space formulation is given in reference 50. The states, however, are the focus of this 

study. The state space vector is composed of vectors of states with different units. The state 

vector may be defined as shown in equation (10),

()  {W(), WU

(), V#(), V$()}> (10)

where W() u v× is a vector of modal coordinate displacements, WU

() u v× is a vector of 

modal coordinate velocities, V#() u v× is a vector of aerodynamic lag states and 

V$() u v&× may be formed from a vector of actuator accelerations, velocities, and 

displacements. The modal filter may be used to sense some or all of the states in the W()

vector, as shown in equation (3). Any of these states (or combinations of these states) can be

used to form the output matrix if the appropriate sensor is utilized. In turn, the output matrix is 

used during control design.

Modal Coordinate Sensor Output Matrix

The output matrix must be modified to incorporate the modal filter in the control design. The 

output matrix  is a matrix of row vectors relating the output sensor to the state vector . Since 

the modal filter directly measures some or all of the W state, the form of the output matrix may 

be cast into the form shown in equation (11), assuming all modal coordinates are measured.

  [3× 0× 0× 0×&] (11)

Measuring all modes is not always required. Higher-order modes which do not significantly 

contribute to the overall modal deformation in the system may sometimes be ignored and the 

row dimension of the output matrix may be reduced.
15

Accelerometer Sensor Output Matrix

The output matrix may also be modified to account for accelerometer measurements. Rather 

than measuring directly any part of the state vector, accelerometers measure linear 

combinations of modal coordinate accelerations. The relationship between the accelerometers 

and the state of the system is often modeled as shown in equation (12).

()  U() = (() + O()) (12)

The matrix  u vV×@ has as many rows as accelerometers, , and as many columns as the 

state space vector size. The matrix  is defined as shown in equation (13) (ref. 53).

  [0V× fV×(:, : ) 0V× 0V×&] (13)

where fV×(:, : ) is the displacement modal matrix row indexed by :.

Modal Filtering Control Design

The above section clarified the way in which the modal filter could be implemented into the 

state space model. This takes care of the modeling side of things. But questions linger over 

which modes should be selected to update the output matrix in equation (11). It is implied 

through the use of the modal filter that distributed sensing will be utilized. But it is not clear if 

some sensors should be excluded or not. There are multiple types of modal filtering techniques, 

some of which cater to particular types of sensors, and the determination of which sensor 

platform to use should be based on the application. There are many options for design of the 

controller with the modal filter. The following ten steps were loosely followed in this study:

1. Identify which modes are significant to the response of the model based on a percent 

modal mass calculation and the V-g analysis.

2. Update the output matrix with the identified modes as in equation (11).

3. Place or select sensors, either using an optimal sensor placement technique or an

intuition of which modes are being measured.

4. Set up the modal matrix and the pseudo-inverse of the modal matrix corresponding to 

sensor locations :, in order to use equations (3)-(5) for modal coordinate or displacement 

estimation.

5. Reduce the order of the aeroservoelastic plant.

6. Specify the performance requirements for the control design.

7. Design a controller for the plant and verify that performance requirements are met.

If they are not, iterate steps 1-6.

Optional steps for virtual deformation tracking:

8. Upgrade the plant with an integral tracking state and design a modal tracking or 

deformation tracking controller.

9. Define optimal or desired reference deflections for virtual deformation tracking.

10. Verify that the selected modes for tracking produce an indirect deformation tracking 

controller, with minimal deformation error due to residual modes. If the error is too high,

track more residual modes or redesign the controller or the actuator placement so that 

the residual modes are not strongly excited.

A few of these steps are discussed because they may help to illuminate some of the paths 

taken below and they may also provide space for complementary work. Liberties were taken 
16

during this study, and the steps were not precisely implemented as listed. Step 1 has many 

options. It is probable that a modal contribution percentage (see ref. 54) to flutter is the best way 

to define the significant modes, although this specific technique was not used directly in this 

study. Step 3 was followed for both accelerometer and fiber optic sensor placement; however, 

the fiber was placed using intuition and the accelerometers were placed using optimal sensor 

placement (ref. 55). Step 9 was not implemented in the present study. Finding desirable shapes 

of the aircraft wing/body may be an entire research topic in itself, likely requiring computational 

fluid dynamics studies. Step 10 was not implemented because the number of control surfaces 

was limited to two; thus, only two modes could be tracked at one time.

Design

This section focuses on the design of the modal-filter-type controller, its comparison to a 

controller with accelerometer inputs, and virtual deformation case studies.

To ensure that the instability of the model is indeed flutter, the wing model is simulated in an 

open-loop study when perturbed from equilibrium. The wing model is upgraded with either 

accelerometers or a modal filter (with fiber optic inputs), and plants [see Eqs. (9) and (11)-(13)]

are designed for both systems. Both plants are different only in that one is using a modal filter 

and the other is using accelerometers for output. The plants are reduced in order and 

differences are noted for each.

An /0 optimal regulator is designed for each plant with the objective of being robustly stable 

to input and output multiplicative uncertainty and having good disturbance rejection properties. 

The simulation is presented for time-history analysis and the performance during a disturbance 

is demonstrated, showing that controllers with either modal coordinate input or accelerometer 

input can be used for GLA. The same controllers are also shown to be capable of AFS.

The modal coordinate controller is updated with integral tracking states corresponding to its 

measured modal coordinates. Two methods of virtual deformation tracking are presented. The 

first virtual deformation tracking case study uses a modal approximation of the deformation 

tracking signal. The second virtual deformation tracking case study uses the modal filter to 

predict the deformations [see eq. (1)] of the plant model, in order to achieve improved 

deformation tracking. The open-loop flutter of the wing model is presented first.

Open-Loop Flutter

The aeroelastic wing model is in an open-loop flutter condition, based on the prediction made 

by the V-g analysis (see fig. 3) at the freestream velocity of 80 m/s and altitude of 305 m. To 

observe the characteristics of the flutter instability, the model was perturbed from equilibrium at 

time 0 by a unit deflection command to the control surfaces which lasted for 0.01 s. The time 

history of the modal coordinates is shown in figure 5.
17

Figure 5. Wing model in open-loop flutter.

The modal amplitudes oscillate at 4.49 Hz, which was almost predicted by the V-


(see fig. 4). The damping ratio for this mode (from a controls perspective) was -5.3 percent,

making it more unstable than what was predicted in the V-g analysis (see fig. 3). The fluttering

mode is predominantly a contribution from the first wing bending and first wing torsion mode. 

The frequencies of the two modes have coalesced into that of a single frequency, which is 

typical of a flutter mode. The time to double amplitude is 0.46 s. The bending mode reaches 

higher amplitudes than the torsion mode, which is expected due to its lower natural frequency. 

The torsion mode is slightly out of phase with the bending mode. From the above results 

(see figs. 3 and 5), it is apparent that the wing will experience strong flutter at the chosen flight 

condition.

Sensor System Design

Two regulators and corresponding estimators for the fluttering wing model are now 

described. The regulators are given accelerometer inputs and modal coordinate inputs,

respectively. Several objectives were defined to guide the controller designs so that the 

controllers could be qualitatively compared. The first objective was to stabilize the plant. As 

indicated above, the wing experiences strong open-loop flutter at the flight condition. The 

second objective was to reduce the controller order as far as possible. This has the effect of 

later improving numerical stability of the estimator and reducing the computational burden in the 

flight computer. A third objective was for the controller to have good disturbance rejection

properties, especially from low-frequency turbulence. Passenger comfort can be improved by 

actively rejecting gust disturbances (ref. 56). The fourth objective of the controller design was

that it be robust to modeled uncertainty. A controller designed about a linearized model rarely 

performs the same way in practice as it does in the laboratory (ref. 57). Below, the control 

design is described in detail, beginning with accelerometer placement and selection.
18

Accelerometer Placement

The first controller was designed with two accelerometers. The accelerometers were placed 

using Kammer’s Effective Independence (EI) procedure (ref. 55), in which only the first wing 

bending and first wing torsion modes were maintained in the modal matrix. The EI procedure 

begins by forming the sensor projection matrix, / = f(f>f)wf> from the modal matrices. 

Large diagonals, *++ of the projection matrix / correspond to relatively important sensor 

locations - on the finite-element wing model. The EI sensor reduction procedure begins by 

removing the row of the modal matrix, f corresponding to the smallest *++. The reduced 

projection matrix is then recalculated and the row corresponding to the smallest *++ is again 

removed. This reduction process can be iterated to the desired number of sensors, which are 

then used to form the index : used in equation (13). The EI procedure tends to select sensors 

that carry the highest amount of the desired modal information. The resulting sensor placements 

are shown in figure 6.

Figure 6. Accelerometer placement on wing model.

By down-selecting the rows of the modal matrix to the wing tips, the EI procedure indicates 

that the wing tips (both leading edge and trailing edge) carry the most modal information.

“The most modal information” means that the first wing bending and wing torsion modes had the 

highest deformation at these points; also see figures 2(a) and 2(b).

The EI procedure is only a first step and was used for its simplicity. The EI procedure 

contains flaws in that it does not attempt to optimize sensor layout for residual modes. It has 

been shown that residual modes tend to corrupt the sensor signal, leading to observation 

spillover (ref. 16). Thus, optimization procedures, such as the modified EI procedure (ref. 58),

have been developed to place sensors on a BWB-type aircraft while minimizing residual mode 

information. The EI procedure is also sensitive to structure with many nodes, because nodes 

tend to be very close together and many good locations may fall in the same spot. The severity 
19

of this problem can be reduced by using correlation matrices to select sensor locations that

maximize modal information without redundancy (ref. 59).

Modal Coordinate Selection

The second controller is designed with modal coordinate feedback. The modes which 

dominate the deformation of the structure and can be easily controlled and observed should be 

selected for feedback. The modal mass, which may be calculated for each mode, is an indicator 

for whether a mode dominates the structural deformation. The percent of the total modal mass 

for the -

$. mode may be calculated as shown in equation (14),

G44+ = 1

4>

e+(, , )>?OA

e+(, , )>?e+(, , ) (14)

where 4> is the total mass of the structure, and OA is an o×1 reference vector of unit 

deflections and rotations. Modes with the highest percent modal mass should be selected for 

feedback, since they may contribute heavily to the response (ref. 39). A good rule of thumb is to 

include modes which sum to approximately 90 percent of the total mass of the structure. The 

percent of the modal mass for each mode as calculated by equation (14) is given in figure 7.

Figure 7. Percent modal mass per mode shape.

The percent modal mass of the first bending mode is clearly the highest at 85 percent,

indicating that it will be a mode which both contributes significantly to the modal response and 

can likely be easily controlled and observed (ref. 52). It is difficult to state an exact measure of 

observability and controllability of modes, as the measure of observability and controllability will 

be determined by the placement of the control effectors and sensors (see refs. 30 and 60).

Thus, another way must be found to decide which modes should be included for measurement.
20

Recall that the airflow interacts with the structure (see fig. 3) near flutter so that some modes 

tend to dominate the structural deformation more than others. The flutter analysis at the 

selected flight condition indicates that the torsion mode will also be highly mobile near flutter. As 

such, based on the observation of the high modal mass of mode 1 and the flutter interaction due 

to mode 2, both first wing-bending and first wing-torsion modal coordinates were selected for 

feedback. For more complicated structures it might be beneficial to utilize the percent 

contribution of each mode to flutter as a criterion. Pak shows how to expand the flutter mode as 

a linear combination of the natural modes and calculate the percent contribution of each mode

to flutter (ref. 54).

Fiber Optic Sensor Placement

The means of estimating the modal coordinates is probably one of the most important 

aspects of the modal coordinate feedback controller design. The sensors could come in the 

form of piezoelectric materials (ref. 43), fiber optics (ref. 22), strain gages (ref. 21), or,

potentially, photogrammetry (ref. 61). Each sensor type has different characteristics, which may 

make some sensors more appropriate for certain systems than others. Since fiber optic sensors

had already been tested on aircraft (see ref. 15) this type of sensor was selected for the true 

sensor feedback to the modal filter. The modeled layout on the wing model is shown in 

highlighted green in figure 8.

Figure 8. Fiber optic sensor placement on wing.

At the locations where the fiber is placed, the deflections are assumed to be directly 

measured during the simulations. Several methods have been developed for the purpose of 

estimating deflection from strain. Equation (5) shows how strain can be directly used to achieve

this objective. Intuition was used to place the fibers so that necessary modal information could 

be estimated. The selected modes for feedback include strong bending and torsion effects. To 

capture sufficient bending information, the sensors are placed span-wise along the entire wing. 

To capture torsional effects, three fibers are placed chord-wise. The spacing between each 
21

location was set at one-half-inch intervals, which is the same spacing used on the NASA Ikhana

Predator B unmanned aircraft (ref. 15). The use of a strain-shape algorithm need not be 

required to use the fiber optic sensors for modal coordinate estimation. Work by Kang et al.

(ref. 22) has shown that fiber optic strain measurements can be utilized to estimate modal 

coordinates. But instead of a modal matrix f, a strain mode matrix i is formed, which can then

be utilized as shown in equation (4).

Controller Design

After sensor locations are selected for both sensor types: accelerometers and fiber optics, 

the process of control design may begin. Here this is demonstrated by reducing the order of the 

plant, respectively for each sensor type. The controllers are then designed around each reduced 

order plant. The robust stability is analyzed for each closed-loop system with two structured 

uncertainty cases. The model reduction approach is presented first.

Model Reduction

There is a multitude of literature on the topic of reducing the order of a controller; two

references are given here (refs. 62 and 63). Some researchers reduce the plant and design the 

controller around the reduced-order plant, others reduce the controller after it has been

designed around the full plant. It is typical to preserve the dominant eigenvalues in the 

reduced-order model. A common method of order reduction is to first balance the plant and then 

reduce it based on the Hankel singular values (ref. 59). This method was selected for the 

current research; a short description is provided here. The Hankel singular values 

b+ = ^+(IIJ),- = 1...D are derived from the eigenvalues {^+,- = 1...D} of the square root of 

the product of the controllability Gramian I and observability Gramian IJ. The Gramians are 

found from a solution of algebraic equations known as Lyapunov equations, shown in 

equation (15) (ref. 59):

I + I> + > = 0

>IJ + IJ+>=0 (15)

The Gramians give a degree of relative observability and controllability if the plant is internally 

balanced. To balance the plant a transformation on the states  = L may be found so that the 

controllability and observability Gramians are both diagonal and equal. The diagonality means 

that each state has its own independent measure of controllability and observability. The 

equality of the Gramians indicates that each state is equally controllable and observable

(is excited to the same degree to which it is sensed). The transformation is found by 

decomposing the solutions of equation 15: I and IJ of the unbalanced system using a

singular-value decomposition (SVD). The left singular matrix 7 of I may be multiplied with left 

singular matrix 8 of IJ, as in reference 59:

6 = 87 (16)

Another SVD of 6 may be performed to arrive at 6 = SZQ>. The transformation matrix may then 

be found as discussed in reference 59 and presented in equation (17).

L = 7QZw

 (17)
22

The transformed state space matrices are given by the method shown in equation (18).

 = LwL

 = Lw

 = L

(18)

To reduce the order of the system, states (rows or columns) of the balanced system may be 

removed which correspond to relatively low Hankel singular values. From the reduced-order 

model, the reduced-order controller and estimator may be designed as it would be from the 

original plant matrices.

The discussion above on the topic of control order reduction illustrates the manner in which

sensor selection can affect the plant order reduction and resulting controller order reduction. 

The transformation matrix L is directly dependent on the SVD of 6. The matrix 6 is, in turn, 

directly dependent on the SVD of IJ. Equation (15) shows a direct relationship between the 

output matrix  and IJ. The use of either modal coordinates or accelerometers affects the form 

of the output matrix and thus will affect the relative Hankel singular values through the 

eigensolution. To illustrate this effect more clearly, the relative unit-normalized Hankel singular 

values of the balanced systems with accelerometer outputs and modal coordinate outputs are 

given in figure 9.

Figure 9. Comparison of normalized Hankel singular values for two plants.

The first plant with accelerometer outputs has significant unit-normalized Hankel singular 

values out to state 14. The Hankel singular values for the second plant with modal coordinate 

outputs show a very steep drop-off after the fifth state. It is not precisely known why this occurs. 

The steep drop-off may be accounted for by the fact that no relationship in the output matrix is 

given for modes past the first two modal states. Since the higher modes are not as observable 

to the system, their input-output contribution may be less, which seems to indicate that modal 
23

filtering may focus the observability in the system. This assumption has merit, since the modal 

filter was first introduced to reduce observation spillover.

The presentation of the Hankel singular values in this form may indicate that the controller

order could be reduced based only on the relative magnitude of the singular values. It was 

found, however, that proceeding thus blindly could lead to an unstable controller. A more 

rigorous approach was taken, by reducing the order of the controller by removing states 

corresponding to the lowest Hankel singular values until the reduced-order controller 

performance diverged significantly from the performance of the original full controller

(see ref. 59). The pole comparison of the two plants is given in figures 10(a) and 10(b).

Figure 10. Order reduction: (a) plant with accelerometer output; and (b) plant with modal 

coordinate output.

The plant with accelerometer outputs was not reduced as far as was the plant with modal 

coordinate outputs. The final order of the plant with accelerometer outputs was 27. The final 

order of the plant with modal coordinate outputs was 5. The plant poles shown in figure 10 

qualitatively compare well to many structures, in which lightly-damped modes are very near the 

imaginary axis (ref. 39). Sometimes these lightly-damped modes can become unstable due to

interaction with a flight control system, described previously as SMI. For the present case, 

aerodynamic coupling (see fig. 3), is the cause of the pole migration to the right half-plane 

(RHP). Simulation results in figure 5 indicated that two modes are unstable in the model. The

poles in figure 10 which have moved into the RHP characterize this instability as a flutter

instability. One of the primary objectives of the present work is to actively suppress this flutter 

mode. That process is described below.

  Optimal Control

Many choices for control design are available once a state space model has been defined as 

in equation (9). The /0 optimal controller (ref. 57) was chosen for this study because it has a 

wide range of applicability. One excellent feature is that it is not assumed that the disturbances 
24

are collocated with the control inputs. The locations of controlled outputs are not necessarily 

collocated with the location of system performance as they are in the LQG controller design

(ref. 59). The /0 optimal controller was designed for both reduced-order plants shown in figures

10(a) and 10(b). The reduced-order plants were cast into the state-space realization shown in 

equation (19) (ref. 59):

U

() = () + T() + O()

() = () + O()

() = () + T()

(19)

where  u v@×@ is the reduced-order state matrix,  u v@×% is the disturbance matrix, 

 u v@× is the control input matrix,  u v×@ is the state-regulated goal matrix,  u v×"

is the control-regulated goal matrix,  u v×@ is the measurement matrix, and  u v×%

is the measurement noise matrix. The reduced-order states (t) u v@× are driven by the 

disturbances T() u v%× and the control inputs O() u v×.

The goal of the /0 optimal control methodology is to find the controller ' which minimizes 

the /0 norm of the transfer function %&(') from disturbance T to regulated output , over all 

possible controllers (ref. 59). Recall that the /0 norm of a transfer function is defined as the 

supremum or least upper bound of the transfer function over all frequencies. So, more simply 

stated, the /0 synthesis routine results in a controller which best suppresses the peak of 

%&(), where  is the Laplace variable.

The suboptimal solution of the /0 problem requires that two algebraic Ricatti equations be 

solved in which the observer and the controller matrices are coupled by an inequality constraint 

on the spectral radius a. Another constraint is also included to assure that the Hamiltonian 

matrices do not have eigenvalues on the imaginary axis, which may cancel poles or zeros on 

the imaginary axis and lead to instability. Typically, the objective function is minimized with a

local optimization technique such as the bi-section method, which generally performs well since 

the objective function is convex (ref. 57). The resulting /0 controller may then be represented 

by equation (20) (ref. 59),

XU() = (+aw

>M0  '  ')X() + '()

O() = 'X() (20)

where, M0 u v@×@ is the solution to the control algebraic Ricatti equation, ' is the filter gain 

matrix, and X(t) u v@× is the estimated state vector of the reduced-order plant. For more 

information on the /0 problem formulation and solution, refer to references 57 and 59.

For control design the two reduced-order plants were subjected to the same disturbance 

input matrix, . The matrix was formed from a gust model and process noise weights. The gust 

model representing the interaction of a wind gust with the modal velocities was derived from a

sinusoidal gust column (ref. 64). The sinusoidal gust column is initialized from 0 m/s wind speed 

at the leading edge of the wing model building in strength toward the trailing edge of the wing.

The derived gust basis represents the physical effect of the gust in modal space and is used in 

the weighting scheme as well as for simulation later on. The first column of disturbance 

weighting matrix  was represented as the velocity basis of the gust weighted with a sustained

gust velocity T!() of 5 m/s. The second column of  was represented by the acceleration 

basis of the gust weighted with a sustained gust acceleration TU

!() of 9.81 m/s2

. The values of 
25

the gust weights were selected arbitrarily, but it is speculated that the values represent what a 

passenger aircraft could experience when passing through rough air. The weighting matrix 

was also augmented with unit vectors characterizing the presence of process noise.

For control design on both reduced-order plants, the goal state matrix  was modified so 

that the first two modes would receive highest weights. The first two modes were weighted 

highest since figure 3 indicates that these two modes will have a flutter interaction. By giving the 

first two modes higher weightings, the optimization technique emphasizes the reduction of the 

peaks of the first two modes due to the gust disturbance across the frequency range of %&()

as much as possible. The control-regulated goal matrix  was given equal weightings which 

penalized high control surface movement.

The measurement matrix  was set for each controller respectively to either equation (11) 

for the controller with modal coordinate input or equation (13) for the controller with 

accelerometer input. The controller’s sensitivity to measurement noise was also reduced by 

modifying the weighting matrix  with moderately high weights corresponding to each sensor.

Robust Stability

The /0 synthesis does not always result in a controller which meets performance 

specifications; thus the controller design process is an iterative one. To bound the design 

process, objectives were defined for both control designs. Assuming an initially-stable control 

design, two objectives for the controllers were that each should be robust to structured

uncertainty and also have good disturbance rejection properties. More specifically, the first goal 

for both controllers was that they be robust to at least 5 percent multiplicative uncertainty on the 

inputs or outputs of the plant at a low frequency of 1 Hz. They should also be robust to at least 

25 percent multiplicative uncertainty near the higher flutter frequency of 4.49 Hz. Secondly, 

each controller was designed to mitigate the modeled sinusoidal gust disturbance described 

above.

A quantitative measure of the robust stability margin of both controllers is the structured 

singular values (SSV) or μ. The SSV is defined as shown in equation (21) (ref. 57),

`(F) 

1

4-D {9| 
(39F[) = 0 for structured [, bc([)  1} (21)

where 9 is the stability margin defined as 9 = 1/`(F), F is the lower linear fractional 

transformation (LFT) of the generalized plant G, [ represents a structured uncertainty block,

and the maximum singular value of [ is defined by bc([). The value of `=1 occurs when there 

is a perturbation with bc([)  1, which is just large enough to make 3  F[ singular. A larger

value of ` is undesirable, as it means that a smaller perturbation makes 3  F[ singular. The 

“generalized small gain theorem” states the robust stability (RS) condition, as shown in 

equation (22) (ref. 57).

LM ¡ `¢F(12j)£ < 1 D
 bc¢[(1j2 )£ < 1, ¥j (22)

To verify RS for both plants, both reduced-order plants were subjected to multiplicative input

or output uncertainty. The class of all generalized plants for multiplicative input uncertainty is 

given as shown in equation (23) (ref. 57).
26

() = (3 + I[) (23)

The class of all generalized plants for multiplicative output uncertainty is given as shown in 

equation (24) (ref. 57).

 () = (3+I [ ) (24)

The structure of the input and output weights is defined to be diagonal; that is,

I  
-{T+(), - = 1 ... }. For I, b is a scalar equal to the dimension of the inputs. 

For I ,  is set equal to the scalar dimension of the outputs. The uncertainty block [ for both 

input and outputs is defined to be diagonal, [  
-{\+, - = 1 ... }, where  is set accordingly 

for inputs and outputs. The uncertainties on the diagonal \+ are also defined to be < 1, so that 

the RS condition shown in equation (22) always holds. Generally, uncertainty is greater at 

higher frequencies, so the uncertainty is made to vary with frequency by weight functions, as 

shown in equation (25) (ref. 57).

T+() =

d + KJ

¦ d

K0§+1 (25)

The constant KJ is the relative uncertainty magnitude at steady state; and 1/d is approximately 

the frequency where the relative uncertainty reaches 100 percent. The constant K0 is the 

magnitude of the weight at higher frequencies. The constants were selected so that 

the uncertainty would be greater than 5 percent at 1 Hz and greater than 25 percent near the 

flutter frequency of 4.49 Hz.

The plant with input multiplicative uncertainty is modeled as that given in figure 11.

Figure 11. Generalized plant with structured input uncertainty.

For the inputs [P, O]>and the outputs [OP, R]>, the generalized plant with multiplicative input 

uncertainty defined above may be shown to be as presented in equation (26).

G =  ̈ 0 I

 © (26)

The F structure may be formed from a lower LFT of G and ' represented as (G, '). By 

carrying out the matrix operations it can be shown that F = I'(3 + ')w = IN,

where N is the input complementary function. The `(F) for the /0 controllers with multiplicative 

control input uncertainty is given in figure 12.
27

Figure 12. Structured singular-value vector analysis with input multiplicative uncertainty.

The stability boundary is defined to be 1 for ` as is a requirement for RS from equation (22).

The maximum ` for the controller with modal coordinate inputs is approximately 0.36 at 

2.14 Hz. The maximum ` for the controller with accelerometer inputs was approximately 0.35 at 

2.23 Hz. The ` for both controllers was bell-shaped across the frequency range. The 

/0 controller with accelerometer inputs resulted in a closed-loop system which was slightly 

more robust to input uncertainty. The difference is not substantial, however. Both controllers 

seemed to experience a peak in ` near 2.2 Hz. One might expect the peak to occur at the 

open-loop flutter frequency of 4.49 Hz; however, the frequencies correspond to the closed-loop 

pole locations. From this analysis, it was determined that both controllers meet and exceed 

expectations with respect to input multiplicative uncertainty.

Output uncertainty is also of interest, since different measurement systems are being utilized. 

The plant with output multiplicative uncertainty is modeled as that given in figure 13.

Figure 13. Generalized plant with multiplicative output uncertainty.

For inputs, [P, O]>and outputs [OP, R]>, the generalized plant with multiplicative output 

uncertainty defined above may be shown to be as presented in equation (27).
28

G =  ̈ 0 I 

3  © (27)

The F structure may be formed from a lower LFT of G and ' represented by (G , '). By 

carrying out the matrix operations, it can be shown that F = I (3 + ')w' = I N, where 

N is the output complementary sensitivity function. The `(F) calculated for both /0 controllers 

with multiplicative measurement output uncertainty is given in figure 14.

Figure 14. Structured singular-value vector analysis with output multiplicative uncertainty.

The maximum ` for the controller with modal coordinate inputs is approximately 0.47 at 

2.10 Hz. The ` for the controller with modal coordinate inputs experienced a second peak near 

a frequency of 3.38 Hz. The maximum ` for the controller with accelerometer inputs was 

approximately 0.46 at 2.08 Hz. The ` for the controller with accelerometer inputs was nearly 

bell-shaped across most of the frequency range, and descended until a frequency of 3.54 Hz, at 

which point it climbed for a short time. Overall, the characteristics of the ` for both controllers 

indicate that the /0 controller with accelerometers resulted in a closed-loop system which was 

slightly more robust to output uncertainty. The relative stability margin difference between the 

two controllers, however, is negligible. Both controllers meet and exceed expectations with 

respect to output multiplicative uncertainty.

Simulation

The time simulation of controllers is a reliable way to diagnose performance and make 

comparisons. Several case studies were selected to be performed with different objectives. The 

first two case studies pertain to GLA and AFS. The next two case studies focused on the use of 

modal filtering for virtual deformation estimation and tracking on the wing. The original plants 

were upgraded with “integral of modal position” states and reduced. New controllers were then

derived using the same methodology presented for the regulators. The first controller tracks 
29

virtual deformations from a modal command; the second controller tracks the same virtual 

deformations with a virtual deformation modal approximation.

Gust Disturbance

The regulators described above were tested in a simulation environment, with a gust 

disturbance input. The simulation structure that was used to model the gust disturbance for the 

controller with modal coordinate inputs is given in figure 15.

Figure 15. Control simulation with modal filter for gust modeling.

The simulation structure for the accelerometer inputs is the same except that the connection 

after the gust inputs is input directly into the summing block with the reference input. The 

exogenous inputs to the system are [D", T, DA]> corresponding to input multiplicative noise, gust 

disturbance states, and additive measurement noise. Zero mean multiplicative Gaussian noise,

D", with a standard deviation of 0.1, is modeled on each control input for both control systems, 

making ?" in figure 15 the identity matrix of size 	.

The standard 1  cos () gust profile (see ref. 64) is modeled to characterize the transient 

shape of the gust disturbance T = T!() TU

!()

> which lasts for 1.6 s. The gust velocity and 

acceleration at time 0 are both initialized to zero. The gust is shaped to achieve a maximum 

velocity, T!() of 5 m/s and a maximum acceleration TU

!() of 9.81 m/s2

. The signals, T!() and

TU

!() are input to the LTI gust system !"#$ which acts as a disturbance on the output of the 

plant .

For the controller with modal coordinate inputs, zero mean Gaussian noise DA with a

standard deviation of 1 cm was added to the measurement signals. Deformations are used 

instead of strain because strain was not available directly in the model. When accelerometers 

were used in place of the deformation measurements, it was assumed that the additive noise 

had a standard deviation of 1.0 m/s2

. For the controller with modal coordinate inputs, the noise 

was added to the simulated deformations, shown in figure 8. Accelerometer measurements tend 

to be somewhat noisy, whereas fiber optic measurement systems are expected to produce 

measurements with a very high signal-to-noise ratio (ref. 65).

This model being a simulation model, true displacement measurements were not available. 

The controller with modal coordinate inputs makes use of a deformation simulation, by 
30

multiplying the modal matrix f indexed at measurement index stations : with the true modal 

coordinates H. The modeled displacement information and additive displacement noise is input 

into the least-squares modal filter introduced in equation (3). The estimated modal coordinates 

are then indexed (?+@AB = {1,2}) to obtain H. This signal is then used to form the control 

signal input to the /0 controller '. Since gust alleviation is the objective of the controller, the 

reference on each measurement is set to 0. The gust simulation results are presented in 

figure 16 for the controllers with acceleration inputs and modal coordinate inputs respectively.

Figure 16. Controller performance in a gust: a) controller with accelerometer inputs, modal 

amplitude time history; b) controller with accelerometer inputs, control surface time history; 

c) controller with modal coordinate inputs, modal amplitude time history; and d) controller with 

modal coordinate inputs, control surface time history.
31

The modal response, figure 16(a), to the gust for the controller with accelerometer inputs 

indicates that the first two modes (first wing torsion and first wing bending) responded most to 

the gust as expected. The peak amplitude of the bending mode was approximately -2.2, which 

corresponds to roughly a 44-cm-upward bending deflection at the wing tip. A negative bending 

modal coordinate corresponds to a positive wing tip deflection, figure 2(a). The twist mode 

moves to a maximum amplitude of 0.4, which corresponds to approximately 10 deg of positive 

wing twist, leading-edge up. Over the gust time history, the control surfaces move to counteract 

the effect of the gust, figure 16(b). As the wing experiences a lift increase, the control surfaces 

rotate upward to reduce the angle of attack of the wing and reduce lift. The rotation of the 

control surfaces stayed well within the bounds of reason for wing control surface rotations.

The gust disturbance rejection performance of the controller with modal coordinate inputs 

was comparable to that of the controller with accelerometer inputs, shown in figure 16(c); the 

peak amplitude of the bending mode was slightly higher at -2.4. The torsion angle was nearly 

the same at approximately 10 deg. Little can be said as to which controller has better 

disturbance rejection. The differences were negligible. The controllers rejected the specified 

gust disturbance adequately.

Flutter Suppression

A major theme in this study has been to demonstrate that the controller with the modal 

coordinate inputs may be used for flutter suppression. Figure 5 shows that the model is 

open-loop unstable, resulting in modal motions characteristic of flutter. The same simulation 

structure (see fig. 15) that was used for the gust disturbance modeling was used for flutter 

suppression with very small changes. The gust model inputs T were set to zero. A small control 

input at time 0 was introduced to perturb the wing model from its trim state. The modal 

amplitudes of the model are allowed to increase without control input until 3.5 s. The controller 

was linearly phased-in from 3.5 s to 4.5 s. The controller was not turned on to full instantly at 

3.5 s to avoid large oscillations due to the output magnitudes being far from the reference 

condition of zero. The simulation results are presented in figure 17 for the controllers with 

acceleration inputs and modal coordinate inputs respectively.
32

Figure 17. Controller performance in a flutter suppression: a) controller with accelerometer 

inputs, modal amplitude time history; b) controller with accelerometer inputs, control surface 

time history; c) controller with modal coordinate inputs, modal amplitude time history; and 

d) controller with modal coordinate inputs, control surface time history.

The time history for the controller with accelerometer inputs is examined first in figure 17(a).

The modal amplitudes oscillate with a frequency of 4.49 Hz increasing in amplitude until 

approximately 4.2 s. At this time, the controller force begins to remove a sufficient amount of 

energy from the flutter mode to begin to reduce the amplitudes of the modes. The oscillations 

die out quickly at approximately 5 s.
33

The time history of the controller matches what would physically be required to reduce flutter 

in the wing. As the wing bends upward, figure 17(a), the control surfaces rotate upward,

figure 17(b), to reduce the angle of attack of the wing and reduce lift. The net aerodynamic force 

has the effect of moving the wing downward. As the wing moves down, the control surfaces 

rotate downward to increase the lift on the wing. This counterbalancing effect performs work and 

removes energy from the flutter mode. The overall effect asymptotically stabilizes the structure. 

After the flutter mode stabilizes, the control surface movements appear to oscillate at low 

frequency and the modal coordinates remain near zero. The movement from equilibrium is in 

response to the additive noise on the accelerometers.

The controller with modal coordinate inputs performed similarly to the controller with 

accelerometer inputs, as before with the gust inputs. Figure 17(c) shows that the modal 

coordinates begin to flutter up to 3.5 s and are slowly damped out once the controller is 

enabled. As before, the control surface movements worked to extract energy from the flutter 

mode as seen in figure 17(d). The modal coordinates also stay near zero as expected. From 

time analyses it was clear that both controllers performed well in meeting the primary objective 

to suppress the flutter mode at the selected flight condition.

So far, the actual modal coordinate time histories are presented in each plot, representing 

the motion of the model in either a gust or fluttering condition. To satisfy curiosity, the modal 

coordinates which were given to the controller for feedback during the flutter suppression H are 

presented in figure 18. Recall that zero mean Gaussian noise with a standard deviation of 1 cm

was added to all deflections that were used to estimate the modal coordinates. This means that 

the error can likely go up approximately 3 cm for the deformations some of the time, assuming 

that some data points will fall roughly 3 standard deviations away. All of the deflections with 

error were put through a least-squares modal filter [see fig. 15 and equation (3)]. The

measurement error of the modal coordinates indicates that a typical least-squares smoothing

has taken place, as the modal amplitudes measurement error tended to stay near 0.01.

Although this smoothing is not substantial enough to raise eyebrows, it does show that the 

errors tend to average out when many sensors are utilized in forming the least-squares 

estimates. It is also clear that the increasing amplitude of the modal coordinates did not affect 

the modal coordinate estimation error, even during open-loop flutter.

Figure 18. Modal coordinate measurement error.
34

Virtual Deformation Control (Case Study: Modal Tracking)

Until this point, much of the focus of this study has been to compare regulators with 

accelerometer inputs or modal coordinate inputs. It was observed that the controllers had similar 

performance during GLA and AFS. In this section, shape control is demonstrated. This type of 

research has been pursued in other works as well; see references 11 and 66.

Shape control on the aeroelastic model is implemented here making use of the modal filter.

Further work with accelerometers is not continued. While the controller with accelerometers 

might be able to track deformations by double-integrating the accelerations, it is not a natural fit;

the deformations predicted with accelerometers may start to drift and require deformation 

updates.

The modal controller may be a suitable match for shape control since modes are 

linearly-related to deflections [see eq. (1)]. If a set of reference deflections 
 are known at 

specific locations :, perhaps from a flight computer, these deformations may be transformed to 

modal coordinate reference values, H, indicating that the reference deformations need not be 

at the points of measurement on the model. These deflections can be referred to as virtual 

deformations, and the control of these deformations as virtual deformation tracking. The 

simulation structure for deformation control through modal reference tracking is presented in 

figure 19.

Figure 19. Control simulation for virtual deformation control: modal control.

To achieve zero steady-state tracking error, “integrator of modal position” states are added to 

the controller. All modes are estimated by the modal filter and indexed (that is, ?+@AB = {1,2})

at the first two locations to give, H. The modal measurements are input to a differencing 

junction with the modal reference and sent through a single continuous-time integrator. 

To simulate a small torsion angle command, 
 = {N<, ;<} was set to {1 44, 1 44}>,

which represents a leading-edge-down rotation. Since small deformations are used as 

references, the noise was adjusted accordingly, so that the standard deviation of the 

multiplicative control noise was set to 0.001. The standard deviation on the deformation 

measurements was assumed to be 1 mm. These noise settings allows one to better see what 

the controller is doing in the resulting plots; this was not done to simulate actual sensor noise 

characteristics. The simulation results are presented in figures 20(a) through 20(c).
35

Figure 20. Virtual deformation control: modal commands: a) modal amplitudes; b) deformations 

and predicted deformations at wing tip; c) control surface rotations.

Figure 20(a) shows that the first torsion modal coordinate overlays the reference torsion 

modal coordinate within 50 s. The bending modal coordinate reference is near zero and is also 

tracked within 50 s. The other eight modal coordinate time histories are also plotted so that the 

effects of residual modes may be observed. The second bending mode becomes highly excited. 

The deformations achieved through modal tracking are presented in figure 20(b). The 

deflections achieve what would be a torsional angle with the leading edge down, but the net 

deformations of both are up approximately 2 mm. There is significant error between the desired 
36

deformations of {1 mm, -1 mm} and what is achieved {3.6 mm, 2.2 mm}. The prominent second 

bending modal coordinate is clearly to blame for this error. From desktop simulations it was 

observed that to reduce this error, the second bending mode, figure 2(c), could be tracked if 

more actuators were available. Note that the deformations to be tracked were at the same 

locations as where the accelerometers were placed. They were not measured by the fiber optics 

themselves. The action of tracking a deformation which is not directly measured is observed 

here.

Figure 20(c) indicates that to achieve small deformations, very large control surface rotations 

were required, almost up to 40 deg. The large rotations are a result of either high stiffness in the 

wing or potentially low control surface steady-state effectiveness. For this reason, the deflection 

references were kept small to ensure the control surfaces rotated within reasonable limits. 

These results indicate that wings with low torsional modal mass (see fig. 7) may be difficult to 

structurally morph using aerodynamic effectors.

Virtual Deformation Control (Case Study: Predicted Deformation Tracking)

In the previous case study, the deformation command was transformed to a reduced modal 

command and the modes were tracked. Due to the effect of residual modes, the wing 

deformation reference command was tracked poorly. To reduce this effect, the modal filter can 

also be used to form a predicted estimate of the deformation of the structure at any point by 

including residual modes into the estimate. To prepare the controller, the output matrix in the 

state space may be defined to have the form shown in equation (28),

  [f(:,:) 0× 0× 0×&] (28)

where K is the number of deformations (or virtual deformations) desired to be tracked. This

method of definition has the effect of making the outputs of the plant equivalent to the 

deformations. A similar transformation is used to model the accelerometers, where the sensors 

are assumed to measure linear combinations of the modal states [see eqs. (12) and (13)].

The simulation scheme used for virtual deformation tracking is presented in figure 21.

The simulation for tracking deformations estimated by modal coordinates is similar to the modal 

tracking simulation presented above (see fig. 19). The difference is that after the 

modal coordinates are estimated and indexed, the modal matrix is again multiplied by the modal 

matrix indexed at desired virtual measurement locations, :. Since all modes are utilized to 

estimate the virtual deflections, H = H#$. Alternatively, it might be stated that

?+@AB = {1,2,3,4,5,6,7,8,9,10}. The noise levels in the previous simulation are also used, as 

were used previously for modal tracking.
37

Figure 21. Control simulation for virtual deformation tracking: predicted deformation control.

The same reference values of {1mm,-1mm} were used for the deformation reference and the 

simulation results are presented in figures 22(a) through 22(c).
38

Figure 22. Virtual deformation control: predicted deformation: a) modal amplitudes; 

b) deformations and predicted deformations at wing tip; c) control surface rotations.

The time history of the modal coordinates in figure 22(a) shows that the torsion modal 

coordinate moved comparative to the way it did previously, seen in figure 20(a). The second 

wing bending modal coordinate also moved positively; however, this time, the first wing bending 

moves from zero to a large positive value, which has the effect of offsetting the second 

wing-bending effects. The net effect of this was that the actual virtual deformations and 

reference virtual deformations were overlaid, seen in figure 22(b).
39

Figure 22(c) indicates that the control surface movements were lower than they were 

previously, figure 20(c). The outboard control surface moved to approximately 30 deg and the 

inboard surface moved to -25 deg. The use of either the tracking strategy presented here or 

that presented above may depend on the application. If a multitude of points on the wing are 

required to be tracked or moved to a particular shape, then the strategy first presented may be 

more useful. The reduction of the reference signals to a few modal coordinates may alleviate 

the control design effort. If only a few virtual deflections at a few points on the structure (that is,

2-4) are required to be tracked, then the strategy presented here may be more applicable. The 

selection of the appropriate tracking strategy will be application-dependent.

Conclusions and Future Work

The findings described herein suggest that the modal filter may be suitable for integration 

into flexible aircraft control systems. A comparison of regulators with modal filters and regulators 

with accelerometers indicated that the stability and performance of the regulators during gust 

load alleviation and active flutter suppression changes very little with use of the modal filter. 

During the control design, an unexpected finding was that when using balanced truncation, the 

order of the plant with modal filter outputs, could be reduced substantially. A lower order plant 

(and thus controller) may facilitate the development of a digital controller.

Modal filters were shown to support the capability for wing-shape control. It was 

demonstrated in two case studies, using two different architectures, how deformations (a few or 

many) at any point on the wing could be controlled. The choice of architectures, it is believed, 

will be application-dependent. The virtual deformation control architecture with modal control will 

be the most generally applicable and further research into limiting the impact of residual modes 

is needed. During the virtual deformation control simulation studies it was discovered that high 

control rotations may be required if wing torsional modal mass is very low.

The model used for the study had a few shortcomings which limited the analysis. The first 

shortcoming was that the structure was notional and not based on an operating or experimental 

aircraft. The second shortcoming was that the deformation measurements were essentially 

perfect before noise was added. The accuracy of the modal estimation may also be called into 

question. All of these modes in the state space simulation model were incorporated into the 

basis of the modal filter, so the bias of residual modes was not present in the modal coordinate 

estimates. Additionally, aircraft require guidance systems and rigid-body mode control; the 

interacting effects of the two control systems with the modal controller were not seen here. 

Further work will likely be to revisit the implementation of the modal filter and potentially its 

variations. A deformation tracking scheme may also be incorporated onto a model that is more 

representative of an aircraft.

Acknowledgments

This research was approved by Steve Jacobson, Branch Chief of the controls branch at the 

Dryden Flight Research Center, Edwards, California, and funded by the National Aeronautics 

and Space Administration Aeronautics Research Mission Directorate Fixed Wing program.

Steve Jacobson; John Bosworth, the Dryden X-56A chief engineer; Chris Regan, the former 

Dryden X-56A controls lead; John Ryan, Dryden X-56A controls lead; and Dryden aeroelasticity 

expert Marty Brenner; have been excellent advisors in the development of the results of this 

research.
40

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