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Introducing a Combined Equation/Output Error Approach

in Parameter Estimation

E. Özger1

EADS Defense & Security - Military Air Systems, 85077 Manching, Germany

This investigation presents a new approach in the field of parameter estimation. The well

known equation and output error estimation methods are combined in a flexible manner

into one estimation approach where the model response and measurement comprise both

state and non-state parameters of the system under test. Various simulation tests are

conducted based on linear, non-linear, flight mechanically unstable aerodynamics with and

without noisy signals. The new approach shows a well balanced matching performance both

in the equation and output error domains. Furthermore, it is more stable in the iteration

process than the output error method due to the equation error ingredients. The estimated

parameters of the pure output error approach converge most closely towards the true ones

but this approach is most sensible and prone to non-convergent solutions. In contrast to this,

the combined equation/output error approach gives flexibility in the estimation process

where by means of a suitable weighting among the state and non-state observation vector

elements any estimator behavior can be emulated from a pure equation error to a pure

output error approach.

Nomenclature

cl = rolling moment coefficient

cl0 = rolling moment offset

clp = roll damping

clr = roll-yaw damping

cl = lateral stability

cl = aileron efficiency

cl = roll-rudder efficiency

cm = pitching moment coefficient

cm0 = zero pitching moment

cm = static stability

cm = elevator efficiency

cmq = pitch damping

cn = yawing moment coefficient

cn0 = rolling moment offset

cnp = yaw-roll damping

cnr = yaw damping

cn = directional stability

cn = yaw-aileron efficiency

cn = rudder efficiency

f = system state function

F = process noise distribution matrix

g = system observation function

G = measurement noise distribution matrix

I = inertia matrix

J = cost function

l = reference cord

v = measurement noise

V = freestream velocity

w = process noise

wpi = diagonal elements of W

W = normalization matrix

x = state vector

x0 = initial condition of state vector

y = simulated observation vector

z = measured observation vector

= angle of attack

= angle of sideslip

= elevator deflection

max = maximum value (=104)

= signal to noise ratio

= unknown parameters to be estimated

= update to parameter estimate

ci = r.m.s. error of parameter ci

= angular rate vector

= aileron deflection

= rudder deflection

= functional

= domain of real number

Subscripts/Superscripts

0 = static offset

ADM = aerodynamic model

1 Flight Test Engineer, Flight Test, Rechliner Str., 85077 Manching, Germany, and AIAA Senior Member.

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-34

Copyright © 2010 by Erol Ozger. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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m = size of unknown parameter vector

M

= moment vector

N = maximum number of time samples

p = size of observation vector

p,q,r = roll, pitch, and yaw rate

qdyn = dynamic pressure

s = half span

S = reference area

t = time

t0 = initial time

u = control input vector

ctrl = control surface deflections

Engine = engine related

Desired = desired derivative

Dest = destabilized derivative

EqnError = equation error

EqnOutpError= combined equation/output error

G = gravity related

i,j,n = i-th, j-th, n-th sample

k = kth time sample

OutpError = output error

T = transposed

~ = simulated parameter

I. Introduction

HE field of parameter estimation1-9

is so wide that it is impossible to cover it in all aspects in such a paper.

Briefly, parameter estimation can be seen as defining a probability density function describing the difference

between system model response and measured system response. The parameters of the model to be estimated give

the highest probability to match the measured system response. Apart from the usually unknown behavior of a

system two ingredients further complicate the estimation process, namely process and measurement noise. Both

noise contribution have in common that they can change the weighting in the probability density functions. Process

noise are gusts and other atmospheric turbulences whereas measurement noise are for example uncertainties in the

measurement performance of vanes or inertial measurement units.

The first restriction of this investigation is the focus on dynamic systems in the time domain. Here, two

estimation approaches can be distinguished for dynamic systems depending on what system characteristics are used

for the probability density function. In case that the system outputs are taken such as angle of attack, angular rates

etc., which are integrated during a simulation and thus represent basically the system states, the estimation approach

is called output error estimation. For the system states, the information of their past history is necessary for

determining the actual value. Here usually, process noise is negligible whereas measurement noise gives a weighting

on the system output. If both process and measurement noise have to be accounted for, the system states are

stochastic so that they cannot be propagated by normal integration. To overcome this problem a filter error approach

by means of a Kalman filter has to be applied.

It is also possible to utilize system non-state parameters for the probability density function such as force and

moment coefficients, which are not integrated during a simulation, and thus, are determined without the knowledge

of their past history. This parameter domain is known as equation error, where the assumption is that measurement

noise is negligible and process noise leads to a weighting of the non-state parameters.

Each approach, be it in the output or equation error domain, has its pros and cons. The motivation for this

investigation lies in the parameter estimation activity in connection with Eurofighter high angles of attack testing.

The author had the strange experience that estimated derivatives in the equation error domain did not lead to a

similar matching in the output error domain and vice versa, particularly when the errors tended to be highly non-

linear. This strange experience led to the idea whether it is possible to combine equation and output error estimation

in one shot so that it is guaranteed to have a estimated parameter combination with an adequate matching quality in

both domains.

There has been attempts of combining the two methodologies into one algorithm. But either the equation error

estimation is used for defining the starting values of the subsequent output error estimation or supposedly diverging

state variables are exchanged with their measured counterparts. The main difference between above mentioned

methods with the new one proposed in this investigation is that the difference in system model and measured

response comprise not only usual state variables (p,q,r,, etc.) or only non-state variables (cl,cm,cn, etc.) but both of

them.

In this investigation the new combined approach of equation/output error estimation is presented with extensive

testing by means of a 6-dof simulation of a Dornier 328-100 aircraft. The simulation10

is based on a linear

aerodynamic model which can be non-linearized and destabilized. Furthermore, the addition of noise on any

simulated parameter is possible. The performance of the new estimator is compared with the performance of the

equation and output error estimation methods both in the equation and output error domains. This comprises also

comparing of the estimated parameters of each method with the known true values. By means of this approach an

T

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overview of the estimation performance of the three estimators is given for a variety of estimation environments

starting from the most simple linear models towards non-linear, unstable and noisy systems.

II. Comparing Existing Parameter Estimation Techniques with New Approach

The following section describes the existing parameter estimation techniques, namely equation error and output

error estimation together with the combined equation/output error estimation to be introduced in this investigation.

A. General System Model

The mathematical representation of a state space model can be summarized as follows

00 xtx (1a)

twFtutxftx ,, (1b)

kkkk vGuxgz ,, (1c)

where kk tzz , kk txx , kk tuu for the sake of brevity. (1d)

Here x is the state vector with x0 as initial condition, t is the time and k is the sampled time index, u is the known

control input vector, zk and yk are the measured and simulated observation vectors at time sample k, f and g are the

system state and observation functions, is the vector of unknown parameters, F is the process noise distribution

matrix, G is the measurement noise distribution matrix, w is the process noise, and v is the measurement noise.

B. Cost Function

Under the assumption that the measurement noise is negligible, and after application of the negative logarithm to

the probability density function, the resulting cost function of the maximum likelihood estimator is

2ln2

detln22

1

1

1 pNFF

NyzFFyzJ T

N

kkk

TTkk

. (2a)

When only the process noise is omitted the resulting output error cost function is

2ln2

detln22

1

1

1 pNGG

NyzGGyzJ T

N

kkk

TTkk

(2b)

where for both approaches 1, pkk yz is valid. In this investigation both process and measurement noise are

not considered, namely ppTT IGGFF , and the second and third terms of equation 2 are not considered

anymore. These assumptions will not impair the generality of the subsequent approach. Thus, both equation and

output error approach differ only in the vectors zk and yk, where for the equation error estimation these are usually

the non-state force and moment coefficients and for the output error estimation the states such as angle of attack,

angular rates etc.

C. Solving the Cost Function

The well known iterative Gauss-Newton algorithm is applied for the cost function J() with

LL 1 (3a)

JJ1

2

2

. (3b)

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The terms in (3b) are defined as follows:

N

kkk

Tk yz

yJ

1 (4a)

N

k

kT

kN

kkk

T

kkT

k yyyz

yyyJ

112

2

2

2

(4b)

with the Jacobian matrix mpky

(4c)

mmm JJ

2

21,

(4d)

where the unknown parameter vector is defined in this investigation as

Tnnnnrnpnmqmmmllllrlpl cccccccccccccccc 000 (5a)

with 1, m . (5b)

The Jacobian matrix is usually calculated numerically by means of finite differences with respect to elements .

D. Equation Error Estimation Approach

For the equation error estimation the parameters to be observed are defined here as follows

Tknmlk cccz (6a)

and Tknmlk cccy ~~~ . (6b)

where the tilde denotes the simulated observation vector elements in contrast to the measured ones. The

measured observation vector zk is determined by means of the equations of motion.

GEngineFT MMMIdt

d

(7a)

with

sSqc

lSqc

sSqc

M

dynn

dynm

dynl

FT

(7b)

where (S, l, s) are the reference surface, mean chord and half span, I is the aircraft inertia, EngineM

is the

engine contribution to the moments and GM

the gravitational contribution to the moments. The aircraft motion

and dynamic pressure qdyn are measured during flight. The output of the equations of motion are the inertia based

moments FTM

around the aircraft centre of gravity.

The simulated observation vector yk contains the elements of to be estimated and measured contributions

during flight such as angle of attack , angle of sideslip , control deflections (, , ). Here, this is a linear model

be estimated and validated (see equation 8).

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llllrlpll cccrcpccc 0~ (8a)

qcccccc mqmmmmm 0~ (8b)

nnnnrnpnn cccrcpccc 0~ (8c)

The Jacobian matrix can be easily evaluated analytically in this case since the aerodynamic model is linear in the

parameters to be estimated.

kk

kk

kk

EqnErr

k

p

q

py

1000000

0001000

0000001

. (9)

E. Output Error Estimation Approach

For the output error estimation the parameters to be observed are defined here as follows

Tkk rqpz (10a)

and Tkk rqpy ~~~~~ . (10b)

where again the tilde denotes the simulated observation vector elements in contrast to the measured ones. The

measured observation vector zk contains the measured angle of attack , angle of sideslip and angular rates (p,q,r)

at time sample k. The modeled observation vector yk is determined in the frame of a non-linear 6-dof simulation

which contains the aerodynamic model of equation 8 to be estimated.

F. Combined Equation/Output Error Estimation Approach

For the new combined equation/output error estimation the parameters to be observed are defined here as follows

Tknmlk cccrqpz (11a)

and Tknmlk cccrqpy ~~~~~~~~ (11b)

where again the tilde denotes the simulated observation vector elements in contrast to the measured ones. The

measured observation vector zk contains measured parameters such as angle of attack , angle of sideslip and

angular rates (p,q,r) on the one side and on the other side 'quasi' measured moment coefficients from the equations

of motion (cl, cm, cn) of equation 7.

The modeled observation vector yk contains on the one side the simulated angle of attack, angle of sideslip and

angular rates by the 6-dof nonlinear simulation. And on the other side it contains also the moment coefficients of the

aerodynamic model to be estimated which are described by equation 8.

The Jacobian matrix of the combined equation/output error approach has then the following form

EqnErr

k

OutpErr

k

EqnOutpErr

k

y

y

y

(12)

G. Normalization of Estimation Process

Since the various elements in the observation error vector (zk-yk) have different order a normalization is

performed in order to guarantee that the mismatch in each element is decreased in a comparable way during the

Gauss-Newton algorithm. The normalization matrix is defined as

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pw

w

w

W

00

00

00

2

1

(13a)

with ppTT WWWW (13b)

where the elements of the matrix W are defined as

max

,,

,minmax

1min

pikpikpi

zzw for ppi 1 , (13c)

where max defines an upper boundary that can be chosen to reasonable values. In this investigation it is max =

104. Consequently, the estimation algorithm changes to

N

kkk

TT

k yzWWyJ

1 (14a)

and

N

k

kTT

k yWW

yJ

12

2

. (14b)

The normalization procedure of the difference in the observation vector implies the following cost function

N

kkk

TTkk yzWWyzJ

12

1 (15)

which means that in consequence the normalization matrix W has a similar impact on the estimation procedure

as the process or measurement noise covariance matrices. In case that process or measurement noise is considered

the corresponding covariance matrix can be multiplied with the normalization matrix to give an overall weighting of

the cost function.

In this investigation the estimation of output and combined equation/output estimation are performed including

the normalization matrix. For the equation error approach the impact of this normalization approach was negligible,

so that it was not applied there.

III. Experimental Setup

A. Experimental Approach

The three estimation approaches, which are described above, are tested in the frame of a 6-dof simulation. For

this test, maneuvers are performed. The matching performance of the estimation methods both in the equation error

and output error domain are compared. Moreover the estimated parameters are compared with the true ones, a

luxury that is usually out of reach in real life.

The simulation can be performed in various ways. Basically a linear aerodynamic model of a Dornier 328-100 is

implemented10

. Additionally non-linear terms can be superposed which are a function of the relevant aerodynamic

input parameters (angle of attack, angle of sideslip, flap deflections, angular rates). Moreover the model can be

destabilized in longitudinal, lateral and directional stability and damping derivatives. In order to prevent the aircraft

from departure, the flight vehicle is stabilized by means of control laws. For all variants of simulation a noise

generator can be put on top of each signal.

Simulations performed under various settings (linear, non-linear, destabilized aerodynamics, noise input) serve

as measurement and the true dataset. On the other side a second set of simulation with the true aerodynamics in the

force coefficients but a linear aerodynamics in the moment coefficients representing the model, that is to be

estimated, is performed.

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Only the linear moment derivatives are estimated within this investigation, even if the measured true

aerodynamics simulation is based non-linear aerodynamics. The reason for this lies in the question how well the

estimators perform if the estimation model does not comprise unknown non-linearities. As starting values of the

estimation process the initial values of the parameters to be estimated are equal to the true (linear destabilized)

derivatives times a factor of 0.7, simulating an a-priori knowledge 30% off the true value. The additional non-linear

terms are not accounted for in the initial conditions, complicating this estimation task further. The following Table 1

displays the simulation experiments with the estimator approaches where a plus symbol denotes the property of the

simulation where case A to D is with linear aerodynamics and case E to F is with non-linear aerodynamics.

experiment identifier A B C D E F G H

basic linear model + + + + + + + +

plus non-linear terms + + + +

plus destabilization + + + +

plus noise + + + +

Table 1. Simulation experiments with the three estimator types

B. Simulation Model

A simple simulation of a Dornier 328-100 in cruise configuration10

is used. The aerodynamic model refers to the

cruise condition and is basically linear. In the following Table 2 all relevant data is displayed that is useful for the

subsequent investigation.

S = 40 m² l = 2.04 m s = 10.4 m m = 10500 kg Vtrim = 144 m/s Thrust = 10 KN

Ix = 103000

kgm²

Iy = 158000

kgm²

Iz = 240000

kgm²

Ixz = 12500 kgm² - -

cL0 = 0.221 cL = 6.00 cA = 0.349 - cLq = 4.39 -

cD0 = 0.027 cD = -0.002 cD = 0 - - -

cm0 = 0.127 cm = -1.678 cm = -1.93 - cmq = -26 -

cY0 = 0 cY = -1.195 cY = 0 cY = 0.389 cYp = 0.109 cYr = 0.576

cl0 = 0 cl = -0.285 cl = -0.299 cl = 0.099 clp = -1.118 clr = 0.295

cn0 = 0 cn = 0.366 cn = -0.0065 cn = -0.392 cnp = -0.140 cnr = -0.619

Table 2. Reference data, inertia data and body axis linear aerodynamic model of Do 328-100 in cruise

configuration

The non-linearization of aerodynamics is performed in three variants, namely

1. by putting on top of the basic linear force and moment derivatives a sinusoidal increment, for example:

cl + cl0 sin(2/0)

2. by putting on top of the linear moment derivatives polynomial increments, for example:

cl + cl + cl + cl2 ² + cl2 ² + cl2 ² + cl + cl + cl

3. same as 2. but all polynomial increments multiplied with a factor of two.

The destabilization is done by exchanging the following linear derivatives shown in Table 3. In order to

guarantee departure free simulation, control laws are implemented which are based on a modified eigen-structure

assignment. For the pitch axis angle of attack and pitch rate q are fed back to the elevator increment on top of

the pitch stick commanded elevator deflection (see equation 16a). For the roll and yaw axis the approach is quite

similar but accounts for the coupling. Here, angle of sideslip , roll and yaw rates (p,r) are fed back to the aileron

and rudder increments (,) that are put on top of the commanded deflections (,), see equation 16b-c.

V

sq

c

cc

c

cc

m

DestmqDesiredmq

m

DestmDesiredm

,,,, (16a)

rcc

pccB

cc

ccB

DestnrDesirednr

DestlpDesiredlp

DestnDesiredn

DestlDesiredl

,,

,,1

,,

,,1

(16b)

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with

nn

ll

cc

ccB . (16c)

where the values of the desired derivatives are equal to the corresponding ones of the stable linear basic model in

Table 2. The consequence of this exercise is that the control surface deflections become more correlated with the

aircraft motion.

cm,Dest =50.5 cmq,Dest = -5 cl,Dest = 0.12 clp,Dest = 0.5 cn,Dest = -0.5 cnr,Dest = 0.12

Table 3. Destabilized stability and damping derivatives

White noise is added to the signals by means of a random number generator. Here, on all signals noise is applied

according to Table 4 where the approximate maximum signal to noise ratio =NoiseAmplitude/SignalAmplitude is

given.

Signal Signal

V ~0.5% cx ~0.4%

~5% cy ~1%

~10% cz ~0.5%

, , ~5% cl ~20%

p, q, r ~2.5% cm ~1%

nx, ny, nz ~5% cn ~2%

Table 4. Noise addition on various signals of the simulation

C. Maneuvers

Eight types of maneuvers are performed in the frame of this investigation for each simulation type (basic linear,

non-linearized, noisy signals, feedback controlled). Various sinusoidal and box car type inputs are applied

consecutively or partly overlapping in the pitch, roll and yaw axis during the maneuver. Due to the large amount of

combinations of maneuver types and simulation types (all in all 8 maneuver types and 16 simulation types equals to

128 maneuver simulations) a more detailed description by means of figures or tables is omitted here. It can be said

that for the unstable and controlled configurations the correlation among the signals is larger than for the naturally

stable ones since angular rates and angles of incidence are used in the feedback loop for the control surface

deflections.

IV. Results

In the following the quantitative outcome of the simulations together with the parameter estimation task will be

presented. To get a good overview mean error of the matching parameters in the equation (cl,cm,cn) and the output

error (,,p,q,r) domains are presented together with the mean error of the elements of the derivative vector (see

equation 5) to be estimated. The mean error is determined according to

e.g.

N

kklll cc

Nc

1

6~10

for equation error parameters, (17a)

e.g.

N

kkN 1

~1 for output error parameters, (17b)

e.g.

lN

kk

N

kklk

l c

c

c

1

2

1,

for estimated parameters, (17c)

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where cl() denotes all aerodynamic model terms with angle of sideslip involved (for the other derivatives this

should be their respective parameter). The other derivatives are treated accordingly, where for the zero moments

only the absolute difference is shown. A certain amount of estimations did not lead to a converged solution,

probably due to the non-optimal initial conditions, so that the subsequent simulations with the estimated parameters

underwent a departure. These departed simulations are taken out of the statistics since they would affect the root

mean square errors too much. Nevertheless, the number of their appearance is counted to give the reader a feeling

how robust the three estimation methods perform.

For each experiment identifier (A,B, etc.) eight different maneuvers are performed in case that the basic

aerodynamics are linear. In case of non-linear aerodynamics with noise-free signals eight different maneuvers for

three variants of non-linear aerodynamics are performed so that 24 maneuvers are performed all in all. In case of

noise the respective simulations are performed five times so that 40 simulations in the linear aerodynamic and 120 in

the non-linear aerodynamic case are performed. For each experiment identifier the mean value with respect to the

estimated parameters, equation and output error parameters is determined over all performed maneuvers. These data

are summarized in Tables 5 for all estimators.

A. Basic Linear Model

The simplest parameter estimation task is perfectly fulfilled by each estimation approach and thus not included

in Table 5. All root mean square errors are zero giving confidence that all algorithms are implemented properly.

Here, it becomes also clear that a linear error in the aerodynamics can be tackled by any estimation algorithm with

adequate quality.

B. Basic Linear Model including Noisy Signals

This is the first case with non-zero errors in the investigated parameters. As becomes clear in the comparison

between equation and output error estimation each of the approaches produces an optimal matching in the domain it

works where the combined approach lies in between with respect to the equation and output error. The estimated

parameters of the pitch axis are best determined by the equation error approach (maximum error in cm = 0.214 and

cm = 0.042) followed by estimations of the combined and the pure output error approaches, maximum error in cm

= 0.455 and cm = 0.132. In the roll and yaw axis the output error approaches performs best with maximum error in

cl = 0.047 and cn = 0.037 followed by the performance of the combined and pure equation error estimators, the

maximum error being cl = 0.134 and cn = 0.155.

C. Basic Linear Model plus Destabilization

As it is in case A, all estimators perform perfectly both in the equation and output error matching as well as in

the estimation of the derivatives approach and thus data are not included in Table 5. All root mean square errors are

zero. Thus, a feedback controlled configuration poses principally no problem as long as the underlying

aerodynamics are linear and the signals free of noise.

D. Basic Linear Model plus Destabilization with Noisy Signals

This case has a further complexity in form of noisy signals compared to case C. All estimation approaches are

fully departure free in the simulations. The matching performance in the equation and output error domain are

similar for the estimators as in the cases B with the equation error estimation showing best match in the equation

error domain and the output error estimator showing best match in the output error domain, the combined approach

lying in between.

In the pitch axis the derivative error is highest for the equation error approach (maximum error in cm = 37.95

and cm = 1.435) followed by the combined approach. The pure output error approach shows here best performance

with maximum error in cm = 4.919 and cm = 0.183. In the roll and yaw axis the pure output error approach

produces the lowest derivative errors with cl = 0.183 and cn = 0.124 versus cl = 0.395 and cn = 0.39 for the other

pure equation error approach and the combined approach lying in between.

E. Basic Linear Model plus Non-Linear Terms

Introducing significant non-linearities into the aerodynamic model that cannot be estimated has a severe impact

on the estimations. Out of 24 maneuvers the equation error and combined approach produces none, the output error

produces 7 departures or non-convergent simulations.

The mean error in the equation error domain is best matched by the equation error estimator, followed closely by

the combined and the pure output error approach. In the output error domain the combined approach shows the best

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overall matching performance followed by the pure equation and output error estimators. This seems to depend on

the convergence problems of the output error approach compared to the other more robust approaches.

In the pitch axis derivatives the pure output error approaches produces the closest estimations to the true values,

maximum error in cm = 0.165 and cm = 0.163, the other approaches producing an error up to cm = 0.22 and cm =

0.296. In the roll and yaw axis no approach is significantly better in all derivatives than the others, the maximum

error being cl = 0.276 and cn = 0.409.

F. Basic Linear Model plus Non-Linear Terms with Noisy Signals

Here an additional complexity compared to case E in form of noisy signals is introduced. As it was in case E

only the output error estimators produced departures, namely 26 compared to none for the other estimators.

The matching characteristics in the equation and output error domains are similar to the cases B, with the

equation error approach producing best match in the equation error domain, the output error approach producing

best match in the output error domain and the combined approach in between in both cases.

In the pitch axis derivatives the pure equation error approach shows the best performance (maximum error cm =

0.288 and cm = 0.303) followed by the combined and pure equation error approach. The error in the roll and yaw

axis derivatives is similar for all approaches showing a maximum error of cl = 0.195 and cn = 0.369.

G. Basic Linear Model plus Non-Linear Terms and Destabilization

Further increasing the complexity level of the estimation task produces a larger amount of departures for all

estimators, namely 10 for the equation error and combined approach, and 13 for the pure output error approach. The

equation error match is best for the pure equation error estimator followed by the combined and pure output error

estimators. In the output error domain the combined method shows the best performance.

The errors in the pitching moment derivative are lowest for the pure output error approach (maximum error cm

= 6.385 and cm = 0.262) followed by the combined and equation error estimators. A similar performance is seen in

the roll and yaw axis derivative errors with the pure output error method showing all in all the lowest errors.

H. Basic Linear Model plus Non-Linear Terms and Destabilization and Noisy Signals

Additionally to case G the signals are noisy. As it was in case G the equation error and combined approach

produced the lowest amount of departures, 50 respectively 51 against 65 departures for the pure output error. The

equation error is best matched by the pure equation error approach followed by the combined and pure output error

estimators. In the output error domain the combined approach shows best matching followed by the other two

estimators.

The output error estimator produces lowest errors in the pitch axis derivatives (maximum error cm = 12.14 and

cm = 0.439) followed by the combined and pure equation error approach. In the roll and yaw axis again the pure

output error approach has all in all the lowest derivative errors. To give the reader a feeling of the estimation task

Figure 1 to Figure 6 present an example of a simulation with the matching quality in equation and output error

domains and the estimation performance of the three approaches.

V. Discussion

In this investigation various influencing factors on the parameter estimation are tested for the pure equation, pure

output and combined equation/output error estimators which are non-linearity, feedback controlled configurations,

and noisy signals. In the following the discussion is separated between the linear and non-linear world.

A. The Linear World

In case of linear aerodynamics with and without feedback control and with noise free signals any estimation

approach leads to a perfect match with true estimated derivatives. Introducing noisy signals and a feedback control

makes the estimations of all tested estimators deviate from the true values.

The matching characteristics and estimation performance of the three approaches are quite regular. As it can be

expected, the pure equation and pure output error approaches perform best in the matching of their respective

domain, where the matching performance of the combined approach is always in between. In average, the estimation

quality produced by the pure output error approach is best followed closely by the other two approaches.

B. The Non-Linear World

Trying to match unknown non-linear aerodynamics with a linear correction model is a very unfair task, though it

is reality with high angle of attack flight testing where not all aerodynamic effects can be investigated in a wind-

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tunnel due to physical and financial reasons. Here, the assumed model structure is only a rough simplification of the

true one, representing a lower order equivalent system neglecting also coupling and interference terms between the

estimated parameters. Moreover, one has to be very careful in interpreting the results, since matching the true mean

derivatives may not be necessarily the best solution.

As it is reported in the linear world, additional feedback and noisy signals impairs the estimation task decreasing

matching and estimation quality. Here again, the pure output error approach produces the closest matching to the

averaged derivatives also with non-linear aerodynamics followed by the pure equation error and the new combined

approach. In terms of the matching in the equation error domains, the pure equation error estimator produces the

best matching, followed closely by the new combined method with the largest deviations produced by the pure

output error method. In the output error domain, the new combined approach shows the best matching performance,

where the pure output error approach produces also more non-convergent simulations or departures. It becomes

clear that the equation error contribution makes the estimation process more robust towards adverse system

conditions.

Within this investigation the weighting between equation error and output error for the combined approach was

selected to be 50/50 which is surely not carved in stone. Any appropriate and sensible weighting can be chosen

which poses the best trade-off between accuracy of estimates and robustness in the estimation process, thus to fulfill

the estimation task in the most efficient manner. Also important, the resulting matching in the output and equation

error domain is always well balanced for the new combined equation/output error approach since both domains are

accounted for in the cost function.

VI. Conclusion

This investigation presents a new estimation approach where both state and non-state parameters of a system are

included as elements in the observation vector of the cost function. Normally, state variables are utilized in the pure

output error approach, and non-state variables are utilized in the equation error approach. The combined

equation/output error approach gives flexibility in the estimation process where by means of a suitable weighting

among the state and non-state observation vector elements any estimator behavior can be emulated from a pure

equation error to a pure output error approach.

In the frame of this investigation the pure equation error, the pure output error and the new combined

equation/output error approach are tested under various system conditions such as linear, non-linear aerodynamics,

noisy signals and feedback control. The matching performance both in the equation error and in the output error

domains are best achieved by the new combined approach. Generally, the pure output error approach gives the

closest estimations to the true values but this approach also suffers from the most non-convergent solutions. In

contrast to this, the equation error approach is quite stable but the estimations are sometimes less accurate. The best

trade-off between estimation robustness and estimation accuracy can be achieved by the new combined

equation/output error approach.

Although not tested within this investigation this flexibility between equation error and output error behavior can

be utilized during the estimation process by varying the weighting from begin of the estimation process towards the

end so that an optimal estimation result can be achieved.

References 1Eykhoff, P., "System identification", John Wiley & Sons, 1974 2Maine, R. E., and Iliff, K. W., "Identification of Dynamic Systems", NASA RP 1138, 1985 3Klein, V., "Estimation of Aircraft Aerodynamic Parameters from Flight Data", Prog. Aerospace Sci., Vol. 26, pp. 1-77, 1989 4Iliff, K. W., "Aircraft Parameter Estimation", NASA TM 88281, 1987 5Iliff, K. W., and Maine, R. E., "Uses of Parameter Estimation in Flight Test", J. Aircraft, Vol. 20, No. 12, December 1983 6Özger, E., "Aerodynamic Model Validation of Eurofighter Aircraft", presented in the AIAA Atmospheric Flight Mechanics

Conference and Exhibit, August 2007, AIAA-2007-6718 7Morelli, E. A., "Practical Aspects of the Equation-Error Method for Aircraft Parameter Estimation", presented in the AIAA

Atmospheric Flight Mechanics Conference and Exhibit, August 2006, AIAA-2006-6144 8Jategaonkar, R., V., "Flight Vehicle System Identification – A Time Domain Methodology", AIAA Progress in Astronautics

and Aeronautics Vol. 216, 2006 9Jategaonkar, R., V., Thieleke, F., "Evaluation of Parameter Estimation Methods for Unstable Aircraft", Journal of Aircraft,

Vol. 31, No. 3, 1994, pp. 510-519 10Brockhaus, R., "Flugregelung", Springer-Verlag, 1994

American Institute of Aeronautics and Astronautics

12

Exp.

Iden-

tifier

B D E F G H

Eqn

Error

Outp

Error

Eqn/

Outp

Error

Eqn

Error

Outp

Error

Eqn/

Outp

Error

Eqn

Error

Outp

Error

Eqn/

Outp

Error

Eqn

Error

Outp

Error

Eqn/

Outp

Error

Eqn

Error

Outp

Error

Eqn/

Outp

Error

Eqn

Error

Outp

Error

Eqn/

Outp

Error

cl0 0 0 0 0 0 0 0 0.001 0 0 0.001 0 0 0 0 0 0 0

clp 0.449 0.191 0.392 0.102 0.108 0.114 0.416 0.303 0.4 0.586 0.408 0.569 0.121 0.173 0.58 0.164 0.229 0.491

clr 0.11 0.057 0.094 0.157 0.097 0.144 0.108 0.149 0.129 0.162 0.139 0.151 0.157 0.163 0.543 0.176 0.185 0.337

cl 0.134 0.047 0.114 0.395 0.183 0.376 0.089 0.276 0.178 0.144 0.195 0.137 0.266 0.341 0.498 0.447 0.398 0.396

cl 0.109 0.053 0.1 0.174 0.081 0.164 0.114 0.086 0.196 0.152 0.115 0.174 0.11 0.079 0.18 0.122 0.112 0.162

cl 0.043 0.024 0.04 0.079 0.026 0.077 0.117 0.059 0.081 0.139 0.075 0.236 0.081 0.028 0.208 0.061 0.045 0.136

cm0 0.003 0.009 0.006 0.097 0.013 0.073 0.023 0.013 0.019 0.024 0.018 0.023 0.024 0.017 0.021 0.082 0.031 0.065

cm 0.214 0.455 0.305 37.95 4.919 28.35 0.166 0.165 0.22 0.288 0.942 0.589 10.85 6.385 9.421 33.45 12.14 26.71

cm 0.042 0.132 0.094 1.435 0.183 1.068 0.296 0.163 0.262 0.303 0.242 0.311 0.426 0.262 0.38 1.231 0.439 0.988

cmq 0.634 1.365 0.97 2.513 0.909 1.432 0.587 1.395 1.676 0.906 2.045 2.279 2.032 3.284 2.518 2.066 3.764 1.923

cn0 0 0 0 0 0 0 0 0 0.001 0 0 0.001 0 0 0 0 0 0

cnp 0.494 0.142 0.166 0.096 0.05 0.139 0.408 0.802 0.528 0.414 0.808 0.43 0.131 0.182 0.947 0.173 0.175 0.506

cnr 0.127 0.041 0.052 0.123 0.062 0.033 0.121 0.267 0.188 0.14 0.242 0.162 0.085 0.109 0.189 0.124 0.145 0.112

cn 0.155 0.037 0.062 0.39 0.124 0.234 0.09 0.409 0.227 0.177 0.369 0.197 0.193 0.326 1.058 0.27 0.25 0.506

cn 0.118 0.032 0.052 0.156 0.04 0.059 0.109 0.181 0.246 0.115 0.179 0.409 0.067 0.083 0.289 0.061 0.075 0.146

cn 0.052 0.031 0.041 0.082 0.034 0.063 0.133 0.155 0.314 0.137 0.152 0.325 0.051 0.03 0.198 0.044 0.054 0.085

cl 169 309 183 143 471 160 396 907 627 375 990 537 247 554 390 285 751 372

cm 493 851 671 3580 8280 4616 1017 748 2216 1173 1233 2378 2840 3370 3027 4550 7997 5417

cn 222 376 297 158 455 282 552 2426 1213 640 2324 1362 273 516 644 337 752 849

[°] 0.01 0.01 0.01 0.05 0.01 0.03 0.62 0.15 0.13 0.62 0.08 0.47 0.24 0.09 0.05 0.16 0.1 0.26

[°] 0.09 0.05 0.05 0.26 0.05 0.17 1.84 0.44 0.49 1.89 0.35 1.19 0.43 0.65 0.23 0.3 0.71 0.46

p[°/s] 0.26 0.11 0.16 2.6 0.11 1.31 2.46 4.03 2.16 2.51 2.74 5.04 11.58 5.92 0.61 6.74 6.09 1

q[°/s] 0.14 0.11 0.11 3.46 0.11 1.29 2.89 4.89 0.91 2.95 1.49 2.24 12.57 7.92 0.8 11.35 7.5 1.32

r[°/s] 0.58 0.1 0.15 3.53 0.1 1.4 12.04 14.03 4.77 12.11 6.7 5.39 10.5 15.59 2.22 5.66 15.61 4.26 Depar-

tures 0/40 0/40 0/40 0/40 0/40 0/40 0/24 7/24 0/24 0/120

26/

120 0/120 10/24 13/24 10/24

50/

120

65/

120

51/

120

Table 5. Comparison of estimator statistics (for the sake of brevity case A and C is not shown, since all estimators performed in these cases perfectly with

zero errors), departures refer to non-convergent simulation producing an unstable behavior

American Institute of Aeronautics and Astronautics

13

Figure 1. Matching performance in equation error domain where measurement is compared with equation

error, output error and equation/output error estimator results (case H)

Figure 2. Matching performance in output error domain where measurement is compared with equation

error, output error and equation/output error estimator results (case H)

American Institute of Aeronautics and Astronautics

14

Figure 3. Matching performance in output error domain where measurement is compared with equation

error, output error and equation/output error estimator results (case H)

Figure 4. Estimation quality of rolling moment derivatives where true model is compared with estimates of

equation error, output error and equation/output error estimators (case H)

cl0 cl()

cl()

cl()

cl(p)

cl(r)

American Institute of Aeronautics and Astronautics

15

Figure 5. Estimation quality of pitching moment derivatives where true model is compared with estimates of

equation error, output error and equation/output error estimators (case H)

Figure 6. Estimation quality of yawing moment derivatives where true model is compared with estimates of

equation error, output error and equation/output error estimators (case H)

cn()

cn()

cn()

cn(p)

cn(r)

cn0

cm0

cm(q)

cm()

cm()