NASA/TM—2011–215966 (Corrected Copy)

Data Driven Model Development for the 

SuperSonic SemiSpan Transport (S4

T) 

Sunil L. Kukreja

Dryden Flight Research Center, Edwards, California

 

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NASA/TM—2011–215966 (Corrected Copy)

Data Driven Model Development for the 

SuperSonic SemiSpan Transport (S4

T)

Sunil L. Kukreja

Dryden Flight Research Center, Edwards, California

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National Aeronautics and

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Dryden Flight Research Center

Edwards, CA 93523-0273

March 2011
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1

Abstract

In this report, we will investigate two common approaches to model development for robust control 

synthesis in the aerospace community; namely, reduced order aeroservoelastic modelling based on 

structural finite-element and computational fluid dynamics based aerodynamic models, and a data-driven 

system identification procedure. It is shown via analysis of experimental SuperSonic SemiSpan Transport

(S4T) wind-tunnel data that by using a system identification approach it is possible to estimate a model at 

a fixed Mach, which is parsimonious and robust across varying dynamic pressures.

Nomenclature

a parameter of the output

 A(q) polynomial in a

 AIC Akaike’s information criterion

 ASE aeroservoelastic

 ARMA AutoRegressive, Moving Average

 ARMAX AutoRegressive, Moving Average eXogenous

 ARX AutoRegressive eXogenous input

 b parameter of the input

 B(q) polynomial in b

 BJ Box Jenkins

 c parameter of the noise model numerator

 C(q) polynomial in c

 D(q) polynomial in d

e(n) an unobservable white-noise disturbance

 F(q) polynomial in f

 FE finite element

 FIR finite impulse response function

 FPE final prediction error

 G(q) B(q)/A(q)

 H(q) 1/A(q)

H∞ H-infinity

 MDL minimal description length

na previous values of the output

nb previous values of the input

N number of data points

p number of model parameters

 PEI prediction error identification

 POD proper orthogonal decomposition

Q dynamic pressure

 ROM reduced order model

S4T SuperSonic SemiSpan Transport

 TPWL trajectory piecewise linearization

u(n) an accessible input

V the prediction error or the residual sum of squares

y measured system output

yˆ predicted model output

y(n) measured output

 %QF percent quality of fit
2

1. Introduction

Undesirable aeroservoelastic (ASE) interactions are a major concern in modern aircraft design. ASE 

interactions between aircraft structure, aerodynamics, and flight control systems can lead to divergent 

oscillations resulting in catastrophic failure (ref. 1). As such, analytical model development is an 

important step in the design and certification of aircraft. Accurate models allow for robust control design,

which is critical for aircraft safety, gust-load alleviation, ride quality, et cetera.

Finite element (FE) based models are used in the design process of aircraft to aid in the description of 

complex elastic and structural components (refs. 2–3). FE based models that accurately characterize the 

aerodynamic and structural components are of very high order (for example, thousands of degrees of 

freedom) and computationally intensive. To reduce the ASE model order, a modal approach is used. This 

approach can reduce the state-space model order to several tens of states. The robustness of this model 

type highly depends on the accuracy of the FE structural and aerodynamic models, and on the number of 

states applied to the modelling. Models of high complexity inhibit their use for control synthesis because 

their real-time implementation is difficult or not possible (ref. 4). This difficulty has led to considerable 

activity in the areas of model and controller reduction techniques in the last decade. The literature is rich 

with many reduced order model (ROM) techniques, which would require a review paper to properly 

discuss them. As such, in this report, we limit ourselves to a handful of approaches.

A leading strategy to FE based model reduction is the proper orthogonal decomposition (POD), which 

is also known as the Karhunen-Loeve procedure (refs. 5). The so-called POD technique is well known in 

the statistical literature as principal-component analysis (ref. 6). The reduced basis method was first 

proposed in references 7 and 8 for structural analysis, and it has been used for structural problems in

references 9 through 11. This technique uses synthetic data from a high fidelity FE based model to 

capture the dominant characteristic information utilising an orthogonalisation process. This allows the 

POD approach to accurately describe a system using a few basis terms, which gives it an advantage 

compared to other numerical procedures (ref. 12). These reasons have allowed POD to become a popular

technique for the implementation of real-time control (ref. 13). Moreover, POD has been successfully

used in a variety of fields including signal analysis and pattern recognition (ref. 14), fluid dynamics and 

coherent structures (refs. 15–17), control theory (refs. 18–20), civil engineering (ref. 21) and inverse 

problems (ref. 22).

More recently the ROM techniques utilising POD were developed for aeroelastic systems analysis 

(ref. 23). This methodology was introduced to the aerospace community for the reduction of aeroelastic 

equations (ref. 24). As a follow-on to this technique, frequency-domain approaches were developed which 

efficiently compute POD basis functions for linearised aeroelastic systems (refs. 25–26). Due to its 

popularity and utility, POD methodology has been proposed and implemented for static and dynamic 

continuous-time nonlinear aeroelastic problems. Subsequent developments lead to extensions to 

encompass discrete Euler equations (refs 27–28). Successful application of this approach has been 

demonstrated for the analysis of limit-cycle oscillation of an airfoil with a nonlinear structural coupling in 

the transonic regime (ref. 29).

Nevertheless, it has been observed that standard POD procedures are less robust for nonlinear 

problems and typically require more basis functions as the function complexity increases (refs. 23 and 

30). To address this limitation, a linearisation strategy, the trajectory piecewise linearisation (TPWL), was 

proposed (refs. 31–32). The TPWL technique combines reduced-order modelling with linearisation of the 

governing equations as a solution to this problem (ref. 23).
3

Although POD offers a significant reduction of the full FE based model, it is often too large to lend 

itself to efficient control design. An alternative approach is to use data driven techniques to let the data 

dictate what the optimal model should be. Such a procedure is commonly known as system identification.

This area, as with ROM methods, also has an extensive base of literature for both linear and nonlinear

system identification techniques and is too large to give a proper review here. As such we refer the reader 

to an often-cited authoritative treatise in the area, which provides an excellent overview and references 

(ref. 33). Below, we provide a brief and incomplete introduction.

There are two broad classes of techniques that can be pursued to accomplish the task of system 

identification: (1) nonparametric and (2) parametric methods. The finite impulse response function (FIR) 

has been widely used for modelling linear time-invariant systems. This type of system description is 

known as nonparametric because it is a numeric representation of the system’s impulse response or kernel 

(refs. 33–35). Although nonparametric methods can be used to represent many classes of systems, they do 

so at the expense of introducing an excessive number of unknown coefficients, which must be estimated. 

Most expansions map the past inputs into the present output and so require a very large number of 

coefficients to characterize the process. Moreover, the parameters are not readily linked to the underlying 

system, except in special cases where significant a priori knowledge of the system has been assumed.

Due to this shortcoming, parametric identification methods have been developed for use in the design 

of better control systems. Parametric models have some advantages in applications. They: (1) are easier to 

understand and interpret, (2) can simplify forecasts (e.g., obtaining forecast intervals), and (3) model 

comparison in a parametric context (i.e., parameter estimates, model order, and model structure) has been 

well studied. Hence, the difficulty of model comparison encountered using nonparametric tools can be 

avoided (ref. 36).

A parametric model consists of a set of differential or difference equations describing the system 

dynamics. Such equations usually contain a “small” number of parameters, which can be varied to alter 

the behavior of the equation. In this report, we only consider the discrete-time case since in any practical 

experimental situation the data available to the experimenter is in discrete-time. As such, most systems 

for identification purposes are represented in discrete-time. In addition, we assume the ROM with which 

we compare our data-driven model is available and as such we do not discuss its development.

The organization of this report is as follows. In section 2 we formulate the identification problem 

addressed here. Section 3 describes the experimental SuperSonic SemiSpan Transport (S4T) test-bed

(ref. 37) and methods used for model development. Section 4 illustrates the results of our study on three 

flight conditions; namely, model robustness to fixed Mach but varying dynamic pressure. Section 5 

provides a discussion of our findings, and section 6 summarises the conclusions of our study.

2. Problem Statement

System identification is the process of developing or improving a mathematical representation of a 

physical system based on observed data. Often the observed data consists of an external user selected 

input, used to perturb the system and elicit an output response. In any experimental situation the system 

output is a sum of the true unknown system output and observation or measurement noise. Given this 

paradigm there are several model structures that can be explored to model a system’s dynamics and noise.

2.1 Model Structures

After preprocessing the recorded data, the first step in system identification is to select a model 

structure to describe the observations. There are a number of model forms to select from when developing 
4

a data-driven model (ref. 33). However, the problem often dictates the model form(s) that are reasonable 

to consider. This insight may come from a priori knowledge of the physical process or previous

morphological modelling efforts. In this report, we use knowledge gained from previous work to limit the 

model sets considered; namely, linear, time-invariant processes where both input-output are available to 

the user (38–39).

2.1.1 ARX Model Structure

The simplest input-output polynomial model is the AutoRegressive eXogenous (ARX) input model, 

represented in equation (1) as (refs. 33, 40, and 41):

y(n) = − a1y(n −1) −− ana y(n − na )

+ b1u(n −1) + b2u(n − 2) ++ bnb u(n − nb )

+ e(n) (1)

where y(n) is the measured output, u(n) is an accessible input, and e(n) is an unobservable white-noise 

disturbance. The current output depends on nb previous values of the input, na previous values of the 

output, and the current disturbance.

This structure can be represented more compactly as shown in equations (2), (3), and (4):

A(q) y(n) = B(q)u(n) + e(n) or (2)

y(n) = G(q)u(n) + H (q) e(n) where (3)

G(q) = B(q)

A(q)

, H (q) = 1

A(q) (4)

where A(q) = 1+ a1q−1 ++ ana q−na , B(q) = b1q−1 ++ bnb q−nb ,q−1 is the backward shift operator,

and a’s and b’s are the parameters of the output and input, respectively.

When there is evidence of significant noise in the system a more flexible noise model may be 

required to model the dynamics and noise using different polynomials.

2.1.2 ARMAX Model Structure

For linear systems, the relationship between input-output and noise can be written as a linear 

difference equation, and is shown in equation (5):

y(n) = − a1y(n −1) −− ana y(n − na )

+ b1u(n −1) + b2u(n − 2) ++ bnb u(n − nb )

+ e(n) + c1e(n −1) ++ cnc e(n − nc ) (5)
5

This is known as the AutoRegressive, Moving Average eXogenous (ARMAX) model. In this model 

structure the current output, y(n) , depends on an exogenous input, u(n), an innovation process, e(n), 

and past values of the output. This structure can be represented more compactly as shown in equation (6):

A(q) y(n) = B(q)u(n) + C(q) e(n) (6)

Substituting equation (6) into equation (3) yields equation (7):

G(q) = B(q)

A(q)

, H (q) = C(q)

A(q) (7)

where C(q) = 1+ c1q−1 ++ cnc q−nc , and the c’s are parameters of the noise model numerator. The 

extra polynomial, C(q), gives the ARMAX structure additional flexibility to model additive disturbance. 

When additional complexity is needed to model noise H (q) can be fully parameterised independent of 

the system dynamics.

2.1.3 Box–Jenkins Model Structure

A natural development of the ARMAX model structure is to parameterise the noise process as an

AutoRegressive, Moving Average (ARMA) model as shown in equation (8):

y(n) = B(q)

F(q)

u(n) +

C(q)

D(q)

e(n) (8)

where F(q) = 1+ f

1q−1 ++ fn f q

−n f , and D(q) = 1+ d1q−1 ++ dnd q−nd model the poles of the 

system and noise separately. Note that, n f = na . This model structure is known as the Box-Jenkins (BJ) 

model due to their seminal work proposing this model form (ref. 42).

2.2 Model Order Selection

Once a model structure is selected, the dynamic order of the system needs to be determined. Although 

there are many techniques that offer a solution to this problem, below we describe three commonly used 

techniques to estimate model order.

2.2.1 Final Prediction Error

The Final Prediction Error (FPE) measure estimates the error in model fit when it is used to predict 

new outputs (ref. 43). The FPE defines an optimal model as one that minimises as shown in equation (9):

FPE = V 1+

2 p

N − p

⎛

⎝

⎜

⎞

⎠

⎟ (9)

where N is the number of data points, V is the prediction error, or the residual sum of squares, and p is the 

number of model parameters.
6

2.2.2 Akaike’s Information Criterion

Akaike’s Information Criterion (AIC) is a weighted estimation error based on the unexplained 

variation of a given time series with a penalty term when exceeding the optimal number of parameters to 

represent the system (ref. 44). Utilising AIC, an optimal model is defined as one that minimises as shown 

in equation (10):

AIC = V 1+

2 p

N

⎛

⎝

⎜ ⎞

⎠

⎟ (10)

According to Akaike’s theory, the most accurate model has the smallest prediction error.

2.2.3 Minimal Description Length

Rissanen’s Minimal Description Length (MDL) approach is based on V plus a penalty for the number 

of terms used (ref. 45). With MDL, an optimal model is one that minimises as shown in equation (11):

MDL = V 1+ p ln N

N

⎛

⎝

⎜ ⎞

⎠

⎟ (11)

A model that minimises the MDL allows the shortest description of measured data.

2.3 Cross–Validation

Model validation is an important step in developing strategies for robust control. This step is typically 

preceded by system identification. Model validation is concerned with assessing whether a given nominal 

model can reproduce data from a plant, collected after some initial experiments to obtain estimation data 

(ref. 46). The model validation problem is really one of model invalidation since a given model can only 

be said to be not invalidated with the current evidence. Future evidence may invalidate the model.

We cross-validate the parameter estimates of the model dynamics using a 1-step-ahead predictor 

(ref. 33). Model goodness is assessed by computing the percent quality of fit (%QF) as shown in equation 

(12):

%QF = 1−

1

N yn − yˆ ( n )

2

n=1

N ∑

1

N y( n )

2

n=1

N ∑

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

×100 (12)

where y is the measured system output, and yˆ the predicted model output.

In the sequel, we show that by implementing these well-known identification strategies it is possible 

to develop a model, which is parsimonious and a robust predictor of measured data and, hence, represent 

the physical process more accurately.

3. Experimental S4T Wind–Tunnel Data

The modelling and identification techniques were applied to experimental wind-tunnel data from the 

S4T project conducted at NASA Langely Research Center (Ref. 37). Figure 1 shows a scale model of the 

S4T test bed in the wind tunnel.
7

Figure 1. S4T model.

The data analysed for this study used horizontal tail position input and structural accelerometer response 

output.

3.1 Data Collection

Wind-tunnel data was gathered during transonic clearance of the S4T. At each flight condition the 

aircraft model was perturbed with a log sine sweep input which had a frequency content of 0.5–25 Hz, 

mean value of 3.5 deg, and ± 0.3 deg amplitude. The inputs were applied to the horizontal tail. The 

accelerometer output was collected from a sensor located at the nacelle inboard aft position. Wind-tunnel

tests were conducted at subsonic, transonic, and supersonic conditions; and varying dynamic pressures. 

The input-output were antialiasing, filtered by an eighth-order Bessel filter with a cut-off at 200 Hz and 

recorded at 1,000 Hz. Data was collected for a range of Mach numbers from M = 0.6 to 1.2 and dynamic 

pressures Q = 20 to 65 psf.

3.2 Data Analysis

In this study, the objective was to develop a parsimonious model for control synthesis to improve 

gust-load alleviation and ride quality. As such, we focused our identification efforts on data collected at 

Mach 0.80, 0.95, and 1.10; and Q = 30, 55, 60, and 65 psf. At each Mach we used Q = 30 psf as the 

estimation data; and Q = 55, 60, and 65 psf as the cross-validation data. At M = 0.95 data was not 

collected at Q = 65 psf, and, therefore, the estimated model could only be cross-validated at Q = 55 and

60 psf. This approach of modeling and validation was taken to assess the model’s predicative capability 

and robustness.

Data was 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 and allowed for an 

accurate representation of the system dynamics.
8

3.3 Identification Procedures

The identification process was performed in four stages to assess the full range of models discussed in 

section 2. In all cases the model order was estimated using the FPE, AIC, and MDL approaches. The 

optimal model order was deemed as one that produced the lowest prediction error of the aforementioned 

techniques. Once the model order was fixed, prediction error identification (PEI) was used to estimate the 

unknown parameters, and its predictive capability was computed for a cross validation set as described 

previously (see section 2.3).

1. The ARX structure’s ability to model wind-tunnel data was assessed because it is the simplest 

input-output model invoking the principle of parsimony. For the ARX model the order was 

determined by preselecting a range of model orders to search over. Specifically, 

na = 2 − 5, nb = 1− 5 and nk = 1−10 was chosen as the search range, where nk denotes input 

delay. This range was selected to allow sufficient model complexity whilst maintaining an 

efficient system description.

2. Further complexity was added to the ARX model to assess whether the same model structure could 

account for significantly greater output variance with na = 2 −10, nb = 1−10 and nk = 1− 20 .

3. Using the best-fit ARX model order to fix na and nb , we searched for an ARMAX order, namely, 

nc = 1− 20 that provided the smallest prediction error as the optimal ARMAX order.

4. The same procedure was followed to assess whether the BJ model structure could account for more 

of the output variance than the ARMAX model with nd = 1− 20 .

4. Results

4.1 ARX Model

Two ARX model orders were evaluated. Specifically, a model with order na = 5, 

nb = 1, nk = 1 and na = 10, nb = 1, nk = 1 were used for model development. The model order was 

estimated as discussed in section 2.2. This analysis allowed for us to assess whether adding complexity to 

the same structure could account for significantly more of the output variance. The models were estimated 

using wind-tunnel data measured at Mach 1.10, Q = 30 psf which yielded structures of the form shown in 

equations (13) and (14).

ARX5:

Aˆ (q) y(n) = Bˆ (q)u(n) + e(n) where

 Aˆ (q) = 1+ aˆ

1q−1 + aˆ2q−2 + aˆ3q−3 + aˆ4q−4 + aˆ5q−5 and

 Bˆ (q) = ˆ

b1q−1

(13)
9

ARX10:

A (q) y(n) = B (q)u(n) + e(n) where

 A (q) = 1+ a

1q−1 + a2q−2 + a3q−3 + a4q−4 + a5q−5

 + a6q−6 + a6q−6+ a6q−6 + a6q−6 + a

10q−10 and

 B (q) = 

b1q−1

(14)

Figures. 2a–2c shows representative results for the ARX structure’s (eqs. (13) and (14)) ability to 

represent S4T wind-tunnel data by evaluating it with cross-validation data at Mach 1.10, and Q =55, 60,

65 psf.

Figure 2(a). Cross-validation fit: Mach 1.10, Q = 55 psf.

Figure 2. Measured and predicted output for ARX5 and ARX10 models at Mach 1.10; (a): Q = 55 psf, 

(b): Q = 60 psf, and (c): Q = 65 psf. Solid line (“—”) measured output. Dash-dash line (“– –”) predicted 

ARX10 model output. Dot-dot line (“...”) predicted ARX5 model output.
10

Figure 2(b). Cross-validation fit: Mach 1.10, Q = 60 psf.

Figure 2(c). Cross-validation fit: Mach 1.10, Q = 65 psf.

Figure 2. Concluded.

This figure compares %QF of the predicted output for the ARX5 and ARX10 models superimposed 

on top of measured data. The %QF’s obtained for the ARX5 model at Q =55, 60, 65 psf are 82.97%, 
11

84.94% and 84.94%, respectively. For the ARX10 model the QF’s percentage are 86.61%, 88.24 and 

88.05%.

Figures 2a–2c demonstrate that adding complexity to the ARX model improves the %QF. As such, 

we deem that the ARX10 model (eq. (14)) is the better fit model due to its predicative capability.

4.2 ARMAX Model

Next, we used the ARX10 model order as a starting point to develop a more complex ARMAX model 

to assess whether it could account for more output variance whilst maintaining an efficient model 

description. The model was estimated at Mach 1.10, Q = 30 psf and yielded a structure of the form shown 

in equation (15):

ARMAX:

A (q) y(n) = B (q)u(n) + C (q) e(n) where 

 C (q) = 1+ c

1q−1 + c2q−2 + c3q−3 + c4q−4 + c5q−5

 + c6q−6 + c7q−7 + c8q−8 + c9q−9 + c

10q−10 (15)

Figures 3a–3c shows representative results for the ARMAX structure’s (eq. (15)) ability to represent 

S4T wind-tunnel data by evaluating it with cross-validation data at Mach 1.10, and Q =55, 60, 65 psf.

Figure 3(a). Cross-validation fit: Mach 1.10, Q = 55 psf.

Figure 3. Measured and predicted output for ARMAX model at Mach 1.10; (a): Q = 55 psf, (b): 

Q = 60 psf, and (c): Q = 65 psf. Solid line (“—”) measured output. Dot-dot line (“...”) predicted ARMAX 

model output.
12

Figure 3(b). Cross-validation fit: Mach 1.10, Q = 60 psf.

Figure 3(c). Cross-validation fit: Mach 1.10, Q = 65 psf.

Figure 3. Concluded.

Figures 3a–3c illustrate %QF of the predicted output for the ARMAX model superimposed on top of 

measured data. The %QF’s obtained for the ARMAX model at Q =55, 60, 65 psf are 87.38%, 88.86%,
13

and 88.67%, respectively. Notice that although this model structure adds complexity, it accounts for 

incrementally more of the output variance.

4.3 Box-Jenkins Model

Lastly, we used the ARMAX model (eq. (15)) order as a starting point to develop a more complex BJ 

model to assess whether it could account for more output variance whilst maintaining an efficient model 

description. The model was estimated at Mach 1.10, Q = 30 psf, and yielded a structure of the form

shown in equation (16):

BJ:

y(n) = B (q)

F (q)

u(n) +

C (q)

D (q)

e(n) where

F (q)  A (q) and

D (q) = 1+ d1q−1 + d2q−2 + d3q−3 + d4q−4 + d5q−5

 + 

d6q−6 + d7q−7 + d8q−8 + d9q−9 + d10q−10 (16)

Figures 4a–4c shows representative results for the BJ structure’s (eq. (16)) ability to represent S4T 

wind-tunnel data by evaluating it with cross-validation data at Mach 1.10, and Q =55, 60, 65 psf.
14

Figure 4(a). Cross-validation fit: Mach 1.10, Q = 55 psf.

 

Figure 4(b). Cross-validation fit: Mach 1.10, Q = 60 psf.

Figure 4. Predicted output of the BJ model at Mach 1.10 superimposed on top of measured output; 

(a): Q = 55 psf, (b): Q = 60 psf, and (c): Q =65 psf. Solid line (“—”) measured output. Dot-dot line (“...”) 

predicted BJ model output.
15

Figure 4(c). Cross-validation fit: Mach 1.10, Q = 65 psf.

Figure 4. Concluded.

Figures 4a–4c illustrate %QF of the predicted output for the BJ model superimposed on top of 

measured data. The %QF’s obtained for the BJ model at Q =55, 60, 65 psf are 87.17%, 88.70%, and 

88.48%, respectively. Although the BJ model offers greater complexity to model the observed data, it 

offers a slightly lower %QF. Using validation data, if the fit of a higher order deteriorates, it is an 

indication that the model complexity is too high (ref. 33). Table 1 summarises the findings of our 

analysis.

Table 1. Summary of cross-validation results with Mach number and dynamic pressure versus model 

structure.

Quality of fit, %

Mach

Dynamic 

pressure, psf ARX5 ARX10 ARMAX BJ

55 86.04 87.63 88.26 88.05

0.80 60 86.81 88.25 88.87 88.61

65 86.16 87.33 87.79 87.66

55 83.31 85.56 86.80 86.05

0.95 60 83.60 85.97 87.11 86.36

65 X X X X

55 82.97 86.61 87.38 87.17

1.10 60 84.94 88.24 88.86 88.70

65 84.94 88.05 88.67 88.48

The results from table 1 illustrate that model fit increases with added complexity for the ARX model 

structure, improves incrementally for the ARMAX structure, but decreases for the BJ. From these results 

we conclude that the higher-order ARX model may be sufficient for our purposes, and as such we 
16

compare the predictive capability of this model to the FE based ASE model used for control design during 

wind-tunnel test (refs. 38–39).

4.4 Comparison of Tenth–Order ARX and ASE Models

Lastly, we compared the ARX10 (eq. (14)), and ASE model’s ability to predict measured data. The 

original FE based model had 17,196 degrees of freedom. Using a modal approach an eightieth-order ASE 

model was developed and used for comparison (refs. 38–39). The ASE model contained 60 structural and 

20 unsteady aerodynamic lag states.

Figures 5a–5h illustrate results of how accurately, as %QF, the two models correspond to wind-tunnel 

data at Mach 0.80, 0.95, 1.1 and, Q =55, 60, 65 psf.

Figure 5(a). Cross-validation fit: Mach 0.80, Q = 55 psf.

Figure 5. Predicted outputs of the ASE and ARX10 model superimposed on top of measured output; 

(a-c): Mach 0.80, Q = 55-65 psf; (d–e): Mach 0.95, Q = 55-60 psf; and (f–h): Mach 1.10, Q = 55-65 psf. 

Solid line (“—”) measured output. Dot-dot line (“...”) predicted ARX10 model output. Dash-dot line 

(“–..”) predicted ASE model output.
17

Figure 5(b). Cross-validation fit: Mach 0.80, Q = 60 psf.

Figure 5(c). Cross-validation fit: Mach 0.80, Q = 65 psf.

Figure 5. Continued.
18

Figure 5(d). Cross-validation fit: Mach 0.95, Q = 55 psf.

Figure 5(e). Cross-validation fit: Mach 0.95, Q = 60 psf.

Figure 5. Continued.
19

Figure 5(f). Cross-validation fit: Mach 1.10, Q = 55 psf.

Figure 5(g). Cross-validation fit: Mach 1.10, Q = 60 psf.

Figure 5. Continued.
20

Figure 5(h). Cross-validation fit: Mach 1.10, Q = 65 psf.

Figure 5. Concluded.

Figures 5a–5h shows the predicted output for the ARX10 and ASE models superimposed on top of 

measured data. The %QF’s obtained for the ARX10 model, at all Mach numbers and Q’s, range from 

85% to 88%. For the ASE model the %QF’s, for all Mach numbers and Q’s, range from 2% to 7%. 

Although the FE ASE has many more degrees of freedom, it was not able to provide better predictive 

capability than the data-driven ARX model. Clearly, the tenth-order ARX model obtained using system 

identification methodology outperforms the FE ASE model.

5. Discussion

This study explored the utility of system identification techniques to develop robust models for 

control synthesis. Initially, at each Mach number, an ARX model with maximum order of five was posed 

to the AIC, FPE and MDL techniques to estimate optimal lag and dynamic order. Next the ARX models 

were allowed greater flexibility with a maximum order of ten to assess whether the same structure could 

explain more of the output variance. Analysis of these results indicated that the tenth-order ARX models 

provided sufficient improvement in model fit to justify the added complexity. Therefore, at each Mach 

number the tenth-order ARX models were used as a starting point for ARMAX model development. The

best fit ARMAX models were then used to develop BJ models.

The results show whilst the ARMAX models provide an improved fit, it was less than 1% for each 

case. The predictive capability of the BJ models was less than that of the ARMAX structures, which is a 

symptom of too much complexity. As such, these results indicate that the tenth-order ARX models were 

optimal in terms of parsimony and predictive capability to describe the dynamics of recoded wind-tunnel 

data.

A comparison of the tenth-order ARX and 80th-order ASE model’s ability to explain the output 

variance revealed that the more complex ASE model was not able to achieve as high of a fit. The ARX 
21

model was able to attain a higher fit because the model was developed using measured data, which often 

contains dynamics that are not captured in a FE based model. Conversely, the ASE model was developed 

from idealised assumptions about mass, damping, et cetera, which are idealisation and often do not 

closely hold with observations under experimental conditions.

Although a different ASE model was developed for each Mach and Q, they yielded significantly 

reduced fits to measured data compared with the tenth-order ARX model. Notice that the ARX model at 

each Mach number was developed with estimation data at Q =30 psf but was able to accurately predict 

the measured data at higher Q’s, namely, Q = 55-65 psf. This is the power of utilising a data driven 

approach to model development.

Often a model, which yields a minimum system description yet provides good predictive capabilities,

is deemed as the best or optimal model. This is a trade-off between the ability to describe the system 

behaviour and parsimony. If two models can describe the system behaviour almost equally well, why 

choose the more complex one? For example, in H∞ -control synthesis the order of the controller is equal 

to the order of the model plus the order of the performance weights. Of course, it is not as 

computationally demanding to compute a controller of lower order. Moreover, selecting a model with 

greater complexity may lead to numerical issues if the model order is too high. Hence, minimum 

complexity often renders control synthesis more tractable (33, 47).

A future study will utilise these data-driven models to develop control laws to assess model and 

controller design robustness. Initially, this study will be performed in a simulation environment but with 

data collected from the wind-tunnel tests described in this report. The successful demonstration of this 

work may lead to comparison of the robustness of control law design based on the two approaches, 

FE ASE and system identification models, on a supersonic flight test vehicle.

6. Conclusion

This study demonstrates the application of system identification techniques to develop parsimonious 

and robust models directly from data with excellent predictive capability. The results show that a 

data-driven model is capable of predicting observed data for a larger operating point. This robust 

predictive power was demonstrated with models developed at Q = 30 psf, but able to accurately predict 

the measured output at higher Q’s. This superior predictive capability allows for simpler control solutions 

and reduced modelling effort, whilst traditional ASE based control strategies rely on computationally 

expensive models with little predictive power rendering control law design more expensive.
22

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Data Driven Model Development for the SuperSonic SemiSpan Transport 

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14. ABSTRACT

In this report, we will investigate two common approaches to model development for robust control synthesis in the aerospace 

community; namely, reduced order aeroservoelastic modelling based on structural finite-element and computational fluid dynamics

based aerodynamic models, and a data-driven system identification procedure. It is shown via analysis of experimental SuperSonic 

SemiSpan Transport (S4T) wind-tunnel data that by using a system identification approach it is possible to estimate a model at a 

fixed Mach, which is parsimonious and robust across varying dynamic pressures.

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Aerodynamic models, Model development, Structural finite-element, System identification

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