NASA/TM–2012–216045

Sensor-Only System Identification

for Structural Health Monitoring

of Advanced Aircraft

Sunil L. Kukreja

NASA Dryden Flight Research Center

Dennis S. Bernstein

Department of Aerospace Engineering, University of Michigan

August 2012
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NASA/TM–2012–216045

Sensor-Only System Identification

for Structural Health Monitoring

of Advanced Aircraft

Sunil L. Kukreja

NASA Dryden Flight Research Center

Dennis S. Bernstein

Department of Aerospace Engineering, University of Michigan

National Aeronautics and

Space Administration

NASA Dryden Flight Research Center, Edwards, California 93524-0001

August 2012
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Abstract

Environmental conditions, cyclic loading, and aging contribute to structural wear and degradation, and thus potentially catastrophic events.

The challenge of health monitoring technology is to determine incipient

changes accurately and efficiently. This project addresses this challenge

by developing health monitoring techniques that depend only on sensor

measurements. Since actively controlled excitation is not needed, sensorto-sensor identification (S2SID) provides an in-flight diagnostic tool that

exploits ambient excitation to provide advance warning of significant

changes. S2SID can subsequently be followed up by ground testing to

localize and quantify structural changes.

The conceptual foundation of S2SID is the notion of a pseudo-transfer

function, where one sensor is viewed as the pseudo-input and another

is viewed as the pseudo-output. This approach is less restrictive than

transmissibility identification and operational modal analysis since no

assumption is made about the locations of the sensors relative to the

excitation.

1 Purpose

The safety and reliability of aircraft depend on the health of structural

components. Environmental conditions, cyclic loading due to takeoff and

landing, and aging all contribute to structural wear and degradation,

leading to potentially catastrophic events.

Structural health monitoring (SHM) techniques address this need

[1]. SHM typically involves ground-based testing, scheduled according

to flight history, that is, flight hours and takeoff/landing cycles [2]. This

approach allocates maintenance based on statistical models of wear and

aging that predict incipient failure modes. However, anomalous failure

modes may be difficult to detect between scheduled maintenance.

To overcome these shortcomings, the goal of this project is to develop

a new approach to analyzing flight data for SHM called sensor-to-sensor

system identification (S2SID). This technique can decrease the need for

costly maintenance that takes the aircraft out of service, while providing

the means for detecting potential failure events that may occur between

traditionally scheduled testing.

2 Background

In standard system identification, measurements of input and output signals are used to fit a model of a chosen structure and dynamic order. In

many applications, however, the input signal is unknown, and thus sensor measurements are the only available data. If a statistical description

of the unknown input is available, then sensor-only (also called “blind”)

1
identification techniques can be used to detect changes in the dynamics

of the system [3–6]. For structural dynamics applications, sensor-only

identification is known as operational modal analysis (OMA) [7–11].

In contrast to blind identification, S2SID requires no knowledge of the

statistical properties of the input. For applications where the excitation

is unknown and only sensor measurements are available, S2SID designates one measurement as the pseudo-input and another measurement

as the pseudo-output. The identified pseudo transfer function (PTF)

typically captures information about only the zeros (anti-resonances)

of the structure. Although pole locations are generally not estimated,

S2SID has the advantage of not requiring knowledge of the system excitation. In fact, the unknown ambient system excitation plays the necessary role of providing excitation that can be used to identify PTFs within

S2SID [12, 13]. Extensions to multi-input, multi-output PTF identification, which is necessitated by non-scalar excitation, is considered in [14].

S2SID is related to transmissibility identification [9, 15, 16]. In the

simplest situation, measurements of transmissibilities assume that one

of the sensors is colocated with the controlled displacement excitation.

The resulting transmissibility involves both resonance (pole) and antiresonance (zero) features of the structure. However, transmissibility

identification can also be performed with arbitrary arrangements of sensors, without regard to the location of the external excitation. In this

case, only anti-resonance (zero) information is captured, and the goal

is to construct a transmissibility that relates one set of measurements

to another set so that the resulting transmissibility is independent of

the forcing function. This objective is thus a specialization of S2SID to

the case of identical sensors (for example, all accelerometers). For the

case of more than two sensors, transmissibility identification is described

in [15]; however, the construction of the PTF in [15] is incorrect since

the structure of the PTF does not correctly cancel the unknown forcing to obtain a transmissibility that is independent of the details of the

forcing function. A correct construction is given in [14]. Furthermore,

S2SID is more general than transmissibility identification since S2SID

is applicable to arbitrary collections of sensors, such as accelerometers,

displacements sensors, and strain gauges. Finally, like transmissibility

identification, S2SID does not perform modal analysis as in the case of

operational modal analysis [7–11]. Instead, S2SID can be applied without knowledge of the structural geometry as long as the sensors are in

close enough proximity to facilitate PTF identification.

Impact and Benefits to NASA or Aeronautics

This work is aligned with the Aviation Safety Program (AvSP), whose

goals are to predict and prevent safety issues, to monitor for safety issues

in-flight and mitigate against them should they occur, to analyze and

design safety issues out of complex system behaviors, and to constantly

analyze designs and operational data for potential hazards. Moreover,

AvSP strives to advance state-of-the-art design tools to detect, avoid,

2
and protect against loss-of-control due to potential adverse events including atmospheric and vehicle system factors, and develops advanced

capabilities for detection and mitigation of aging-related hazards before they become critical. These goals are addressed in our research

by providing in-flight health monitoring without relying on controlled

excitation.

This research will provide significant cost and time savings for aircraft

health monitoring while improving safety and reliability. In addition, this

work will contribute to the following NASA and national goals.

National Security & Homeland Defense R&D Goals & Objectives.

Demonstrate innovative airframe structural concepts for efficient highaltitude flight.

Aviation Safety R&D Goals & Objectives.

Develop technologies to reduce accidents and incidents through enhanced

vehicle design, structure, and subsystems.

3 Approach

The approach taken for this investigation was a government-academia

collaboration that consisted of a partnership between NASA Dryden

Flight Research Center (DFRC) and the Department of Aerospace Engineering at the University of Michigan. Both partners provided the

theoretical analysis of the problem, algorithmic development, and validation. In addition, NASA facilitated the strategic research task and

provided flight-test data from the SOFIA (Stratospheric Observatory for

Infrared Astronomy) aircraft for algorithmic validation.

4 Summary of Research

This project focused on theoretical, algorithmic, and implementation issues that are critical to making sensor-to-sensor system identification

(S2SID) SHM a viable technology for aircraft SHM. The theoretical and

algorithmic issues in S2SID-SHM are described in Aim 1. Aim 2 focuses on validating S2SID-SHM based on simulated data. Aim 3 applies

S2SID-SHM to flight data for further validation. These descriptions are

taken from the Phase I proposal.

Aim 1: Theoretical and Algorithmic Extensions of S2SID-SHM.

Issue 1: Persistency and identifiability. Since S2SID depends on

freely available and unknown ambient excitation, it is necessary to ascertain that this excitation is sufficiently persistent (despite being otherwise

unknown) to facilitate estimation of key parameters. In addition, key parameters must be identifiable, that is, estimated without ambiguity. We

will address both issues through analysis of the algorithm based on the

aircraft’s flight envelope and the expected ambient excitation spectrum.

3
Status: This issue was addressed by analysis of SOFIA data. We

performed coherence, correlation, and detrend analysis of the data, and

we discovered a direct relationship between correlation and proximity of

sensors. Although this finding was not surprising, it demonstrated the

integrity of the data, building confidence in its usefulness and limitations

for PTF identification. With regard to persistency and identifiability, the

data was found to be rich in spectral content, and thus sufficient for use

in PTF identification.

Issue 2: Sensor noise. Identification accuracy depends on the ability

to obtain consistent estimates of key parameters, that is, asymptotically

vanishing estimation bias as increasingly larger data sets are used. A

technique that ensures consistency was demonstrated in [17]; however,

[17] assumes knowledge of the autocorrelation (coloring) of the sensor

noise. Our goal is to alleviate the need for this assumption.

Status: Significant effort was devoted to this issue, which is challenging due to the fact that sensor noise is unknown and affects all

sensors. Estimation of the impulse response is a well-studied problem;

we compared various techniques with regard to their sensitivity to sensor noise in [18]. Analysis of the SOFIA data indicated measurement

quantization as a result of sensor resolution constraints, but otherwise

the noise statistics are unknown. We performed extensive least-squares

fits of sensor data, using both Infinite-impulse-response (IIR) and finiteimpulse-response (FIR) model structures. The main finding is that the

best prediction errors were obtained using an FIR model structure. This

finding is consistent with theoretical results that show that transfer function identification with a white input signal and with noise corrupting

the output sensor yields consistent estimates of the impulse response.

However, in the case of PTFs, the input signal is not white, and sensor

noise may corrupt both the input and output signals.

Issue 3: Nonlinearity. Thus far, S2SID is based on linear models.

Recent work [19] shows that consistent estimates are achievable despite

the presence of certain types of nonlinearities. For S2SID, we will investigate the accuracy of the estimated PTFs by developing modifications

to the algorithm that ensure that the PTF estimates are independent of

the unknown excitation in the presence of nonlinearities. The goal is to

demonstrate the robustness of S2SID to nonlinearities in the structural

response. Extensions to nonlinear PTFs can also facilitate this goal.

Status: We applied techniques for detecting whether the PTF includes significant nonlinearities. In particular, we applied two different

techniques for identifying Hammerstein systems, which are systems involving the cascade of a static input nonlinearity and a linear dynamic

subsystem [19]. The static input nonlinearity provides an indication of

distortion present in the PTF. These techniques showed that no significant nonlinearity was present in the PTFs between the sensors that were

selected.

Aim 2: Validation with Simulated Data. The performance of

4
S2SID-SHM will be assessed and demonstrated by using synthetic data

sets generated from simulated mass-spring-damper truth models of various geometries with simulated sensor noise and nonlinearities. For each

geometry, we will perform statistical analysis of the accuracy of the

method. The estimated PTFs will be compared to the true PTFs to

assess the effectiveness of S2SID-SHM. This phase will provide a means

for quantifying damage by providing estimates of the structural parameters.

Status: We performed extensive tests of synthetic data sets in conjunction with the analysis of SOFIA data. The approach taken was to

apply identification techniques to SOFIA data, and then use synthetic

data sets to probe the underlying model properties. Results of this type

are described later in the next section.

Aim 3: Application to Experimental Flight-Test Data. We will

apply S2SID-SHM to experimental flight-test data from NASA DFRC’s

SOFIA program. The SOFIA data, which were collected to observe

structural loads during missions, is ideally suited for assessing how the

structural dynamics and parameters change during the aircraft’s flight

history. In contrast to simulated data, the truth model is unknown, and

thus successful demonstration of this approach to the analysis of flight

data will serve to validate the utility of S2SID-SHM to NASA for mission

readiness and safety assurance.

Status: We applied S2SID to flight data obtained from the SOFIA

aircraft located at NASA DFRC. SOFIA is a highly modified Boeing

747SP, housing an infrared telescope in the aft fuselage. The telescope

is isolated from the onboard scientific staff and equipment by means of

a pressure dome, thus allowing a door to open and permit astronomical

observations. Due to long flight durations of 10 hours or more and the

stress of opening the telescope door in-flight, structural health monitoring of both the aircraft and telescope are of concern. As such, a suite

of accelerometers have been placed at various locations throughout the

aircraft to provide data for post-flight stress analysis. We obtained three

data sets from SOFIA flights.

In particular, experimental flight data were gathered during a routine flight at 12,192–15,240 m (40,000–50,000 ft) to conduct astronomical

observations onboard SOFIA. Data were collected by the sensors under

ambient conditions during flight. The accelerometer data used in this

study were collected from sensors located at the right horizontal stabilizer

tip, rear spar (vertical direction), and vertical stabilizer front and rear

spars (lateral direction). The sensor data were filtered by a sixth-order

antialiasing Butterworth filter with a cut-off at 1 kHz and recorded at 5

kHz. Data were collected while the aircraft operated in the Mach number range M = 0.4–0.7 and dynamic pressures Q = 260–390 psf (pounds

per square foot). Data were preprocessed to remove the linear trend,

mean, and outliers. The preprocessing step ensured that all unwanted

low-frequency disturbances, offsets, trends, and drifts were removed to

5
enhance the accuracy of the identified models. Coherence between signals was studied with and without preprocessing. We applied S2SID

with and without preprocessing in order to ascertain the effects of these

procedures. We also estimated the level of error in the data due to the

sensor resolution. In particular, the output resolution is the smallest

distance between signal measurements. Dividing the output range by

the output resolution gives the dynamic range, which is the maximum

number of distinct sensor output values over the output range.

5 Accomplishments

Results of PTF Identification PTF identification is based on least squares

optimization in conjunction with specialized model structures. In the

following discussion we assume that the system being identified is linear, and we focus on linear model structures for system identification.

Although there are various ways to represent systems with inputs and

outputs, system identification typically uses discrete-time, time-series

models, where the current output is a linear combination of past inputs

and outputs. Special cases of these models include moving average (MA)

models, autoregressive (AR) models, and autoregressive/moving average

with exogenous input (ARMAX) models. Time series models can be represented as transfer functions, where the numerator coefficients weight

the past inputs, and the denominator coefficients weight the past outputs. In contrast to state space models, time-series models do not involve

an explicit internal state.

Two types of time-series models are used for system identification.

Infinite-impulse-response (IIR) models possess an impulse response that

requires an infinite number of steps to decay to zero. The impulse response of a time-series model consists of numbers called impulse response

parameters, which are denoted by H1, H2, . . . . Impulse response parameters are also called Markov parameters. A specialized form of IIR models

is given by the μ-Markov model [18,20–22], whose numerator coefficients

include a collection of impulse response parameters. Finally, a finiteimpulse-response (FIR) model is a discrete-time, time-series model with

the special property that its impulse response reaches and remains at

zero after a finite number of steps. All of the coefficients of an FIR

model are Markov parameters. μ-Markov models provide a bridge between IIR models and FIR models in the sense that, as μ increases, the

μ-Markov model increasingly mimics the form of an FIR model.

The accuracy of least-squares identification of time-series models depends on various aspects of the problem. For example, the model order

may be unknown, and errors may be incurred by overestimating or underestimating the order. Next, the inputs to the system must be sufficiently

persistent to allow estimation of the model coefficients. Furthermore,

beyond persistency, for the case in which the input to the system is

6
0 50 100 150 200 250 300

15

15.5

16

16.5

17

17.5

18

18.5

19

Output Delay

Prediction Error (PE)

Figure 1. This plot shows the prediction error (cross validation) for FIR

model fits as a function of output delay. Delaying the output relative

to the input improves the accuracy of the model fit as measured by a

prediction error criterion.

stochastic, the statistical properties of the input can affect the accuracy

of the parameter estimates. And, finally, the nature of the noise corrupting the input and output of the system impacts the accuracy of the

parameter estimates. The most challenging situation arises when both

the input and output signals are corrupted by noise that is mutually correlated. (We note that “sensor noise” on the input refers to uncertainty

about the input signal due to actuator noise.)

In [18] we compared least-squares techniques with various time-series

models under different types of inputs and sensor noise. In the case of

a persistent but otherwise arbitrary input signal and in the presence of

noise on both the input and the output, it is shown in [17] that consistent parameters can be obtained if the statistical nature of the input and

output noise is known. However, this knowledge is usually not available

in practice. In the more realistic case in which the statistical properties of the input and output noise are unknown, it is shown in [18] that

consistency of the impulse response parameters can be achieved using

μ-Markov models under more restrictive assumptions, namely, if the input signal is white noise and only output noise is present. This result

motivates interest in using μ-Markov models for system identification.

If, in addition, input noise is present, then it is shown in [19] that semiconsistency can be achieved, where semi-consistency refers to consistency

up to an unknown multiplicative constant. In view of these issues, it is

clear that the challenging aspects of S2SID are 1) the input signal is not

white, 2) the input and output are corrupted by correlated noise, and 3)

the statistical properties of the sensor noise is unknown.

Using SOFIA data to guide the Phase I investigations, we found that

the most accurate model fits, as determined by prediction error (cross

validation) were obtained from least-squares optimization of FIR timeseries models. However, these investigations provided an unexpected

7
0 100 200 300 400 500 600

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Impulse Response Index

Impulse Response Value

Figure 2. This plot shows the PTF impulse response of the FIR model

fit corresponding to a chosen output delay of 298 steps. The surprising

feature of this impulse response is that it has a significant noncausal

component, which appears to the left of the delay of 298 steps.

feature illustrated in Figure 1. Specifically, Figure 1 shows that the prediction error decreases as the output delay is increased relative to the

input. The reason for this surprising effect becomes clear only upon plotting the impulse response of the FIR model. As shown in Figure 2, the

impulse response of the fit model has a significant noncausal component,

plotted to the left of the chosen delay step.

To confirm that the noncausal component of the SOFIA impulse

response is contributing to the prediction error, we remove the noncausal

component and then re-include it one impulse parameter at a time; this

is done by including the impulse response parameters one at a time from

the left of the chosen delay step in Figure 2. Figure 3 shows that the

prediction error decreases as noncausal impulse response parameters are

included in the identified FIR model.

Source of the Noncausal Impulse Response As shown above, the noncausal component of the impulse response estimated by identification

with an FIR model structure provides a significant improvement to the

identification accuracy as measured by the prediction error. This is a surprising observation since it suggests that the PTF between the pseudoinput and pseudo-output is not physically meaningful. To investigate

this issue, we constructed synthetic data sets by simulating the lumped

mass-spring-damper systems shown in Figure 4. Specifically, we excite

this structure with an external force, and we record the velocities of two

masses for use in PTF identification. We consider parameters for two

cases. In the first case, Figure 5 shows that delaying the pseudo-output

does not lead to a noncausal component of the impulse response. However, in the second case, Figure 6 shows that delaying the pseudo-output

does in fact produce a noncausal component of the impulse response.

8
0 50 100 150 200 250 300

18.5

19

19.5

20

20.5

21

21.5

22

Number of Included Noncausal Impulse Response Parameters

Prediction Error (PE)

Figure 3. The prediction error is plotted here as an increasing number

of noncausal impulse response parameters are included in the model.

This plot confirms that the noncausal component of the estimated model

contributes to the prediction accuracy.

m2

k1

m1

k2 k3 k4

m3

x1 x2 x3

c1 c2 c3 c4

f

Figure 4. Mass-spring-damper structure. This example is used to produce synthetic data for developing and testing PTF identification methods. By adjusting the parameters, stable and unstable PTFs can be

constructed.

The key distinction between the PTFs in these two cases is the fact that

in the former case the PTF is stable, whereas, in the latter case, the

PTF is unstable.

Although the mass-spring-damper system is stable, the transfer function from force to a velocity measurement may be nonminimum phase,

that is, it may have zeros outside of the unit circle. In this case, the

PTF may be unstable, although the pseudo-input and pseudo-output

data used to identify the PTF are bounded. Applying system identification to fit the PTF yields an FIR model with a significant noncausal

component. This noncausal component is an artifact of the use of an FIR

model structure. Although an IIR model structure can be used to avoid

the noncausal component, the presence of noise corrupting the sensors

yields inaccurate IIR model fits, whereas FIR model fits are significantly

more accurate.

9
0 10 20 30 40 50 60 70 80

ï0.2

ï0.15

ï0.1

ï0.05

0

0.05

0.1

0.15

0.2

Impulse Response Index

Impulse Response Value

Figure 5. PTF identification is applied to the structure in Figure 4 with

the parameters m1 = 9 kg, m2 = 4 kg, m3 = 2 kg, k1 = 99 N/m, k2 = 28

N/m, k3 = 310 N/m, k4 = 101 N/m, c1 = 0.9 N-sec/m, c2 = 5.1 Nsec/m, c3 = 0.8 N-sec/m, c4 = 5.2 N-sec/m and discretization time step

of 0.2 sec. A random white noise force excitation is applied to m2. The

pseudo-input is the velocity of m3, and the pseudo-output is the velocity

of m1. The estimated PTF impulse response is found to be causal.

10
0 10 20 30 40 50 60 70 80

ï0.2

ï0.15

ï0.1

ï0.05

0

0.05

0.1

0.15

0.2

Impulse Response Index

Impulse Response Value

Figure 6. PTF identification is applied to the structure in Figure 4 with

the parameters as in Figure 5 except that now m3 = 14 kg. In this case,

the estimated impulse response is found to be noncausal. The noncausal

component of the impulse response is due to the fact that one of the

transfer functions from excitation to measurement is nonminimum phase,

and therefore the PTF from the pseudo-input to the pseudo-output is

unstable. The instability of the PTF induces a noncausal component in

the PTF impulse response.

11
6 Next Steps

The proposed Phase II research is aimed at refining and demonstrating

this approach, thereby moving S2SID to an enhanced readiness status

for transition to NASA applications, with potential applications to nonaerospace structures for infrastructure monitoring. The objectives of the

proposed Phase II project are as follows:

Task 1. Refine the prediction error of the identified PTFs. The

approach developed in Phase I indicates the ability to construct models

that provide high-fidelity predictions of the response of one sensor based

on another sensor. Our goal is to continue to improve the accuracy of

these predictions, which is essential to the next two objectives.

Task 2. Develop detection metrics to assess PTF structural changes.

Once system identification is used to construct an empirical PTF, the

next crucial step is to develop metrics for assessing changes to the aircraft

that warrant inspection. Metrics can be based on either changes to the

PTF or its prediction error. This is the primary objective of Phase II.

Task 3. Apply S2SID to flight data to determine threshold criteria.

We plan to continue working with data from SOFIA [23] along with other

aircraft to assess sensitivity to flight conditions and possible long-term

changes in the structural dynamics.

7 Applicable NASA Programs/Projects

This research complements the Vehicle Systems Safety Technologies (VSST)

Project, whose goal is to detect, mitigate, and recover from hazardous

flight conditions, while maintaining airworthiness and health. These

goals will be addressed in our research by providing in-flight health monitoring without relying on controlled excitation. In addition, we envision

our S2SID approach to SHM to be tested and refined inflight on-board

the NASA DFRC flexible MUTT research aircraft.

8 Publications and Patent Applications

Conference Papers Submitted

C1 K. Aljanaideh, A. Ali, M. Holzel, S.L. Kukreja and D.S. Bernstein.

“Sensor-to-Sensor Identification of Hammerstein Systems”, Submitted to In Proc. 51st IEEE Conference on Decision and Control,

pages TBD, Maui, Hawaii, USA, December 2012.

C2 K. Aljanaideh, B.C. Coffer, D.S. Dionne, S.L. Kukreja and D.S.

Bernstein. “Sensor-to-Sensor Identification for the SOFIA Testbed”,

Accepted to In Proc. AIAA Guidance, Navigation, and Control

Conference, pages TBD, Minneapolis, Minnesota, August 2012.

12
C3 A. V. Morozov, A. A. Ali, A. M. D’Amato, A.J. Ridley, S. L. Kukreja

and D. S. Bernstein. “Retrospective-Cost-Based Model Refinement

for System Emulation and Subsystem Identification”, In Proc. 50th

IEEE Conference on Decision and Control and European Control

Conference, volume 50, pages 2142 - 2147, Orlando, Florida, December 2011.

C4 A.J. Brzezinski, S.L. Kukreja, J. Ni and D.S. Bernstein. “Identification of Sensor-Only MIMO Pseudo Transfer Functions,” In

Proc. 50th IEEE Conference on Decision and Control and European Control Conference, volume 50, pages 2154 - 2159, Orlando,

Florida, December 2011.

Journal Papers in Progress

J2 A.J. Brzezinski, B.C. Coffer, S.L. Kukreja, J. Ni, and D.S. Bernstein.

Sensor-to-Sensor Identification of Pseudo Transfer Functions. In

preparation for Automatica.

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