NASA/TM-2007-214609

Tailored Excitation for Frequency Response 

Measurement Applied to the X-43A Flight 

Vehicle

Ethan Baumann

NASA Dryden Flight Research Center

Edwards, California

January 2007
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NASA/TM-2007-214609

Tailored Excitation for Frequency Response 

Measurement Applied to the X-43A Flight 

Vehicle

Ethan Baumann

NASA Dryden Flight Research Center

Edwards, California

January 2007

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273
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Available from:

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7115 Standard Drive

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(301) 621-0390
ABSTRACT

An important aspect of any flight research project is assessing aircraft stability and flight 

control performance. In some programs this assessment is accomplished through the estimation 

of the in-flight vehicle frequency response. This estimation has traditionally been a lengthy task 

requiring separate swept sine inputs for each control axis at a constant flight condition. Hypersonic 

vehicles spend little time at any specific flight condition while they are decelerating. Accordingly, 

it is difficult to use traditional methods to calculate the vehicle frequency response and stability 

margins for this class of vehicle. A technique has been previously developed to significantly reduce 

the duration of the excitation input by tailoring the input to excite only the frequency range of 

interest. Reductions in test time were achieved by simultaneously applying tailored excitation 

signals to multiple control loops, allowing a quick estimate of the frequency response of a particular 

aircraft. This report discusses the flight results obtained from applying a tailored excitation input to 

the X-43A longitudinal and lateral–directional control loops during the second and third flights. The 

frequency responses and stability margins obtained from flight data are compared with preflight 

predictions.

Nomenclature

6-DOF six-degree-of-freedom

AMW all-moving wing

DFRC NASA Dryden Flight Research Center

dB decibels

FFT Fast Fourier Transform

GNC guidance, navigation, and controls

Gxx input autospectral density

Gxy cross-spectral density

Gyy output autospectral density

HXLV Hyper-X Launch Vehicle

HXRV Hyper-X Research Vehicle

IPT integrated product team

Km scale factor of mth input

k discrete frequency variable

LaRC NASA Langley Research Center

MIMO multiple-input/multiple-output

n f number of frequencies

PID parameter identification

Rmax maximum desired input rate


t 	 	 	 time

Um final mth input signal

um intermediate mth input signal

u input signal derivative

γ xy

 magnitude squared coherence

λm wavelength of mth signal

Φm phase shifting of mth signal

Introduction

When flight-testing a new vehicle such as the X-43A it is desirable to estimate the in-flight 

frequency response and stability margins of the vehicle. This information has traditionally been 

used to clear an aircraft to a new portion of the flight envelope and to validate preflight models. In 

addition, a multiple-input/multiple-output (MIMO) system analysis can yield valuable information 

concerning the robustness of the vehicle control system. For the X-43A, also known as the Hyper‐X 

Research Vehicle (HXRV), this information was desired for post-flight analysis and to aid in model 

validation.

A shortcoming in performing in-flight frequency response analysis has been the long duration 

of the excitation signal required to obtain this information. Traditionally, a swept sinusoid is used 

as the excitation signal. The swept sinusoid is often input by the pilot and requires a relatively long 

dwell time at a given flight condition (ref. 1). The HXRV rapidly descends in altitude and in Mach 

number following the completion of the scramjet engine test. This rapid descent made it desirable 

to utilize an excitation method that would quickly excite the control loops, before the vehicle had 

moved on to a significantly different flight condition. 

The use of tailored excitation input signals provides a measurement of the system frequency 

response over a specified range of frequencies. The duration of the input signal can be minimized 

by utilizing only the frequency range required for stability margin verification. Additionally, the 

simultaneous application of the excitation signal to all of the vehicle control inputs allows for 

short-duration tests. These simultaneous inputs consist of uncorrelated frequency components. By 

using phase shifting as described by Schroeder (ref. 2), it is possible to minimize the magnitude 

of these tailored excitation signals. This technique has been shown to be superior to traditional 

methods (ref. 3).

Simultaneous tailored inputs allow for rapid estimation of the vehicle frequency response 

in each control loop. Chirp-Z transformation algorithms can be used to extract the response at 

the specifically excited frequencies. A Fast Fourier Transform (FFT) can also be used to extract 

frequency response information provided the individual frequencies of the input signal sufficiently 

approximate a continuous frequency range. In addition, the singular values of the MIMO system 

can be estimated. The methodology and feasibility of using tailored excitation input signals for 

frequency response estimation were explored in a simulation environment by Bosworth and 

Burken (ref. 4). The work done by Bosworth and Burken applied only to the longitudinal axis of
the X‐31A. A similar technique has been applied to the lateral–directional axes of the NASA 

F‐15 ACTIVE research aircraft for aerodynamic parameter estimation (ref. 5). The tailored inputs 

used for the X‐43A simultaneously excited the longitudinal and lateral–directional axes in flight, 

which is the first time this simultaneous excitation has been attempted at the NASA Dryden Flight 

Research Center (DFRC), Edwards, California.

This report discusses the in-flight application of simultaneous tailored input excitation signals 

to the HXRV. The flight-data-derived frequency response and high-frequency stability margin 

measurements are presented, along with a comparison with linear model predictions.

Project Description

The Hyper-X program was an experimental flight research program intended to demonstrate 

advanced hypersonic technologies (ref. 6). The primary research objective was to flight-test an 

airframe-integrated scramjet engine and validate the vehicle design tools. Hypersonic research is 

intended to enable the development of high-speed airbreathing technologies, which could be used 

in future high-speed aircraft and reusable launch vehicles. The NASA Langley Research Center 

(LaRC), Hampton, Virginia was the Hyper-X program lead, and NASA DFRC led the flight-test 

effort. Each flight test consisted of two vehicles: a launch vehicle and a research vehicle. The 

Hyper‐X Launch Vehicle (HXLV) was provided by Orbital Sciences Corporation of Chandler, 

Arizona. An industry team consisting of Alliant Technosystems, Inc., GASL Division (Tullahoma, 

Tennessee) and Boeing Phantom Works (Seal Beach, California) constructed the HXRV.

Hyper-X Research Vehicle Description

As shown in figure 1, the HXRV was an unmanned autonomous vehicle that measured 

approximately 12 ft long and 5 ft wide, and weighed approximately 3000 lb. The scramjet engine 

was attached to the underside of the HXRV. A cowl door on the leading edge of the engine controlled 

airflow through the engine. The HXRV had four control surface effectors: a left and right allmoving wing (AMW) and twin rudders. The rudders moved symmetrically, and the AMWs moved 

symmetrically and differentially for pitch and roll control. Symmetric AMW deflection is defined 

as elevator deflection, and differential AMW deflection is defined as aileron deflection.
All-moving wing

Rudder

Engine

144 in. 050589

148 in.

60 in.

19 in.

26 in.

30 in.

Length: 12 ft 4 in.

Width: 5 ft 0 in.

Height: 2 ft 2 in.

Weight: 3000 lb max

050589

Figure 1. Hyper-X Research Vehicle three-view.

Hyper-X Research Vehicle Flight Test

Three HXRVs were built for the Hyper-X program. Two of the vehicles were intended 

for missions at Mach 7, and one vehicle was intended for a mission at Mach 10. The HXRVs 

collectively received the designation of X-43A. All three vehicles have the same outer mold line; 

the primary difference among the vehicles is the internal engine flowpaths. Flight 1 was intended 

to reach Mach 7 and was attempted on June 2, 2001. The HXLV lost control shortly after launch, 

resulting in the loss of both the HXLV and HXRV (ref. 7). During Flight 2, flown to Mach 6.8 

on March 27, 2004, the HXRV successfully demonstrated the in-flight operation of the scramjet 

(ref. 8). All of the goals for that mission were achieved, including positive acceleration of the 

vehicle by the scramjet. The third and final flight was flown to Mach 9.6 on November 16, 2004 

(ref. 9). During both successful missions, the HXRV was in controlled autonomous flight from the 

point of separation to the splashdown in the Pacific Ocean.

Hyper-X Research Vehicle Mission Description

The HXRV mission timelines were similar for Flight 2 and Flight 3. The principal difference 

between the two flights is the higher Mach number attained during Flight 3, which resulted in a 

correspondingly longer descent trajectory. Figure 2 shows an overview of Flight 3. The launch, 

experiment, and descent portions of the mission were conducted off the coast of southern California. 

The HXRV was boosted to the test condition by the HXLV. The HXRV was connected to the HXLV 

by an adapter module. The entire configuration, known as the X-43A stack and shown in figure 3, 

was carried under the wing of the NASA DFRC NB-52B, ship number 008.
B-52B

 carriage

Time since drop, s

Altitude,

ft

0 5 ~ 90 ~ 112

0

5

11 x 104

~ 700

Mission

complete Terminal glide

Decel-descent

Booster ignition

Pull-up

Accel-ascent

Push-over

Primary

mission Power-on test

Power-off tare

Cowl closes

Cowl open

Stage separation

Recovery maneuver

PID maneuvers

 to desired impact area

Fuel off

Fuel on

Power-off tare

Booster burnout

050590

050590

Figure 2. Hyper-X Research Vehicle Flight 3 overview.

050591

Figure 3. The X-43A stack configuration.


For both missions, the HXLV propelled the X-43A stack to a separation altitude of 

approximately 100 000 ft and a dynamic pressure of approximately 1000 psf. The HXRV separated 

at Mach 6.9 during Flight 2 and at Mach 9.7 during Flight 3. Several seconds after separation, 

the primary experiment of the scramjet engine test was conducted. The primary mission portion 

lasted approximately 45 s for Flight 2 and 30 s for Flight 3. Following the primary mission, an 

unpowered trajectory was flown to a splashdown into the Pacific Ocean. During the descent, a 

series of parameter identification (PID) maneuvers were performed at every integer Mach number 

down to Mach 2. These maneuvers began at Mach 5 during Flight 2 and at Mach 8 during Flight 3. 

The HXRV was constantly decelerating throughout the PID maneuvers, so the Mach number was 

not constant. The PID maneuvers consisted of control surface inputs designed by the aerodynamics 

Integrated Product Team (IPT) (ref. 10) and guidance, navigation, and controls (GNC) IPT. Figure 4 

shows the Mach 5 PID input set for the elevator loop from Flight 3. The aerodynamics IPT’s inputs 

were intended to characterize the rigid body aircraft characteristics and consisted of step inputs 

for both Flights, and frequency sweeps for Flight 3. The aerodynamics IPT’s input set occurred 

at the beginning and end of the PID maneuver set, and consisted of inputs that excited the vehicle 

over a lower frequency range than did the GNC IPT’s inputs. The GNC IPT’s inputs consisted 

of tailored excitation inputs to the control loops intended to characterize the vehicle open-loop 

frequency response. All inputs were added directly to the control loop commands generated by the 

flight control laws.

Time, s

3

2

1

0

–1

–2

–3

Step inputs

only at

Mach 5

and below

Aerodynamics frequency sweep G&C frequency sweep

Elevator loop input, deg

050592

050592

Figure 4. Mach 5 elevator loop parameter identification maneuver from Flight 3.
Input Excitation Signal

The simultaneous tailored input excitation signals used on the HXRV consisted of a series 

of summed sine waves. A summed sine wave has the advantage of reducing the excitation time 

when compared with a traditional swept sine wave. The method used to generate the input signals 

generally follows that described by Bosworth and Burken (ref. 4), but a modification was made to 

the method to prevent control surface rate limiting.

The simultaneous tailored excitation signals were generated following the method described 

by Bosworth and Burken (ref. 4). The frequency range was selected such that the preflight gain 

and phase crossover frequency predictions for the HXRV were within the selected range. Once the 

frequency range had been selected, 35 distinct frequencies for each control loop were calculated. 

Table 1 details the input frequencies for each control loop. The method for selecting the input 

frequencies resulted in a smaller spacing between frequencies toward the upper end of the range. 

This spacing resulted in the input sines being sufficiently close enough in frequency range for the 

FFT method to work well above approximately 0.9 Hz.

Equations (1) through (4) below were used to construct the input excitation signals.

The input excitation signal for the mth input is

um = sin(λm (n)t + Φm (n))

n=1

n f

3

∑

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

where λm ( ) n is the wavelength of each frequency and Φm ( ) n is the phase shifting used to reduce 

the peak factor of the excitation while maintaining the same frequency content (ref. 2).

For the HXRV, each of the input signals was scaled to limit the control surface input. This scaling 

was necessary to prevent actuator rate limiting so as to retain actuator margin for maneuvering. 

A value of 50 deg/s was used as the maximum allowable rate limit when designing the inputs for 

the HXRV. This value was chosen to keep the commanded surface position rate change for each 

control surface within the actuator limit of approximately 100 deg/s, while retaining an adequate 

maneuvering margin.

The derivative of each input signal is calculated to find the maximum rate of change of the 

input signal:

um = λm cos(λm (n)t + Φm (n))

n=1

n f

3

∑

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(1)

(2)
Table 1. Excitation input frequencies.

Elevator Aileron Rudder

frequency, Hz

0.25 0.19 0.32

0.45 0.38 0.51

0.64 0.57 0.70

0.83 0.76 0.89

1.02 0.95 1.08

1.21 1.15 1.27

1.40 1.34 1.46

1.59 1.53 1.66

1.78 1.72 1.85

1.97 1.91 2.04

2.16 2.10 2.23

2.36 2.29 2.42

2.55 2.48 2.61

2.74 2.67 2.80

2.93 2.86 2.99

3.12 3.06 3.18

3.31 3.25 3.37

3.50 3.44 3.56

3.69 3.63 3.76

3.88 3.82 3.95

4.07 4.01 4.14

4.26 4.20 4.33

4.46 4.39 4.52

4.65 4.58 4.71

4.84 4.77 4.90

5.03 4.97 5.09

5.22 5.16 5.28

5.41 5.35 5.47

5.60 5.54 5.67

5.79 5.73 5.86

5.98 5.92 6.05

6.17 6.11 6.24

6.37 6.30 6.43

6.56 6.49 6.62

6.75 6.68 6.81
A scale factor ( Km ) is then found by dividing the maximum desired rate ( Rmax ) by the 

maximum value of the derivative of the input signal:

Km = Rmax

max(um )

The input signals are then scaled by Km to minimize the control surface rate change:

U K m m = um

The end result is three input signals that are now uncorrelated, periodic, and appropriately 

scaled.

Input Signal Used for X-43A

The excitation inputs were embedded in the X-43A flight control laws and were activated 

during the descent PID maneuvers. Table 1 details the frequencies used for the elevator, aileron, 

and rudder loop excitation frequencies. Figure 5 shows the time history trace for each of the three 

inputs. The same inputs were used at each PID maneuver set during the descent.

2

0

–2

2

0

–2

2

0

–2

0 2 4 6 8 10 12 14 16

Time, s

Rudder input,

deg

Aileron input,

deg

Elevator input,

deg

050557

050557

Figure 5. Hyper-X Research Vehicle tailored excitation signal time histories.

(3)

(4)
10

Figure 6 is a block diagram showing how the excitation signals are applied to each control 

axis. The inputs are added to the commands generated by the flight control system. The flight 

control system consists of a longitudinal and lateral–directional controller (ref. 11). The openloop transfer function is then calculated between the tailored excitation signal and the output of 

the longitudinal and lateral–directional controllers. Both the input and output data for the transfer 

function are obtained as 100-Hz signals.

Control system

Control

surface

mixer

Rudder

command

Left AMW command

Right AMW command

050558

Left Rudder command

Right Rudder command

Aileron

command

Elevator

command

PID

generation

Longitudinal

controller

Vehicle

dynamics

Lateral –

directional

controller

+

+

+

050558

Figure 6. Block diagram detailing the application of the tailored excitation input signals to the 

Hyper-X Research Vehicle.

Response Estimation

Both Chirp-Z and Fast Fourier transforms were applied to the HXRV flight data to calculate 

the in-flight frequency responses. A Chirp-Z transform can be used to match the input and output 

frequencies. Only those frequencies present in the input are examined in a Chirp-Z transform. 

The FFT method is a computationally efficient way to implement the Discrete Fourier Transform, 

which is used to transform discrete data from the time domain to the frequency domain.

The FFT method also provides an indication of the quality of the input–output relationship. 

This quality is determined by examining the magnitude squared coherence function shown in 

equation (5).

γ xy

xy

xx yy

k

G k

G k G k





( ) = ( )

( ) ( ) (5)
11

A linear, noise-free system would yield a coherence of 1. System nonlinearities, input and 

output noise, and secondary inputs would produce a coherence below 1. For the purposes of this 

report, a coherence value of 0.8 indicates an acceptable response measurement.

Linear Model and Six-Degree-of-Freedom Nonlinear Simulation

The Hyper-X program developed a high-fidelity six-degree-of-freedom (6-DOF) nonlinear 

simulation model of the X-43A along with a simplified linear model used for control law design and 

analysis. The 6-DOF simulation was based on the standard DFRC simulation architecture (ref. 12) 

and contained detailed subsystem models of the flight control system, vehicle aerodynamics, 

actuators, mass properties, engine, and sensors. The linear model was constructed in MATLAB®

and Simulink® (both registered trademarks of The MathWorks, Inc., Natick, Massachusetts) and 

contained linearized versions of these subsystems.

The tailored input excitations were used in the nonlinear simulation as part of the frequency 

response comparison with the linear analysis tool. Figures 7 and 8 show sample elevator and 

rudder loop comparisons for the Mach 5 PID frequency response predictions for Flight 2. The FFT 

and Chirp-Z-derived frequency responses are shown compared with linear model predictions at 

representative flight conditions; the start, mean, and end flight conditions during the GNC portion 

of the PID maneuvers. The aileron loop comparison was similar to that of the elevator loop. The 

elevator and aileron loop results of the simulation and linear models were found to be in excellent 

agreement and were used prior to Flight 2 to predict the in-flight performance of the tailored 

excitation input signals. The rudder loop results showed discrepancies with the simulation results, 

which are discussed in detail in the following section.

0.5

1.0

0

–400

–200

0

200

–50

0

50

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050559

Simulation data, 

 fast Fourier transform

Simulation data,

 Chirp-Z transform

Linear tool predictions

+

050559

Figure 7. Comparison of elevator frequency responses for the Flight 2 Mach 5 parameter 

identification maneuver from the six-degree-of-freedom simulation and the linear analysis tool.
12

0.5

1.0

0

–400

–40

–200

0

200

–20

0

20

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050560

Simulation data, 

 fast Fourier transform

Simulation data,

 Chirp-Z transform

Linear tool predictions

+

050560

Figure 8. Comparison of rudder frequency responses for the Flight 2 Mach 5 parameter identification 

maneuver from the six-degree-of-freedom simulation and the linear analysis tool.

Problems Identified Prior to Flight

A few months prior to Flight 2, simulation testing uncovered several problems with the tailored 

excitation signals. Rudder loop frequency response values generated from simulation data showed 

very poor matches with the linear model. In addition, the coherence value for the rudder loop was 

below acceptable levels. The problem was discovered too late to implement a solution for Flight 2, 

especially considering that the descent portion of the mission was of secondary importance. Data 

from Flight 2, presented below, also showed poor rudder loop frequency responses.

Shortly after Flight 2, a number of approaches were investigated to improve the rudder 

loop frequency response results. The most effective approach was to decouple the rudder and 

aileron excitations. Simulation results indicated that this modification produced good comparisons 

between the linear analysis tools and 6-DOF simulation for the rudder loop, as shown in figure 9. 

This approach was adopted for Flight 3, resulting in the rudder and aileron control loop inputs 

being applied at alternating Mach numbers. The improvement in the rudder loop comparisons 

when applied without the aileron loop inputs indicates a correlation between the rudder and aileron 

loop responses that is not understood at this time.
13

0.2

0.6

1.0

0

–500

–100

–300

0

–50

–30

10

0

–10

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050561

Simulation data, 

 fast Fourier transform

Simulation data,

 Chirp-Z transform

Linear tool predictions

+ + + + + + + + + +

+ + + + + + + +

050561

Figure 9. Rudder loop frequency response comparison between six-degree-of-freedom simulation 

and linear analysis tools when input decoupled from aileron loop for the Flight 3 Mach 4 parameter 

identification maneuver.

Another possible explanation for the poor rudder loop comparisons is that the magnitude of 

the rudder excitation was insufficient to move beyond the rudder actuation system nonlinearities. 

The excitation input magnitude for the rudder was increased in the simulation with mixed results. 

The response improved slightly, but was still inadequate.

As a result of the poor rudder frequency response estimation, MIMO analysis was not 

conducted. A MIMO analysis requires valid transfer function values from all control axes. The 

poor characteristics of the rudder loop frequency response data precluded attempting any MIMO 

analysis on Flight 2 data. In addition, Flight 3 did not simultaneously excite the aileron and rudder 

loops at the same flight condition. Valid aileron and rudder frequency responses were never obtained 

at the same flight condition, thereby precluding any flight-data-derived MIMO stability analysis 

for the X-43A for either flight.

The simulation comparisons with the linear analysis predictions displayed another problem 

with the input signals. The elevator frequency responses for the Mach 2 PID did not match linear 

predictions as well as did those at the higher Mach numbers. Figure 10 shows the Flight 2 elevator 

loop simulation and linear analysis comparison for the Mach 2 PID. The excitation signals occur 

near the end of the PID maneuver set, so they finished near or in the transonic regime. The 

longitudinal aerodynamics of the vehicle change rapidly as the vehicle maneuvers through the 

transonic regime; the poor matches are not unexpected since the linear analysis results are at a 

specific flight condition.
14

0.5

1.0

0

–500

0

500

– 40

– 20

0

20

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050562

Simulation data, 

 fast Fourier transform

Simulation data,

 Chirp-Z transform

Linear tool predictions

+

050562

Figure 10. Comparison of elevator frequency responses for the Flight 2 Mach 2 parameter 

identification maneuver from the six-degree-of-freedom simulation and the linear analysis tool.

Flight Results

For both Flights 2 and 3, the HXRV achieved scramjet operation and successfully flew an 

unpowered descent into the Pacific Ocean. Figure 11 shows the altitude profile for the two flights. 

As can be seen, the two flights flew different descent profiles in the common Mach regions. In 

addition, the control system used for Flights 2 and 3 utilized different gain schedules. These 

differences render direct comparisons between the two flights impossible. It is possible, however, 

to examine the quality of the linear values to flight-derived frequency response comparisons 

between the two flights.

Flight 3 included an unusual event after the engine test that affected the vehicle flight 

characteristics. It is believed that air started flowing through the engine shortly after the initiation 

of the recovery maneuver. The cowl door had been commanded closed at the end of the engine 

test, and this closing had shut off the majority of the air flowing through the engine as expected. 

An unknown mechanism allowed significant quantities of air to flow through the vehicle shortly 

after the engine test. Just prior to the Mach 8 PID maneuvers, the internal engine pressures 

indicated a decrease in the amount of air flowing through the vehicle. The vehicle aerodynamics 

was significantly different with air flowing through the engine than during cowl-closed operations. 

The nonlinear simulation and linear analysis tools lack the capability to model air flowing through 

the engine during the descent. It is not known to what extent the differences between the in-flight 

and linear analysis values for the frequency responses and stability margins are attributable to this 

phenomenon.
15

0

11 x 104

~ Mach 7 ~ Mach 1

Decreasing Mach number

Mach 7 mission

Mach 10 mission

Inertial altitude, ft

050563

050563

Figure 11. Post-engine-test inertial altitude comparison between Flights 2 and 3.

Linear analysis predictions were generated for comparison with the flight-data-derived 

frequency responses. Representative flight conditions at the start, mean, and end of the tailored 

excitation inputs were used to generate the linear analysis responses. Onboard vehicle data from 

each flight was used to generate the flight conditions for the linear analysis. The Mach number and 

dynamic pressure were estimated onboard with the inertial velocity measurements and a reference 

atmosphere. The angle of attack is the inertial in-flight measured angle of attack.

The high-frequency stability margins are calculated from the frequency response data by 

linear interpolation. The MATLAB margin function was used to perform this calculation.

The coherence values for each control loop and Mach number combination were examined. 

In general, the coherence values were near 1 for the elevator and aileron loops for both flights over 

the frequency range of interest. The rudder loop coherence was poor for Flight 2 and acceptable 

for Flight 3. The rudder loop frequency response measurements were found to compare well with 

the linear analysis predictions when the coherence value was above 0.8.

Elevator Loop Results

Figure 12 shows a Bode plot comparing the elevator loop in-flight frequency response for the 

Mach 5 PID maneuver of Flight 2 with that of the linear models. The comparison seen in figure 12 

is representative of the elevator loop frequency response comparisons for both Flights 2 and 3. The 

comparisons are excellent in the frequency range excited by the input signal.
16

0.5

1.0

0

–300

–100

–200

0

100

–20

0

20

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050564

Flight data, 

 fast Fourier transform

Flight data,

 Chirp-Z transform

Linear tool predictions

+

050564

Figure 12. Elevator loop frequency response comparisons for the Flight 2 Mach 5 parameter 

identification maneuver.

The elevator loop stability margin values for the HXRV during Flight 2 can be seen in 

figure 13. The gain and phase margin measurements are generally within 1.6 dB and 5 deg of 

the linear model predictions. The exception is the phase margin prediction for the Mach 2 PID 

maneuver where the phase margin determined by the Chirp-Z method is 6 to 16 deg different from 

the linear analysis predictions. The elevator loop frequency response for the Mach 2 PID maneuver 

is shown in figure 14. The FFT method shows a lower coherence and did not return a gain crossing 

for phase margin calculation for the Mach 2 PID maneuver. At the corresponding flight conditions, 

the linear analysis predictions vary by as much as 10 deg between the start, mean, and end flight 

conditions. This difference indicates that the vehicle aerodynamics was changing rapidly and it is 

therefore not surprising that the flight-derived value is different from the linear predictions.

Figure 15 shows the stability margin values from Flight 3 for the elevator loop. The in-flight 

gain margin values for the Mach 8 through Mach 5 PID maneuvers are consistently higher with 

the greatest difference being 2.5 dB. This difference in gain margin is possibly attributable to 

the unmodeled effects of air flowing through the engine. The phase margin predictions at these 

Mach numbers match the linear predictions within 2 deg. The greatest spread in the linear analysis 

predictions is again seen during the Mach 2 PID maneuver set since the vehicle aerodynamics are 

rapidly changing as the maneuver ends in the transonic regime.
17

50

100

150

10

15

20

25

0

5 4 3 2

Mach number PID

Phase margin, deg Gain margin, dB

050565

Flight data, fast Fourier 

 transform

Flight data, Chirp-Z

 transform

Linear tool predictions

050565

Figure 13. Elevator loop stability margin estimates for Flight 2.

0.5

1.0

0

–200

0

200

–40

–20

0

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050566

Flight data, 

 fast Fourier transform

Flight data,

 Chirp-Z transform

Linear tool predictions

+

050566

Figure 14. Elevator loop frequency response comparisons for the Flight 2 Mach 2 parameter 

identification maneuver.
18

50

100

150

0

8 7 6 5 4 3 2

Mach number PID

Phase margin, deg Gain margin, dB

050567

10

15

20

25 Flight data, fast Fourier 

 transform

Flight data, Chirp-Z

 transform

Linear tool predictions

050567

Figure 15. Elevator loop stability margin estimates for Flight 3.

Rudder Loop Results

The rudder loop stability margin values for the HXRV during Flight 2 were not calculated 

because of the problem with obtaining an acceptable rudder loop frequency response, as discussed 

above. The coherence value was very poor, indicating little correlation between the input and 

output signals. In addition, the Chirp-Z results were not considered believable because they did 

not match predictions. As previously discussed, this result is believed to be because of excessive 

correlation between the aileron and rudder control loops. Figure 16 shows the flight-derived 

frequency response compared with linear model predictions for Flight 2 during the Mach 5 PID 

maneuver set.

As can be seen in figure 17, the rudder loop frequency responses for Flight 3 matched linear 

analysis predictions better than did those of Flight 2. This comparison at Mach 8 is slightly worse 

than for the Mach 6 and Mach 4 sets. This result is believed to be because of air flowing through 

the engine during the Mach 8 PID. The rudder loop frequency responses obtained from Flight 3 

showed considerable improvement over those of Flight 2, however they did not compare as 

favorably with linear predictions as did the elevator and aileron loop responses. In particular, the 

FFT-derived coherence values for the rudder loop are acceptable, but not as high as for the elevator 

or aileron loops. Separating the aileron and rudder loop inputs for Flight 3 helped to improve the 

flight-derived rudder frequency response.
19

0.5

1.0

0

–300

–200

–100

0

–10

–20

–30

0

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050568

Flight data, 

 fast Fourier transform

Flight data,

 Chirp-Z transform

Linear tool predictions

+

050568

Figure 16. Rudder loop frequency response comparisons for the Flight 2 Mach 5 parameter 

identification maneuver.

0.5

1.0

0

–400

–200

0

–60

–40

–20

0

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050569

Flight data, 

 fast Fourier transform

Flight data,

 Chirp-Z transform

Linear tool predictions

+

050569

Figure 17. Rudder loop frequency response comparisons for the Flight 3 Mach 8 parameter 

identification maneuver.
20

The rudder loop stability margin comparisons for Flight 3 are presented in figure 18. The gain 

margin predictions are generally within 2.5 dB of the flight-derived values. The largest gain margin 

difference of nearly 6.0 dB was seen during the Mach 8 PID, which is when air was believed to be 

flowing through the engine. An infinite rudder loop phase margin was measured in flight because 

no in-flight gain crossover frequency was identified since the crossover frequency was generally at 

the lower end of the frequency range used to generate the tailored excitation input signals.

50

100

150

0

8 7 6 5 4 3 2

Mach number PID

Phase margin, deg Gain margin, dB

050570

10

15

20

25 Flight data, fast Fourier 

 transform

Flight data, Chirp-Z

 transform

Linear tool predictions

050570

Figure 18. Rudder loop stability margin estimates for Flight 3.

Aileron Loop Results

Figure 19 shows a sample aileron loop frequency response for Flight 2. This response for the 

Mach 2 PID maneuver showed the lowest coherence and worst comparison with linear analysis 

predictions. As discussed above, this result is believed to be at least partially attributable to 

correlation between the rudder and aileron loops. For the Mach 2 comparison only, a shift in the 

gain response is seen between the in-flight frequency responses and those predicted by the linear 

analysis tools. This shift in the gain response is possibly indicative of less aileron loop control 

power at this flight condition than predicted.

The aileron loop frequency responses for Flight 3 are qualitatively similar to those of Flight 2, 

with slightly better coherence values. The slightly better aileron loop coherence values seen in 

the Flight 3 data are possibly because of the aileron and rudder loops not being simultaneously 

excited. Figure 20 shows a representative Bode plot detailing the Mach 3 PID frequency response 

comparison with the linear model predictions for Flight 3.
21

0.5

1.0

0

–400

–300

–200

–100

–20

0

20

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050571

Flight data, 

 fast Fourier transform

Flight data,

 Chirp-Z transform

Linear tool predictions

+

050571

Figure 19. Aileron loop frequency response comparisons for the Flight 2 Mach 2 parameter 

identification maneuver.

0.5

1.0

0

–300

–100

–200

0

–20

0

20

10–1 100

Frequency, Hz

Coherence Phase angle, deg Magnitude, dB

050572

Flight data, 

 fast Fourier transform

Flight data,

 Chirp-Z transform

Linear tool predictions

+

050572

Figure 20. Aileron loop frequency response comparisons for the Flight 3 Mach 3 parameter 

identification maneuver.
22

The aileron loop stability margin values for the HXRV during Flight 2 can be seen in 

figure 21. The gain and phase margin measurements for the Mach 5 through the Mach 3 PID are 

consistently higher than the linear predictions with differences generally up to 1.4 dB and 10 deg. 

The magnitude of these differences indicates a good correlation between the linear predictions and 

the in-flight measurements. The gain and phase margin measurements are as much as 4.25 dB and 

12 deg higher during the Mach 2 PID than the predictions. This greater difference seen during the 

Mach 2 PID maneuver is caused by the shift in the gain response already discussed.

50

100

150

5

10

15

20

0

5 4 3 2

Mach number PID

Phase margin, deg Gain margin, dB

050573

Flight data, fast Fourier 

 transform

Flight data, Chirp-Z

 transform

Linear tool predictions

050573

Figure 21. Aileron loop stability margin estimates for Flight 2.

Figure 22 shows the stability margin comparisons for Flight 3. The Mach 7 PID showed 

gain margin differences of approximately 2.5 dB and phase margin differences of approximately 

7 deg. The comparisons improved for the Mach 5 PID, while the Mach 3 PID shows a significant 

difference in the phase margin predictions. The coherence during this last aileron PID maneuver 

shows a slight decrease in the gain crossover region and the linear analysis tools predict a slightly 

different phase response in this area. The lower coherence estimate and slightly different phase 

response explains the increased differences when compared with linear analysis values.
23

50

100

150

0

7 6 5 4 3

Mach number PID

Phase margin, deg Gain margin, dB

050574

10

15

20

25 Flight data, fast Fourier 

 transform

Flight data, Chirp-Z

 transform

Linear tool predictions

050574

Figure 22. Aileron loop stability margin estimates for Flight 3.

Conclusion

Tailored input excitation signals were used in the X-43A flight test program to obtain 

control loop frequency responses during the post-engine test descent. The short time period of 

the excitation signals allowed for frequency responses and stability margins to be extracted at 

multiple flight conditions during the descent of the X-43A to the ocean. A problem with obtaining 

data for the rudder loops did not allow for valid frequency response data for each control loop at 

the same flight condition and thus precluded a multiple-input/multiple-output system analysis. 

The original premise of simultaneously applying excitation signals to multiple control loops still 

appears to be valid, because data were successfully obtained from the elevator and aileron control 

loops. The cause of the rudder and aileron loop interference is currently under investigation. The 

valid in-flight frequency response and stability margin values obtained for Flights 2 and 3 matched 

linear analysis predictions well, indicating a good understanding of the Hyper-X Research Vehicle 

aerodynamics and subsystems.
24

REFERENCES

Bosworth, John T., and Johnny C. West, “Real-Time Open-Loop Frequency Response Analysis 

of Flight Test Data,” AIAA-1986-9738, 1986.

Schroeder, M. R., “Synthesis of Low-Peak-Factor Signals and Binary Sequences With Low 

Autocorrelation,” IEEE Transactions On Information Theory, January 1970.

Young, Peter, and Ronald J. Patton, “Comparison of Test Signals for Aircraft Frequency

Domain Identification,” Journal of Guidance, Control, and Dynamics, Vol. 13, No. 3, 1990, 

pp. 430–438.

Bosworth, John T., and John J. Burken, Tailored Excitation for Multivariable StabilityMargin Measurement Applied to the X-31A Nonlinear Simulation, NASA Technical 

Memorandum 113085, 1997.

Morelli, Eugene A., “Multiple Input Design for Real-Time Parameter Estimation in the 

Frequency Domain,” 13th IFAC Conference on System Identification, Paper REG-360, 2003.

Reubush, David E., Luat T. Nguyen, and Vincent L. Rausch, “Review of X-43A Return to 

Flight Activities and Current Status,” AIAA-2003-7085, 2003.

X-43A Mishap Investigation Board, Report of Findings: X-43A Mishap, Vol. 1, 2003.

Marshall, Laurie A., Griffin P. Corpening, and Robert Sherrill, “A Chief Engineer’s View of the 

NASA X-43A Scramjet Flight Test,” AIAA-2005-3332, 2005.

Marshall, Laurie A., Catherine Bahm, Griffin P. Corpening, and Robert Sherrill, “Overview 

With Results and Lessons Learned of the X-43A Mach 10 Flight,” AIAA-2005-3336, 2005.

Morelli, Eugene A., Stephen D. Derry, and Mark S. Smith, “Aerodynamic Parameter Estimation 

for the X-43A (Hyper-X) from Flight Data,” AIAA-2005-5921, 2005.

Davidson, J., F. Lallman, J. D. McMinn, J. Martin, J. Pahle, M. Stephenson, J. Selmon, and 

D. Bose, “Flight Control Laws for NASA’s Hyper-X Research Vehicle,” AIAA-1999-4124, 

1999.

Norlin, Ken A., Flight Simulation Software at NASA Dryden Flight Research Center, NASA 

Technical Memorandum 104315, 1995.

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11.

12.
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

1. REPORT DATE (DD-MM-YYYY)

08-01-2007

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Tailored Excitation for Frequency Response Measurement Applied to the 

X-43A Flight Vehicle

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Baumann, Ethan

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

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 REPORT NUMBER

H-2679

10. SPONSORING/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

Baumann, Dryden Flight Research Center. Presented at the 44th Aerospace Sciences Meeting and Exhibit, January 9–12, 2006, 

Reno, Nevada. 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 08 Availability: NASA CASI (301) 621-0390 Distribution: Standard

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

An important aspect of any flight research project is assessing aircraft stability and flight control performance. In some programs this 

assessment is accomplished through the estimation of the in-flight vehicle frequency response. This estimation has traditionally been 

a lengthy task requiring separate swept sine inputs for each control axis at a constant flight condition. Hypersonic vehicles spend 

little time at any specific flight condition while they are decelerating. Accordingly, it is difficult to use traditional methods to 

calculate the vehicle frequency response and stability margins for this class of vehicle. A technique has been previously developed to 

significantly reduce the duration of the excitation input by tailoring the input to excite only the frequency range of interest. 

Reductions in test time were achieved by simultaneously applying tailored excitation signals to multiple control loops, allowing a 

quick estimate of the frequency response of a particular aircraft. This report discusses the flight results obtained from applying a 

tailored excitation input to the X-43A longitudinal and lateral-directional control loops during the second and third flights. The 

frequency responses and stability margins obtained from flight data are compared with preflight predictions.

15. SUBJECT TERMS

Frequency sweep, Hyper-X, Stability margin estimate, Tailored excitation, X-43A

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(301) 621-0390

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