![Nonlinear model identification of airplanes using noisy flight data
6
Kivonat
A repülıgépek irányításának problémája egy kiterjedt és rohamosan fejlıdı ága a
szabályozástechnikának. A repülıgépek fedélzeti számítógépeire implementált
szabályozások nemcsak a robotpilóta funkciót valósítanak meg, hanem komoly
szerepet játszanak a pilóta munkájának támogatásában is. Ezek a rendszerek az ún.
stabilitás (SAS) és szabályozás (CAS) kiterjesztı rendszerek. Ezen eljárásoknak számos
civil alkalmazása van azonban továbbra is a katonai alkalmazások motiválják a terület
fejlıdését. A precíz szabályozások tervezéséhez elengedhetetlenül fontos, hogy
ismerjük a repülıgép pontos nemlineáris dinamikus modelljét. A modell matematikai
alakja egzakt módon elıáll, azonban a modell paramétereinek meghatározása komoly
méréstechnikai és jelfeldolgozási feladat.
A nemlineáris modell paramétereinek kiszámítása a rendszerek identifikációjának
részét képezı tudományághoz tartozik. Jelen munkában feldolgozásra és bemutatásra
kerülnek a frekvencia és idıtartománybeli Maximum Likelihood (ML) módszerek,
különös tekintettel a kimeneti hiba (OE) módszerekre. A módszerek demonstrálása
során felhasználásra kerülnek a SIDPAC identifikációs Toolbox [1] függvényei.
Ezután virtuális tesztrepülések során MATLAB/Simulink környezetben adatok
győjtésére kerül sor. A szimulált adatsorokhoz zajokat és egyéb zavaró hatásokat adok
a valós repülési körülmények modellezése érdekében. Végül a fent említett szőrési és
identifikációs eljárások alkalmazása történik a szimulált adatsorokra. A dolgozatot a
kapott eredmények és a jövıbeli fejlesztések összefoglalása zárja.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-6-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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physical system based on observations are called identification methods. In order
reducing the adverse effects of noises require the introduction of statistical theory
and methods. The most important requirement for a mathematical model is to be
applicable in many ways. It must be simple enough to be useful and at the same
time complex enough to represent perfectly the physical world. Hence, system
identification takes a significant part in the field of control engineering.
In this MSc thesis, several identification methods such as the time domain
Maximum Likelihood (ML) and frequency domain methods are reviewed, based on
[1]. Particular focus is placed on the output-error method based on [2] and [1].
Some important methods for filtering and smoothing of signals playing important
roles in identification [1] are also studied. Methods mentioned above are
demonstrated in MATLAB environment using the SIDPAC Toolbox applying the
flight data from the literature [1]. In the following, simulated data sequences are
gathered by virtual test flights applying the nonlinear dynamic model of a DHC-2
Beaver aircraft. The sense of this experiment is to verify the applied identification
techniques and to ensure about its applicability in the case of data sequences
gathered on a real aircraft. In order to model real life situations, additional noises
and disturbances are added to signals. Then, filtering methods written in [1] are
used to eliminate the adverse effects of the noises. Time and frequency domain
output-error methods are then applied to determine the parameters of the Beaver
aircraft. Furthermore, given results are compared with the real parameter values.
Eventually, a summary and a conclusion about the results and the future research
plans close this MSc thesis work.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-12-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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2 Review of time-domain and frequency domain Maximum
Likelihood identification methods
In this chapter the time domain Maximum Likelihood (ML) and the frequency domain
ML identification methods are summarized for later application. Also the fundamentals
of influential directions are summarized. Particular focus is placed on the output-error
(OE) method. The theory written in this chapter is based on the descriptions and
definitions can be found in [1] and [2].
2.1 Why the Maximum Likelihood methods
During aircraft system identification, the applied measured time series are generated
by nonlinear dynamic systems and corrupted by noises. Because of this, a very simple
idea is used: assuming a model that depends on the unknown parameters, then
minimizing a cost function in order to find the values of the parameters that make the
outputs of the identified model most likely to correlate to the measured data.
2.2 Outline of estimation theory
The parameter estimation consists in finding adequate values of unknown model
parameters θ in an assumed model, based on noisy measurements z . Since the
estimator computes approximate values of the parameter vector ˆθ based on noisy
measured data z , hence ˆθ is a random variable. In order to develop an estimator, four
specifications are required:
(i) A model structure with unknown parameters;
(ii) Measurements z ;
(iii) A mathematical model of the process;
(iv) Assumptions about the uncertainties in the measurement noise v , and in
the unknown model parameters.
Observations can be linear or nonlinear functions of the parameters. A model is called
linear in the parameters if its output y is given by
y Hθ= , (2.1)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-13-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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where the matrix H is assumed to be known. The measurement equation then can be
expressed as
z H vθ= + . (2.2)
A model is nonlinear in the parameters, if its measurement equation is given by
( )z h vθ= + , (2.3)
where ( )h θ is assumed to be a known function. Generally, there are 0n measured
outputs, a measured data vector consisting of N data points. Three models of
uncertainties in the parameters and the observations are considered. They are known
as the Bayesian model, the Fisher model and the least-squares model. The parameters
and measurements are formed as follows:
Bayesian model
θ is a vector of random variables with probability density ( )p θ ;
v is a random noise sequence with a probability density ( )p v .
Fisher model
θ is a vector of unknown constant parameters;
v is a random noise sequence with probability density ( )p v .
Least-squares model
θ is a vector of unknown constant parameters;
v is a random vector of measurement noise.
The time domain ML and frequency domain ML estimator were developed for
the Fisher model. θ is the vector of unknown constant parameters and v has an
expected value or mean that equal to 0, and a variance R . This can be written in the
form (0, )Rℕ . The Fisher model is based on the Fisher estimation theory. Detailed
description can be found in [1]. The concept of the likelihood function is used,
where ( ; ) ( )L z p zθ θ= . The parameter vector θ is assumed to be a vector of unknown
constants. Hence, the most common estimator of the Fisher model is to maximize the
likelihood function, which is equal to the value of the parameter vector that maximizes
( ; )L z θ for given z . If ( )p z is Gaussian, the likelihood function has the form
1
12
1
( ; ) (2 ) exp ( ) ( )
2
N T
L z R z H R z Hθ π θ θ
− −
= − − −
. (2.4)
where i denotes the determinant.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-14-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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Then the estimate
ˆ max ( ; )L z
θ
θ θ= , (2.5)
minimizes
11
( ) ( ) ( )
2
T
J z H R z Hθ θ θ−
= − − . (2.6)
Or in the case of nonlinear measurement equation
11
( ) ( ( )) ( ( ))
2
T
J z h R z hθ θ θ−
= − − . (2.7)
The Fisher information matrix M is defined as
2
ln ln ln
( ) ,
T
T
L L L
M E Eθ
θ θ θ θ
∂ ∂ ∂
= = −
∂ ∂ ∂ ∂
(2.8)
This equality is discussed in detail in [1].
2.3 Time-domain ML methods
It can be seen below that in the case of a linear measurement equation z H vθ= + the
ML estimator becomes a simple least-square estimator with weighting equal to the
inverse of the noise covariance matrix. Furthermore, in the case of a stochastic
dynamic system with process noise, the measurements are nonlinear functions of the
parameters. Thus, the general form of the measurement equation is ( )z h vθ= + .
The solution combines a state estimator and a nonlinear parameter estimator.
The models to be estimated are in state-space form. Hence, the state estimator is a
Kalman filter which is necessary because of the process noise in the dynamic equations.
This method is also known as the filter-error method. The nonlinear parameter
estimator is required because of the nonlinear measurement equation, which means
that there is nonlinear connection between model parameters and measured outputs.
However, the combined estimation of parameters and states induce a difficult nonlinear
estimation problem. For further practical implementation, three simplifications are
made:
(i) The Gauss-Newton method, also known as the modified Newton-
Raphson method is used for minimizing the cost function instead of the
Newton-Raphson method;](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-15-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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(ii) In the case of linear dynamic models, the general Kalman filter is
replaced by its steady-states form;
(iii) The unknowns are divided into three subsets: states, model parameters
and the elements of the noise covariance matrix.
In the following section, the filter-error method is introduced briefly. It is
followed by the properties of ML methods. Properties of ML estimators are also
introduced. For further practical applications, the assumption of no process noise is
made. In this case, the states can be derived by direct numerical integration. The
resulting ML estimator is known as the output-error method. This method is a
generalization of the nonlinear least squares estimator presented in [1]. Flight testing
is done preferably on a day with calm air in order to eliminate the process noise in the
model. Finally, the equation-error method is presented. If the state variables are
measured without error, the equation-error version of the ML estimator can be used
that is identical to linear regression.
2.3.1 Filter-error method
In most cases of practical system identification problems, such as the aircraft dynamic
modeling problem, discrete-time measurements made on a continuous-time dynamic
system are given. A stochastic forcing term is added to the system to represent
inadequacies of the linear model structure. The model equations then can be written
as:
( ) ( ) ( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ),
1,2,..., ,
wx t A x t B u t B w t
y t C x t D u t
z i y i v i
i N
θ θ θ
θ θ
= + +
= +
= +
=
ɺ
(2.9)
[ ]
[ ][ ]{ }
0
0 0 0
(0) ( ),
(0) ( ) (0) ( ) ( ),
T
E x x
E x x x x P
θ
θ θ θ
=
− − =
(2.10)
where ( )w t is the process noise and ( )v i is the measurement noise, and they are white
noise sequences.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-16-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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[ ]
[ ]
( ) 0,
( ) 0,
( ) ( ) ( , ) ( ),
( ) ( ) ( , ) ,
T
i j i i j
T
ij
E w t
E v i
E w t w t Q t t t
E v i v j R i
θ δ
θ δ
=
=
= −
=
(2.11)
where θ is the unknown parameter vector.
The random vectors ( )w t and ( )v i are assumed to be white and uncorrelated. The
Q and R matrices are usually diagonal. The likelihood function for a sequence of
measurements will be denoted ( ; )NL Z θ that means [ ](1), (2),..., ( )
T
NZ z z z N= . The
expression of the likelihood function can be derived by successive applications of the
Bayes’s rule.
[ ] [ ]
[ ]
1 1
1 2 2
1
1
( ; ) (1), (2),... ( ), ( ) ; ;
( ) ; ( 1) ; ; ...
( ) ; .
N N N
N N N
N
i
i
L Z z z z N L z N Z L Z
L z N Z L z N Z L Z
L z i Z
θ θ θ θ
θ θ θ
θ
− −
− − −
−
=
= =
= − =
= ∏
(2.12)
As was mentioned below, in order to find an optimal parameter vector, the likelihood
function has to be maximized. However, for computational purposes, it is
advantageous to minimize the negative logarithm of the likelihood function instead.
Thus, the ML estimator can be written in the form of
[ ] { }1 1
11
ˆ max ; max ( ) ; min ( ) ;
N N
N i i
ii
L Z L z i Z ln L z i Z
θθ θ
θ θ θ θ− −
==
= = = − ∑∏ . (2.13)
If ( )w t and ( )v i are normally distributed and independent, then ( )z i will also have the
same properties.
The following notation will be used: ˆ( 1)x i i − is the best estimate of the state
ˆ( )x i using measurements [ ](1), (2),..., ( 1)z z z i − . Then [ ]1( ) ; ( );iL z i Z L z iθ θ− = will
be determined by the mean and its covariance matrix. These quantities can be
expressed as
[ ]
[ ] { }
ˆ( ); ( 1)
ˆ ˆ( ); ( ) ( 1) ( ) ( 1)
( ) ( ) ( ),
T
T
E z i y i i
Cov z i E z i y i i z i y i i
i i i
θ
θ
= −
= − − − −
= = v v BBBB
(2.14)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-17-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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where
ˆ( ) ( ) ( 1)i z i y i i= − −v , (2.15)
is the vector of innovations and ( )iBBBB is the innovation covariance matrix. Hence, the
probability density of the innovation approaches a Gaussian distribution, as the
sampling rate increases. In the case of a sufficiently high sample rate, the likelihood
function has the form of
[ ]
1
1
2
1
( ); (2 ) ( ) exp ( ) ( ) ( ) .
2
N T
L z i i i i iθ π
− −
= −
v vB BB BB BB B (2.16)
Then, the negative log-likelihood function can be expressed as
( ) 1 0
1
1
; ( ) ( ) ( ) ( ) (2 ),
2 2
N
T
N
i
Nn
ln L Z i i i ln i lnθ π−
=
− = + + ∑ v vB BB BB BB B (2.17)
where 0n is the number of output variables. It can be seen that the last term on the
right side is a constant, and it has no effect on the optimization. Thus, it can be
neglected, which results in
( ) 1
1
1
; ( ) ( ) ( ) ( ) .
2
N
T
N
i
ln L Z i i i ln iθ −
=
− = + ∑ v vB BB BB BB B (2.18)
In order to determine a negative log-likelihood function the mean value and the
covariance matrix of the innovation have to be calculated. These quantities can be
derived using a Kalman filter. The filtering has two steps, consisting of prediction and
measurement update. The following equations are derived for linear time-invariant
system.
Initial condition:
[ ]
[ ][ ]{ }
0
0 0 0
(0) (0) ( ),
(0) (0) (0) ( ),
T
x E x x
P E x x x x P
θ
θ
= =
= − − =
(2.19)
Prediction:
ˆ ˆ( 1) ( ) ( 1) ( ) ( ),
( 1) ( ) ( 1) ( 1) ( ) ( ) ( ),
( 1) ,
T T
w w
d
x t i A x t i B u t
dt
d
P t i A P t i P t i A B QB
dt
i t t i t
θ θ
θ θ θ θ
− = − +
− = − + − +
− ∆ ≤ ≤ ∆
(2.20)
Measurement Update:](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-18-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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[ ]
1
ˆ ˆ( ) ( 1) ( ) ( ),
( ) ( 1) ( ) ( 1),
( ) ( ) ( ) ( 1),
T
x i i x i i K i i
K i P i i C i i
P i i I K i C P i i
θ
θ
−
= − +
= − −
= − −
v
BBBB (2.21)
where
( 1) ( ) ( 1) ( ) ( ),
ˆ( ) ( ) ( ) ( 1) ( )( ),
T
i i C P i i C R
i z i C x i i D i
θ θ θ
θ θ
− = − +
= − − −v
BBBB
(2.22)
Optimization algorithm
The ML parameter estimates are obtained by minimizing the negative log-likelihood
function
1
1 1
1 1
( ) ( ) ( ) ( ) ( ).
2 2
N N
T
i i
J i i i iθ −
= =
= +∑ ∑v vB BB BB BB B (2.23)
In order to solve this optimization problem, generally, the Newton-Raphson method is
used. To determine a solution to the optimization-problem given in (2.23), the cost
function θ is expanded in a Taylor-series around a nominal parameter estimation 0θ .
The choice of this nominal parameter estimation plays an important role in both the
convergence and computational time of the optimization. The Taylor-series expansion
is given by
0 0 0
1
( ) ( ) '( ) "( ) ...,
2
T T
J J J Jθ θ θ θ θ θ θ= + ∆ + ∆ ∆ + (2.24)
We approximate ( )J θ with the second-order expansion of its Taylor-series,
0 0 0
1
( ) ( ) '( ) "( ) ,
2
T T
J J J Jθ θ θ θ θ θ θ≈ + ∆ + ∆ ∆ (2.25)
where '( )J θ is the Jacobian, "( )J θ is the Hessian of the cost function at the nominal
parameter estimation and θ∆ is the vector of changes inθ .
( )
(1)
( )
( )
(2)'( ) ,
( )
( )p
J
J
J
J
J
n
θ
θ
θ
θ
θθ
θ
θ
θ
∂
∂
∂
∂ ∂= = ∂
∂
∂
⋮
(2.25)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-19-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
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2 2 2
2 2 2
2
2 2 2
( ) ( ) ( )
(1) (1) (1) (2) (1) ( )
( ) ( ) ( )
( )
(2) (1) (2) (2) (2) ( )"( ) .
( ) ( ) ( )
( ) (1) ( ) (2) ( ) ( )
p
pT
p p p p
J J J
n
J J J
J
nJ
J J J
n n n n
θ θ θ
θ θ θ θ θ θ
θ θ θ
θ
θ θ θ θ θ θθ
θ θ
θ θ θ
θ θ θ θ θ θ
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂
∂
∂ ∂ ∂ ∂ ∂ ∂= =
∂ ∂
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
⋯
⋯
⋮ ⋮ ⋱ ⋮
⋯
(2.26)
The necessary condition to minimize ( )J θ becomes
0 0'( ) "( ) 0.J Jθ θ θ+ ∆ = (2.27)
The solution of this equation gives an estimate for the vector of parameter changes,
[ ]
1
ˆ "( ) '( ).J Jθ θ θ
−
∆ = − (2.28)
We assumed that the Hessian is nonsingular. Since the first and second order gradient
of the cost function are computed around a nominal parameter estimation 0θ , the
updated parameter vector estimation ˆθ is computed by,
0
ˆ .θ θ θ= + ∆ (2.29)
Because of the approximation in the Taylor series, it is necessary to repeat the
estimation using the estimated vector ˆθ as the new nominal value 0θ . That is
important because the outputs are nonlinear functions of the parameters, thus the cost
function is more complicated than quadratic. In the case of linear least-squares
problems, which are discussed in detail in [1], the output depends linearly on the
parameters. Thus, the cost function is a quadratic function of the parameters and the
term given in (2.25) is an exact equality. Accordingly, the parameter estimates could
be derived in one single iteration. In this case, the repeated applications of the
quadratic cost to the nonlinear function on the parameters are used to iteratively
arrive to the solution. This iterative process will be arrived at when the convergence
criteria are satisfied. Convergence criteria that have been used in practice are
introduced later in this section.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-20-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
21
Minimization of the negative-log likelihood function for the filter-error problem is a
very difficult optimization problem. Computing the second order gradients in the
Hessian matrix is an extremely time demanding task and is not robust enough. Hence,
for practical applications the modified Newton-Raphson method is used. Details of this
approach are introduced later. A block diagram of the filter-error method is shown
below in Figure.2.1.
Figure 2.1 Block diagram of filter-error method
Simplification possibilities
In order to obtain simplifications of this optimization problem, several solutions are
introduced in [1] and [2]. Comparisons and summaries of these simplifications are
also outlined in [1]. The ML estimation algorithm for a time-invariant stochastic
linear dynamic system can be simplified through replacing the Kalman filter by its
steady-states form.
Initial condition:
[ ]
[ ][ ]{ }
0
0 0 0
(0) (0) ( ),
(0) (0) (0) ( ),
T
x E x x
P E x x x x P
θ
θ
= =
= − − =
(2.30)
Prediction:
ˆ ˆ( 1) ( ) ( 1) ( ) ( ),
( 1) ,
d
x t i A x t i B u t
dt
i t t i t
θ θ − = − +
− ∆ ≤ ≤ ∆
(2.31)
Measurement update:
1
ˆ ˆ( ) ( 1) ( ),
( ) ,T
x i i x i i K i
K PC θ −
= − +
=
v
BBBB
(2.32)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-21-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
22
where
( ) ( ) ( ),
ˆ( ) ( ) ( ) ( 1) ( ) ( ).
T
C PC R
i z i C x i i D u i
θ θ θ
θ θ
+
= − − −v
B =B =B =B =
(2.33)
In this case, the steady-state form of the Riccati equation can be solved for P.
1
( ) ,T T T T
w wP P PC CPC R CP Q−
= Φ − + Φ + Γ Γ (2.34)
Φ and wΓ can be written in the form:
1
,
( ) .
A t
A t
w w
e
A e I B
∆
− ∆
Φ =
Γ = −
(2.35)
This means that using a steady-state filter brings simplification to the equation for the
first order gradients of the cost function, because the partial derivatives of the cost
function with respect the unknown parameter vector are calculated from a linear
matrix equation. Henceforward, the estimation algorithm has some difficulties.
Further simplification to algorithm can be obtained by recasting the vector of
unknown parameters. The elements of K and BBBB are included into the parameter
vector, instead of the elements of Q and R. This leads to a new cost function in which
an approximation of the covariance matrix is used according to (2.36) that is derived in
[1].
1
1ˆ ( ) ( ).
N
T
i
i i
N =
∑B =B =B =B = v v (2.36)
The resulting cost can be minimized by using a nonlinear optimization technique, such
as the Newton-Raphson method as it was mentioned earlier. This is also an iterative
optimization procedure that continues until selected convergence criteria are satisfied.
In this case, the unknown parameters are estimated in subsets. First, the parameters
contained in A,B,C,D and K held constant while the estimation of the parameters in BBBB
is in progress, and then the parameters contained in BBBB are held constant while the
parameters in A,B,C,D and K are estimated. The estimate of the measurement
covariance matrix can be derived from (2.32-33):
ˆ ˆˆ ˆ( )R I CK= − B.B.B.B. (2.37)
This method carries also many problems in the computation of wB and Q in (2.32-33).
Because of these difficulties in the simplified algorithms a third formulation was](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-22-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
23
proposed. It is advantageous to recast the unknown parameters as the elements of A,
B, wB , C, D, Q and BBBB .
The resulting optimization algorithm has six steps:
(i) Assembling the system and covariance matrices A, B, wB , C, D, Q and
R using the nominal parameter values;
(ii) Computing the covariance matrix of the innovations (2.36);
(iii) Estimating of the unknown parameters in the state and covariance
matrices, using a nonlinear optimization technique;
(iv) For ˆ=B BB BB BB B , compute ˆK and ˆR from (2.32) and (2.37);
(v) Update the nominal values of the unknown parameters;
(vi) Repeat steps 1-5 until convergence criteria are satisfied.
The filter-error method is the most general estimation method for practical
aircraft system identification problems because it allows the presence of both the
measurement and process noise in the model. However, the main practical problem
with this method is that there are a lot of parameters to be estimated which often leads
to difficulties.
2.3.2 Properties of ML methods
The accuracy of the estimation concerns itself with the properties of the estimates.
Using the definitions introduced in [1].
(i) ML estimates of dynamic system parameters are asymptotically unbiased;
ˆ( ) ,
.
E
N
θ θ→
→ ∞
(ii) ML estimates are consistent;
ˆ ,
.N
θ θ→
→ ∞
(iii) ML estimates are asymptotically efficient;
1ˆ( ) ,
,
Cov M
N
θ −
→
→ ∞
where M the Fisher information matrix defined earlier by (2.8).](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-23-320.jpg)
 ( ) ( ) .T
ijCov v i E v i v i Rδ = =
Furthermore, because of the lack of process noise, Q is zero and the state equations are
deterministic. The Kalman gain K is also zero, so the Kalman filter can be replaced by
a simple integration of the state equations. The innovations in the method become:
( ) ( ) ( ) ( ) ( ) ( ) ( ), 1,2,..., .z i y i z i C x i D u i i Nθ θ− = − − =v = (2.39)
Practically, they are equal to output errors. The negative log-Likelihood function then
can be written as:
( ) -1 0
1
1
; ( ) ( ) (2 ),
2 2 2
N
T
N
i
NnN
ln L Z i R i ln R lnθ π
=
− = + + ∑v v (2.40)
The two last terms of (2.40) can be dropped because they do not depend on the
unknown parameter vector θ . The unknown parameters are elements of state matrices
as can be seen above and the initial condition vector. Solving (2.40) with respect to R is
done by differentiating with respect to R. The result then is set equal to zero and
solved for R. That gives (2.41) which is similar to (2.36).
1
1ˆ ( ) ( ).
N
T
i
R i i
N =
= ∑v v (2.41)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-24-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
25
Generally, only the diagonal elements of the R matrix are estimated from (2.40)
assuming that the measurement noises are uncorrelated with one another. For a given
ˆR the cost function can be written as:
[ ] [ ]1 1
1 1
1 1
( ) ( ) ( ) ( ) ( ) ( ) ( ) .
2 2
N N
TT
i i
J i R i z i y i R z i y iθ − −
= =
= = − −∑ ∑v v (2.42)
This cost is minimized by using a relaxation technique in which first θ held fixed
while computing ˆR from (2.41), then minimizing the cost with respect to θ for a
given fixed R, where ˆR R= . These two steps are repeated until convergence criteria
are satisfied. Optimization of the cost in (2.42) can be done using the Newton-Raphson
method as discussed earlier. The first and second order gradient can be calculated by
(2.25-26). The first order gradient has the following form:
1 1
1 1
( ) ( ) ( )ˆ ˆ( ) ( ),
T TN N
i i
J i y i
R i R i
θ
θ θ θ
− −
= =
∂ ∂ ∂
= = −
∂ ∂ ∂
∑ ∑
v
v v (2.43)
where the elements of the
( )J θ
θ
∂
∂
vector can be written as
1
1
( ) ( ) ˆ ( ), 1,2,... .
TN
p
ij j
J y i
R i j n
θ
θ θ
−
=
∂ ∂
= − =
∂ ∂
∑ v (2.44)
The elements of the second order gradient then have the form of
2 2
1 1
1 1
( ) ( ) ( ) ( )ˆ ˆ ( ), , 1,2,...
TN N
p
i ij k j k j k
J y i y i y i
R R i j k n
θ
θ θ θ θ θ θ
− −
= =
∂ ∂ ∂ ∂
= =
∂ ∂ ∂ ∂ ∂ ∂
∑ ∑- v (2.45)
The term on the far right side can be neglected. Thus, the resulting optimization
method is called Gauss-Newton method or modified Newton-Raphson method. This
simplification is made for practical reasons. Obtaining the second-order gradient is
computationally expensive. Furthermore, it is susceptible to numerical errors. The
parameter vector change then can be written using the approximate second-order
gradient:
1
1 1
1 10 0 0
( ) ( ) ( )ˆ ˆ ˆ ( ) ,
TN N
i i
y i y i y i
R R iθ
θ θ θ
−
− −
= =
∂ ∂ ∂
∆ =
∂ ∂ ∂
∑ ∑ v (2.46)
where
y
θ
∂
∂
is an 0 pn n× matrix called the output sensitivity matrix and its elements
are called the output sensitivities. The elements of this matrix quantify the changes in
the outputs due to changes in the parameters. 1ˆR−
is usually diagonal. According to
(2.46), it can be seen that output sensitivities must be linearly independent and](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-25-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
26
nonzero for a good matrix inversion. When this condition is held, each model
parameter has a unique and significant impact on the model outputs. Henceforward, in
the case when the approximation for the second-order gradient of the cost function is
used, the Fisher information matrix can be written as
2
1
1
ln ( , ) ( ) ( )ˆ( ) .
TN
N
T
i
L Z y i y i
M E R
θ
θ
θ θ θ θ
−
=
∂ ∂ ∂
= =
∂ ∂ ∂ ∂
∑ (2.47)
Then the ML estimator has the form of:
1 0
0 0
0
( )ˆ ( ) .
J
M
θ
θ θ θ
θ
− ∂
= −
∂
(2.48)
The parameter covariance matrix satisfies the Cramer-Rao inequality
1ˆ( ) ( ) ,Cov Mθ θ −
≥ (2.49)
which means that the inverse of the Fisher information matrix gives the lower bound
for the parameter covariance matrix. A block diagram of the output-error method can
be seen in Fig 2.2.
Figure 2.2 Block diagram of output-error method
Convergence criteria, computing sensitivities and nearly singular information
matrix
It was shown in [1] that the practical application of ML estimator to flight data
sometimes leads difficulties. For instance, the dependence in the likelihood function
can be far from quadratic. Furthermore, the information matrix can be singular or
nearly singular. The optimization method for the output-error method discussed
earlier can be summarized as:](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-26-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
28
Some convergence criteria that worked well for aircraft parameter estimation
problems in [1] are introduced here.
5
1( ) ( ) 1 10 , 1,2,... ,j k j k pj nθ θ −
−− < × = (2.51a)
or
1
1
ˆ ˆ
0.001,
ˆ
k k
k
θ θ
θ
−
−
−
< (2.51b)
where k denotes the kth iteration and denotes the Euclidien norm. This can also
be written:
1
1
1
0,
1
( ) ( )
0.001,
( )
ˆ ˆ( ) ( )
0.05, 1,2,...
ˆ( )
k k
k
jj k jj k
jj k
J J
J
r r
j n
r
θ θ
θ
−
−
−
−
−
<
−
< =
(2.52)
where ˆjjr denotes the estimate of the jth diagonal element of ˆR . Furthermore, the
magnitude of the changes can be varied by directly applying a multiplicative term in
θ∆ :
0 0
ˆ ( ) ( ).cM gθ θ θ∆ = − (2.52)
Two methods can be used to compute the elements of the sensitivity matrix. These
two methods are the analytical approach and the numerical approach.
The analytical approach uses the output equations of the system with respect to the
unknown parameters , 1,2,...j pj nθ = . In the case of the linear system given by (2.38)
this can be written as:
, 1,2,... .p
j j j j
y x C D
C x u j n
θ θ θ θ
∂ ∂ ∂ ∂
= + + =
∂ ∂ ∂ ∂
(2.53)
The states can be derived from the state equations of the dynamic system:
0
,
(0).
x Ax Bu
x x
= +
=
ɺ
(2.54)
The state sensitivities are computed by solving the state sensitivity equations, which
are obtained by differentiating the state equations with respect to the unknown
parameters:
, 1,2,... .p
j j j j
d x x A B
A x u j n
dt θ θ θ θ
∂ ∂ ∂ ∂
= + + = ∂ ∂ ∂ ∂
(2.55)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-28-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
31
1 1
0( ) ,M M kA− −
= + (2.63)
here 0M is the original information matrix, k is a positive nonzero scalar parameter
and A is positive definite matrix. Generally, A is chosen as the identity matrix,
1 1
0( ) .M M kI− −
= + (2.64)
The scalar parameter k can be obtained by an iterative procedure. An initial value for k
is recommended, as 0.01k = .
Bayes-like method
This method improves the conditioning of the information matrix by combining the
measured data with prior estimates of some or all of the parameters in the model. The
resulting cost function is similar to that for the Bayesian estimator [1].
1 1
1
1 1
( ) ( ) (i) + ( ) ( ),
2 2
N
T T
p p p
i
J i Rθ θ θ θ θ− −
=
= − ∑ −∑v v (2.65)
where pθ is the vector containing the prior parameter values, and p∑ is a positive
semidefinite covariance matrix that represents the confidence in the parameters of pθ .
θ∆ then can be computed by using the modified Newton-Raphson method:
1 1 1
0 0
ˆ ( ( ) ) ( ) ,p p pM gθ θ θ θ− − −
∆ = − + ∑ + ∑ − (2.66)
in the cases when the information matrix is nearly singular. In other words, it is
singular only at certain points in the progression towards the global optimum. In
these cases, the methods discussed above can be used temporarily. The rank deficiency
method is the easiest to use in this way. Computing the inverse of the information
matrix is always done by calculating the SVD. The singular value ratios are checked
against the criteria (2.62). If any singular values are too small, their corresponding
terms will be dropped from the inverse according to (2.60-61).
2.3.4 Equation-error method
Another parameter estimation method can be obtained by assuming that
(i) the dynamic model is in linear states-space form with process noise,
(ii) the parameters to be estimated are dimensional stability and control
derivatives,
(iii) all of the states can be measured without error.
The model then has the form of](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-31-320.jpg)
 0,
( ) ( ) .T
ij
E i
E i j Rδ
=
v
v v
(2.68)
The negative log-likelihood function then can be written in the form of
[ ] [ ]1
1
0
1
L(Z ; ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2
(2 ).
2 2
TN
N
i
ln z i A x i B u i R z i A x i B u i
NnN
ln R ln
θ θ θ θ θ
π
−
=
− − − − −
+ +
∑
(2.69)
When ˆR is known, the cost function to be minimized has the form of
[ ] [ ]1
1
1 ˆ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .
2
TN
i
J z i A x i B u i R z i A x i B u iθ θ θ θ θ−
=
− − − −∑ (2.70)
In this case, the innovations are the equation errors,
( ) ( ) ( ) ( ) ( ) ( ), 1,2,... .i z i A x i B u i i Nθ θ= − − =v (2.71)
It is simple to use output equations instead of dynamic state equations. In (2.67) can be
changed to use measured output in the state-space formulation:
( ) ( ) ( ) ( ) ( ) ( ), 1,2,... .z i C x i D u i v i i Nθ θ= + + = (2.72)
As a matter of fact, the measurements are always corrupted by errors; hence, the
resulting parameters are biased estimates of the unknown model parameters in the
state matrices A,B,C and D. A block diagram of the equation-error method is shown
below Figure 2.3.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-32-320.jpg)
 ( ) ( ) ,j t
x t x x t e dtω
ω
∞
−
−∞
= = ∫ɶFFFF (2.73)
where 1j = − and ω is the angular frequency in radian per second. ( )x ωɶ is called
the Fourier transform of ( )x t . The ( )x ωɶ can be transformed back to the time-domain
by using the inverse Fourier-transformation.
1
( ) ( ) ,
2
j t
x x e dω
ω ω ω
π
∞
−∞
= ∫ɶ ɶ (2.74)
These are also known as the Fourier transform pair. The Fourier transformation of a
function on a finite time interval is called the finite Fourier transform. The expressions
of the finite Fourier transform have the form
0
( ) ( ) ,
T
j t
x x t e dtω
ω −
= ∫ɶ (2.75)
or
2
0
( ) ( ) ,
T
j ft
x f x t e dtπ−
= ∫ɶ (2.76)
where
2 fω π= (2.77)
When the function ( )x t is sampled at discrete and evenly spaced time intervals t∆ , the
finite Fourier transform can be approximated by
1
2
0
( ) ( ) ,
N
j fi t
i
x f t x i e π
−
− ∆
=
≈ ∆ ∑ɶ (2.78)
where the time length of measured data is ( 1)T N t= − ∆ and
/ ( 1),
( ) ( ) ( ),
.
i
i
t T N
x i x t x i t
t i t
∆ = −
= = ∆
= ∆
(2.79)
In the case of conventional Fourier transform, the frequencies are chosen as](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-34-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
35
, 0,1,2..., 1k
k
f k N
N t
= = −
∆
(2.80)
2 2 , 0,1,2,... 1k k
k
f k N
N t
ω π π= = = −
∆
(2.81)
Using these frequencies mentioned below, the approximation of the Fourier transform
can be written
1
(2 / )
0
( ) ( ) , 0,1,2,..., 1
N
j k N i
i
x k t x i e k Nπ
−
−
=
= ∆ = −∑ɶ ɶ (2.82)
and its inverse
1
(2 / )
0
1
( ) ( ) , 0,1,2,..., 1
N
j k N i
k
x i x k e i N
N t
π
−
=
= = −
∆
∑ ɶ (2.83)
When the sampling intervals in (2.82) and (2.83) are normalized, the result is called
the discrete Fourier transform (DFT).
1
(2 / )
0
( ) ( ) , 0,1,2,..., 1
N
j k N i
i
X k x i e k Nπ
−
−
=
= = −∑ (2.84)
1
(2 / )
0
1
( ) ( ) , 0,1,2,..., 1
N
j k N i
k
x i X k e i N
N
π
−
=
= = −∑ (2.85)
Computing directly the DFT using (2.84) is not economical computationally. An
efficient method to compute the DFT is the fast Fourier transform (FFT). In practical
aircraft model estimation problems, there is no need to compute the Fourier
transformation in the whole frequency band [ ]0, Nf . A method based on the chirp z-
transform that can be compute the finite Fourier transformation with very high
accuracy at arbitrarily selected frequencies is discussed in the next chapter.
2.4.1 Frequency domain ML estimator for linear dynamic model
In order to derive frequency domain ML methods, it is first necessary to transform the
linear dynamic model to the frequency domain. Model equations for linear continuous-
time dynamic system with process noise and discrete-time measurements have the
form:
[ ]
0
( ) ( ) ( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) ( ), ( ),
(0) 0,
(0) (0) , 1,2,... .
w
T
x t A x t B u t B w t
z i C x i D u i v i
E x
E x x P i N
θ θ θ
θ θ
= + +
= + +
=
= =
ɺ
(2.86)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-35-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
36
where noises ( )w t and ( )v i are assumed to be independent and white. They have the
following properties:
[ ]
[ ]
( ) 0,
( ) 0,
( ) ( ) ( ),
( ) ( ) .
T
j i
T
ij
E w t
E v i
E w t w t Q t t
E v i v i R
δ
δ
=
=
= −
=
(2.87)
The unknown parameter vector θ occurs in the system and covariance matrices. As
was shown in the previous section, estimation of all of the parameters is extremely
difficult. This problem can be simplified by replacing the Kalman filter with its steady-
state version. Using terms from (2.14) and (2.30-33), the modified filter equations can
be written in the form
ˆ ˆ( 1) ( ) ( 1) ( ) ( ), ( 1) ,
ˆ ˆ( ) ( 1) ( )
d
x t i A x t i B u t i t t i t
dt
x i i x i i K (i),
θ θ
θ
− = − + − ∆ ≤ ≤ ∆
= − + v
(2.88)
where the innovations
ˆ( 1) ( ),(i)= z(i)-Cx i i Du i− −v (2.89)
with
[ ] 0,
.T
ij
E (i)
E (i) (i) δ
=
=
v
v v BBBB
(2.90)
The elements of the unknown parameter vector are the elements of A,B,C,D,K and B.
Then, the equations are written in the form of their Fourier series expansion. The
components of ( )x i can be written by (2.78) as
1
(2 / )
0
( ) ( ) , 0,1,2,... 1
N
j k N i
i
x k t x i e k Nπ
−
−
=
= ∆ = −∑ɶ (2.91)
As was mentioned earlier, the finite Fourier transform can be approximated by (2.91)
and for practical applications it can be replaced by a high-accuracy calculation
presented in the next chapter. Dynamic equations now can be written as
( ) ( ) ( ) ( ) ( ) ( ),kj x k A x k B u k K kω θ θ= + + ɶɶ ɶ ɶ v (2.92)
or
1 1
( ) ( ( )) ( ) ( ) ( ( )) ( )
( ) ( ) ( ) , 0,1,2,... 1
k kx k j I A B u k j I A K (k),
z(k)= C( )x k D u k (k) k N
ω θ θ ω θ θ
θ θ
− −
= − + −
+ + = −
ɶɶ ɶ
ɶɶ ɶɶ
v
v
(2.93)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-36-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
37
where kω is the kth frequency. The transformed innovations ( )kɶv are uncorrelated
Gaussian random variables.
[ ] 0,
,T vv
E (k)
S
E (k) (k)
N
=
=
ɶ
ɶ ɶ
v
v v
(2.94)
where vvS is a real matrix with diagonal elements equal to the power spectral densities
of the elements of ( )iv , and †
(i)ɶv is the complex conjugate transpose of ( )iv . The left
side of (2.93) requires that the trend and bias term be removed from the sequence ( )x t ,
since
[ ]
0 0
( ) ( ) ( ) ( ) (0)
( ) ( ) (0),
T T
j t j t j T
j T
x t x t e dt j x t e dt x T e x
j x x T e x
ω ω ω
ω
ω
ω ω
− − −
−
= = + −
= + −
∫ ∫ɺ ɺ ɺ
ɶ
FFFF
(2.95)
It can be written from (2.93)
1
1
1 2
( ) ( )( ( )) ( ) ( ) ( )
( )( ( )) ( )
( , ) ( ) ( , ) ),
k
k
z k C j I A B D u k
C j I A K I (k)
= G k u k G k (k
θ ω θ θ θ
θ ω θ θ
θ θ
−
−
= − +
+ − +
+
ɶɶ
ɶ
ɶɶ
v
v
(2.96)
where G1 and G2 are the system transfer matrices.
1
1
1
2
( , ) ( )( ( )) ( ) ( ) ,
( , ) ( )( ( )) ( ) .
k
k
G k C j I A B D
G k C j I A K I
θ θ ω θ θ θ
θ θ ω θ θ
−
−
= − +
= − +
(2.97)
According to the Kalman filter representation in (2.88) is invertible; thus G2 is
nonsingular, and the innovations can be written as:
1 1
2 2 1( , ) ( ) ( , ) ( , ) ( ),(k)= G k z k G k G k u kθ θ θ− −
−ɶ ɶɶv (2.98)
In order to develop the likelihood function, a NV vector that consists of all innovations
up to and including the frequency for which 1k N= − is introduced.
[ ](0) (1) ... ( 1) ,NV N= −ɶ ɶ ɶv v v (2.99)
( )kɶv has a probability density ( )p ɶɶ v , then it can be written applying the definition of
probability densities:
1 1( ) ( 1) ( ).N N Np V p N V p V− −= − ɶv (2.100)
Repeated use of (2.100) leads to the expression of the likelihood function:](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-37-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
38
[ ]1 2 1
( ; ) ( )
( 1) ( 2) ... (1) (0) .
N N
N N
L V p V
p N V p N V p V p
θ θ
− −
=
= − − ɶ ɶ ɶ ɶv v v v
(2.101)
The distribution of ( )kɶv is Gaussian, thus the distribution of kV is also Gaussian.
Hence, the probability density for this complex Gaussian vector is
0
0
† 1
( ) exp ( ) ( ) .
n
k vvn
vv
N
p k V N k S k
Sπ
−
= − ɶ ɶ ɶv v v (2.102)
Combining (2.101) and (2.102), the negative log-likelihood function ( ; )NL V θ is
[ ]
1
† 1
0
0
( ; ) ( ) ( ) ( ) .
N
N vv vv
k
ln L V J N k S k Nln S Nn ln
N
π
θ θ
−
−
=
− = = + +
∑ ɶ ɶv v (2.103)
The term on the far right side can be neglected, because it is unaffected by θ , the cost
function to be minimized then can be written as
1
† 1
0
( ) ( ) ( ) .
N
vv vv
k
J N k S k Nln Sθ
−
−
=
= +∑ ɶ ɶv v (2.104)
The ML parameter vector estimate is the value of θ that minimizes the (2.104). The
estimate of ˆ
vvS has the form of
1 1 †
†
0 0
ˆ ˆ ˆ( ) ( ) ( ) ( , ) ( ) ( , ) ,
N N
vv
k k
S k k z k y k z k y kθ θ
− −
= =
= = − − ∑ ∑ɶ ɶ ɶ ɶɶ ɶv v (2.105)
where ˆθ is an estimate of the parameter vector θ . The initial state of ˆ
vvS can be
computed by using 0θ that is the nominal starting value of the unknown parameter
vector θ . The estimates of the remaining parts can be done by using the modified
Newton-Raphson method. For a given ˆ
vvS the cost function is
1
† 1
0
( ) ( ) ( ),
N
vv
k
J k S kθ
−
−
=
= ∑ ɶ ɶv v (2.106)
that is minimized by iteratively computing
0
ˆ ,θ θ θ= + ∆ (2.107)
where the parameter vector update is
1
0
ˆ ( ) '( ),M Jθ θ θ−
∆ = − (2.108)
and M is the Fisher information matrix:
2 2
.N
T T
lnL(V ; ) J( )
M E E
θ θ
θ θ θ θ
∂ ∂
= − = ∂ ∂ ∂ ∂
(2.109)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-38-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
40
2.4.2 Output-error method
As was done in the time-domain analysis, the ML in the frequency domain can be
simplified when there is no process noise. Hence, the resulting system is deterministic.
[ ]
†
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) ( ),
( ) ( , ) ( ) ( ),
( ) 0,
( ) ( ) ,
k
vv
j x k A x k B u k
y k C x k D u k
z k G k u k k
E k
S
E k k
N
ω θ θ
θ θ
θ
= +
= +
= +
=
=
ɶ ɶ ɶ
ɶ ɶ ɶ
ɶɶɶ
ɶ
ɶ ɶ
v
v
v v
(2.115)
where
1
1( , ) ( , ) ( )( ( )) ( ) ( ),
2 / , 0,1,2,..., 1k
G k G k C j I A B D
k T k N
θ θ θ ω θ θ θ
ω π
−
= = − +
= = −
(2.116)
and vvS the spectral density matrix of ( )iv . The innovations are reduced to the output
errors
( , ) ( ) ( , ) ( ) ( , ) ( ),k z k y k z k G k u kθ θ θ= − = −ɶ ɶ ɶɶ ɶv (2.117)
The negative log-likelihood function and ˆ
vvS then can be written
1
† 1
0
( ) ( , ) ( , ) .
N
vv vv
k
J N k S k Nln Sθ θ θ
−
−
=
= +∑ ɶ ɶv v (2.118)
1 1 †
†
0 0
ˆ ˆ ˆ( , ) ( , ) ( ) ( , ) ( ) ( , ) .
N N
vv
k k
S k k z k y k z k y kθ θ θ θ
− −
= =
= = − − ∑ ∑ɶ ɶ ɶ ɶɶ ɶv v (2.119)
In order to estimate the unknown parameters, the modified Newton-Raphson method
is used. The elements of the first and second-order gradients are computed by
simplifying (2.112) and (2.113), where G2 = I and G1 = G:
†1
1
0
( ) ˆ ˆ ˆ2Re ( ) ,
N
vv zu uu
kj j
GJ
Tr S S GS
θ
θ θ
−
−
=
∂∂
= − −
∂ ∂
∑ (2.120)
†2 1
1 1
0
( ) ˆ ˆ2Re ,
N
ij vv uu
ki j j i
G GJ
M Tr S S
θ
θ θ θ θ
−
−
=
∂ ∂∂
= =
∂ ∂ ∂ ∂
∑ (2.121)
Estimates of the input spectral density matrix and the cross spectral density matrix
have the same form given in (2.114). The negative log-likelihood function, its gradient
and the information matrix can be expressed as](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-40-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
41
1
† 1
0
ˆ( ) ( , ) ( , ),
N
vv
k
J N k S kθ θ θ
−
−
=
= ∑ ɶ ɶv v (2.122)
1
† 1
0
( ) ˆ2 Re ( ) ( , ) ,
N
vv
k
J
N S k S k
θ
θ
θ
−
−
=
∂
= − ∂
∑ ɶv (2.123)
2 1
† 1
0
( ) ˆ( ) 2 Re ( ) ( ) ,
N
vvT
k
J
M N S k S S k
θ
θ
θ θ
−
−
=
∂
= = ∂ ∂
∑ (2.124)
where ( )S k is the output sensitivity matrix with dimensions 0 pn n× given by
( , ) ( )
( ) .
G k u k
S k
θ
θ
∂
=
∂
ɶ
(2.125)
The modified Newton-Raphson step is given by (2.107) and (2.108).
When the measured data is in the form of frequency response functions, the output-
error parameter estimation algorithm can be easily modified. In this particular case,
the cost function can be written
[ ] [ ]
1
†† 1
0
( ) ( ) ( ) ( , ) ( ) ( , ) ( ),
N
E vv E
k
J N u k G k G k S G k G k u kθ θ θ
−
−
=
= − −∑ ɶ ɶ (2.126)
where ( , )G k θ is matrix with transfer functions as elements, and ( )EG k is a matrix of
experimentally determined frequency responses. Cost functions in (2.122) or (2.126)
then can be minimized with respect to unknown parameter vector θ . When the
dynamic system has only one input, the output-error cost function can be written as
[ ] [ ]
1
†* 1
0
( ) ( ) ( ) ( , ) ( ) ( , ) ( ),
N
E vv E
k
J N u k G k G k S G k G k u kθ θ θ
−
−
=
= − −∑ ɶ ɶ (2.127)
In order to optimize the cost functions, it is not necessary to include all of the
frequencies. In the next chapter, a method that allows using arbitrary frequencies is
introduced. This method is called the high-accuracy finite Fourier transform. Using
arbitrary frequencies, automatically discards some of the power in the time-domain
data. Generally, this is equivalent to a zero-lag, low-pass filtering procedure, which
improves the convergence and accuracy of the parameter estimates. The weighting
matrix in the cost function is the spectral density matrix, which is ideally estimated as
a function of frequency index k. Nevertheless, in most practical cases, the frequencies
that correspond to the bandwidth of the dynamic system modes are used; which is also
the bandwidth where most of the residual power resides in the case of colored
residuals. Hence, using a constant vvS over frequencies used in the analysis is a good
approximation of the spectral density of the residual power. Accordingly, the assumed](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-41-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
42
noise model matches well to reality, and it is not necessary to compute the estimated
error bounds for colored residuals when parameter estimates are done in the frequency
domain.
2.4.4 Equation-error method
Assuming that all of the state variables can be measured without errors, the equation-
error model can be written
( ) ( ) ( ) ( ) ( ) ( ),k k k k w kj x A x B u B wω ω θ ω θ ω ω= + +ɶ ɶ ɶ ɶ (2.128)
where process noise is included. Considering the state derivatives at discrete
frequencies 2 / ,k k Tω π=
( ) ( ), 2 / .k kz k j x k k Tω ω π= =ɶɶ (2.129)
The model equation then can be written as
( ) ( ) ( ) ( ) ( ) ( ), 1,2,3,..., 1z k A x k B u k v k k Nθ θ= + + = −ɶ ɶ ɶɶ (2.130)
where ( ) ( )wv k B w k=ɶ ɶ are the equation errors, with
[ ]
†
( ) 0,
( ) ( ) .vv
E v k
S
E v k v k
N
=
=
ɶ
ɶ ɶ
(2.131)
The cost function of the equation-error method has the form
[ ] [ ]
1
† 1
0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,
N
vv
k
J N z k A x k B u k S z k A x k B u kθ θ θ θ θ
−
−
=
= − + − +∑ ɶ ɶ ɶ ɶɶ ɶ (2.132)
where the innovations are equation errors:
( ) = ( ) ( ) ( ) ( ) ( ).k z k A x k B u kθ θ− +ɶ ɶ ɶɶv (2.133)
Estimation of the elements in the unknown parameter vector θ is obtained by
minimizing the cost function given in (2.132). Assuming that the errors in the
equations are mutually uncorrelated, this procedure is applied separately to each state
equation, corresponding to each row in (2.128). Similarly to the time-domain the case,
the equation-error method can be used to estimate dimensional or nondimensional
aerodynamic parameters.
Equations of the measured output can also be used to estimate aerodynamic
parameters. In that case, the vector equation has the form
( ) ( ) ( ) ( ) ( ) ( ), 0,1,2..., 1z k C x k D u k v k k Nθ θ= + + = −ɶ ɶ ɶɶ (2.134)
The innovations are](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-42-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
43
( ) ( ) ( ) ( ) ( ) ( ).k z k C x k D u kθ θ= − +ɶ ɶ ɶɶv (2.135)
For instance, output equations for measured accelerometer outputs can be used to
estimate aerodynamic force parameters.
2.5 Accuracy of parameter estimates
In the last two sections the time-domain and frequency domain ML parameter
estimation methods were introduced. In this section the accuracy and confidence
intervals of these methods are investigated. In the case of a nonlinear estimator, the
developing of the parameter accuracy is much more difficult then in the case of a linear
estimator. However, in order to compute the accuracy of a nonlinear ML estimator,
the Cramer-Rao lower bound can be used, which can be obtained from the main
diagonal elements of the inverse information matrix. In the development of ML
estimator, it was assumed that the measurement noise ( )v i is a random white
sequence. Hence, the power of the noise is evenly spread over the frequency range
[ ]0, Nf , where ( )1/ 2Nf t= ∆ the Nyquist-frequency. The Nyquist-frequency is a
property of the dynamic system. In other words, all of the important signal
components are within the frequency interval of [ ]0, Nf .
The relation between the one-sided power spectral density of the noise and the
covariance matrix is
,
2
N
PSD
R PSDf
t
= =
∆
(2.136)
where PSD is a diagonal matrix with the power spectral densities of the elements of
the noise vector on the main diagonal. The Cramer-Rao inequality for the parameter
covariance matrix of the output error estimator has two alternative forms
1
1
1
ˆ( ) ( ) ( ) ,
N
T
i
Cov S i R S iθ
−
−
=
≥
∑ (2.137)
1
1
1
1ˆ( ) ( )( ) ( ) .
2
N
T
i
Cov S i PSD S i
t
θ
−
−
=
≥ ∆
∑ (2.138)
In practical cases, the residuals are colored. Namely, the power spectral densities
of the residuals are concentrated in a particular frequency band which corresponds to
the rigid-body aircraft dynamics. However, in the development of the ML estimator an
assumption that the residuals are white was made, because the power spectral densities](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-43-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
45
and R are from the conventional time-domain ML estimator calculations; thus, they
are the same quantities as in (2.41), (2.50) and (2.140). Accordingly, (2.141) and (2.142)
can be used as a residual postprocessor after conventional time-domain ML parameter
estimation, which corrects the parameter covariance matrix. As was discussed earlier,
in the case of the frequency domain ML estimator, there is no need to compute a
correction for the parameter covariance matrix.
In the case of linear parameter estimation, the confidence interval can easily be
computed. The confidence level associated with this interval expresses the probability
of finding the adequate parameter value within the interval. Therefore, in the case of
nonlinear parameter estimation, the confidence ellipsoid is just a quadratic
approximation of the confidence region. Furthermore, the parameter estimates in
nonlinear case do not have Gaussian distribution, the approximation of the confidence
level is based on the F-distribution. In [1] a method of computing confidence intervals
for ML estimators is presented. That method has not been widely used, because the
algorithm is very time-demanding.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-45-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
46
3 Data preprocessing
In most practical aircraft identification cases, the measurements are corrupted by
noises or any other kind of disturbance, which can be caused serious problems in the
parameter estimation. Hence, it is very important to apply methods to separate
deterministic signals from random noise for a measured time series. In this chapter
some data preprocessing methods are presented. First, the global smoothing in
frequency domain is introduced followed by the numerical differentiation. Finally, a
practical method for computing the finite Fourier transform based on the chirp z-
transform is discussed. The theory presented in this chapter is based on [1].
3.1 Filtering and smoothing
Filtering is a common approach to separate deterministic parts from noise. Filters can
be implemented by a circuit using hardware elements or by a computer-implemented
transfer function. Furthermore, filters have their own dynamics; thus, they represent
an additional dynamic between the physical system and the signal of interest. Filtering
can modify phase and amplitude of the signal resulting in a false modeling result.
Filtering applications use only the current and past data. Furthermore, filtering
methods assume that the analyst knows the frequency range of the physical system.
Contrarily, in the case of identification application, it is not always clear what this
frequency range should be, and a good choice of the frequency band is the same for
different measured sequences. To address these problems, global smoothing method
based on Fourier analysis can provide information about the adequate frequency band
to be used for system identification. Global smoothing method in the frequency
domain is presented in the next section.
Smoothing applications use subsequent data; thus, these methods can only be
used post-flight.
3.2 Global smoothing in the frequency domain
As was mentioned above, global smoothing has several beneficial properties in the case
of aircraft identification problems. Contrarily with the local smoothing methods which](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-46-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
47
are described in [1] in detail, global smoothing concerns the smoothing problem in
the global sense. Since it works in the frequency domain, it is based on the Fourier
analysis.
Consider a noisy measured signal ( )z t sampled at constant intervals t∆ ,
( ) ( ), 0,1,2,..., 1z i z i t i N= ∆ = − (3.1)
The aim is to distinguish signal and noise, and then separate the two components. The
measured sequence then can be written
( ) ( ) ( ), 0,1,2,..., 1z i y i v i i N= + = − (3.2)
The way for determining the noise sequence ( ), 0,1,2,..., 1v i i N= − is to find the signal
components in the frequency domain, invert the transformation to get the signal in the
time domain ( )y i , then compute the noise ( )v i from (3.2).
( ) ( ) ( ), 0,1,2,..., 1v i z i y i i N= − = − (3.3)
Fourier series expansion has an assumption that the signal under investigation is
periodic. Furthermore, most measured data from flight tests have discontinuities in the
amplitude and in first and higher order derivatives of the endpoints. A typical flight
test measurement is depicted in Figure 3.1. This data sequence is from [5] and [6].
These discontinuities can cause difficulties in determining an adequate representation
of the signal in the frequency domain, because the Fourier series expansions of signals
with these discontinuities have much slower convergence than a signal with no
discontinuities. This means that the magnitudes of the Fourier coefficients decrease
asymptotically with 1
k−
in the case of signals with discontinuities, and with 3
k−
in the
case of signals with no discontinuities, where k denotes the number of terms in the
Fourier expansion. To address this problem, it is possible to remove the endpoint
discontinuities from any time series by subtracting the linear trend from the original
measured time series. That makes the amplitudes at the endpoints equal to zero.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-47-320.jpg)
 ( 2) ... ( 1) (0) (1) ... ( 1) ,g g N g N g g g g N= − + − + − − (3.5)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-48-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
51
Finally, the linear trend removed from the signal by (3.3) has to be restored to
the smoothed signal. The smoothed signal then can be written as
( 1) (0)
ˆ ˆ( ) ( ) ( ) (0) , 1,2,3,..., 1
1
s s
z N z
y i z i g i z i i N
N
− −
= = + + = − −
(3.17)
The endpoints of the original measured sequence are excluded from the computing of
the smoothed signal. It can cause serious errors in the case of very noisy data. To
address this problem, the endpoints have to be smoothed locally by using a local
smoothing method presented in details in [1]. The smoothed signal is depicted in
Figure3.3.
Eventually, the noise sequence can be computed by using (3.3). Note here, that
the noise sequence computed by applying a Wiener filter is a wideband random white
noise.
Figure 3.3 Smoothed pitch rate measured time series
3.3 Numerical differentiation
To compute an adequate numerical derivative for a measured sequence is a common
problem in system identification applications, especially in the case of aircraft system
identification, when there are no sensors to measure some signals which are very
important in model parameter determination. For instance, generally, there is no
sensor to measure body-axis angular accelerations, so the values of , ,p q rɺ ɺ ɺ must be
computed by numerically differentiating the body-axis angular rate measurements.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-51-320.jpg)
 ( ) cos( ) sin( )j t
x t e x t t j tω
ω ω−
= − . This
integrand oscillates as the time variable t changes causing inaccuracies in the
approximation. The oscillation increases as the frequency increases. There is
the same problem when t∆ gets larger.
It can be seen that using the fast Fourier transformation causes inaccuracies as the
data sequence decreases, leading to a loss of detail in the frequency domain.
Furthermore, modal frequencies for a rigid-body dynamics of a full-scale aircraft are
usually below 2Hz. For such a small frequency band, most of the processing in the
transformation rests outside the range of interest.
To address these problems in [1], the defining integrand of the Fourier
transformation given by (3.24) is approximated by using an approach based on the
chirp z-transform, where the frequencies can be chosen arbitrarily regardless of the
length of the data sequence. Applying this method for the frequencies of the frequency
band of interest, a high accuracy approximation of the Fourier transform can be
achieved with excellent computational efficiency.
The discrete Fourier transformation in (3.26-27) can be written as
1
0
( ) ( ) ,
N
ki
i
X k x i AW
−
−
=
= ∑ (3.29)
where
2
1, .
j
N
A W e
π
= = (3.30)
k
AW for 0,1,2,..., 1k N= − represents equally spaced points on the unit circle in the
complex plane. Hence, the k
AW is a complex transform variable z , the selection given
in (3.30) is a z-transform of the data in the time-domain ( ), 0,1,...,x i i N= . A and W
can be changed to achieve different contours in the complex plane. This method is](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-54-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
55
called the chirp z-transform. In fact, angular steps are determined by W , and the value
of A can be selected as a starting point in the complex plane. So the chirp z-transform
is a discrete Fourier transform with arbitrarily selectable frequency range and
resolution.
When the contour in the complex plane is along the unit circle, W and A have the
forms of
0
,
,
j
j
A e
W e
θ
θ∆
=
=
(3.31)
where 0 0 / tω θ= ∆ is the lower limit of the frequency band in the chirp z-transform in
rad/s and / tω θ∆ = ∆ ∆ is the frequency resolution. Values of 0θ and θ∆ can be
chosen arbitrarily. The only limitations for the resolution and the lower limit is both
must be in the range [ ]0, / tπ ∆ , where / tπ ∆ is the Nyquist angular frequency in
rad/s. This chirp z-transform can be used to calculate the finite Fourier transform
when the selected frequencies are regularly spaced in the frequency range of interest.
In the case of aircraft system identification, evenly spreading the frequencies is the
preferred approach to capture all important features in the frequency domain.
The chip z-transform evaluated at M arbitrarily frequencies kf in Hz, where
the selected frequency range is 0 k Nf f≤ ≤ yields
1
2
0
( ) ( ) ,
0 , 0,1,2,..., 1
k
N
j f i
i
k N
X k x i e
f f k M
π
−
−
=
=
≤ ≤ = −
∑ (3.32)
The number of frequencies can be chosen with the limitation that M cannot be larger
then 2N to avoid numerical problems:
2 .k kt f tθ ω π= ∆ = ∆ (3.33)
The expression for the high-accuracy calculation can be written as
{
]}
0 1 2 3
/ * *
0 1
* *
2 3
( ) ( ) ( )
( ) (0) ( ) (1) ( ) (2) ( ) (3)
( ) ( 1) ( ) ( 2)
( ) ( 3) ( ) ( 4) ,
j T t
x t W X k
x x x x
e x N x N
x N x N
θ
θ θ
γ θ γ θ γ θ γ θ
γ θ γ θ
γ θ γ θ
∆
≈ ∆
+ + + +
+ − + −
+ − + −
ɶ
(3.34)
where 0 1 2( ), ( ), ( ), ( )W θ γ θ γ θ γ θ and 3 ( )γ θ are the weights calculated by analytically
evaluating the finite Fourier transform integral in (3.34) using cubic Lagrange](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-55-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
56
interpolation applied to the time-domain data. According to [1], the resulting
expressions for the weights have the form:
4 6
2 4 6
0
2 4 6
11 23
( ) 1 ,
720 15120
2 1 103 169
( )
3 45 15120 226800
2 2 8 86
,
45 105 2835 467775
W
j
θ θ θ
γ θ θ θ θ
θ θ θ θ
≈ − +
≈ − + + −
− + + +
(3.35a)
2 4 6
1
2 4 6
2 4 6
2
2 4 6
2 4 6
3
2
7 7 5 7
( )
24 180 3456 259200
7 1 11 13
,
72 168 72576 5987520
1 1 5 1
( )
6 45 6048 64800
7 1 11 13
,
90 210 90720 7484400
1 1 5 1
( )
24 180 24192 259200
7 1
360 840
j
j
j
γ θ θ θ θ
θ θ θ θ
γ θ θ θ θ
θ θ θ θ
γ θ θ θ θ
θ θ
≈ − + −
− − + −
≈ − + − +
− − + − +
≈ − + −
− − 4 611 13
.
362880 29937600
θ θ
+ −
(3.35b)
Furthermore, it is advantageous to select small distance between frequencies in the
frequency range of interest, to ensure that all details of the time-domain data are
captured.
The terms to compute the finite Fourier transform applying the chirp z-transform
have the form:
0 , 0,1,2,..., 1kf f k f k M= + ∆ = − (3.36)
1 0( )
.
( 1)
f f
f
M
−
∆ =
−
(3.37)
The steps of the procedure now can be written
(i) Choose three of the four values in (3.37) and then compute the
remaining value. This determines the frequencies for the analysis.
(ii) Calculate the M values of θ from (3.33).
(iii) Use the chirp z-transform to compute the discrete Fourier transform
for the selected frequencies applying (3.29-30).
(iv) Compute the high accuracy finite Fourier transform by using (3.34-
35).](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-56-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
57
4 Demonstration of time domain ML and OE methods
using test data from known data bases
In this chapter the identification ability of time-domain output-error method presented
in Chapter 2 is demonstrated in MATLAB environment. To execute the identification
method, the SIDPAC Toolbox developed by NASA is used. The SIDPAC is a Toolbox
software for [1]. In this chapter three case-studies are presented. This chapter is based
on the theory written in [1] and [4] and data sequences can be found in the SIDPAC
Toolbox.
According to the previous chapter the following algorithm is used here:
(i) Assembling the system matrices A,B,C,D using the nominal parameter
values;
(ii) Computing the covariance matrix of the innovations (2.41);
(iii) Minimizing the cost function given by (2.42) using the modified Newton-
Raphson method;
(iv) Update the parameter values in the system matrices;
(v) Repeat steps (i)-(iv) until the convergence criteria (2.51) are satisfied.
In the first example the measured data are from a lateral maneuver run1 of the
NASA Twin Otter aircraft, which is a twin-engine turboprop commuter aircraft
depicted on Picture 4.1. In the case of the output-error parameter estimation method,
where the equations of motion are used together, the lateral aerodynamic coefficients
are estimated at the same time. There are two data sequences named as run1 and run2.
Measured data from run1 is used to estimate the model parameters, and then flight
data from run2 is used to validate the results. Measured data sequences of the
maneuver run1 and run2 can be seen in Figure 4.1 and 4.2.
Picture 4.1 De Havilland DHC-6 Twin Otter](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-57-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
59
Detailed aerodynamic equations can be found in [1] , [4] and [8]. Linearized
equations have the following form:
0
0 0
0
0 0
0
0
0 0
0
0 0
0
2
cos
sin cos ,
,
2 2
,
2 2
tan
r r
p r a r
p r a r
Y Y Y r
x xz l l l l a l r p
x xz n n n n a n r r
q S rb
C C C
mV V
g
p r b
V
pb rb
I p I r q Sb C C C C C b
V V
pb rb
I r I p q Sb C C C C C b
V V
p r
β δ
β δ δ
β δ δ
β
β β δ
θ
α α φ
β δ δ
β δ δ
φ θ
= + +
+ − + +
+ = + + + + +
+ = + + + + +
= + +
ɺ
ɺ
ɺ
ɺ
ɺ ɺ
ɺ ɺ
ɺ ,bφ
(4.1)
0
0
,
2r yr
y Y Y Y r a
q S rb
a C C C b
mg Vβ δ
β δ
= + + +
(4.2)
These equations are linear dynamic equations for the lateral states , ,p rβ , which are
also measured outputs of the aircraft. (4.2) is an equation for the lateral acceleration. In
(4.1-2), values with subscript o denote the measured trim values of the maneuver.
These are equal to the starting trim values or to the mean values computed from
measured data sequence. All of the dynamic equations include a bias term. As it was
described in Chapter 3, there are several methods to remove the bias term from a
measured data sequence. This is for any measurement biases not removed from the
data sequences. Equations (4.1-2), which are dynamics and measurements, can be
written in state-space form
0
1 0 1 2 0
0
3 3 2 3 2
3 3 2 3 2
0
1
3 3
cos
sin ( cos )1 0 0 0
0 0 0
0 0
0
0 0 0 1
0 1 tan 0
0
−
− − =
− −
+
ɺ
ɺ
ɺ
ɺ
ɺ
r
p r
p r
r
a r
Y Y
xx xz
l l l
xz zz
n n n
Y
l l
g
k C k C k
V
I I pp k C k C k k C k
I I rr
k C k C k k C k
k C b
k C k C b
β
β
β
δ
δ δ
β
θ
α α ββ
φφ
θ
3 3
,
1
0 0
ɺ
ɺ
ɺ
a r
a
p
r
n n rk C k C b
b
δ δ
φ
δ
δ
(4.3a)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-59-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
66
Table 4.2 Parameter estimation results for lateral maneuver using data sequence
run1
Parameter ˆθ ˆ( )s θ white residuals
(2.137)
ˆ( )corrs θ colored
residuals (2.141)
ˆ( )
ˆ( )
corrs
s
θ
θ
YC β
-8.6457e-001 2.8260e-003 8.82e-003 3.15
rYC 9.3313e-001 1.6697e-002 6.04e-002 3.65
r
YC δ
3.7999e-001 8.0588e-003 3.15e-002 3.95
lC β
-1.1959e-001 6.0201e-004 1.89e-003 3.37
plC -5.8590e-001 2.5898e-003 8.23e-003 3.44
rlC 1.8888e-001 1.8195e-003 8.89e-003 5.19
a
lCδ
-2.2818e-001 9.1234e-004 3.03e-003 3.60
r
lCδ
4.2212e-002 7.4475e-004 6.03e-003 8.47
nC β
8.6376e-002 2.1507e-004 1.53e-003 7.36
pnC -6.4414e-002 9.8548e-004 6.41e-003 6.76
rnC -1.9059e-001 6.2567e-004 4.06e-003 6.68
a
nC δ
-2.9796e-003 3.6329e-004 2.23e-003 6.35
r
nC δ
-1.3548e-001 2.7071e-004 1.86e-003 7.00
In the third example dimensional stability and control derivatives are estimated
for a longitudinal dynamic state-space model using longitudinal flight test data
sequences measured on the same aircraft as in the case of the first and second example.
Based on [4], the state-space structure is considered as
( )0 0 0
,
,
1
1 0 0 0
0 1 0 0 ,
1
1
e
e
e
biasq e
q bias
e
z
q z bias
Z ZZ Z
M Mq q M M
q
q
a V V V
Z Z Z a
g g g
δα
α δ
α δ
α α δ
α
α δ
= +
= +
−
ɺ
ɺ
(4.5)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-66-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
67
where the elements of the unknown parameter vector are
,e eq bias q bias z biasZ Z Z Z M M M M aα δ α δθ = . Detailed description of this
model structure can be found in [1], and in [4]. The states of the system are α and
q , which are the angle of attack and pitch rate, respectively. The outputs are the α , q
and za , where za is the acceleration along the z-axis. 0V and g denote the absolute
velocity and gravitational acceleration, respectively. Measured outputs of the
longitudinal dynamic system can be seen in Figure 4.7. Estimating the unknown
parameters is performed using output-error method. Initial value of the unknown
parameter vector θ is computed by using equation-error method in one step. The
output-error solution then took 69 modified Newton-Raphson iterations.
0 1 2 3 4 5 6 7 8
-0.02
0
0.02
0.04
alpha(rad)
Measured Outputs
0 1 2 3 4 5 6 7 8
-0.1
-0.05
0
0.05
0.1
q(rps)
0 1 2 3 4 5 6 7 8
-1.4
-1.2
-1
-0.8
-0.6
az(g)
time (sec)
Figure 4.7 Measured outputs for a longitudinal maneuver
Estimates of the bias parameters have been omitted. The results can be seen in Table
4.3. Since, the residuals are colored due to modeling error, hence Cramer-Rao bounds
must be corrected for colored residuals. The residuals can be seen in Figure 4.8.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-67-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
68
Table 4.3 Parameter estimation results for longitudinal maneuver with
corrected error bounds
0 1 2 3 4 5 6 7 8
-2
0
2
x 10
-3
alpharesiduals(rad)
Residuals
0 1 2 3 4 5 6 7 8
-5
0
5
x 10
-3
qresiduals(rps)
0 1 2 3 4 5 6 7 8
-0.05
0
0.05
azresiduals(g)
time (sec)
Figure 4.8 Output-error residuals for longitudinal maneuver
The estimated system matrices then can be written in the form of
-1.9693 0.9927 -0.0936 0.0042
, B= .
-9.4361 -3.0755 -12.1453 -0.0344
A
=
(4.6)
Parameter Estimate Standard Error in % 95% Confidence Interval
Zα
- 1.969e+000 4.896e-002 [ -2.067 , -1.871 ]
qZ 9.927e-001 1.476e-002 [ 0.963 , 1.022 ]
e
Zδ
-9.360e-002 2.957e-002 [ -0.153 , -0.034 ]
Mα
-9.436e+000 1.715e-001 [ -9.779 , -9.093 ]
qM -3.076e+000 6.835e-002 [ -3.212 , -2.939 ]
e
Mδ
-1.215e+001 1.595e-001 [ -12.464 , -11.826 ]](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-68-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
69
5 Demonstration of frequency domain ML and OE
methods using test data from known data bases
In this chapter the identification ability of frequency domain output-error method
presented in Chapter 2 is demonstrated using MATLAB software environment. To
execute the identification method and the operations in the frequency domain, the
SIDPAC Toolbox developed by NASA is used, as was done in Chapter 4. In this
chapter two case-studies are presented, to demonstrate of the efficiency of frequency
domain ML method. This chapter is based on the theory written in [1] and [4] and
data sequences can be found in the SIDPAC Toolbox.
According to the chapter 2 the following algorithm is used here:
(i) Transform the necessary data sequences into frequency domain using the
chirp-z transform presented in subsection 3.4;
(ii) Assembling the system matrices A,B,C,D according to (2.115);
(iii) Computing the covariance matrix of the innovations in the frequency
domain (2.119);
(iv) Minimizing the cost function (2.127) by applying the modified Newton-
Raphson method;
(v) Update the parameter values in the system matrices;
(vi) Repeat steps (i)-(v) until the convergence criteria (2.51) are satisfied.
In the first example, the measured flight test data sequences are gathered from the
same aircraft as in the preceding chapter. Measured inputs and outputs can be seen in
Figure 4.1-2. Then all relevant data time series are transformed into frequency domain
using the high accuracy finite Fourier transform described in Chapter 3. The
frequency vector [ ]0.10,0.12,0.14,...1.5
T
f = Hz includes all significant spectral
components. The output-error cost gradient and information matrix are computed
from (2.123) and (2.124) and the parameter estimate updates are computed from
(2.108). Output sensitivities are computed numerically using central finite differences
from (2.125). Estimation results can be seen in Table 5.1 where the results in the
frequency domain are listed along with time domain results of the first example in
Chapter 4. It can be seen that the parameters computed in the frequency domain are
close to those that were computed in the time-domain. However, in the frequency
domain, parameters do not require a re-estimation of biases, because they were
eliminated before the Fourier transformation. Figure 5.1 shows the model fitted to
measured outputs in the frequency domain.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-69-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
72
equations for short period motion described in [1] and [4], the longitudinal LOES
model can be written:
1
( ),
e
e
q
e
q
LL L
t
M Mq q M
ηα
α η
α α
η τ
− −
= + −
ɺ
ɺ
(5.1)
where the input is now a longitudinal pilot stick deflection, with its time delay.
Assuming that 0qL ≈ and 0e
Lη ≈ for low angle of attack, and then performing the
Laplace transform,
01
,
e
s
e
q
s L
e
MM s M q
α τ
ηα
α
η −
+ −
+ − −
ɶ
ɶ
ɶ
(5.2)
then the transfer function of the system can be expressed as
1 0
2 2
1 0
( ) ( )
,
( ) ( )
e
s s
e q q
M s L e b b eq
s L M s M L M s a s a
τ τ
η α
α α αη
− −
+ +
= =
+ − − + + +
ɶ
ɶ
(5.3)
Estimating the time delay parameter τ is problematic in the time-domain [1]. To
address this problem, the data will be analyzed in the frequency domain.
Flight data used in this part is measured on a Tu-144L supersonic transport
aircraft. Data sequences of the longitudinal maneuver are shown in Figure 5.2-3.
There are also two data sequences, run1 and run2 for the same reason as in first
example of Chapter 4. Nominal starting parameter vector for output-error method is
computed by using equation-error method. The equation-error method is much more
robust to starting values of the parameters [1]. The elements of the unknown
parameter vector are the coefficients of the transfer function given in
(5.3), 1 0, 1 0, , , .b b a aθ τ = Measured data are transformed into frequency domain for a
selected frequency range by using the high accuracy finite Fourier transformation
described in Chapter 3. The cost function given by (2.122) has to modified
1
* 1
0
( ) ( , ) ( , ),
M
vv
k
J M k S kθ θ θ
−
−
=
= ∑ ɶ ɶv v (5.4)
where 95M = denotes the number of data points in the frequency domain. The
innovations are
[ ]1 0
2
1 0
( , ) ( ) ( ) .
( )
kk j
e
k k
j b b
k q k k e
j j a a
ω τω
θ η
ω ω
−+
= −
+ +
ɶ ɶv (5.5)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-72-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
76
6 Flight test data determination using simulation
In this chapter, simulated flight data is gathered using a known non-linear model of an
aircraft [7]. The use of this experiment is to verify our identification methods. Firstly,
a brief review about aircraft modeling is given. Then, measured flight data sequences
are gathered virtually using the nonlinear model of a DHC-2 Beaver aircraft.
Additional noises and other disturbances are added to the signals, in order to create
lifelike simulation circumstances.
6.1 Outline of aircraft modeling
In this subsection, necessary information about aircraft modeling is given based
mainly on [1], [4] and [7]. The FDC-1.2 Simulink Toolbox [7] was used for the
simulations. In this toolbox, a well parameterized nonlinear dynamic model of the De
Havilland DHC-2 Beaver can be found. The aircraft is depicted on Figure 6.1.
Figure 6.1 De Havilland DHC-2 Beaver
This work, does not deal how to build a nonlinear dynamic model for an
aircraft; therefore the results and notations as in [4] and [7] are used to describe
aerodynamic parameters, angular rates/velocities/accelerations and the orientation of
the aircraft. The state and the input vector of the Beaver are
[ ]
T
e ex V p q r x y Hα β ψ θ ϕ= (6.1)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-76-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
78
moment and navigational equations are used [4]. Furthermore, the given model can
be divided into two decoupled subsets [4]. One of them describes the longitudinal
while the other one describes the lateral motion. The longitudinal state variables are
the angle of attack (α ), absolute velocity (V), pitch rate (q ) and the Euler pitch angle
(θ ).The variables describing the lateral motion are the sideslip angle ( β ), roll rate
( p ), yaw rate (r ), Euler roll angle (ϕ ) and Euler heading angle (ψ ).
In the following, non-dimensional aerodynamic parameters will be identified.
So a particular focus is placed on the aerodynamic force equation. Aerodynamic forces
and moments depend always on the flight condition. In the case of Beaver aircraft, it
has the following form
1( , , )aero aeroF dp x x u= ɺ (6.3)
where aeroF is a vector of aerodynamic forces, 1p is a polynomial vector-function
containing the non-dimensional aerodynamic coefficients and d converts the non-
dimensional coefficients to dimensional forces and moments. Detailed description
concerning the place of these force/torque components in the nonlinear dynamic
equations of an aircraft can be found in Appendix A. Note the polynomial function
depends on xɺ too. It is only taken into account by βɺ in the case of aerodynamic side
force coefficient ( aY ). 1p describes the aerodynamic forces and moments coefficients in
ABC. It has the following form
2 3
0 2
0
3 2
0 2
3
3
0
2 2 2
2
= + + + + +
+ +
+
+ +
= + + + + +
= + + + + + +
= + ++
ɺ
X X X X X X X X f X fq r f f
Y Y Y Y Y Y a Y r Y rp r a r r
Z Z Z Z Z Z e Z E Z f Z fq e f fe
l l l
ra
Ya
a
l la p r
qc
C C C C CC
V
C C C C C
V V V
q
C C C
V
C
C C C
pb r
C C
V
b b
C C C C
c
C C C C C C
pb
C C
α δ δ αδα α
β δ δ δ α β
α δ δ αδα δ β
β
α α α δ δ αδ
β
β δ δ δ α
α α δ δ β δ αδ
β
2 2
0 2 2
3
30
2
2
2 2
+ +
= + + + + + +
= + + + +
+
+ ++
l l
m m m m ma
n n n
a l
a a
m m m e X fq re f
n n n ap r r q Yna a
r n
rb
C
c rb
C
V
q
C C
V V
q
C C C C C
V
C C C C C C C
pb rb c
C
V V
C C C
δ δ α
α δ δα β
β δ δ β
δ δ α
α α δ β δ
β δ δ β
(6.4)
where the subscript a stands for aero. That means these represent only the
aerodynamic effects in the nonlinear equations. There is also 2 ( , )=prop tF dp x dp [1]
that describes the contribution of the engine to nonlinear equations. In this work I did
not concern the identification of the coefficients in 2p .](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-78-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
80
6.2 Virtual test flights
In order to estimate the values of the coefficients in 1p (6.3), necessary data sequences
are obtained by virtual test flights using FDC 1.2 Toolbox [7] and the results of a
previous MSc Thesis [12] in MATLAB/Simulink environment. As was mentioned
before, the state variables can be divided into two decoupled subsets. Hence, the
simulations are done in two phases as well. One of them performs a lateral maneuver
while the other a longitudinal maneuver. Implicitly, variables belong to the other
maneuver are held a constant trim values during the maneuver. In both cases, two
simulations are performed in order to obtain a teaching part and a testing part of the
measured data. In the following, the teaching data will play a role in parameter
estimation while the testing data will be served to validate our estimation results.
Furthermore, measurement noises are added to the signals in order to model real
flight circumstances. The applied noise model can be seen in the next subsection.
Figure 6.5 - 6.12 show the simulated teaching and testing data sequences of a
lateral maneuver with additional measurement noises. Teaching and testing simulated
data sequences of a longitudinal maneuver can be seen on Figure 6.13 - 6.18. Note that
the only difference between teaching and testing sequences is the sign of the input
doublets. Table 6.2 shows us the used trim values during simulation. Trim values were
subtracted from all of the data sequences.
Table 6.2 Applied trim values during the lateral maneuver
State Trim value
V 45 m/s
α 0.14724 rad
β -0.011814 rad
p -0.009.7782 rad/s
q 0.032429 rad/s
r 0.080425 rad/s
ϕ 0 rad
θ 0.11229 rad
ψ 0.38328 rad
ex , ey 0 m
H 1000 m](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-80-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
88
0 5 10 15 20 25 30
-6
-4
-2
0
2
4
6
el(deg)
0 5 10 15 20 25 30
-30
-20
-10
0
10
20
the(deg)
time (sec)
Figure 6.18 Testing data from a longitudinal maneuver (elevator and pitch
angles)
6.3 The applied noise model
In this subsection, the applied noise model is presented based mainly on noise model
can be found in [1]. Taking into account a standard instrumentation the measurement
error has the following form:
( ) (1 ) ( ) ( ) 1,2,...z i y i b v i i Nλ= + + + = (6.5)
where ( )z i is the ith measured output from the sensor, ( )y i is the true value of the
output, ( )v i is a random measurement noise, λ is a constant scale factor error
parameter and b is a constant bias error parameter. The noise model can be seen on
Figure 6.19.
Figure 6.19 Instrumentation error model](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-88-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
91
7 Parameter estimation using time domain ML and OE
methods
In this chapter identification of the lateral and longitudinal aerodynamic parameters
are done applying noisy simulated data sequences and time domain ML and OE
methods. In both cases, identification is done by using the teaching part of the
gathered simulated data. As was mentioned in the previous subsection, prefiltering and
smoothing of the signals were done before identification. Teaching and testing part of
the simulated lateral data sequences can be seen on Figure 6.5-12. Teaching and
testing part of the applied longitudinal simulated data sequences are depicted on
Figure 6.13-18. During the identification of the coefficients related to the lateral state
variables, the ones that related to the longitudinal coefficients are held on a constant
trim value; similarly, in the case of the identification of the longitudinal state variable
coefficients. Finally, testing part is used to validate the identification.
Identification of the lateral aerodynamic force coefficients
Firstly, lateral non-dimensional aerodynamic parameters are estimated which means
the coefficients of the second, fourth and sixth row from 6.4 ( , ,Y l nC C C ). In the case of
output-error method, all of the coefficients are estimated at the same time. Note that
these algorithms depend highly on the initial parameter values. Several experiments
were made using different starting values; most of them diverged. Hence, results of the
equation-error and linear regression methods were used as a starting value. These
linear regression methods were in the focus of an earlier work [10] that was made for
a student science conference. Modified Newton-Raphson method was used to minimize
the cost function in (2.42). The estimated parameter covariance matrix was computed
from (2.50) as the Cramer-Rao lower bound 1
0
ˆ( ) ( ).Cov Mθ θ−
≥ Note that the residuals
are assumed to be white. The same model structure is used during identification as in
the first example of Chapter 4 (4.1-4.3). Time domain output-error model fit to the
testing part of the measured data is shown in Figure 7.1. It can be seen that a quite
good prediction is obtained by using the estimated parameters. The solution took 37
Newton-Raphson iterations. Parameter estimation results are summarized in Table
7.1-7.3.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-91-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
94
Identification of the longitudinal aerodynamic force coefficients
Secondly, non-dimensional longitudinal aerodynamic parameters are estimated. The
used model structure came from [1]. Similarly to the first example in chapter 3 a
linear perturbation model is used. In (7.1-2), values with subscript o denote the
measured trim values of the maneuver.
0
0 0
0 0
0
0
0 0 0
0 0 0
0 0
0
0 0
2
sin
cos ( ) ,
2
sin cos
( ) ,
,
2
∆
= − + ∆ + +
− ∆ − ∆ − ∆ +
∆
= − + ∆ + + +
+ − ∆ − ∆ − +
∆
= + ∆ + + +
ɺ
ɺ
ɺ
ɺ
ɺ
ɺ
ɺ
V q
V q
V q
D D D D e
V
L L L L e
m m m m e q
y
q S V qc
V C C C C
m V V
T
g b
m
q S V qc
C C C C
mV V V
g T
q b
V mV
q Sc V qc
q C C C C b
I V V
α δ
α δ
α δ
α
α δ
α
γ θ α α
α α δ
γ α
θ α
α δ
,= qθ
(7.1)
In order to use these equations for output-error parameter estimation, it has to be
rewritten in the state-space form (7.2). Equation (7.4) describes the connection
between (6.4) and the coefficients can be found in (7.1). Nominal parameter values are
computed by using the parameter estimation results of ,X ZC C and mC [10] and
equation (7.4). Detailed description of the calculation of the nominal parameters can be
found in Appendix B. Note that in [10] the parameters were estimated by applying
linear regression method. In the case of longitudinal parameter estimation, only the
first 16 seconds of the gathered longitudinal data were used in the identification.
0 0
1 1 0 1 2 0
0 0 0
3 3 0 3 2
0
4 4 4 2
1
3
4
sin
( cos ) cos
sin sin
( cos ) 1
0
0 1 1 0
0
− + − − − + + =
+
ɺ
ɺ
ɺ
ɺ
V q
V q
V q
D D D
L L L
m m m
D V
L
m q
T
k C k C g k C k g
m VV
T g
k C k C g k C k
m V qq
k C k C k C k
k C b
k C b
k C b
b
α
α
α
δ
δ
δ
α
θ
α
γ γ
α γ αα γ
θθ
,
1
eδ
(7.2a)](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-94-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
96
8 Parameter estimation using frequency domain ML and
OE methods
Similarly to the previous section identification of the lateral aerodynamic parameters is
done. In this case, frequency domain ML and OE methods are used to determine the
unknown parameters. Identification is done by using the teaching part of the gathered
simulated data. Before identification, filtering and smoothing methods were applied to
the simulated data sequences. Teaching and testing part of the simulated lateral data
sequences can be seen in Figure 6.5-12. During the identification of the coefficients of
the lateral model, the parameters used only in the longitudinal model are held on a
constant trim value.
Identification of the lateral aerodynamic force coefficients
Estimation of the same lateral non-dimensional aerodynamic parameters as in the
previous chapter is done by applying frequency domain output-error method. Results
from equation-error linear regression [1] done in frequency domain were used as
initial parameters values. All relevant time series were transformed into frequency
domain using the high-accuracy finite Fourier transformation described in subsection
3.4 with the frequency vector [ ]0.10 0.12 0.14 ... 1.5
T
f Hz= . All significant
spectral components caught by applying this frequency range [1]. The same linear
dynamic model (4.3) was used to estimate the non-dimensional aerodynamic
coefficients. The cost gradient and the information matrix were computed using
(2.123) and (2.124), respectively. It can be seen on Figure 8.1 that predicted outputs
match quite well to the testing part of the measured data. Note that in the case of
frequency domain methods, constant parameters cannot be estimated because biases
and trends were removed before Fourier transformation. The constant parameters and
other biases must be estimated in time domain. Furthermore, parameter estimation
results are also written in Table 8.1-8.3.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-96-320.jpg)
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100
In the case of lateral non-dimensional aerodynamic coefficients in frequency domain:
>> FD_OE_Beaver_Lat_ID
In the case of longitudinal non-dimensional aerodynamic coefficients in time domain:
>> TD_OE_Beaver_Long_ID
Description of the elaborated scripts and applied Simulink models
Simulink models
The applied Simulink models can be seen on Figure 9.1 and 9.2. Observe the input
selection switches in the models. In both cases, Input Run 1 is used to generate the
teaching data sequences while Input Run 2 is applied to produce the testing part of the
data. Furthermore under the mask named Non-linear Beaver, there is the non-linear
model of the Beaver which is part of the FDC Toolbox [7].
Concerning the file structure, the following set-up is used
MSc_Thesis – Contains the main scripts that perform the identification methods
MSc_ThesisVirtual_Test_Flight. – Contains the scripts and Simulink models that
perform the virtual test flights, it also contains data necessary to the simulations and
functions that perform data saving. In order to execute simulations, SetPath.m has to
be run first. The library also contains the FDC Toolbox (public).
The applied SIDPAC functions
smoo – Optimal global Fourier smoothing.
lesq – Least squares linear regression
fint – High-accuracy finite Fourier integral for arbitrary
frequencies
oe – Output-error parameter estimation in time domain
fdoe - Output-error parameter estimation in frequency domain
r_colores – Parameter covariance for colored residuals from
linear regression.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-100-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
104
fnlatss_beaver - Frequency-domain lateral state-space model file
using non-dimensional parameters.
Usage: [Y,X,A,B,C,D] = fnlatss(p,U,w,x0,c);
Input:
p = parameter vector.
U = matrix of input vectors in the frequency domain.
w = frequency vector, rad/sec.
x0 = initial state vector.
c = vector of constants.
Output:
Y = model output vector or matrix.
X = model state vector or matrix.
A,B,C,D = system matrices.
tnlatss_beaver - Time-domain lateral state-space model of the
Beaver aircraft file using non-dimensional parameters. Usage:
[y,x,A,B,C,D] = tnlatss(p,u,t,x0,c);
Input:
p = parameter vector.
u = input vector or matrix.
t = time vector.
x0 = initial state vector.
c = vector of constants.
Output:
y = model output vector or matrix time history.
x = model state vector or matrix time history.
A,B,C,D = system matrices.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-104-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
105
beaver_params - Assembles constants for dimensional state-space
models.
Usage: css = beaver_params;
Input:
fdata = flight test data array in standard configuration.
Output:
css = vector of constants for a dimensional state-space model.
plot4ident_lat and plot4ident_long - Generates the figures that
can be found in the subsection 6.2.
tnlonss_beaver - Time-domain longitudinal state-space model of
the Beaver aircraft file using non-dimensional parameters.
Usage: [y,x,A,B,C,D] = tnlonss(p,u,t,x0,c);
Input:
p = parameter vector.
u = input vector or matrix.
t = time vector.
x0 = initial state vector.
c = vector of constants.
Output:
y = model output vector or matrix time history.
x = model state vector or matrix time history.
A,B,C,D = system matrices.
long_inic – Initialize the identification method of the
longitudinal model and computes the nominal parameter values.
Usage: p0 = long_inic(xinco, fdata, selevator, sV, salpha, sq,t);
Input:
xinco = trim value of the states.
fdata = standard SIDPAC data structure.
selevator = smoothed data sequence of the elevator.
sV = smoothed data sequence of the absolute velocity.
salpha = smoothed data sequence of alpha.
sq = smoothed data sequence of q](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-105-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
107
10 Summary and concluding results
In this MSc thesis I reviewed and analyzed several parameter estimation methods such
as the time-domain and frequency domain Maximum Likelihood (ML) methods.
Particular focus was placed on the output-error method. The parameter estimations
are based on noisy flight data sequences, so some of the essential smoothing and
filtering methods were presented, in order to decrease the blast of the measurement
noises. A high accuracy approximation of the finite Fourier transformation was also
described to execute an adequate transformation of the measured data sequences
between time-domain and frequency domain. In Chapter 4 and 5, identification ability
of time-domain and frequency domain parameter estimation methods were
demonstrated using the SIDPAC Toolbox in MATLAB environment.
The identification methods studied in the preceding chapters were applied to
reconstruct the dynamic model of a known aircraft from flight data. Simulated data
sequences were determined by using the nonlinear model of the de Havilland DHC-2
Beaver commercial aircraft in MATLAB/Simulink environment. These data sequences
were corrupted artificially by noises and by other kind of disturbances. The aircraft
sensor noise model found in [1] was used. All of the parameters of the Beaver aircraft
were known; thus, it was possible to validate the applied parameter estimation and
smoothing methods. Several functions were elaborated to SIDPAC Toolbox in order
to perform the identification of the parameters of the nonlinear dynamic model of the
Beaver. To sum up, it could be seen that the estimated parameters obtained by
applying time and frequency domain Maximum Likelihood and output-error methods
were equal or very close to the real parameter values. These results are convenient for
controller design techniques. Hence, the methods will be able to use in the case of any
other kind of flight data, also for an unknown aircraft. Note that in reality there will be
more disturbing factors in the parameter estimation; for instance important signals are
not measurable, hence state estimation and sensor fusion is needed.
Future research goal may be the identification of a real aircraft based on real
flight data. In order to use the algorithms tested here, it is necessary to develop the
fusion of GPS, IMU and magnetometer data and solve the state estimation problem.
The methods presented in this work solved the first part of this large program and
documented the applicability of the parameter estimation methods.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-107-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
108
11 Appendix
A. State equations of an aircraft
State equations of an aircraft can be obtained considering rigid body equations of
motions, completed by kinematic and dynamic equations including aerodynamic and
engine effects. Equations showed here came from [1], [4] and [8]. Hence, state
equations can be written in a compact form:
Force equation in body frame:
sin ,
cos sin ,
cos cos .
= − + − +
= − + +
= − + +
ɺ
ɺ
ɺ
T
X
Y
Z
FqS
u rv qw C g
m m
qS
v pw ru C g
m
qS
w qu pv C g
m
θ
θ φ
θ φ
Force equation in wind-axes frame:
cos cos (cos cos sin cos
sin cos sin sin cos cos ),
sin
tan ( cos sin )
cos cos
(cos cos cos sin sin ),
cos
sin cos cos sin cos
sin
( cos sin
= − + +
+ −
= − + − + −
+ +
= + − +
+
ɺ
ɺ
ɺ
W
W
T
D
T
L
Y
FqS
V C g
m m
FqS
C q p r
mV mV
g
V
qS q
C p r
mV V
g
V
α β φ θ α β
φ θ β θ α β
α
α β α α
β β
φ θ α θ α
β
β α α β φ θ
β
α θ
cos
sin cos cos ).− + TF
g
m
α
α φ θ
Moment equations:
2 2
( )
,
( )
( ) ,
( )
.
−
− = − +
−
= − − − + Ω
−
− = − + − Ω
ɺ ɺ
ɺ
ɺ ɺ
z yxz xz
l
x x x x
px z xz
m p
y y y y
y x pxz xz
n p
z z z z z
I II IqSb
p r C qr qp
I I I I
II I IqSc
q C pr p r r
I I I I
I I II IqSb
r p C pq qr q
I I I I I](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-108-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
109
Acceleration sensor equations:
1
( ),
1
( ),
1
( ),
= +
=
=
x X T
y Y
z Z
a qSC F
m
a qSC
m
a qSC
m
where
cos sin ,
cos sin ,
= − +
= − −
L Z X
D X Z
C C C
C C C
α α
α α
and
cos sin ,
cos sin .
= − +
= − −
W
W
D D Y
Y Y D
C C C
C C C
β β
β β
here the notations are the same as that is used in subsection 6.1. Furthermore,
, , ,x y z pI I I I and xzI are inertia while , ,S m c and b (wing area, mass, mean chord line
and wingspan) are physical parameters of the aircraft. Exact values of these parameters
for the Beaver aircraft came from [7].](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-109-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
110
B. Computing nominal parameter values for longitudinal model
identification
In this part, calculation of the nominal parameter values for the Maximum Likelihood
output-error identification of the longitudinal non-dimensional aerodynamic
parameters is described.
According to [1]:
0
0
,
,
= ∆ +
= ∆ +
X X X
D Z Z
C C C
C C C
cos sin ,
cos sin ,
= − +
= − −
L Z X
D X Z
C C C
C C C
α α
α α
It can be obtained by combining these equations:
0 0
0 0
0 0 0 0
0 0
0 0 0 0
1 1
0 0 0 0 0
cos sin ,
( )cos( ) ( )sin( )
( )(cos cos sin sin ) ( )(sin cos cos sin )
... cos sin cos sin ( sin
∆ ∆
= − −
= − ∆ + + ∆ − ∆ + + ∆
= − ∆ + ∆ − ∆ − ∆ + ∆ + ∆
≈ − − − ∆ − ∆ + −
D X Z
D X X Z Z
X X Z Z
X Z X Z X Z
C C C
C C C C C
C C C C
C C C C C C
α α
α α
α α α α
α α α α α α α α
α α α α α
0 0
0 0
0 0 0
0
0 0
0 0 0 0
1 1
0 0 0 0
cos ) ,
cos sin ,
( )cos( ) ( )sin( )
( )(cos cos sin sin ) ( )(sin cos cos sin )
... cos sin cos sin (
∆ ∆
∆
= − +
= − ∆ + + ∆ + ∆ + + ∆
= − ∆ + ∆ − ∆ + ∆ + ∆ + ∆
≈ − + − ∆ + ∆ +
L Z X
L Z Z X X
Z Z X X
Z X Z X Z
C C C
C C C C C
C C C C
C C C C C
α α
α α
α α
α α α α
α α α α α α α α
α α α α 00 0sin cos ) .+ ∆XCα α α
Then substituting (6.4) can be taken into consideration. Note that nonlinear terms are
dropped. We have to use also the results of the engine model of the Beaver aircraft
[7].
1 2 3
,
1
2
,
.
= +
=
=
p dpt
p dpt
X X
X Z
P
dpt C C
V
C C dpt
C C dpt
where P is the engine power and dpt is the non-dimensional propeller slipstream.
Derivative of dpt according to V is then calculated to obtain the form that contains
0
∆V
V
.](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-110-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
117
E. References
[1] Vladislav Klein and Eugene A. Morelli, Aircraft System Identification: Theory
and Practice, 1st ed.: AIAA, 2006
[2] Micheal Verhaegen and Vincent Verdult, Filtering and System Identification: A
Least Squares Approach, Cambridge, 2007
[3] Micheal Verhaegen, Vincent Verdult and Niek Bergboer, Filtering and System
Identification: An Introduction to using Matlab Software, TU Delft, 2007
[4] Brian L. Stevens, Frank L. Lewis, Aircraft Control and Simulation, John Wiley
and Sons, 1992
[5] Hungarian Academy of Sciences, Computer and Automation Research Institute,
Systems and Control Laboratory, Flight data sequences measured on
UltraStick-25
[6] Bauer Péter, Háromüzemmódú kibıvített Kalman szőrık repülıgép
orientációjának becslésére, 2010 Repüléstudományi közlemények
[7] Marc Rauw, FDC 1.2 A Simulink Toolbox for Flight Dynamics and Control
Analysis, 2001,
[8] Béla Lantos, Lırinc Márton, Nonlinear Control of Vehicles and Robots,
Springer, 2011
[10] Gerlai Tibor, Vízer Dániel, Repülıgép nemlineáris dinamikus modelljének
identifikációja szenzorfúzióval, BME-IIT, TDK dolgozat, II. díj, 2010](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-117-320.jpg)
![Nonlinear model identification of airplanes using noisy flight data
118
[11] Vízer Dániel, Nonlinear dynamic model identification for aircraft applying time
and frequency domain identification techinques, BME-IIT, TDK dolgozat, III.
díj, 2011
[12] FöldesiTamás, Repülıgépek robusztus irányítási módszerei, Diplomamunka,
BME-IIT, 2009](https://image.slidesharecdn.com/9bf6f625-aac2-492e-bbe6-407156428576-160920101512/85/Vizer_MSc_Thesis_2011-118-320.jpg)
Vizer_MSc_Thesis_2011
- 1. Budapest University of Technology and Economics Department of Control Engineering and Information Technology École Nationale Supérieure d’Ingénieurs de Poitiers Laboratoire d’Automatique et d’Informatique Industrielle Nonlinear model identification of airplanes using noisy flight data MSc. Thesis Written by: Dániel Vízer Consultants: Prof. Emer. Dr. Béla Lantos, BUTE CEIT Assoc. Prof. Dr. Guillaume Mercère, ENSIP LAII December, 2011
- 2. Nonlinear model identification of airplanes using noisy flight data 2 DIPLOMA THESIS TASK Dániel Vizer finalist in MSc Electrical Engineering Nonlinear model identification of airplanes using noisy flight data The nonlinear dynamic model of airplanes is strongly nonlinear as the result of numerous (mechanical, aerodynamical) effects containing many unknown parameters. The determination of the nonlinear model and its parameters are parts of the field of system identification. In order to simplify the problem, it will be assumed that the flight test data for identification can be obtained in simulation investigations of an airplane having known dynamic model. The goal is to become acquainted with time domain and frequency domain identification methods and validate whether the known model parameters can be reconstructed by using these methods. To the investigations the SIDPAC Toolbox developed by NASA can be applied and completed with own software components in MATLAB R2006a and Toolboxes environment. The proposed tasks of this thesis are: Part I: 1. Review of time domain Maximum Likelihood (ML) and frequency domain (FD) identification methods summarizing the fundamentals of influential theoretical directions, with special regard to the Output-Error (OE) methods. 2. Development of methods for prefiltering and robust differentiation of signals playing role in the identification. Use the advantageous properties of data analysis methods like global smoothing, discrete Fourier-transformation and chirp z-transform. 3. Develop the algorithms of time domain ML and OE methods. Demonstrate the identification ability of these methods based on flight test data available in known databases of the literature. 4. Develop the algorithms of frequency domain ML and OE methods. Demonstrate the identification ability of these methods based on flight test data available in known databases of the literature. Part II: 5. Determine flight test data using simulation of the Beaver airplane having known nonlinear dynamic model. Add noises to the signals. The flight data has to be divided into teaching and testing parts. Apply prefiltering/differentiation to the signals. 6. Based on the teaching data of Beaver airplane and using the time domain ML and OE methods, develop the identification of the airplane model and perform its validation. 7. Based on the teaching data of Beaver airplane and using the frequency domain ML and OE methods, develop the identification of the airplane model and perform its validation. 8. Give the user documentation of the developed software and suggestions for the further developments for similar applications. External consultant (Part I): Assoc. Prof. Dr Guillaume Mercere Place of thesis practice (Part I): Universite de Poitiers, LAII University consultant: Prof. emer. Dr. Béla Lantos, BUTE CEIT Place of thesis practice (Part II): Intelligent Control Laboratory, BUTE CEIT Deadline: December 9, 2011 Final exam subjects: 1. Theory of Robot and Control Systems BMEVIIIM127 2. Nonlinear and Robust Control BMEVIIIM211 Budapest, January 17, 2011 Prof. Dr. László Szirmay-Kalos
- 3. Nonlinear model identification of airplanes using noisy flight data 3 Nyilatkozat Alulírott Vízer Dániel, a Budapesti Mőszaki és Gazdaságtudományi Egyetem hallgatója kijelentem, hogy ezt a szakdolgozatot meg nem engedett segítség nélkül, saját magam készítettem, és a szakdolgozatban csak a megadott forrásokat használtam fel. Minden olyan részt, melyet szó szerint, vagy azonos értelemben, de átfogalmazva más forrásból átvettem, egyértelmően a forrás megadásával megjelöltem. 2011. december 09. Vízer Dániel ....................................
- 4. Nonlinear model identification of airplanes using noisy flight data 4 So what is control engineering? “Control is the intelligence behind how things work!” /Tariq Samad, fellow president of IEEE CSS/
- 5. Nonlinear model identification of airplanes using noisy flight data 5 Acknowledgements First of all, I would like to thank my consultants, Prof. Emer Dr. Béla Lantos for his tremendous amount of support during my studies and Assoc. Prof. Dr. Guillaume Mercère who helped me to write the first part of this work in Poitiers. Special thanks to Keith Veizer for proofreading my thesis. Last but not least, I would like to thank my family and my friends who have always been beside me.
- 6. Nonlinear model identification of airplanes using noisy flight data 6 Kivonat A repülıgépek irányításának problémája egy kiterjedt és rohamosan fejlıdı ága a szabályozástechnikának. A repülıgépek fedélzeti számítógépeire implementált szabályozások nemcsak a robotpilóta funkciót valósítanak meg, hanem komoly szerepet játszanak a pilóta munkájának támogatásában is. Ezek a rendszerek az ún. stabilitás (SAS) és szabályozás (CAS) kiterjesztı rendszerek. Ezen eljárásoknak számos civil alkalmazása van azonban továbbra is a katonai alkalmazások motiválják a terület fejlıdését. A precíz szabályozások tervezéséhez elengedhetetlenül fontos, hogy ismerjük a repülıgép pontos nemlineáris dinamikus modelljét. A modell matematikai alakja egzakt módon elıáll, azonban a modell paramétereinek meghatározása komoly méréstechnikai és jelfeldolgozási feladat. A nemlineáris modell paramétereinek kiszámítása a rendszerek identifikációjának részét képezı tudományághoz tartozik. Jelen munkában feldolgozásra és bemutatásra kerülnek a frekvencia és idıtartománybeli Maximum Likelihood (ML) módszerek, különös tekintettel a kimeneti hiba (OE) módszerekre. A módszerek demonstrálása során felhasználásra kerülnek a SIDPAC identifikációs Toolbox [1] függvényei. Ezután virtuális tesztrepülések során MATLAB/Simulink környezetben adatok győjtésére kerül sor. A szimulált adatsorokhoz zajokat és egyéb zavaró hatásokat adok a valós repülési körülmények modellezése érdekében. Végül a fent említett szőrési és identifikációs eljárások alkalmazása történik a szimulált adatsorokra. A dolgozatot a kapott eredmények és a jövıbeli fejlesztések összefoglalása zárja.
- 7. Nonlinear model identification of airplanes using noisy flight data 7 Abstract Aircraft control is one of the most progressive fields of the modern control techniques. Control methods implemented on embedded flight computers enhance not only robot pilot functions but also support the pilot. These are known as stability and control augmentation systems. There are also several civil applications of these methods; for instance, in the case of vehicles of commercial and personal air transport. However, the military is the most motivated to improve these applications and methods. It is indispensable to have an accurate nonlinear dynamic model of the aircraft to achieve an adequate control that satisfies all of the requirements. A mathematical model of the aircraft can be derived using kinematic, dynamic and navigational equations of the rigid body. Contrarily, it is a more difficult process to determine its parameters. The determination of the nonlinear model and its parameters are parts of the field of system identification. In this MSc thesis, several identification methods such as the Maximum Likelihood (ML) method both in the time domain and in the frequency domain are reviewed. Particular focus is placed on the output error method. A brief outline of filtering and smoothing the signals playing important roles in the model parameter determination is also presented. Identification methods mentioned above are demonstrated applying the SIDPAC Toolbox developed by NASA. Then, simulated data sequences are gathered in MATLAB/Simulink environment using a nonlinear model of a Beaver aircraft. Noises and other disturbances are added to the signals in order to model a real flight situation. Eventually, filtering methods are applied to the gathered data sequences and the application of identification methods mentioned above are done following a conclusion of the results.
- 8. Nonlinear model identification of airplanes using noisy flight data 8 Résumé La commande des avions est l'un des champs applicatifs les plus émergents des techniques modernes de commande. Les algorithmes de commande embarqués réalisent non seulement une fonction importante d'auto-pilotage mais permettent également d'aider le pilote dans la plupart des tâches qui lui incombent. Ces techniques sont généralement regroupés sous la dénomination SAS (stability augmentation system) ou CAS (control augmentation system). Il existe un grand nombre d'applications civiles à ce type de méthodes. On peut citer le transport public ou commercial par avion. Cependant, il semble que le secteur militaire soit le plus demandeur pour développer et améliorer les techniques de commande actuelles. Pour la synthèse de lois de commande qui satisfassent les nouvelles contraintes liées aux avions modernes, des modèles dynamiques non linéaires précis sont désormais nécessaires L'obtention de tels modèles mathématiques peut être réalisée en appliquant les lois de la cinématique, de la mécanique des corps rigides... Malheureusement, le calcul des paramètres de tels modèles est généralement très complexe . La modélisation expérimentale (ou identification) est une solution intéressante pour estimer les paramètres d'un modèle non-linéaire du comportement d'un avion. Au sein de ce mémoire de thèse de Master, une étude bibliographique et comparative de plusieurs techniques d'identification est proposée. Les méthodes de maximum de vraisemblance dans le domaine temporel et fréquentiel sont plus précisément considérées. Une attention particulière est portée aux méthodes d'erreur de sortie. Une description des techniques de filtrage des signaux d'entrée-sortie de l'avion est également présentée. Le mémoire se focalise ensuite sur l'application des techniques citées précédemment à l'aide de la boîte à outils SIDPAC développée par la NASA. Ensuite, des vols virtuels sont performés dans l’environement de MATLAB /Simulink en ajoutant des bruits aux signaux d’entrée-sortie pour modeliser les situations réels. Les méthodes de filtrage sont ensuite utilisées pour éliminer des effects nuisibles des bruits. Finalement, les methodes d’identification sont exécutées. La conclusion des resultats et des perspectives de futurs travaux concluent ce manuscrit.
- 9. Nonlinear model identification of airplanes using noisy flight data 9 Table of contents 1 Introduction.......................................................................................................... 11 2 Review of time-domain and frequency domain Maximum Likelihood identification methods ................................................................................................ 13 2.1 Why the Maximum Likelihood methods.................................................................... 13 2.2 Outline of estimation theory ........................................................................................ 13 2.3 Time-domain ML methods........................................................................................... 15 2.3.1 Filter-error method............................................................................................................16 2.3.2 Properties of ML methods .................................................................................................23 2.3.3 Output-error method .........................................................................................................24 2.3.4 Equation-error method ......................................................................................................31 2.4 Frequency domain ML methods................................................................................... 33 2.4.1 Transforming between time-domain and frequency domain.............................................34 2.4.1 Frequency domain ML estimator for linear dynamic model.............................................35 2.4.2 Output-error method .........................................................................................................40 2.4.4 Equation-error method ......................................................................................................42 2.5 Accuracy of parameter estimates................................................................................. 43 3 Data preprocessing.............................................................................................. 46 3.1 Filtering and smoothing......................................................................................... 46 3.2 Global smoothing in the frequency domain ........................................................ 46 3.3 Numerical differentiation....................................................................................... 51 3.4 Finite Fourier Transform....................................................................................... 53 4 Demonstration of time domain ML and OE methods using test data from known data bases ......................................................................................................... 57 5 Demonstration of frequency domain ML and OE methods using test data from known data bases................................................................................................ 69 6 Flight test data determination using simulation............................................. 76 6.1 Outline of aircraft modeling.................................................................................. 76 6.2 Virtual test flights................................................................................................... 80 6.3 The applied noise model ........................................................................................ 88 6.4 Prefiltering the signals........................................................................................... 90 7 Parameter estimation using time domain ML and OE methods................... 91 8 Parameter estimation using frequency domain ML and OE methods.......... 96 9 Description of the applied software components and Simulink models ..... 99 10 Summary and concluding results..................................................................... 107 11 Appendix ............................................................................................................. 108 A. State equations of an aircraft................................................................................... 108 B. Computing nominal parameter values for longitudinal identification............... 110
- 10. Nonlinear model identification of airplanes using noisy flight data 10 C. Lateral parameter identification results both TD and FD issues in common tables.................................................................................................................................... 112 D. List of figures and tables .......................................................................................... 114 E. References................................................................................................................... 117
- 11. Nonlinear model identification of airplanes using noisy flight data 11 1 Introduction Aircraft control is one of the most progressive fields of the modern control techniques. Embedded on board control systems enhance not only robot pilot functions but also support the pilot. In the past five decades the huge growth in aircraft development has focused on the improvement of modern control methods. This began in the late 60s when digital computers become available. Nowadays, most modern warplanes, also known as fourth generation jet fighters, require not only accurate dynamical models to obtain very precise control but also high speed computers on board. Furthermore, there are not any direct connections between control surface actuators and the pilot’s instruments of control (pilot stick, pedals…). These aircraft are also known as “fly-by wire” controlled aircraft. Most modern commercial aircraft are also fly-by-wire controlled. These airplanes cannot be controlled without an on board computer because they are not stable aerodynamically. Several control methods exist that cannot be controlled by the pilot such as canards and flaperons. These additional control surfaces are controlled by embedded control systems in order to obtain the required stability. Thus, control and stability augmentation systems are indispensable. Consequently, having an accurate nonlinear dynamic model of the aircraft is necessary in order to develop an adequate control that satisfies all of the requirements. Several types of measurements and theoretical background are available for the technicians in order to compose a nonlinear dynamic model of the aircraft. For many applications, an aircraft can be assumed as a rigid body, whose motion is governed by Newtonian physics. The navigational equations play also an important role in building the mathematical model of the aircraft. These equations enable transformations between the different coordinate-systems. Measured data can be gathered from ground measurements (wind-tunnel tests) or directly from the aircraft instrumentation. These measurements include input and output data sequences. In practice, measurements are corrupted by noises. In this work, I only concerned with noisy flight data Building a mathematical model for real physical systems based on noisy measured data is considerably old and one of the most fundamental parts of all human scientific pursuits. Methods aiming to reach an adequate mapping of a
- 12. Nonlinear model identification of airplanes using noisy flight data 12 physical system based on observations are called identification methods. In order reducing the adverse effects of noises require the introduction of statistical theory and methods. The most important requirement for a mathematical model is to be applicable in many ways. It must be simple enough to be useful and at the same time complex enough to represent perfectly the physical world. Hence, system identification takes a significant part in the field of control engineering. In this MSc thesis, several identification methods such as the time domain Maximum Likelihood (ML) and frequency domain methods are reviewed, based on [1]. Particular focus is placed on the output-error method based on [2] and [1]. Some important methods for filtering and smoothing of signals playing important roles in identification [1] are also studied. Methods mentioned above are demonstrated in MATLAB environment using the SIDPAC Toolbox applying the flight data from the literature [1]. In the following, simulated data sequences are gathered by virtual test flights applying the nonlinear dynamic model of a DHC-2 Beaver aircraft. The sense of this experiment is to verify the applied identification techniques and to ensure about its applicability in the case of data sequences gathered on a real aircraft. In order to model real life situations, additional noises and disturbances are added to signals. Then, filtering methods written in [1] are used to eliminate the adverse effects of the noises. Time and frequency domain output-error methods are then applied to determine the parameters of the Beaver aircraft. Furthermore, given results are compared with the real parameter values. Eventually, a summary and a conclusion about the results and the future research plans close this MSc thesis work.
- 13. Nonlinear model identification of airplanes using noisy flight data 13 2 Review of time-domain and frequency domain Maximum Likelihood identification methods In this chapter the time domain Maximum Likelihood (ML) and the frequency domain ML identification methods are summarized for later application. Also the fundamentals of influential directions are summarized. Particular focus is placed on the output-error (OE) method. The theory written in this chapter is based on the descriptions and definitions can be found in [1] and [2]. 2.1 Why the Maximum Likelihood methods During aircraft system identification, the applied measured time series are generated by nonlinear dynamic systems and corrupted by noises. Because of this, a very simple idea is used: assuming a model that depends on the unknown parameters, then minimizing a cost function in order to find the values of the parameters that make the outputs of the identified model most likely to correlate to the measured data. 2.2 Outline of estimation theory The parameter estimation consists in finding adequate values of unknown model parameters θ in an assumed model, based on noisy measurements z . Since the estimator computes approximate values of the parameter vector ˆθ based on noisy measured data z , hence ˆθ is a random variable. In order to develop an estimator, four specifications are required: (i) A model structure with unknown parameters; (ii) Measurements z ; (iii) A mathematical model of the process; (iv) Assumptions about the uncertainties in the measurement noise v , and in the unknown model parameters. Observations can be linear or nonlinear functions of the parameters. A model is called linear in the parameters if its output y is given by y Hθ= , (2.1)
- 14. Nonlinear model identification of airplanes using noisy flight data 14 where the matrix H is assumed to be known. The measurement equation then can be expressed as z H vθ= + . (2.2) A model is nonlinear in the parameters, if its measurement equation is given by ( )z h vθ= + , (2.3) where ( )h θ is assumed to be a known function. Generally, there are 0n measured outputs, a measured data vector consisting of N data points. Three models of uncertainties in the parameters and the observations are considered. They are known as the Bayesian model, the Fisher model and the least-squares model. The parameters and measurements are formed as follows: Bayesian model θ is a vector of random variables with probability density ( )p θ ; v is a random noise sequence with a probability density ( )p v . Fisher model θ is a vector of unknown constant parameters; v is a random noise sequence with probability density ( )p v . Least-squares model θ is a vector of unknown constant parameters; v is a random vector of measurement noise. The time domain ML and frequency domain ML estimator were developed for the Fisher model. θ is the vector of unknown constant parameters and v has an expected value or mean that equal to 0, and a variance R . This can be written in the form (0, )Rℕ . The Fisher model is based on the Fisher estimation theory. Detailed description can be found in [1]. The concept of the likelihood function is used, where ( ; ) ( )L z p zθ θ= . The parameter vector θ is assumed to be a vector of unknown constants. Hence, the most common estimator of the Fisher model is to maximize the likelihood function, which is equal to the value of the parameter vector that maximizes ( ; )L z θ for given z . If ( )p z is Gaussian, the likelihood function has the form 1 12 1 ( ; ) (2 ) exp ( ) ( ) 2 N T L z R z H R z Hθ π θ θ − − = − − − . (2.4) where i denotes the determinant.
- 15. Nonlinear model identification of airplanes using noisy flight data 15 Then the estimate ˆ max ( ; )L z θ θ θ= , (2.5) minimizes 11 ( ) ( ) ( ) 2 T J z H R z Hθ θ θ− = − − . (2.6) Or in the case of nonlinear measurement equation 11 ( ) ( ( )) ( ( )) 2 T J z h R z hθ θ θ− = − − . (2.7) The Fisher information matrix M is defined as 2 ln ln ln ( ) , T T L L L M E Eθ θ θ θ θ ∂ ∂ ∂ = = − ∂ ∂ ∂ ∂ (2.8) This equality is discussed in detail in [1]. 2.3 Time-domain ML methods It can be seen below that in the case of a linear measurement equation z H vθ= + the ML estimator becomes a simple least-square estimator with weighting equal to the inverse of the noise covariance matrix. Furthermore, in the case of a stochastic dynamic system with process noise, the measurements are nonlinear functions of the parameters. Thus, the general form of the measurement equation is ( )z h vθ= + . The solution combines a state estimator and a nonlinear parameter estimator. The models to be estimated are in state-space form. Hence, the state estimator is a Kalman filter which is necessary because of the process noise in the dynamic equations. This method is also known as the filter-error method. The nonlinear parameter estimator is required because of the nonlinear measurement equation, which means that there is nonlinear connection between model parameters and measured outputs. However, the combined estimation of parameters and states induce a difficult nonlinear estimation problem. For further practical implementation, three simplifications are made: (i) The Gauss-Newton method, also known as the modified Newton- Raphson method is used for minimizing the cost function instead of the Newton-Raphson method;
- 16. Nonlinear model identification of airplanes using noisy flight data 16 (ii) In the case of linear dynamic models, the general Kalman filter is replaced by its steady-states form; (iii) The unknowns are divided into three subsets: states, model parameters and the elements of the noise covariance matrix. In the following section, the filter-error method is introduced briefly. It is followed by the properties of ML methods. Properties of ML estimators are also introduced. For further practical applications, the assumption of no process noise is made. In this case, the states can be derived by direct numerical integration. The resulting ML estimator is known as the output-error method. This method is a generalization of the nonlinear least squares estimator presented in [1]. Flight testing is done preferably on a day with calm air in order to eliminate the process noise in the model. Finally, the equation-error method is presented. If the state variables are measured without error, the equation-error version of the ML estimator can be used that is identical to linear regression. 2.3.1 Filter-error method In most cases of practical system identification problems, such as the aircraft dynamic modeling problem, discrete-time measurements made on a continuous-time dynamic system are given. A stochastic forcing term is added to the system to represent inadequacies of the linear model structure. The model equations then can be written as: ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ), 1,2,..., , wx t A x t B u t B w t y t C x t D u t z i y i v i i N θ θ θ θ θ = + + = + = + = ɺ (2.9) [ ] [ ][ ]{ } 0 0 0 0 (0) ( ), (0) ( ) (0) ( ) ( ), T E x x E x x x x P θ θ θ θ = − − = (2.10) where ( )w t is the process noise and ( )v i is the measurement noise, and they are white noise sequences.
- 17. Nonlinear model identification of airplanes using noisy flight data 17 [ ] [ ] ( ) 0, ( ) 0, ( ) ( ) ( , ) ( ), ( ) ( ) ( , ) , T i j i i j T ij E w t E v i E w t w t Q t t t E v i v j R i θ δ θ δ = = = − = (2.11) where θ is the unknown parameter vector. The random vectors ( )w t and ( )v i are assumed to be white and uncorrelated. The Q and R matrices are usually diagonal. The likelihood function for a sequence of measurements will be denoted ( ; )NL Z θ that means [ ](1), (2),..., ( ) T NZ z z z N= . The expression of the likelihood function can be derived by successive applications of the Bayes’s rule. [ ] [ ] [ ] 1 1 1 2 2 1 1 ( ; ) (1), (2),... ( ), ( ) ; ; ( ) ; ( 1) ; ; ... ( ) ; . N N N N N N N i i L Z z z z N L z N Z L Z L z N Z L z N Z L Z L z i Z θ θ θ θ θ θ θ θ − − − − − − = = = = − = = ∏ (2.12) As was mentioned below, in order to find an optimal parameter vector, the likelihood function has to be maximized. However, for computational purposes, it is advantageous to minimize the negative logarithm of the likelihood function instead. Thus, the ML estimator can be written in the form of [ ] { }1 1 11 ˆ max ; max ( ) ; min ( ) ; N N N i i ii L Z L z i Z ln L z i Z θθ θ θ θ θ θ− − == = = = − ∑∏ . (2.13) If ( )w t and ( )v i are normally distributed and independent, then ( )z i will also have the same properties. The following notation will be used: ˆ( 1)x i i − is the best estimate of the state ˆ( )x i using measurements [ ](1), (2),..., ( 1)z z z i − . Then [ ]1( ) ; ( );iL z i Z L z iθ θ− = will be determined by the mean and its covariance matrix. These quantities can be expressed as [ ] [ ] { } ˆ( ); ( 1) ˆ ˆ( ); ( ) ( 1) ( ) ( 1) ( ) ( ) ( ), T T E z i y i i Cov z i E z i y i i z i y i i i i i θ θ = − = − − − − = = v v BBBB (2.14)
- 18. Nonlinear model identification of airplanes using noisy flight data 18 where ˆ( ) ( ) ( 1)i z i y i i= − −v , (2.15) is the vector of innovations and ( )iBBBB is the innovation covariance matrix. Hence, the probability density of the innovation approaches a Gaussian distribution, as the sampling rate increases. In the case of a sufficiently high sample rate, the likelihood function has the form of [ ] 1 1 2 1 ( ); (2 ) ( ) exp ( ) ( ) ( ) . 2 N T L z i i i i iθ π − − = − v vB BB BB BB B (2.16) Then, the negative log-likelihood function can be expressed as ( ) 1 0 1 1 ; ( ) ( ) ( ) ( ) (2 ), 2 2 N T N i Nn ln L Z i i i ln i lnθ π− = − = + + ∑ v vB BB BB BB B (2.17) where 0n is the number of output variables. It can be seen that the last term on the right side is a constant, and it has no effect on the optimization. Thus, it can be neglected, which results in ( ) 1 1 1 ; ( ) ( ) ( ) ( ) . 2 N T N i ln L Z i i i ln iθ − = − = + ∑ v vB BB BB BB B (2.18) In order to determine a negative log-likelihood function the mean value and the covariance matrix of the innovation have to be calculated. These quantities can be derived using a Kalman filter. The filtering has two steps, consisting of prediction and measurement update. The following equations are derived for linear time-invariant system. Initial condition: [ ] [ ][ ]{ } 0 0 0 0 (0) (0) ( ), (0) (0) (0) ( ), T x E x x P E x x x x P θ θ = = = − − = (2.19) Prediction: ˆ ˆ( 1) ( ) ( 1) ( ) ( ), ( 1) ( ) ( 1) ( 1) ( ) ( ) ( ), ( 1) , T T w w d x t i A x t i B u t dt d P t i A P t i P t i A B QB dt i t t i t θ θ θ θ θ θ − = − + − = − + − + − ∆ ≤ ≤ ∆ (2.20) Measurement Update:
- 19. Nonlinear model identification of airplanes using noisy flight data 19 [ ] 1 ˆ ˆ( ) ( 1) ( ) ( ), ( ) ( 1) ( ) ( 1), ( ) ( ) ( ) ( 1), T x i i x i i K i i K i P i i C i i P i i I K i C P i i θ θ − = − + = − − = − − v BBBB (2.21) where ( 1) ( ) ( 1) ( ) ( ), ˆ( ) ( ) ( ) ( 1) ( )( ), T i i C P i i C R i z i C x i i D i θ θ θ θ θ − = − + = − − −v BBBB (2.22) Optimization algorithm The ML parameter estimates are obtained by minimizing the negative log-likelihood function 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ). 2 2 N N T i i J i i i iθ − = = = +∑ ∑v vB BB BB BB B (2.23) In order to solve this optimization problem, generally, the Newton-Raphson method is used. To determine a solution to the optimization-problem given in (2.23), the cost function θ is expanded in a Taylor-series around a nominal parameter estimation 0θ . The choice of this nominal parameter estimation plays an important role in both the convergence and computational time of the optimization. The Taylor-series expansion is given by 0 0 0 1 ( ) ( ) '( ) "( ) ..., 2 T T J J J Jθ θ θ θ θ θ θ= + ∆ + ∆ ∆ + (2.24) We approximate ( )J θ with the second-order expansion of its Taylor-series, 0 0 0 1 ( ) ( ) '( ) "( ) , 2 T T J J J Jθ θ θ θ θ θ θ≈ + ∆ + ∆ ∆ (2.25) where '( )J θ is the Jacobian, "( )J θ is the Hessian of the cost function at the nominal parameter estimation and θ∆ is the vector of changes inθ . ( ) (1) ( ) ( ) (2)'( ) , ( ) ( )p J J J J J n θ θ θ θ θθ θ θ θ ∂ ∂ ∂ ∂ ∂= = ∂ ∂ ∂ ⋮ (2.25)
- 20. Nonlinear model identification of airplanes using noisy flight data 20 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) (1) (1) (1) (2) (1) ( ) ( ) ( ) ( ) ( ) (2) (1) (2) (2) (2) ( )"( ) . ( ) ( ) ( ) ( ) (1) ( ) (2) ( ) ( ) p pT p p p p J J J n J J J J nJ J J J n n n n θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θθ θ θ θ θ θ θ θ θ θ θ θ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ (2.26) The necessary condition to minimize ( )J θ becomes 0 0'( ) "( ) 0.J Jθ θ θ+ ∆ = (2.27) The solution of this equation gives an estimate for the vector of parameter changes, [ ] 1 ˆ "( ) '( ).J Jθ θ θ − ∆ = − (2.28) We assumed that the Hessian is nonsingular. Since the first and second order gradient of the cost function are computed around a nominal parameter estimation 0θ , the updated parameter vector estimation ˆθ is computed by, 0 ˆ .θ θ θ= + ∆ (2.29) Because of the approximation in the Taylor series, it is necessary to repeat the estimation using the estimated vector ˆθ as the new nominal value 0θ . That is important because the outputs are nonlinear functions of the parameters, thus the cost function is more complicated than quadratic. In the case of linear least-squares problems, which are discussed in detail in [1], the output depends linearly on the parameters. Thus, the cost function is a quadratic function of the parameters and the term given in (2.25) is an exact equality. Accordingly, the parameter estimates could be derived in one single iteration. In this case, the repeated applications of the quadratic cost to the nonlinear function on the parameters are used to iteratively arrive to the solution. This iterative process will be arrived at when the convergence criteria are satisfied. Convergence criteria that have been used in practice are introduced later in this section.
- 21. Nonlinear model identification of airplanes using noisy flight data 21 Minimization of the negative-log likelihood function for the filter-error problem is a very difficult optimization problem. Computing the second order gradients in the Hessian matrix is an extremely time demanding task and is not robust enough. Hence, for practical applications the modified Newton-Raphson method is used. Details of this approach are introduced later. A block diagram of the filter-error method is shown below in Figure.2.1. Figure 2.1 Block diagram of filter-error method Simplification possibilities In order to obtain simplifications of this optimization problem, several solutions are introduced in [1] and [2]. Comparisons and summaries of these simplifications are also outlined in [1]. The ML estimation algorithm for a time-invariant stochastic linear dynamic system can be simplified through replacing the Kalman filter by its steady-states form. Initial condition: [ ] [ ][ ]{ } 0 0 0 0 (0) (0) ( ), (0) (0) (0) ( ), T x E x x P E x x x x P θ θ = = = − − = (2.30) Prediction: ˆ ˆ( 1) ( ) ( 1) ( ) ( ), ( 1) , d x t i A x t i B u t dt i t t i t θ θ − = − + − ∆ ≤ ≤ ∆ (2.31) Measurement update: 1 ˆ ˆ( ) ( 1) ( ), ( ) ,T x i i x i i K i K PC θ − = − + = v BBBB (2.32)
- 22. Nonlinear model identification of airplanes using noisy flight data 22 where ( ) ( ) ( ), ˆ( ) ( ) ( ) ( 1) ( ) ( ). T C PC R i z i C x i i D u i θ θ θ θ θ + = − − −v B =B =B =B = (2.33) In this case, the steady-state form of the Riccati equation can be solved for P. 1 ( ) ,T T T T w wP P PC CPC R CP Q− = Φ − + Φ + Γ Γ (2.34) Φ and wΓ can be written in the form: 1 , ( ) . A t A t w w e A e I B ∆ − ∆ Φ = Γ = − (2.35) This means that using a steady-state filter brings simplification to the equation for the first order gradients of the cost function, because the partial derivatives of the cost function with respect the unknown parameter vector are calculated from a linear matrix equation. Henceforward, the estimation algorithm has some difficulties. Further simplification to algorithm can be obtained by recasting the vector of unknown parameters. The elements of K and BBBB are included into the parameter vector, instead of the elements of Q and R. This leads to a new cost function in which an approximation of the covariance matrix is used according to (2.36) that is derived in [1]. 1 1ˆ ( ) ( ). N T i i i N = ∑B =B =B =B = v v (2.36) The resulting cost can be minimized by using a nonlinear optimization technique, such as the Newton-Raphson method as it was mentioned earlier. This is also an iterative optimization procedure that continues until selected convergence criteria are satisfied. In this case, the unknown parameters are estimated in subsets. First, the parameters contained in A,B,C,D and K held constant while the estimation of the parameters in BBBB is in progress, and then the parameters contained in BBBB are held constant while the parameters in A,B,C,D and K are estimated. The estimate of the measurement covariance matrix can be derived from (2.32-33): ˆ ˆˆ ˆ( )R I CK= − B.B.B.B. (2.37) This method carries also many problems in the computation of wB and Q in (2.32-33). Because of these difficulties in the simplified algorithms a third formulation was
- 23. Nonlinear model identification of airplanes using noisy flight data 23 proposed. It is advantageous to recast the unknown parameters as the elements of A, B, wB , C, D, Q and BBBB . The resulting optimization algorithm has six steps: (i) Assembling the system and covariance matrices A, B, wB , C, D, Q and R using the nominal parameter values; (ii) Computing the covariance matrix of the innovations (2.36); (iii) Estimating of the unknown parameters in the state and covariance matrices, using a nonlinear optimization technique; (iv) For ˆ=B BB BB BB B , compute ˆK and ˆR from (2.32) and (2.37); (v) Update the nominal values of the unknown parameters; (vi) Repeat steps 1-5 until convergence criteria are satisfied. The filter-error method is the most general estimation method for practical aircraft system identification problems because it allows the presence of both the measurement and process noise in the model. However, the main practical problem with this method is that there are a lot of parameters to be estimated which often leads to difficulties. 2.3.2 Properties of ML methods The accuracy of the estimation concerns itself with the properties of the estimates. Using the definitions introduced in [1]. (i) ML estimates of dynamic system parameters are asymptotically unbiased; ˆ( ) , . E N θ θ→ → ∞ (ii) ML estimates are consistent; ˆ , .N θ θ→ → ∞ (iii) ML estimates are asymptotically efficient; 1ˆ( ) , , Cov M N θ − → → ∞ where M the Fisher information matrix defined earlier by (2.8).
- 24. Nonlinear model identification of airplanes using noisy flight data 24 This means that the main diagonal elements of the inverse information matrix provide the lower bounds on the parameter variances, named the Cramer-Rao bounds. These diagonal elements represent the achievable accuracy for the parameter estimates. (iv) ML estimates are asymptotically normal. The distributions of the estimates are asymptotical approaches a normal distribution with mean θ and variance 1 M − , namely ˆθ is 1 ( , )Mθ − ℕ . 2.3.3 Output-error method Further simplification can be achieved when ML estimation method is applied to a deterministic linear dynamic system. In this case, there is no process noise, and the system can be written in the form of ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ), 1,2,..., , x t A x t B u t y t C x t D u t z i y i v i i N θ θ θ θ = + = + = + = ɺ (2.38) where v is (0, )Rℕ and [ ]( ) ( ) ( ) .T ijCov v i E v i v i Rδ = = Furthermore, because of the lack of process noise, Q is zero and the state equations are deterministic. The Kalman gain K is also zero, so the Kalman filter can be replaced by a simple integration of the state equations. The innovations in the method become: ( ) ( ) ( ) ( ) ( ) ( ) ( ), 1,2,..., .z i y i z i C x i D u i i Nθ θ− = − − =v = (2.39) Practically, they are equal to output errors. The negative log-Likelihood function then can be written as: ( ) -1 0 1 1 ; ( ) ( ) (2 ), 2 2 2 N T N i NnN ln L Z i R i ln R lnθ π = − = + + ∑v v (2.40) The two last terms of (2.40) can be dropped because they do not depend on the unknown parameter vector θ . The unknown parameters are elements of state matrices as can be seen above and the initial condition vector. Solving (2.40) with respect to R is done by differentiating with respect to R. The result then is set equal to zero and solved for R. That gives (2.41) which is similar to (2.36). 1 1ˆ ( ) ( ). N T i R i i N = = ∑v v (2.41)
- 25. Nonlinear model identification of airplanes using noisy flight data 25 Generally, only the diagonal elements of the R matrix are estimated from (2.40) assuming that the measurement noises are uncorrelated with one another. For a given ˆR the cost function can be written as: [ ] [ ]1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) . 2 2 N N TT i i J i R i z i y i R z i y iθ − − = = = = − −∑ ∑v v (2.42) This cost is minimized by using a relaxation technique in which first θ held fixed while computing ˆR from (2.41), then minimizing the cost with respect to θ for a given fixed R, where ˆR R= . These two steps are repeated until convergence criteria are satisfied. Optimization of the cost in (2.42) can be done using the Newton-Raphson method as discussed earlier. The first and second order gradient can be calculated by (2.25-26). The first order gradient has the following form: 1 1 1 1 ( ) ( ) ( )ˆ ˆ( ) ( ), T TN N i i J i y i R i R i θ θ θ θ − − = = ∂ ∂ ∂ = = − ∂ ∂ ∂ ∑ ∑ v v v (2.43) where the elements of the ( )J θ θ ∂ ∂ vector can be written as 1 1 ( ) ( ) ˆ ( ), 1,2,... . TN p ij j J y i R i j n θ θ θ − = ∂ ∂ = − = ∂ ∂ ∑ v (2.44) The elements of the second order gradient then have the form of 2 2 1 1 1 1 ( ) ( ) ( ) ( )ˆ ˆ ( ), , 1,2,... TN N p i ij k j k j k J y i y i y i R R i j k n θ θ θ θ θ θ θ − − = = ∂ ∂ ∂ ∂ = = ∂ ∂ ∂ ∂ ∂ ∂ ∑ ∑- v (2.45) The term on the far right side can be neglected. Thus, the resulting optimization method is called Gauss-Newton method or modified Newton-Raphson method. This simplification is made for practical reasons. Obtaining the second-order gradient is computationally expensive. Furthermore, it is susceptible to numerical errors. The parameter vector change then can be written using the approximate second-order gradient: 1 1 1 1 10 0 0 ( ) ( ) ( )ˆ ˆ ˆ ( ) , TN N i i y i y i y i R R iθ θ θ θ − − − = = ∂ ∂ ∂ ∆ = ∂ ∂ ∂ ∑ ∑ v (2.46) where y θ ∂ ∂ is an 0 pn n× matrix called the output sensitivity matrix and its elements are called the output sensitivities. The elements of this matrix quantify the changes in the outputs due to changes in the parameters. 1ˆR− is usually diagonal. According to (2.46), it can be seen that output sensitivities must be linearly independent and
- 26. Nonlinear model identification of airplanes using noisy flight data 26 nonzero for a good matrix inversion. When this condition is held, each model parameter has a unique and significant impact on the model outputs. Henceforward, in the case when the approximation for the second-order gradient of the cost function is used, the Fisher information matrix can be written as 2 1 1 ln ( , ) ( ) ( )ˆ( ) . TN N T i L Z y i y i M E R θ θ θ θ θ θ − = ∂ ∂ ∂ = = ∂ ∂ ∂ ∂ ∑ (2.47) Then the ML estimator has the form of: 1 0 0 0 0 ( )ˆ ( ) . J M θ θ θ θ θ − ∂ = − ∂ (2.48) The parameter covariance matrix satisfies the Cramer-Rao inequality 1ˆ( ) ( ) ,Cov Mθ θ − ≥ (2.49) which means that the inverse of the Fisher information matrix gives the lower bound for the parameter covariance matrix. A block diagram of the output-error method can be seen in Fig 2.2. Figure 2.2 Block diagram of output-error method Convergence criteria, computing sensitivities and nearly singular information matrix It was shown in [1] that the practical application of ML estimator to flight data sometimes leads difficulties. For instance, the dependence in the likelihood function can be far from quadratic. Furthermore, the information matrix can be singular or nearly singular. The optimization method for the output-error method discussed earlier can be summarized as:
- 27. Nonlinear model identification of airplanes using noisy flight data 27 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 ˆ , ( ) ( ), ˆ( ) ( ), ˆ( ) ( , ) ( , ) , ˆ( ) ( , ) ( ) , ( , ) ( ) , ˆ( ) ( ) ( , ). N T i N T i j k M g Cov M M S i R S i g S i R i y i S i i z i y i θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ − − − = − = = + ∆ ∆ = − ≥ = = ∂ = ∂ = − ∑ ∑ v v (2.50) where the information matrix M has dimensions p pn n× and the sensitivity matrix ( )S i has dimensions 0 pn n× . In order to reach at the global maximum of the likelihood function the modified Newton-Raphson method is used, but the nominal parameter value of θ must be near to the global minimum. The modified Newton-Raphson method is based on this assumption. If the nominal parameter vector is far from the optimal solution, then the result will be either convergence to a local minimum, or divergence. There are several methods to obtain excellent starting values; for instance, regression methods or in particular cases the subspace identification method. The applicability of the subspace identification methods computing a nominal parameter vector is under investigation. Furthermore, there is a method that can be used to eliminate the need of a perfect starting value. This method will be discussed later in this chapter. When there are problems with model structure, model structure determination methods can be used not only to identify an adequate model structure, but also to determine a good starting value. Also, it is necessary to have an initial value for R that can be obtained using 0θ and (2.41). This optimization is finished when convergence criteria are satisfied. There are criteria for both θ and ˆR . The criteria can be: (i) Absolute value of the elements of θ∆ are sufficiently small; (ii) Changes in ( )J θ are sufficiently small; (iii) The absolute values of the elements of the cost gradient are close to zero; (iv) Changes in the elements of ˆR are sufficiently small.
- 28. Nonlinear model identification of airplanes using noisy flight data 28 Some convergence criteria that worked well for aircraft parameter estimation problems in [1] are introduced here. 5 1( ) ( ) 1 10 , 1,2,... ,j k j k pj nθ θ − −− < × = (2.51a) or 1 1 ˆ ˆ 0.001, ˆ k k k θ θ θ − − − < (2.51b) where k denotes the kth iteration and denotes the Euclidien norm. This can also be written: 1 1 1 0, 1 ( ) ( ) 0.001, ( ) ˆ ˆ( ) ( ) 0.05, 1,2,... ˆ( ) k k k jj k jj k jj k J J J r r j n r θ θ θ − − − − − < − < = (2.52) where ˆjjr denotes the estimate of the jth diagonal element of ˆR . Furthermore, the magnitude of the changes can be varied by directly applying a multiplicative term in θ∆ : 0 0 ˆ ( ) ( ).cM gθ θ θ∆ = − (2.52) Two methods can be used to compute the elements of the sensitivity matrix. These two methods are the analytical approach and the numerical approach. The analytical approach uses the output equations of the system with respect to the unknown parameters , 1,2,...j pj nθ = . In the case of the linear system given by (2.38) this can be written as: , 1,2,... .p j j j j y x C D C x u j n θ θ θ θ ∂ ∂ ∂ ∂ = + + = ∂ ∂ ∂ ∂ (2.53) The states can be derived from the state equations of the dynamic system: 0 , (0). x Ax Bu x x = + = ɺ (2.54) The state sensitivities are computed by solving the state sensitivity equations, which are obtained by differentiating the state equations with respect to the unknown parameters: , 1,2,... .p j j j j d x x A B A x u j n dt θ θ θ θ ∂ ∂ ∂ ∂ = + + = ∂ ∂ ∂ ∂ (2.55)
- 29. Nonlinear model identification of airplanes using noisy flight data 29 In the numerical approach, an approximation is used to compute the output sensitivities: 0 0( ) ( ) , 1,2,... .p j j y yy j n θ δθ θ θ δθ + −∂ = = ∂ (2.56) or 0 0( ) ( ) , 1,2,... . 2 p j j y yy j n θ δθ θ δθ θ δθ + − −∂ = = ∂ (2.57) where jθ denotes a perturbation vector for parameter jθ and jδθ is scalar magnitude of jθ . The first equation (2.56) implements forward finite differences and the second (2.57) implements central finite differences. Forward finite differences requires ( 1)pn + steps, whereas the central finite differences requires (2 )pn steps. The sensitivities must be calculated for every iteration of the modified Newton-Raphson method. As was mentioned earlier, the information matrix can be singular. In other words, it can be ill-conditioned. There are three frequent reasons for an ill-conditioned information matrix: (i) There are too many unknown parameters in the information matrix. (ii) There is misspecification in the model. Changes in one or more model parameters produce nearly equivalent change in outputs, or one or more model parameter have little or no effect on outputs. (iii) There is insufficient information in the data. Hence, there are two main consequences about a nearly singular information matrix. (iv) The information matrix M can be negative definite that causes a cost increase jθ for the modified Newton-Raphson step. (v) The step size θ∆ can be large in one or more directions. The latter problem can be seen from the decomposition of the inverse M matrix as follows: 1 1 1 , pn T j j i j M t t λ − = = ∑ (2.58a) 1ˆ ( ) ,j T j jt g tλθ λ ∆ = − (2.58b)
- 30. Nonlinear model identification of airplanes using noisy flight data 30 where jλ are the eigenvalues of M corresponding to eigenvectors jt . Then the next step in the modified Newton-Raphson method can be derived by combining (2.58b) with the second equation in (2.50). Thus, the step will be large for small jλ in a direction of jt . The parameter step θ∆ is therefore dominated by inappropriate information. There are several techniques that can be used to address this problem. Three of them are introduced here. Rank deficient method This method is based on the reduced inverse of M applying the singular value decomposition (SVD). T M UDT= (2.59) The reduced-order inverse then can be computed as 1 2 1 1 1 1 0 1 . 1 0 0 0 p p n m T T j j j j n m M T U t u µ µ µ µ − − = − = = ∑ ⋱ ⋱ (2.60) where jt and ju are the jth column of T and U matrices, respectively. The singular values are arranged from the largest to smallest in magnitude. 1 2 1... ...p p pn m n m nµ µ µ µ µ− − +≥ ≥ ≥ ≥ ≥ ≥ (2.61) The smallest m singular values are then dropped. The selection criteria to drop the singular values is max , j N µ ε µ < (2.62) where N is the number of data points, and ε is the computing machine precision. The rank of the reduced-order inverse of M matrix is pn m− . Levenberg-Marquardt method This method augments the information matrix to improve its conditioning and produce a reasonable inverse. The augmentation term can be written as
- 31. Nonlinear model identification of airplanes using noisy flight data 31 1 1 0( ) ,M M kA− − = + (2.63) here 0M is the original information matrix, k is a positive nonzero scalar parameter and A is positive definite matrix. Generally, A is chosen as the identity matrix, 1 1 0( ) .M M kI− − = + (2.64) The scalar parameter k can be obtained by an iterative procedure. An initial value for k is recommended, as 0.01k = . Bayes-like method This method improves the conditioning of the information matrix by combining the measured data with prior estimates of some or all of the parameters in the model. The resulting cost function is similar to that for the Bayesian estimator [1]. 1 1 1 1 1 ( ) ( ) (i) + ( ) ( ), 2 2 N T T p p p i J i Rθ θ θ θ θ− − = = − ∑ −∑v v (2.65) where pθ is the vector containing the prior parameter values, and p∑ is a positive semidefinite covariance matrix that represents the confidence in the parameters of pθ . θ∆ then can be computed by using the modified Newton-Raphson method: 1 1 1 0 0 ˆ ( ( ) ) ( ) ,p p pM gθ θ θ θ− − − ∆ = − + ∑ + ∑ − (2.66) in the cases when the information matrix is nearly singular. In other words, it is singular only at certain points in the progression towards the global optimum. In these cases, the methods discussed above can be used temporarily. The rank deficiency method is the easiest to use in this way. Computing the inverse of the information matrix is always done by calculating the SVD. The singular value ratios are checked against the criteria (2.62). If any singular values are too small, their corresponding terms will be dropped from the inverse according to (2.60-61). 2.3.4 Equation-error method Another parameter estimation method can be obtained by assuming that (i) the dynamic model is in linear states-space form with process noise, (ii) the parameters to be estimated are dimensional stability and control derivatives, (iii) all of the states can be measured without error. The model then has the form of
- 32. Nonlinear model identification of airplanes using noisy flight data 32 0 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ), (0) , 1,2,... . x t A x t B u t z i A x i B u i v i x x i N θ θ θ θ = + = + + = = ɺ (2.67) ( )v i can include both measurement errors and process noise. Generally, the states cannot be measured directly but instead can be derived by using smoothed numerical differentiation of the measured states. Smoothing and filtering methods will be discussed later in this work. The errors ( )v i in the model assumed to be Gaussian and have the following properties: [ ]( ) 0, ( ) ( ) .T ij E i E i j Rδ = v v v (2.68) The negative log-likelihood function then can be written in the form of [ ] [ ]1 1 0 1 L(Z ; ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 (2 ). 2 2 TN N i ln z i A x i B u i R z i A x i B u i NnN ln R ln θ θ θ θ θ π − = − − − − − + + ∑ (2.69) When ˆR is known, the cost function to be minimized has the form of [ ] [ ]1 1 1 ˆ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . 2 TN i J z i A x i B u i R z i A x i B u iθ θ θ θ θ− = − − − −∑ (2.70) In this case, the innovations are the equation errors, ( ) ( ) ( ) ( ) ( ) ( ), 1,2,... .i z i A x i B u i i Nθ θ= − − =v (2.71) It is simple to use output equations instead of dynamic state equations. In (2.67) can be changed to use measured output in the state-space formulation: ( ) ( ) ( ) ( ) ( ) ( ), 1,2,... .z i C x i D u i v i i Nθ θ= + + = (2.72) As a matter of fact, the measurements are always corrupted by errors; hence, the resulting parameters are biased estimates of the unknown model parameters in the state matrices A,B,C and D. A block diagram of the equation-error method is shown below Figure 2.3.
- 33. Nonlinear model identification of airplanes using noisy flight data 33 Figure 2.3 Block diagram of Equation-error method 2.4 Frequency domain ML methods Switching to frequency domain has many advantages. Several methods for data analysis and system modeling can be formulated in frequency domain. Furthermore, frequency domain analysis requires a smaller number of data points and the results can be applied directly to control system design. The basis for these methods is the finite Fourier transform that is the procedure for transforming time-domain data to frequency domain. Frequency domain ML methods are sensitive for transforming errors. Any errors in the transformation affect the accuracy of the frequency domain data. For parametric modeling in the frequency domain, the Fisher model is considered similarly to time-domain analysis. The Fisher model has the following form: (i) ( )z h θ= +ɶ ɶɶ v ; (ii) θ is a vector of unknown constant parameters; (iii) ɶv is a random noise sequence with probability density p( )ɶv . where hɶ is a known complex vector function, xɶ and zɶ are functions of the transformed states and controls, respectively. The measurement vector zɶ is the Fourier transform of the time-domain data vector z . In the following section, the transformation from time-domain to frequency domain is presented; then, the ML estimator for linear dynamic model with process noise is introduced followed by the frequency domain output-error method where there is no process noise. Finally, equation-error method in the frequency domain is presented.
- 34. Nonlinear model identification of airplanes using noisy flight data 34 2.4.1 Transforming between time-domain and frequency domain The transformation between time-domain and frequency domain is called the Fourier- transformation that is based on the Fourier integral. For a given continuous function in the time domain, the corresponding continuous function in the frequency domain is: [ ]( ) ( ) ( ) ,j t x t x x t e dtω ω ∞ − −∞ = = ∫ɶFFFF (2.73) where 1j = − and ω is the angular frequency in radian per second. ( )x ωɶ is called the Fourier transform of ( )x t . The ( )x ωɶ can be transformed back to the time-domain by using the inverse Fourier-transformation. 1 ( ) ( ) , 2 j t x x e dω ω ω ω π ∞ −∞ = ∫ɶ ɶ (2.74) These are also known as the Fourier transform pair. The Fourier transformation of a function on a finite time interval is called the finite Fourier transform. The expressions of the finite Fourier transform have the form 0 ( ) ( ) , T j t x x t e dtω ω − = ∫ɶ (2.75) or 2 0 ( ) ( ) , T j ft x f x t e dtπ− = ∫ɶ (2.76) where 2 fω π= (2.77) When the function ( )x t is sampled at discrete and evenly spaced time intervals t∆ , the finite Fourier transform can be approximated by 1 2 0 ( ) ( ) , N j fi t i x f t x i e π − − ∆ = ≈ ∆ ∑ɶ (2.78) where the time length of measured data is ( 1)T N t= − ∆ and / ( 1), ( ) ( ) ( ), . i i t T N x i x t x i t t i t ∆ = − = = ∆ = ∆ (2.79) In the case of conventional Fourier transform, the frequencies are chosen as
- 35. Nonlinear model identification of airplanes using noisy flight data 35 , 0,1,2..., 1k k f k N N t = = − ∆ (2.80) 2 2 , 0,1,2,... 1k k k f k N N t ω π π= = = − ∆ (2.81) Using these frequencies mentioned below, the approximation of the Fourier transform can be written 1 (2 / ) 0 ( ) ( ) , 0,1,2,..., 1 N j k N i i x k t x i e k Nπ − − = = ∆ = −∑ɶ ɶ (2.82) and its inverse 1 (2 / ) 0 1 ( ) ( ) , 0,1,2,..., 1 N j k N i k x i x k e i N N t π − = = = − ∆ ∑ ɶ (2.83) When the sampling intervals in (2.82) and (2.83) are normalized, the result is called the discrete Fourier transform (DFT). 1 (2 / ) 0 ( ) ( ) , 0,1,2,..., 1 N j k N i i X k x i e k Nπ − − = = = −∑ (2.84) 1 (2 / ) 0 1 ( ) ( ) , 0,1,2,..., 1 N j k N i k x i X k e i N N π − = = = −∑ (2.85) Computing directly the DFT using (2.84) is not economical computationally. An efficient method to compute the DFT is the fast Fourier transform (FFT). In practical aircraft model estimation problems, there is no need to compute the Fourier transformation in the whole frequency band [ ]0, Nf . A method based on the chirp z- transform that can be compute the finite Fourier transformation with very high accuracy at arbitrarily selected frequencies is discussed in the next chapter. 2.4.1 Frequency domain ML estimator for linear dynamic model In order to derive frequency domain ML methods, it is first necessary to transform the linear dynamic model to the frequency domain. Model equations for linear continuous- time dynamic system with process noise and discrete-time measurements have the form: [ ] 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ), ( ), (0) 0, (0) (0) , 1,2,... . w T x t A x t B u t B w t z i C x i D u i v i E x E x x P i N θ θ θ θ θ = + + = + + = = = ɺ (2.86)
- 36. Nonlinear model identification of airplanes using noisy flight data 36 where noises ( )w t and ( )v i are assumed to be independent and white. They have the following properties: [ ] [ ] ( ) 0, ( ) 0, ( ) ( ) ( ), ( ) ( ) . T j i T ij E w t E v i E w t w t Q t t E v i v i R δ δ = = = − = (2.87) The unknown parameter vector θ occurs in the system and covariance matrices. As was shown in the previous section, estimation of all of the parameters is extremely difficult. This problem can be simplified by replacing the Kalman filter with its steady- state version. Using terms from (2.14) and (2.30-33), the modified filter equations can be written in the form ˆ ˆ( 1) ( ) ( 1) ( ) ( ), ( 1) , ˆ ˆ( ) ( 1) ( ) d x t i A x t i B u t i t t i t dt x i i x i i K (i), θ θ θ − = − + − ∆ ≤ ≤ ∆ = − + v (2.88) where the innovations ˆ( 1) ( ),(i)= z(i)-Cx i i Du i− −v (2.89) with [ ] 0, .T ij E (i) E (i) (i) δ = = v v v BBBB (2.90) The elements of the unknown parameter vector are the elements of A,B,C,D,K and B. Then, the equations are written in the form of their Fourier series expansion. The components of ( )x i can be written by (2.78) as 1 (2 / ) 0 ( ) ( ) , 0,1,2,... 1 N j k N i i x k t x i e k Nπ − − = = ∆ = −∑ɶ (2.91) As was mentioned earlier, the finite Fourier transform can be approximated by (2.91) and for practical applications it can be replaced by a high-accuracy calculation presented in the next chapter. Dynamic equations now can be written as ( ) ( ) ( ) ( ) ( ) ( ),kj x k A x k B u k K kω θ θ= + + ɶɶ ɶ ɶ v (2.92) or 1 1 ( ) ( ( )) ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) , 0,1,2,... 1 k kx k j I A B u k j I A K (k), z(k)= C( )x k D u k (k) k N ω θ θ ω θ θ θ θ − − = − + − + + = − ɶɶ ɶ ɶɶ ɶɶ v v (2.93)
- 37. Nonlinear model identification of airplanes using noisy flight data 37 where kω is the kth frequency. The transformed innovations ( )kɶv are uncorrelated Gaussian random variables. [ ] 0, ,T vv E (k) S E (k) (k) N = = ɶ ɶ ɶ v v v (2.94) where vvS is a real matrix with diagonal elements equal to the power spectral densities of the elements of ( )iv , and † (i)ɶv is the complex conjugate transpose of ( )iv . The left side of (2.93) requires that the trend and bias term be removed from the sequence ( )x t , since [ ] 0 0 ( ) ( ) ( ) ( ) (0) ( ) ( ) (0), T T j t j t j T j T x t x t e dt j x t e dt x T e x j x x T e x ω ω ω ω ω ω ω − − − − = = + − = + − ∫ ∫ɺ ɺ ɺ ɶ FFFF (2.95) It can be written from (2.93) 1 1 1 2 ( ) ( )( ( )) ( ) ( ) ( ) ( )( ( )) ( ) ( , ) ( ) ( , ) ), k k z k C j I A B D u k C j I A K I (k) = G k u k G k (k θ ω θ θ θ θ ω θ θ θ θ − − = − + + − + + ɶɶ ɶ ɶɶ v v (2.96) where G1 and G2 are the system transfer matrices. 1 1 1 2 ( , ) ( )( ( )) ( ) ( ) , ( , ) ( )( ( )) ( ) . k k G k C j I A B D G k C j I A K I θ θ ω θ θ θ θ θ ω θ θ − − = − + = − + (2.97) According to the Kalman filter representation in (2.88) is invertible; thus G2 is nonsingular, and the innovations can be written as: 1 1 2 2 1( , ) ( ) ( , ) ( , ) ( ),(k)= G k z k G k G k u kθ θ θ− − −ɶ ɶɶv (2.98) In order to develop the likelihood function, a NV vector that consists of all innovations up to and including the frequency for which 1k N= − is introduced. [ ](0) (1) ... ( 1) ,NV N= −ɶ ɶ ɶv v v (2.99) ( )kɶv has a probability density ( )p ɶɶ v , then it can be written applying the definition of probability densities: 1 1( ) ( 1) ( ).N N Np V p N V p V− −= − ɶv (2.100) Repeated use of (2.100) leads to the expression of the likelihood function:
- 38. Nonlinear model identification of airplanes using noisy flight data 38 [ ]1 2 1 ( ; ) ( ) ( 1) ( 2) ... (1) (0) . N N N N L V p V p N V p N V p V p θ θ − − = = − − ɶ ɶ ɶ ɶv v v v (2.101) The distribution of ( )kɶv is Gaussian, thus the distribution of kV is also Gaussian. Hence, the probability density for this complex Gaussian vector is 0 0 † 1 ( ) exp ( ) ( ) . n k vvn vv N p k V N k S k Sπ − = − ɶ ɶ ɶv v v (2.102) Combining (2.101) and (2.102), the negative log-likelihood function ( ; )NL V θ is [ ] 1 † 1 0 0 ( ; ) ( ) ( ) ( ) . N N vv vv k ln L V J N k S k Nln S Nn ln N π θ θ − − = − = = + + ∑ ɶ ɶv v (2.103) The term on the far right side can be neglected, because it is unaffected by θ , the cost function to be minimized then can be written as 1 † 1 0 ( ) ( ) ( ) . N vv vv k J N k S k Nln Sθ − − = = +∑ ɶ ɶv v (2.104) The ML parameter vector estimate is the value of θ that minimizes the (2.104). The estimate of ˆ vvS has the form of 1 1 † † 0 0 ˆ ˆ ˆ( ) ( ) ( ) ( , ) ( ) ( , ) , N N vv k k S k k z k y k z k y kθ θ − − = = = = − − ∑ ∑ɶ ɶ ɶ ɶɶ ɶv v (2.105) where ˆθ is an estimate of the parameter vector θ . The initial state of ˆ vvS can be computed by using 0θ that is the nominal starting value of the unknown parameter vector θ . The estimates of the remaining parts can be done by using the modified Newton-Raphson method. For a given ˆ vvS the cost function is 1 † 1 0 ( ) ( ) ( ), N vv k J k S kθ − − = = ∑ ɶ ɶv v (2.106) that is minimized by iteratively computing 0 ˆ ,θ θ θ= + ∆ (2.107) where the parameter vector update is 1 0 ˆ ( ) '( ),M Jθ θ θ− ∆ = − (2.108) and M is the Fisher information matrix: 2 2 .N T T lnL(V ; ) J( ) M E E θ θ θ θ θ θ ∂ ∂ = − = ∂ ∂ ∂ ∂ (2.109)
- 39. Nonlinear model identification of airplanes using noisy flight data 39 Since, ˆθ∆ is a vector with only real elements, and the negative log-likelihood function is also real, the first and second-order gradients of the cost function ( )J θ are also real: 1 † 1 0 1 † 1 0 ( )ˆ( ) 2 Re ( )ˆ2 Re , N vv k N vv k k J N (k)S y k N (k)S θ θ θ − − = − − = ∂ = ∂ ∂ = − ∂ ∑ ∑ ɶ ɶ ɶ ɶ v v v (2.110) 2 †1 1 0 †1 1 0 ( ) ( ) ( )ˆ2 Re ( ) ( )ˆ2 Re . N vvT k N vv k J k k N S y k y k N S θ θ θ θ θ θ θ − − = − − = ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∑ ∑ ɶ ɶ ɶ ɶ v v (2.111) The elements of both cost gradients can be expressed as † †1 † 1 1 1 † 11 2 2 2 1 2 0 ( ) ˆ ˆ ˆ2Re ( ) ( ) ( ) , N vv zu uu kj j j G GJ Tr G S G S G S Tr G θ θ θ θ − − − − − = ∂ ∂∂ = − − − ∂ ∂ ∂ ∑ (2.112) †2 1 † 1 1 11 1 2 2 0 † † 1 12 2 2 2 ( ) ˆ ˆ2Re ( ) ( ) − − − − = − − ∂ ∂∂ = = ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∑ N ij vv uu ki j j i j i G GJ M Tr G S G S G G Tr G G θ θ θ θ θ θ θ (2.113) where the estimates of the input spectral density matrix and the cross spectral density matrix, are respectively † † ˆ ( ) ( ) ( ), ˆ ( ) ( ) ( ). uu zu S k u k u k S k z k u k = = ɶ ɶ ɶɶ (2.114) The final estimates have the same properties given in section 2.3.2.
- 40. Nonlinear model identification of airplanes using noisy flight data 40 2.4.2 Output-error method As was done in the time-domain analysis, the ML in the frequency domain can be simplified when there is no process noise. Hence, the resulting system is deterministic. [ ] † ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ), ( ) ( , ) ( ) ( ), ( ) 0, ( ) ( ) , k vv j x k A x k B u k y k C x k D u k z k G k u k k E k S E k k N ω θ θ θ θ θ = + = + = + = = ɶ ɶ ɶ ɶ ɶ ɶ ɶɶɶ ɶ ɶ ɶ v v v v (2.115) where 1 1( , ) ( , ) ( )( ( )) ( ) ( ), 2 / , 0,1,2,..., 1k G k G k C j I A B D k T k N θ θ θ ω θ θ θ ω π − = = − + = = − (2.116) and vvS the spectral density matrix of ( )iv . The innovations are reduced to the output errors ( , ) ( ) ( , ) ( ) ( , ) ( ),k z k y k z k G k u kθ θ θ= − = −ɶ ɶ ɶɶ ɶv (2.117) The negative log-likelihood function and ˆ vvS then can be written 1 † 1 0 ( ) ( , ) ( , ) . N vv vv k J N k S k Nln Sθ θ θ − − = = +∑ ɶ ɶv v (2.118) 1 1 † † 0 0 ˆ ˆ ˆ( , ) ( , ) ( ) ( , ) ( ) ( , ) . N N vv k k S k k z k y k z k y kθ θ θ θ − − = = = = − − ∑ ∑ɶ ɶ ɶ ɶɶ ɶv v (2.119) In order to estimate the unknown parameters, the modified Newton-Raphson method is used. The elements of the first and second-order gradients are computed by simplifying (2.112) and (2.113), where G2 = I and G1 = G: †1 1 0 ( ) ˆ ˆ ˆ2Re ( ) , N vv zu uu kj j GJ Tr S S GS θ θ θ − − = ∂∂ = − − ∂ ∂ ∑ (2.120) †2 1 1 1 0 ( ) ˆ ˆ2Re , N ij vv uu ki j j i G GJ M Tr S S θ θ θ θ θ − − = ∂ ∂∂ = = ∂ ∂ ∂ ∂ ∑ (2.121) Estimates of the input spectral density matrix and the cross spectral density matrix have the same form given in (2.114). The negative log-likelihood function, its gradient and the information matrix can be expressed as
- 41. Nonlinear model identification of airplanes using noisy flight data 41 1 † 1 0 ˆ( ) ( , ) ( , ), N vv k J N k S kθ θ θ − − = = ∑ ɶ ɶv v (2.122) 1 † 1 0 ( ) ˆ2 Re ( ) ( , ) , N vv k J N S k S k θ θ θ − − = ∂ = − ∂ ∑ ɶv (2.123) 2 1 † 1 0 ( ) ˆ( ) 2 Re ( ) ( ) , N vvT k J M N S k S S k θ θ θ θ − − = ∂ = = ∂ ∂ ∑ (2.124) where ( )S k is the output sensitivity matrix with dimensions 0 pn n× given by ( , ) ( ) ( ) . G k u k S k θ θ ∂ = ∂ ɶ (2.125) The modified Newton-Raphson step is given by (2.107) and (2.108). When the measured data is in the form of frequency response functions, the output- error parameter estimation algorithm can be easily modified. In this particular case, the cost function can be written [ ] [ ] 1 †† 1 0 ( ) ( ) ( ) ( , ) ( ) ( , ) ( ), N E vv E k J N u k G k G k S G k G k u kθ θ θ − − = = − −∑ ɶ ɶ (2.126) where ( , )G k θ is matrix with transfer functions as elements, and ( )EG k is a matrix of experimentally determined frequency responses. Cost functions in (2.122) or (2.126) then can be minimized with respect to unknown parameter vector θ . When the dynamic system has only one input, the output-error cost function can be written as [ ] [ ] 1 †* 1 0 ( ) ( ) ( ) ( , ) ( ) ( , ) ( ), N E vv E k J N u k G k G k S G k G k u kθ θ θ − − = = − −∑ ɶ ɶ (2.127) In order to optimize the cost functions, it is not necessary to include all of the frequencies. In the next chapter, a method that allows using arbitrary frequencies is introduced. This method is called the high-accuracy finite Fourier transform. Using arbitrary frequencies, automatically discards some of the power in the time-domain data. Generally, this is equivalent to a zero-lag, low-pass filtering procedure, which improves the convergence and accuracy of the parameter estimates. The weighting matrix in the cost function is the spectral density matrix, which is ideally estimated as a function of frequency index k. Nevertheless, in most practical cases, the frequencies that correspond to the bandwidth of the dynamic system modes are used; which is also the bandwidth where most of the residual power resides in the case of colored residuals. Hence, using a constant vvS over frequencies used in the analysis is a good approximation of the spectral density of the residual power. Accordingly, the assumed
- 42. Nonlinear model identification of airplanes using noisy flight data 42 noise model matches well to reality, and it is not necessary to compute the estimated error bounds for colored residuals when parameter estimates are done in the frequency domain. 2.4.4 Equation-error method Assuming that all of the state variables can be measured without errors, the equation- error model can be written ( ) ( ) ( ) ( ) ( ) ( ),k k k k w kj x A x B u B wω ω θ ω θ ω ω= + +ɶ ɶ ɶ ɶ (2.128) where process noise is included. Considering the state derivatives at discrete frequencies 2 / ,k k Tω π= ( ) ( ), 2 / .k kz k j x k k Tω ω π= =ɶɶ (2.129) The model equation then can be written as ( ) ( ) ( ) ( ) ( ) ( ), 1,2,3,..., 1z k A x k B u k v k k Nθ θ= + + = −ɶ ɶ ɶɶ (2.130) where ( ) ( )wv k B w k=ɶ ɶ are the equation errors, with [ ] † ( ) 0, ( ) ( ) .vv E v k S E v k v k N = = ɶ ɶ ɶ (2.131) The cost function of the equation-error method has the form [ ] [ ] 1 † 1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , N vv k J N z k A x k B u k S z k A x k B u kθ θ θ θ θ − − = = − + − +∑ ɶ ɶ ɶ ɶɶ ɶ (2.132) where the innovations are equation errors: ( ) = ( ) ( ) ( ) ( ) ( ).k z k A x k B u kθ θ− +ɶ ɶ ɶɶv (2.133) Estimation of the elements in the unknown parameter vector θ is obtained by minimizing the cost function given in (2.132). Assuming that the errors in the equations are mutually uncorrelated, this procedure is applied separately to each state equation, corresponding to each row in (2.128). Similarly to the time-domain the case, the equation-error method can be used to estimate dimensional or nondimensional aerodynamic parameters. Equations of the measured output can also be used to estimate aerodynamic parameters. In that case, the vector equation has the form ( ) ( ) ( ) ( ) ( ) ( ), 0,1,2..., 1z k C x k D u k v k k Nθ θ= + + = −ɶ ɶ ɶɶ (2.134) The innovations are
- 43. Nonlinear model identification of airplanes using noisy flight data 43 ( ) ( ) ( ) ( ) ( ) ( ).k z k C x k D u kθ θ= − +ɶ ɶ ɶɶv (2.135) For instance, output equations for measured accelerometer outputs can be used to estimate aerodynamic force parameters. 2.5 Accuracy of parameter estimates In the last two sections the time-domain and frequency domain ML parameter estimation methods were introduced. In this section the accuracy and confidence intervals of these methods are investigated. In the case of a nonlinear estimator, the developing of the parameter accuracy is much more difficult then in the case of a linear estimator. However, in order to compute the accuracy of a nonlinear ML estimator, the Cramer-Rao lower bound can be used, which can be obtained from the main diagonal elements of the inverse information matrix. In the development of ML estimator, it was assumed that the measurement noise ( )v i is a random white sequence. Hence, the power of the noise is evenly spread over the frequency range [ ]0, Nf , where ( )1/ 2Nf t= ∆ the Nyquist-frequency. The Nyquist-frequency is a property of the dynamic system. In other words, all of the important signal components are within the frequency interval of [ ]0, Nf . The relation between the one-sided power spectral density of the noise and the covariance matrix is , 2 N PSD R PSDf t = = ∆ (2.136) where PSD is a diagonal matrix with the power spectral densities of the elements of the noise vector on the main diagonal. The Cramer-Rao inequality for the parameter covariance matrix of the output error estimator has two alternative forms 1 1 1 ˆ( ) ( ) ( ) , N T i Cov S i R S iθ − − = ≥ ∑ (2.137) 1 1 1 1ˆ( ) ( )( ) ( ) . 2 N T i Cov S i PSD S i t θ − − = ≥ ∆ ∑ (2.138) In practical cases, the residuals are colored. Namely, the power spectral densities of the residuals are concentrated in a particular frequency band which corresponds to the rigid-body aircraft dynamics. However, in the development of the ML estimator an assumption that the residuals are white was made, because the power spectral densities
- 44. Nonlinear model identification of airplanes using noisy flight data 44 of the residuals are not known a priori. To address this problem, the residuals are approximated as a band-limited white noise sequence. This is method can be seen in Figure 2.4. It can now be written , , 2 2 R PSD B R PSD tB t = ⋅ = ∆ ∆ (2.139) and 1 1 1 1ˆ( ) ( ) ( ) , 2 N T i Cov S i R S i tB θ − − = ≥ ∆ ∑ (2.140) from (2.138). It can be seen in (2.140) that in the case of colored noise, the Cramer-Rao bounds can be underestimated by a factor 1/ (2 ) .NtB f B∆ = Figure 2.4 Approximation of colored noise with band-limited white noise Cramer-Rao bounds can be improved by assuming that residuals are colored, including an estimate of the residual coloring in the expression for the Cramer-Rao bound estimate. This approach can be generalized for output-error parameter estimation: 1 1 1 1 1 1 1 ˆ ˆ( ) ( ) ( ) ( ) , N N T vv i j Cov M S i R R i j R S j Mθ − − − − − = = ≥ − ∑ ∑ (2.141) where ( )vvR i j− is the autocorrelation matrix for the output residual vector. This can be calculated by 1 1ˆ ˆ( ) ( ) ( ) ( ), 0,1,2,..., N k T vv vv i R k v i v i k R k k r N − = = + = − =∑ (2.142) where the value of r determines the maximum time difference used in the computation of the residual autocorrelation. This estimate usually calculates only the diagonal elements, because the other elements are zero or close to zero. The elements of S, M
- 45. Nonlinear model identification of airplanes using noisy flight data 45 and R are from the conventional time-domain ML estimator calculations; thus, they are the same quantities as in (2.41), (2.50) and (2.140). Accordingly, (2.141) and (2.142) can be used as a residual postprocessor after conventional time-domain ML parameter estimation, which corrects the parameter covariance matrix. As was discussed earlier, in the case of the frequency domain ML estimator, there is no need to compute a correction for the parameter covariance matrix. In the case of linear parameter estimation, the confidence interval can easily be computed. The confidence level associated with this interval expresses the probability of finding the adequate parameter value within the interval. Therefore, in the case of nonlinear parameter estimation, the confidence ellipsoid is just a quadratic approximation of the confidence region. Furthermore, the parameter estimates in nonlinear case do not have Gaussian distribution, the approximation of the confidence level is based on the F-distribution. In [1] a method of computing confidence intervals for ML estimators is presented. That method has not been widely used, because the algorithm is very time-demanding.
- 46. Nonlinear model identification of airplanes using noisy flight data 46 3 Data preprocessing In most practical aircraft identification cases, the measurements are corrupted by noises or any other kind of disturbance, which can be caused serious problems in the parameter estimation. Hence, it is very important to apply methods to separate deterministic signals from random noise for a measured time series. In this chapter some data preprocessing methods are presented. First, the global smoothing in frequency domain is introduced followed by the numerical differentiation. Finally, a practical method for computing the finite Fourier transform based on the chirp z- transform is discussed. The theory presented in this chapter is based on [1]. 3.1 Filtering and smoothing Filtering is a common approach to separate deterministic parts from noise. Filters can be implemented by a circuit using hardware elements or by a computer-implemented transfer function. Furthermore, filters have their own dynamics; thus, they represent an additional dynamic between the physical system and the signal of interest. Filtering can modify phase and amplitude of the signal resulting in a false modeling result. Filtering applications use only the current and past data. Furthermore, filtering methods assume that the analyst knows the frequency range of the physical system. Contrarily, in the case of identification application, it is not always clear what this frequency range should be, and a good choice of the frequency band is the same for different measured sequences. To address these problems, global smoothing method based on Fourier analysis can provide information about the adequate frequency band to be used for system identification. Global smoothing method in the frequency domain is presented in the next section. Smoothing applications use subsequent data; thus, these methods can only be used post-flight. 3.2 Global smoothing in the frequency domain As was mentioned above, global smoothing has several beneficial properties in the case of aircraft identification problems. Contrarily with the local smoothing methods which
- 47. Nonlinear model identification of airplanes using noisy flight data 47 are described in [1] in detail, global smoothing concerns the smoothing problem in the global sense. Since it works in the frequency domain, it is based on the Fourier analysis. Consider a noisy measured signal ( )z t sampled at constant intervals t∆ , ( ) ( ), 0,1,2,..., 1z i z i t i N= ∆ = − (3.1) The aim is to distinguish signal and noise, and then separate the two components. The measured sequence then can be written ( ) ( ) ( ), 0,1,2,..., 1z i y i v i i N= + = − (3.2) The way for determining the noise sequence ( ), 0,1,2,..., 1v i i N= − is to find the signal components in the frequency domain, invert the transformation to get the signal in the time domain ( )y i , then compute the noise ( )v i from (3.2). ( ) ( ) ( ), 0,1,2,..., 1v i z i y i i N= − = − (3.3) Fourier series expansion has an assumption that the signal under investigation is periodic. Furthermore, most measured data from flight tests have discontinuities in the amplitude and in first and higher order derivatives of the endpoints. A typical flight test measurement is depicted in Figure 3.1. This data sequence is from [5] and [6]. These discontinuities can cause difficulties in determining an adequate representation of the signal in the frequency domain, because the Fourier series expansions of signals with these discontinuities have much slower convergence than a signal with no discontinuities. This means that the magnitudes of the Fourier coefficients decrease asymptotically with 1 k− in the case of signals with discontinuities, and with 3 k− in the case of signals with no discontinuities, where k denotes the number of terms in the Fourier expansion. To address this problem, it is possible to remove the endpoint discontinuities from any time series by subtracting the linear trend from the original measured time series. That makes the amplitudes at the endpoints equal to zero.
- 48. Nonlinear model identification of airplanes using noisy flight data 48 Figure 3.1 Pitch rate measured time series Let ( )g i be the new time series with ( 1) (0) ( 1) 0g N g g N− + = = − = . Then ( )g i can be computed as ( 1) (0) ( ) ( ) (0) , 1 ( ) ( ), 0,1,2,... 1 z N z g i z i z i N g i g i i N − − = − − − − = − = − (3.4) Figure 3.2 shows the new time series using the sequence of Figure3.1. Figure 3.2 Pitch rate measured time series with endpoints discontinuities removed Define the vector [ ]( 1) ( 2) ... ( 1) (0) (1) ... ( 1) ,g g N g N g g g g N= − + − + − − (3.5)
- 49. Nonlinear model identification of airplanes using noisy flight data 49 and its Fourier series expansion 1 1 ˆ( ) ( )sin , 0,1,2,... 1 1 N k i g i b k k i N N π − = = = − − ∑ (3.6) where ˆ( )g i denotes the approximation of ( )g i , and ( )b k are Fourier sine series coefficients. Note that there are only positive values in (3.6), because those values correspond the original measured time sequence. The Fourier sine series coefficients for g can be computed as 2 1 2 ( ) ( )sin , 0,1,2,..., 1 1 1 N i i b k g i k k N N N π − = = = − − − ∑ (3.7) The index 1N − corresponds to the Nyquist frequency, also (0)g and ( 1)g N − are zero. The period of the kth Fourier sine series can be computed by 2( 1) ,k N t T k − ∆ = (3.8) and then the kth frequency is given by , 2( 1) k k f N t = − ∆ (3.9) To achieve an effectively smoothed data, it is necessary that the Nyquist frequency Nf be much higher than the highest frequency components in the signal. The Fourier sine series computed above contains both signal and noise components. Note that, the Fourier series expansion of a coherent signal can be visually distinguished from Fourier series of a noise sequence. Since the signal with discontinuities mentioned above has Fourier series coefficient amplitudes that decrease asymptotically, and the incoherent noise has constant power with relatively small amplitude over the entire frequency range up to the Nyquist frequency. The contrast is so high between components of noise and components of coherent signal that it is often easy to find an adequate separation point visually. Visually determining the cut- off frequency generates inaccuracies. To address this problem the Wiener filter can be used. Consider the Fourier transform of the measured time series given in (3.2) ( ) ( ) ( ).z f y f v f= +ɶ ɶɶ (3.10) The aim is to develop a frequency domain filter that minimizes the squared error between the true signal ( )y fɶ and the estimated signal ˆ( )y fɶ over the entire frequency range of the system. This can also be written as:
- 50. Nonlinear model identification of airplanes using noisy flight data 50 2 0 ˆ( ) ( ) , Nf y f y f df − ∫ ɶ ɶ (3.11) where Nf is the Nyquist frequency. The estimated signal in the frequency domain can be computed by multiplying a filter ( )fΦ with ( )z fɶ . ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) .y f f z f f y f y f = Φ = Φ − ɶ ɶ ɶɶ (3.12) Then, we can substitute (3.12) to (3.11), since ( ) ( )y f v fɶ ɶ is equal to zero. Because of the incoherence of the noise, (3.11) can be written as { }2 2 2 2 2 0 ( ) 2 ( ) ( ) ( ) ( ) ( ) . Nf y f f y f f y f v f df − Φ + Φ + ∫ ɶ ɶ ɶ ɶ (3.13) Solving the equation for ( )fΦ by taking the derivative of (2.15) with respect to ( )fΦ , and setting the result equal to zero gives 2 2 2 ( ) ( ) , 0 ( ) ( ) N y f f f f y f v f Φ = ≤ ≤ + ɶ ɶ ɶ (3.14) This can be written in terms of discrete frequency index k , 2 2 2 ( ) ( ) , 0 1 ( ) ( ) y k k k N y k v k Φ = ≤ ≤ − + ɶ ɶ ɶ (3.15) The resulting optimal filter in the frequency domain given by (3.15) is the Wiener filter. This optimal filter from (3.15) then multiplies the ( )b k in the inverse Fourier transform. Hence, the smoothed signal is computed from 1 1 ˆ ( ) ( ) ( )sin , 0,1,2,..., 1 1 N s k i g i k b k k i N N π − = = Φ = − − ∑ (3.16) The terms where ( ) ( )k b kΦ is sufficiently small can be neglected. Applying a Wiener filter gives room for errors caused by visual determination of the cut-off frequency, since it is near unity at low frequencies, then decreases slowly to zero at high frequencies, removing the power components of the noise. Some noise components can be found in lower frequency range, but there is no method to distinguish them from the signal of interest, which also lies in low- frequency. However, when the signal-to-noise ratio is high, the power of the noise relative to the power of the signal is very small, and it becomes much smaller after removing the noise components from the higher frequency range.
- 51. Nonlinear model identification of airplanes using noisy flight data 51 Finally, the linear trend removed from the signal by (3.3) has to be restored to the smoothed signal. The smoothed signal then can be written as ( 1) (0) ˆ ˆ( ) ( ) ( ) (0) , 1,2,3,..., 1 1 s s z N z y i z i g i z i i N N − − = = + + = − − (3.17) The endpoints of the original measured sequence are excluded from the computing of the smoothed signal. It can cause serious errors in the case of very noisy data. To address this problem, the endpoints have to be smoothed locally by using a local smoothing method presented in details in [1]. The smoothed signal is depicted in Figure3.3. Eventually, the noise sequence can be computed by using (3.3). Note here, that the noise sequence computed by applying a Wiener filter is a wideband random white noise. Figure 3.3 Smoothed pitch rate measured time series 3.3 Numerical differentiation To compute an adequate numerical derivative for a measured sequence is a common problem in system identification applications, especially in the case of aircraft system identification, when there are no sensors to measure some signals which are very important in model parameter determination. For instance, generally, there is no sensor to measure body-axis angular accelerations, so the values of , ,p q rɺ ɺ ɺ must be computed by numerically differentiating the body-axis angular rate measurements.
- 52. Nonlinear model identification of airplanes using noisy flight data 52 Applying the central finite differences method to compute numerical time derivatives for a noisy measurement sequence increases the noise. A better method to compute numerical time derivatives is the local smoothing solution. This method involves fitting a polynomial in time to measured data at the point to be smoothed. The polynomial to be fitted is a second-order function in terms of time that can be written 2 0 1 2 1 . 2 y a a t a t= + + (3.18) The parameters in (3.18) can be identified by using the following method. Let ( )z i denotes the ith measured data point sampled at time i t∆ , then the equations can be written as 2 0 1 2 2 0 1 2 0 2 0 2 2 0 2 1 ( 2 ) ( 2 ) ( 2) 2 1 ( ) ( ) ( 1) 2 ( ) 1 ( ) ( ) ( 1) 2 1 (2 ) (2 ) ( 2) 2 a a t a t z i a a t a t z i a z i a t a t z i a t a t z i + − ∆ + − ∆ = − + −∆ + −∆ = − = + ∆ + ∆ = + + ∆ + ∆ = + (3.19) There are five equations for determining three unknown parameters. These equations can be written in matrix form as ( ) ( ) ( ) ( ) 2 2 0 1 2 2 2 1 2 2 ( 2)1 1 ( 1)2 1 0 0 ,( ) 1 ( 1) 1 2 ( 2) 1 2 2 t t z i t t a z i a z i a z i t t z i t t − ∆ ∆ − −∆ ∆ − = + ∆ ∆ + ∆ ∆ (3.20) Then the resulting normal equations have the following form ( ) ( ) 2 2 0 2 2 2 1 2 2 2 2 0 2 2 5 5 ( ), 10 ( ), 10 17 ( ), i k i i k i i k i a t a z k ta kz k a t a k z k + = − + = − + = − + ∆ = ∆ = + ∆ = ∑ ∑ ∑ (3.21)
- 53. Nonlinear model identification of airplanes using noisy flight data 53 When the local polynomial model given by (3.18) is differentiated with respect to time, which is given by 1 2 ,y a a t= +ɺ (3.22) then substituting zero for t , the resulting value of the local smoothed derivative is equal to the coefficient of the linear term in (3.18). Then this method can be used for each data point. Smoothed numerical derivatives of a signal can also be computed by using the global Fourier smoothing. Equations (3.16) and (3.17) are used to determine the smoothed derivative. In order to obtain the smoothed derivative, the sine series and the linear trend are differentiated before transforming back to time domain. The smoothed derivative of the signal given by (3.17) then can be written as 1 1 ˆ( ) ( 1) (0) ( ) ( ) cos , 1 1 1 0,1,2,..., 1 N k dy i z N z k i k b k k di N N N i N π π − = − − = + Φ − − − = − ∑ (3.23) 3.4 Finite Fourier Transform In section 2.3.1 the finite Fourier transform was considered as 2 0 ( ) ( ) , T j ft x f x t e dtπ− = ∫ɶ (3.24) and it can be approximated by 1 2 0 ( ) ( ) , N j fi t i x f t x i e π − − ∆ = ≈ ∆ ∑ɶ (3.25) Another approximation for the finite Fourier transform can be given by applying discrete frequencies defined in (2.80). 1 (2 / ) 0 ( ) ( ) , 0,1,2,..., 1 N j k N i i x f t x i e k Nπ − − = ≈ ∆ = −∑ɶ (3.26) The summation in (3.26) is the discrete Fourier transform ( )X k , also the finite Fourier transform can be approximated by multiplying the discrete Fourier transform by the sampling interval t∆ . ( ) ( ),x k tX k= ∆ɶ (3.27) where 1 (2 / ) 0 ( ) ( ) , 0,1,2,..., 1 N j k N i i X k x i e k Nπ − − = = = −∑ (3.28)
- 54. Nonlinear model identification of airplanes using noisy flight data 54 In the case of flight data analysis, there are two important disadvantages to using the approximation mentioned above: (i) The spacing of the discrete frequencies for the discrete Fourier transform is equal to the reciprocal of the length of data sequence. Hence, the frequencies are fixed at values defined by N and t∆ . To achieve an efficient and fast Fourier transformation, this particular selection of frequencies has to be applied. Which means, that the frequencies cannot be chosen arbitrarily. (ii) Hence, the approximation of the finite Fourier transform is a zeroth-order Euler approximation of the integrand [ ]( ) ( ) cos( ) sin( )j t x t e x t t j tω ω ω− = − . This integrand oscillates as the time variable t changes causing inaccuracies in the approximation. The oscillation increases as the frequency increases. There is the same problem when t∆ gets larger. It can be seen that using the fast Fourier transformation causes inaccuracies as the data sequence decreases, leading to a loss of detail in the frequency domain. Furthermore, modal frequencies for a rigid-body dynamics of a full-scale aircraft are usually below 2Hz. For such a small frequency band, most of the processing in the transformation rests outside the range of interest. To address these problems in [1], the defining integrand of the Fourier transformation given by (3.24) is approximated by using an approach based on the chirp z-transform, where the frequencies can be chosen arbitrarily regardless of the length of the data sequence. Applying this method for the frequencies of the frequency band of interest, a high accuracy approximation of the Fourier transform can be achieved with excellent computational efficiency. The discrete Fourier transformation in (3.26-27) can be written as 1 0 ( ) ( ) , N ki i X k x i AW − − = = ∑ (3.29) where 2 1, . j N A W e π = = (3.30) k AW for 0,1,2,..., 1k N= − represents equally spaced points on the unit circle in the complex plane. Hence, the k AW is a complex transform variable z , the selection given in (3.30) is a z-transform of the data in the time-domain ( ), 0,1,...,x i i N= . A and W can be changed to achieve different contours in the complex plane. This method is
- 55. Nonlinear model identification of airplanes using noisy flight data 55 called the chirp z-transform. In fact, angular steps are determined by W , and the value of A can be selected as a starting point in the complex plane. So the chirp z-transform is a discrete Fourier transform with arbitrarily selectable frequency range and resolution. When the contour in the complex plane is along the unit circle, W and A have the forms of 0 , , j j A e W e θ θ∆ = = (3.31) where 0 0 / tω θ= ∆ is the lower limit of the frequency band in the chirp z-transform in rad/s and / tω θ∆ = ∆ ∆ is the frequency resolution. Values of 0θ and θ∆ can be chosen arbitrarily. The only limitations for the resolution and the lower limit is both must be in the range [ ]0, / tπ ∆ , where / tπ ∆ is the Nyquist angular frequency in rad/s. This chirp z-transform can be used to calculate the finite Fourier transform when the selected frequencies are regularly spaced in the frequency range of interest. In the case of aircraft system identification, evenly spreading the frequencies is the preferred approach to capture all important features in the frequency domain. The chip z-transform evaluated at M arbitrarily frequencies kf in Hz, where the selected frequency range is 0 k Nf f≤ ≤ yields 1 2 0 ( ) ( ) , 0 , 0,1,2,..., 1 k N j f i i k N X k x i e f f k M π − − = = ≤ ≤ = − ∑ (3.32) The number of frequencies can be chosen with the limitation that M cannot be larger then 2N to avoid numerical problems: 2 .k kt f tθ ω π= ∆ = ∆ (3.33) The expression for the high-accuracy calculation can be written as { ]} 0 1 2 3 / * * 0 1 * * 2 3 ( ) ( ) ( ) ( ) (0) ( ) (1) ( ) (2) ( ) (3) ( ) ( 1) ( ) ( 2) ( ) ( 3) ( ) ( 4) , j T t x t W X k x x x x e x N x N x N x N θ θ θ γ θ γ θ γ θ γ θ γ θ γ θ γ θ γ θ ∆ ≈ ∆ + + + + + − + − + − + − ɶ (3.34) where 0 1 2( ), ( ), ( ), ( )W θ γ θ γ θ γ θ and 3 ( )γ θ are the weights calculated by analytically evaluating the finite Fourier transform integral in (3.34) using cubic Lagrange
- 56. Nonlinear model identification of airplanes using noisy flight data 56 interpolation applied to the time-domain data. According to [1], the resulting expressions for the weights have the form: 4 6 2 4 6 0 2 4 6 11 23 ( ) 1 , 720 15120 2 1 103 169 ( ) 3 45 15120 226800 2 2 8 86 , 45 105 2835 467775 W j θ θ θ γ θ θ θ θ θ θ θ θ ≈ − + ≈ − + + − − + + + (3.35a) 2 4 6 1 2 4 6 2 4 6 2 2 4 6 2 4 6 3 2 7 7 5 7 ( ) 24 180 3456 259200 7 1 11 13 , 72 168 72576 5987520 1 1 5 1 ( ) 6 45 6048 64800 7 1 11 13 , 90 210 90720 7484400 1 1 5 1 ( ) 24 180 24192 259200 7 1 360 840 j j j γ θ θ θ θ θ θ θ θ γ θ θ θ θ θ θ θ θ γ θ θ θ θ θ θ ≈ − + − − − + − ≈ − + − + − − + − + ≈ − + − − − 4 611 13 . 362880 29937600 θ θ + − (3.35b) Furthermore, it is advantageous to select small distance between frequencies in the frequency range of interest, to ensure that all details of the time-domain data are captured. The terms to compute the finite Fourier transform applying the chirp z-transform have the form: 0 , 0,1,2,..., 1kf f k f k M= + ∆ = − (3.36) 1 0( ) . ( 1) f f f M − ∆ = − (3.37) The steps of the procedure now can be written (i) Choose three of the four values in (3.37) and then compute the remaining value. This determines the frequencies for the analysis. (ii) Calculate the M values of θ from (3.33). (iii) Use the chirp z-transform to compute the discrete Fourier transform for the selected frequencies applying (3.29-30). (iv) Compute the high accuracy finite Fourier transform by using (3.34- 35).
- 57. Nonlinear model identification of airplanes using noisy flight data 57 4 Demonstration of time domain ML and OE methods using test data from known data bases In this chapter the identification ability of time-domain output-error method presented in Chapter 2 is demonstrated in MATLAB environment. To execute the identification method, the SIDPAC Toolbox developed by NASA is used. The SIDPAC is a Toolbox software for [1]. In this chapter three case-studies are presented. This chapter is based on the theory written in [1] and [4] and data sequences can be found in the SIDPAC Toolbox. According to the previous chapter the following algorithm is used here: (i) Assembling the system matrices A,B,C,D using the nominal parameter values; (ii) Computing the covariance matrix of the innovations (2.41); (iii) Minimizing the cost function given by (2.42) using the modified Newton- Raphson method; (iv) Update the parameter values in the system matrices; (v) Repeat steps (i)-(iv) until the convergence criteria (2.51) are satisfied. In the first example the measured data are from a lateral maneuver run1 of the NASA Twin Otter aircraft, which is a twin-engine turboprop commuter aircraft depicted on Picture 4.1. In the case of the output-error parameter estimation method, where the equations of motion are used together, the lateral aerodynamic coefficients are estimated at the same time. There are two data sequences named as run1 and run2. Measured data from run1 is used to estimate the model parameters, and then flight data from run2 is used to validate the results. Measured data sequences of the maneuver run1 and run2 can be seen in Figure 4.1 and 4.2. Picture 4.1 De Havilland DHC-6 Twin Otter
- 58. Nonlinear model identification of airplanes using noisy flight data 58 -10 0 10 beta(deg) -20 0 20 40 p(dps) -20 -10 0 10 r(dps) 0 5 10 15 20 -10 0 10 20 phi(deg) time (sec) -10 0 10 ail(deg) -5 0 5 rdr(deg) 0 5 10 15 20 -0.5 0 0.5 ay(g) time (sec) Figure 4.1 Measured input and output data sequences for lateral maneuver run1 -10 0 10 beta(deg) -20 0 20 p(dps) -20 -10 0 10 r(dps) 0 5 10 15 20 -20 -10 0 10 phi(deg) time (sec) -10 0 10 ail(deg) -5 0 5 rdr(deg) 0 5 10 15 20 -0.2 0 0.2 ay(g) time (sec) Figure 4.2 Measured input and output data sequences for lateral maneuver run2 The maneuver is a small perturbation maneuver, thus a linearized model can be used.
- 59. Nonlinear model identification of airplanes using noisy flight data 59 Detailed aerodynamic equations can be found in [1] , [4] and [8]. Linearized equations have the following form: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 cos sin cos , , 2 2 , 2 2 tan r r p r a r p r a r Y Y Y r x xz l l l l a l r p x xz n n n n a n r r q S rb C C C mV V g p r b V pb rb I p I r q Sb C C C C C b V V pb rb I r I p q Sb C C C C C b V V p r β δ β δ δ β δ δ β β β δ θ α α φ β δ δ β δ δ φ θ = + + + − + + + = + + + + + + = + + + + + = + + ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ ,bφ (4.1) 0 0 , 2r yr y Y Y Y r a q S rb a C C C b mg Vβ δ β δ = + + + (4.2) These equations are linear dynamic equations for the lateral states , ,p rβ , which are also measured outputs of the aircraft. (4.2) is an equation for the lateral acceleration. In (4.1-2), values with subscript o denote the measured trim values of the maneuver. These are equal to the starting trim values or to the mean values computed from measured data sequence. All of the dynamic equations include a bias term. As it was described in Chapter 3, there are several methods to remove the bias term from a measured data sequence. This is for any measurement biases not removed from the data sequences. Equations (4.1-2), which are dynamics and measurements, can be written in state-space form 0 1 0 1 2 0 0 3 3 2 3 2 3 3 2 3 2 0 1 3 3 cos sin ( cos )1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 tan 0 0 − − − = − − + ɺ ɺ ɺ ɺ ɺ r p r p r r a r Y Y xx xz l l l xz zz n n n Y l l g k C k C k V I I pp k C k C k k C k I I rr k C k C k k C k k C b k C k C b β β β δ δ δ β θ α α ββ φφ θ 3 3 , 1 0 0 ɺ ɺ ɺ a r a p r n n rk C k C b b δ δ φ δ δ (4.3a)
- 60. Nonlinear model identification of airplanes using noisy flight data 60 44 4 2 0 0 1 2 3 0 4 0 0 0 0 01 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 . 0 0 00 0 0 1 1 00 0 , , , 2 yr r a r Y aY Yy p p r q k C bk C k C ka q S q Sb k k k q Sb k mV V mg δβ β β δ δ φ φ = + = = = = (4.3b) To optimize the cost function given by (2.42) the modified Newton-Raphson method is used. The measurement covariance matrix is computed from (2.41). Output sensitivities are computed by using central finite differences from (2.57). The vector of unknown parameters has the form , , , , , , , , , , , , , , , , , . r r p r a r p r ya r Y Y Y l l l l l p T n n n n n r a C C C b C C C C C b C C C C C b b b β δ β δ δ β δ δ β φ θ = ɺ ɺ ɺɺ (4.4) Identification results can be seen in Figure 4.3. The plot shows an excellent model fit to the measured data sequence. Estimated parameter values can be seen in Table 4.1 To achieve this result, the solution took 48 modified Newton-Raphson iterations. Convergence criteria given by (2.51) are used. The estimated parameter covariance matrix is computed from (2.50) as the Cramer-Rao lower bound 1 0 ˆ( ) ( )Cov Mθ θ− = . In this case, the residuals are assumed to be white.
- 61. Nonlinear model identification of airplanes using noisy flight data 61 -0.2 0 0.2 Output-Error Modeling beta(rad) data model -0.5 0 0.5 p(rps) -0.5 0 0.5 r(rps) -0.5 0 0.5 phi(rad) 0 2 4 6 8 10 12 14 16 18 20 -0.5 0 0.5 ay(g) time (sec) Figure 4.3 Output-error model fit to measured perturbation outputs about trim values for lateral maneuver run1 As was mentioned above, flight data run2 is used to validate the parameter estimation. Figure.4.4 shows results when measured data from run2 is predicted by using estimated values given in Table 4.1. In this case, the bias parameters in (4.1) are re-estimated using output-error parameter estimation. Re-estimate the bias values took 12 modified Newton-Raphson iterations. All other model parameters are held fixed at the values given in Table 4.1.
- 62. Nonlinear model identification of airplanes using noisy flight data 62 -0.1 0 0.1 Prediction beta(rad) data model -0.5 0 0.5 p(rps) -0.2 0 0.2 r(rps) -0.5 0 0.5 phi(rad) 0 2 4 6 8 10 12 14 16 18 20 -0.2 0 0.2 ay(g) time (sec) Figure 4.4 Model prediction for lateral maneuver run2 using re-estimated biases and estimated parameters applying data sequence run1
- 63. Nonlinear model identification of airplanes using noisy flight data 63 Table 4.1 Parameter estimation results for lateral maneuver using data sequence run1 Parameter ˆθ ˆ( )s θ YC β -8.6457e-001 2.8260e-003 rYC 9.3313e-001 1.6697e-002 r YC δ 3.7999e-001 8.0588e-003 lC β -1.1959e-001 6.0201e-004 plC -5.8590e-001 2.5898e-003 rlC 1.8888e-001 1.8195e-003 a lCδ -2.2818e-001 9.1234e-004 r lCδ 4.2212e-002 7.4475e-004 nC β 8.6376e-002 2.1507e-004 pnC -6.4414e-002 9.8548e-004 rnC -1.9059e-001 6.2567e-004 a nC δ -2.9796e-003 3.6329e-004 r nC δ -1.3548e-001 2.7071e-004 In the second example the same data sequence as in the first example is used to demonstrate the estimated parameter covariance corrections in the case of colored residuals. The model parameters are estimated by using output-error method. The model structure is also the same as in the first example given by (4.3). In this particular case, measured input-output data sequences are not converted to perturbations about initial trim values. This can be seen by comparing initial values of ya on Figures. 4.3 and 4.5. This estimation is identical to the results of the preceding case except the biases. Figure 4.5 shows the model fit to the measured output data sequence. Figure 4.6 shows the residuals which are the difference between the measured and estimated data sequence in Figure 4.5. Residuals are colored. In Table
- 64. Nonlinear model identification of airplanes using noisy flight data 64 4.2, the results of output-error parameter estimation can be seen with parameter standard errors using both the white residual assumption computed by (2.137), and the colored residual assumption computed by (2.141). -0.2 0 0.2 Output-Error Modeling beta(rad) data model -0.5 0 0.5 p(rps) -0.5 0 0.5 r(rps) -0.2 0 0.2 phi(rad) 0 2 4 6 8 10 12 14 16 18 20 -0.5 0 0.5 ay(g) time (sec) Figure 4.5 Output-error model fit to measured data using lateral maneuver run1
- 65. Nonlinear model identification of airplanes using noisy flight data 65 -5 0 5 x 10 -3 Output-Error Residuals beta(rad) -0.05 0 0.05 p(rps) -0.01 0 0.01 r(rps) -0.02 0 0.02 phi(rad) 0 2 4 6 8 10 12 14 16 18 20 -0.05 0 0.05 ay(g) time (sec) Figure 4.6 Output-error residuals for lateral maneuver, run1
- 66. Nonlinear model identification of airplanes using noisy flight data 66 Table 4.2 Parameter estimation results for lateral maneuver using data sequence run1 Parameter ˆθ ˆ( )s θ white residuals (2.137) ˆ( )corrs θ colored residuals (2.141) ˆ( ) ˆ( ) corrs s θ θ YC β -8.6457e-001 2.8260e-003 8.82e-003 3.15 rYC 9.3313e-001 1.6697e-002 6.04e-002 3.65 r YC δ 3.7999e-001 8.0588e-003 3.15e-002 3.95 lC β -1.1959e-001 6.0201e-004 1.89e-003 3.37 plC -5.8590e-001 2.5898e-003 8.23e-003 3.44 rlC 1.8888e-001 1.8195e-003 8.89e-003 5.19 a lCδ -2.2818e-001 9.1234e-004 3.03e-003 3.60 r lCδ 4.2212e-002 7.4475e-004 6.03e-003 8.47 nC β 8.6376e-002 2.1507e-004 1.53e-003 7.36 pnC -6.4414e-002 9.8548e-004 6.41e-003 6.76 rnC -1.9059e-001 6.2567e-004 4.06e-003 6.68 a nC δ -2.9796e-003 3.6329e-004 2.23e-003 6.35 r nC δ -1.3548e-001 2.7071e-004 1.86e-003 7.00 In the third example dimensional stability and control derivatives are estimated for a longitudinal dynamic state-space model using longitudinal flight test data sequences measured on the same aircraft as in the case of the first and second example. Based on [4], the state-space structure is considered as ( )0 0 0 , , 1 1 0 0 0 0 1 0 0 , 1 1 e e e biasq e q bias e z q z bias Z ZZ Z M Mq q M M q q a V V V Z Z Z a g g g δα α δ α δ α α δ α α δ = + = + − ɺ ɺ (4.5)
- 67. Nonlinear model identification of airplanes using noisy flight data 67 where the elements of the unknown parameter vector are ,e eq bias q bias z biasZ Z Z Z M M M M aα δ α δθ = . Detailed description of this model structure can be found in [1], and in [4]. The states of the system are α and q , which are the angle of attack and pitch rate, respectively. The outputs are the α , q and za , where za is the acceleration along the z-axis. 0V and g denote the absolute velocity and gravitational acceleration, respectively. Measured outputs of the longitudinal dynamic system can be seen in Figure 4.7. Estimating the unknown parameters is performed using output-error method. Initial value of the unknown parameter vector θ is computed by using equation-error method in one step. The output-error solution then took 69 modified Newton-Raphson iterations. 0 1 2 3 4 5 6 7 8 -0.02 0 0.02 0.04 alpha(rad) Measured Outputs 0 1 2 3 4 5 6 7 8 -0.1 -0.05 0 0.05 0.1 q(rps) 0 1 2 3 4 5 6 7 8 -1.4 -1.2 -1 -0.8 -0.6 az(g) time (sec) Figure 4.7 Measured outputs for a longitudinal maneuver Estimates of the bias parameters have been omitted. The results can be seen in Table 4.3. Since, the residuals are colored due to modeling error, hence Cramer-Rao bounds must be corrected for colored residuals. The residuals can be seen in Figure 4.8.
- 68. Nonlinear model identification of airplanes using noisy flight data 68 Table 4.3 Parameter estimation results for longitudinal maneuver with corrected error bounds 0 1 2 3 4 5 6 7 8 -2 0 2 x 10 -3 alpharesiduals(rad) Residuals 0 1 2 3 4 5 6 7 8 -5 0 5 x 10 -3 qresiduals(rps) 0 1 2 3 4 5 6 7 8 -0.05 0 0.05 azresiduals(g) time (sec) Figure 4.8 Output-error residuals for longitudinal maneuver The estimated system matrices then can be written in the form of -1.9693 0.9927 -0.0936 0.0042 , B= . -9.4361 -3.0755 -12.1453 -0.0344 A = (4.6) Parameter Estimate Standard Error in % 95% Confidence Interval Zα - 1.969e+000 4.896e-002 [ -2.067 , -1.871 ] qZ 9.927e-001 1.476e-002 [ 0.963 , 1.022 ] e Zδ -9.360e-002 2.957e-002 [ -0.153 , -0.034 ] Mα -9.436e+000 1.715e-001 [ -9.779 , -9.093 ] qM -3.076e+000 6.835e-002 [ -3.212 , -2.939 ] e Mδ -1.215e+001 1.595e-001 [ -12.464 , -11.826 ]
- 69. Nonlinear model identification of airplanes using noisy flight data 69 5 Demonstration of frequency domain ML and OE methods using test data from known data bases In this chapter the identification ability of frequency domain output-error method presented in Chapter 2 is demonstrated using MATLAB software environment. To execute the identification method and the operations in the frequency domain, the SIDPAC Toolbox developed by NASA is used, as was done in Chapter 4. In this chapter two case-studies are presented, to demonstrate of the efficiency of frequency domain ML method. This chapter is based on the theory written in [1] and [4] and data sequences can be found in the SIDPAC Toolbox. According to the chapter 2 the following algorithm is used here: (i) Transform the necessary data sequences into frequency domain using the chirp-z transform presented in subsection 3.4; (ii) Assembling the system matrices A,B,C,D according to (2.115); (iii) Computing the covariance matrix of the innovations in the frequency domain (2.119); (iv) Minimizing the cost function (2.127) by applying the modified Newton- Raphson method; (v) Update the parameter values in the system matrices; (vi) Repeat steps (i)-(v) until the convergence criteria (2.51) are satisfied. In the first example, the measured flight test data sequences are gathered from the same aircraft as in the preceding chapter. Measured inputs and outputs can be seen in Figure 4.1-2. Then all relevant data time series are transformed into frequency domain using the high accuracy finite Fourier transform described in Chapter 3. The frequency vector [ ]0.10,0.12,0.14,...1.5 T f = Hz includes all significant spectral components. The output-error cost gradient and information matrix are computed from (2.123) and (2.124) and the parameter estimate updates are computed from (2.108). Output sensitivities are computed numerically using central finite differences from (2.125). Estimation results can be seen in Table 5.1 where the results in the frequency domain are listed along with time domain results of the first example in Chapter 4. It can be seen that the parameters computed in the frequency domain are close to those that were computed in the time-domain. However, in the frequency domain, parameters do not require a re-estimation of biases, because they were eliminated before the Fourier transformation. Figure 5.1 shows the model fitted to measured outputs in the frequency domain.
- 70. Nonlinear model identification of airplanes using noisy flight data 70 0 0.2 0.4 Output-Error Modeling beta(rad) data model 0 0.5 1 p(rps) 0 0.5 1 r(rps) 0 0.25 0.5 phi(rad) 0 0.5 1 1.5 0 0.5 1 ay(g) frequency (Hz) Figure 5.1 Model fitted to measured outputs in the frequency domain
- 71. Nonlinear model identification of airplanes using noisy flight data 71 Table 5.1 Parameter estimation results for lateral maneuver using data sequence run1 Time-domain Frequency domain Parameter ˆθ ˆ( )s θ ˆθ ˆ( )s θ YC β -8.6457e-001 2.8260e-003 -8.6083e-001 7.1768e-003 rYC 9.3313e-001 1.6697e-002 9.5302e-001 4.2914e-002 r YC δ 3.7999e-001 8.0588e-003 3.4170e-001 2.1800e-002 lC β -1.1959e-001 6.0201e-004 -1.1979e-001 9.1951e-004 plC -5.8590e-001 2.5898e-003 -5.9285e-001 4.0330e-003 rlC 1.8888e-001 1.8195e-003 1.7381e-001 2.9350e-003 a lCδ -2.2818e-001 9.1234e-004 -2.3304e-001 1.4608e-003 r lCδ 4.2212e-002 7.4475e-004 2.6060e-002 1.3599e-003 nC β 8.6376e-002 2.1507e-004 8.5469e-002 5.5418e-004 pnC -6.4414e-002 9.8548e-004 -6.6103e-002 2.5351e-003 rnC -1.9059e-001 6.2567e-004 -1.9329e-001 1.6850e-003 a nC δ -2.9796e-003 3.6329e-004 -3.6351e-003 9.4129e-004 r nC δ -1.3548e-001 2.7071e-004 -1.3737e-001 7.7180e-004 In the second example the parameters of a low-order equivalent system (LOES) are estimated by using output-error method in the frequency domain. The LOES model includes closed-loop dynamic response of the aircraft and the control system from the pilot point of view; thus, the inputs for LOES modeling are pilot stick and pedal deflections, and the outputs are aircraft response variables, such as pitch rate or angle of attack. These models are used for many applications, for instance; control law design validation or flying qualities research. LOES models attempt to approximate the total high-order closed-loop aircraft response using a low-order linear dynamic model with a time delay caused by actuator dynamics. Based on the dimensional linear
- 72. Nonlinear model identification of airplanes using noisy flight data 72 equations for short period motion described in [1] and [4], the longitudinal LOES model can be written: 1 ( ), e e q e q LL L t M Mq q M ηα α η α α η τ − − = + − ɺ ɺ (5.1) where the input is now a longitudinal pilot stick deflection, with its time delay. Assuming that 0qL ≈ and 0e Lη ≈ for low angle of attack, and then performing the Laplace transform, 01 , e s e q s L e MM s M q α τ ηα α η − + − + − − ɶ ɶ ɶ (5.2) then the transfer function of the system can be expressed as 1 0 2 2 1 0 ( ) ( ) , ( ) ( ) e s s e q q M s L e b b eq s L M s M L M s a s a τ τ η α α α αη − − + + = = + − − + + + ɶ ɶ (5.3) Estimating the time delay parameter τ is problematic in the time-domain [1]. To address this problem, the data will be analyzed in the frequency domain. Flight data used in this part is measured on a Tu-144L supersonic transport aircraft. Data sequences of the longitudinal maneuver are shown in Figure 5.2-3. There are also two data sequences, run1 and run2 for the same reason as in first example of Chapter 4. Nominal starting parameter vector for output-error method is computed by using equation-error method. The equation-error method is much more robust to starting values of the parameters [1]. The elements of the unknown parameter vector are the coefficients of the transfer function given in (5.3), 1 0, 1 0, , , .b b a aθ τ = Measured data are transformed into frequency domain for a selected frequency range by using the high accuracy finite Fourier transformation described in Chapter 3. The cost function given by (2.122) has to modified 1 * 1 0 ( ) ( , ) ( , ), M vv k J M k S kθ θ θ − − = = ∑ ɶ ɶv v (5.4) where 95M = denotes the number of data points in the frequency domain. The innovations are [ ]1 0 2 1 0 ( , ) ( ) ( ) . ( ) kk j e k k j b b k q k k e j j a a ω τω θ η ω ω −+ = − + + ɶ ɶv (5.5)
- 73. Nonlinear model identification of airplanes using noisy flight data 73 840 850 860 870 880 V(fps) 250 300 350 400 450 alpha(deg) 0 5 10 15 20 -200 -100 0 100 200 q(dps) time (sec) 200 300 400 500 600 the(deg) 0 5 10 15 20 -4 -2 0 2 etae(in) 0 5 10 15 20 -2 -1.5 -1 -0.5 az(g) time (sec) Figure 5.2 Measured input-output data sequences for a longitudinal maneuver, run1 880 881 882 883 884 V(fps) 250 300 350 400 alpha(deg) 0 5 10 15 20 -200 -100 0 100 q(dps) time (sec) 150 200 250 300 350 the(deg) 0 5 10 15 20 -3 -2 -1 0 1 etae(in) 0 5 10 15 20 -1.5 -1 -0.5 az(g) time (sec) Figure 5.3 Measured input-output data sequences for a longitudinal maneuver, run2
- 74. Nonlinear model identification of airplanes using noisy flight data 74 Estimated parameter standard errors are do not require correction, because the linear trend and bias are removed before the Fourier transformation. Output-error parameter estimation took 25 modified Newton-Raphson iterations. Table 5.2 shows the estimation results. The second and third columns of Table 5.2 contain nominal parameter values computed by using equation-error method. Model fit to the measured outputs can be seen in Figure 5.4. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 Frequency-Domain Output-Error Modeling frequency (Hz) data model 0 2 4 6 8 10 12 14 16 18 20 -3 -2 -1 0 1 2 3 time (sec) q(deg/sec) Figure 5.4 LOES model fit to measured data Table 5.2 Tu-144L longitudinal short-period LOES model parameters Nominal starting values Output-error Parameter ˆθ ˆ( )s θ ˆθ ˆ( )s θ 1b 3.9165 8.8292e-002 3.7877 1.6844e-001 0b 4.9213 3.6032e-001 5.1812 2.4910e-001 1a 3.7948 9.7172e-002 3.7091 1.4818e-001 0a 7.1281 3.7642e-001 7.3960 2.5796e-001 τ 2.7024e 3.3495e-003 2.6837 7.6069e-003 As a final step, measured data from a similar maneuver (run2, Figure 5.3) are used to validate the estimated parameters. Estimated LOES model is used to predict
- 75. Nonlinear model identification of airplanes using noisy flight data 75 the pitch rate response for maneuver run2. The results can be seen in Figure 5.5. The model is a good predictor. Note that some high-order effects cannot be captured by this low-order model.
- 76. Nonlinear model identification of airplanes using noisy flight data 76 6 Flight test data determination using simulation In this chapter, simulated flight data is gathered using a known non-linear model of an aircraft [7]. The use of this experiment is to verify our identification methods. Firstly, a brief review about aircraft modeling is given. Then, measured flight data sequences are gathered virtually using the nonlinear model of a DHC-2 Beaver aircraft. Additional noises and other disturbances are added to the signals, in order to create lifelike simulation circumstances. 6.1 Outline of aircraft modeling In this subsection, necessary information about aircraft modeling is given based mainly on [1], [4] and [7]. The FDC-1.2 Simulink Toolbox [7] was used for the simulations. In this toolbox, a well parameterized nonlinear dynamic model of the De Havilland DHC-2 Beaver can be found. The aircraft is depicted on Figure 6.1. Figure 6.1 De Havilland DHC-2 Beaver This work, does not deal how to build a nonlinear dynamic model for an aircraft; therefore the results and notations as in [4] and [7] are used to describe aerodynamic parameters, angular rates/velocities/accelerations and the orientation of the aircraft. The state and the input vector of the Beaver are [ ] T e ex V p q r x y Hα β ψ θ ϕ= (6.1)
- 77. Nonlinear model identification of airplanes using noisy flight data 77 T e a r fu δ δ δ δ = (6.2) An explication of the elements of the state and input vector can be found in the Table 6.1. Table 6.1 Description of the elements of the state and input vector State Description Dimension V absolute velocity m/s α angle of attack rad β sideslip angle rad p roll rate rad/s q pitch rate rad/s r yaw rate rad/s ψ Euler heading angle rad θ Euler pitch angle rad ϕ Euler roll angle rad ex x coord. in NED frame m ey y coord. in NED frame m H altitude m eδ elevator deflection rad aδ aileron deflection rad rδ rudder deflection rad fδ flaperon deflection rad In aircraft control and simulation theory, several coordinate frames are used. Generally, measured data are gathered in the Aircraft Body Coordinate system (ABC) or in the Wind-axis coordinate system. In the case of both, the origin is the center of gravity of the aircraft. The ABC and the Wind frame can be seen in Figure 6.2 and 6.3, respectively. It can be seen on Figure 6.3 that the angle of attack (α ), the sideslip angle ( β ) and the absolute velocity are considered in the Wind-axis coordinate system. Furthermore, the angular rates ( p , q , r ), the Euler angles (ϕ ,θ ,ψ ) are considered in the ABC frame. X,Y,Z are the body-axis components of aerodynamic force acting on the aircraft and L,M,N are the body axis components of aerodynamic moment acting on the aircraft (Figure 6.2). The input vector contains the deflections of the actuators (Figure 6.3). The position of the aircraft is considered in the NED (North-East-Down) coordinate system; in which the origin is on the surface of the Earth, the x axis points to north, the y axis points to east and the z axis points to down. To build the nonlinear dynamic model of an aircraft, kinematic, aerodynamic,
- 78. Nonlinear model identification of airplanes using noisy flight data 78 moment and navigational equations are used [4]. Furthermore, the given model can be divided into two decoupled subsets [4]. One of them describes the longitudinal while the other one describes the lateral motion. The longitudinal state variables are the angle of attack (α ), absolute velocity (V), pitch rate (q ) and the Euler pitch angle (θ ).The variables describing the lateral motion are the sideslip angle ( β ), roll rate ( p ), yaw rate (r ), Euler roll angle (ϕ ) and Euler heading angle (ψ ). In the following, non-dimensional aerodynamic parameters will be identified. So a particular focus is placed on the aerodynamic force equation. Aerodynamic forces and moments depend always on the flight condition. In the case of Beaver aircraft, it has the following form 1( , , )aero aeroF dp x x u= ɺ (6.3) where aeroF is a vector of aerodynamic forces, 1p is a polynomial vector-function containing the non-dimensional aerodynamic coefficients and d converts the non- dimensional coefficients to dimensional forces and moments. Detailed description concerning the place of these force/torque components in the nonlinear dynamic equations of an aircraft can be found in Appendix A. Note the polynomial function depends on xɺ too. It is only taken into account by βɺ in the case of aerodynamic side force coefficient ( aY ). 1p describes the aerodynamic forces and moments coefficients in ABC. It has the following form 2 3 0 2 0 3 2 0 2 3 3 0 2 2 2 2 = + + + + + + + + + + = + + + + + = + + + + + + = + ++ ɺ X X X X X X X X f X fq r f f Y Y Y Y Y Y a Y r Y rp r a r r Z Z Z Z Z Z e Z E Z f Z fq e f fe l l l ra Ya a l la p r qc C C C C CC V C C C C C V V V q C C C V C C C C pb r C C V b b C C C C c C C C C C C pb C C α δ δ αδα α β δ δ δ α β α δ δ αδα δ β β α α α δ δ αδ β β δ δ δ α α α δ δ β δ αδ β 2 2 0 2 2 3 30 2 2 2 2 + + = + + + + + + = + + + + + + ++ l l m m m m ma n n n a l a a m m m e X fq re f n n n ap r r q Yna a r n rb C c rb C V q C C V V q C C C C C V C C C C C C C pb rb c C V V C C C δ δ α α δ δα β β δ δ β δ δ α α α δ β δ β δ δ β (6.4) where the subscript a stands for aero. That means these represent only the aerodynamic effects in the nonlinear equations. There is also 2 ( , )=prop tF dp x dp [1] that describes the contribution of the engine to nonlinear equations. In this work I did not concern the identification of the coefficients in 2p .
- 79. Nonlinear model identification of airplanes using noisy flight data 79 Figure 6.2 The ABC coordinate frame Figure 6.3 The Wind-axis coordinate frame Figure 6.4 The actuators
- 80. Nonlinear model identification of airplanes using noisy flight data 80 6.2 Virtual test flights In order to estimate the values of the coefficients in 1p (6.3), necessary data sequences are obtained by virtual test flights using FDC 1.2 Toolbox [7] and the results of a previous MSc Thesis [12] in MATLAB/Simulink environment. As was mentioned before, the state variables can be divided into two decoupled subsets. Hence, the simulations are done in two phases as well. One of them performs a lateral maneuver while the other a longitudinal maneuver. Implicitly, variables belong to the other maneuver are held a constant trim values during the maneuver. In both cases, two simulations are performed in order to obtain a teaching part and a testing part of the measured data. In the following, the teaching data will play a role in parameter estimation while the testing data will be served to validate our estimation results. Furthermore, measurement noises are added to the signals in order to model real flight circumstances. The applied noise model can be seen in the next subsection. Figure 6.5 - 6.12 show the simulated teaching and testing data sequences of a lateral maneuver with additional measurement noises. Teaching and testing simulated data sequences of a longitudinal maneuver can be seen on Figure 6.13 - 6.18. Note that the only difference between teaching and testing sequences is the sign of the input doublets. Table 6.2 shows us the used trim values during simulation. Trim values were subtracted from all of the data sequences. Table 6.2 Applied trim values during the lateral maneuver State Trim value V 45 m/s α 0.14724 rad β -0.011814 rad p -0.009.7782 rad/s q 0.032429 rad/s r 0.080425 rad/s ϕ 0 rad θ 0.11229 rad ψ 0.38328 rad ex , ey 0 m H 1000 m
- 81. Nonlinear model identification of airplanes using noisy flight data 81 0 5 10 15 20 25 30 35 -8 -6 -4 -2 0 2 4 6 8 ail(deg) 0 5 10 15 20 25 30 35 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 rdr(deg) time (sec) Figure 6.5 Teaching data from a lateral maneuver (aileron, rudder) 0 5 10 15 20 25 30 35 -20 -15 -10 -5 0 5 10 15 p(dps) 0 5 10 15 20 25 30 35 -8 -6 -4 -2 0 2 4 6 8 10 r(dps) time (sec) Figure 6.6 Teaching data from a lateral maneuver (roll and yaw rates)
- 82. Nonlinear model identification of airplanes using noisy flight data 82 0 5 10 15 20 25 30 35 -10 -5 0 5 10 15 phi(deg) 0 5 10 15 20 25 30 35 -15 -10 -5 0 5 10 15 beta(deg) time (sec) Figure 6.7 Teaching data from a lateral maneuver (Euler roll and sideslip angles) 0 5 10 15 20 25 30 35 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 ay(g) time (sec) Figure 6.8 Teaching data from a lateral maneuver (acceleration along the y axis)
- 83. Nonlinear model identification of airplanes using noisy flight data 83 0 5 10 15 20 25 30 35 -8 -6 -4 -2 0 2 4 6 8ail(deg) 0 5 10 15 20 25 30 35 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 rdr(deg) time (sec) Figure 6.9 Testing data from a lateral maneuver (aileron, rudder) 0 5 10 15 20 25 30 35 -15 -10 -5 0 5 10 15 20 p(dps) 0 5 10 15 20 25 30 35 -10 -8 -6 -4 -2 0 2 4 6 8 r(dps) time (sec) Figure 6.10 Testing data from a lateral maneuver (roll and yaw rate)
- 84. Nonlinear model identification of airplanes using noisy flight data 84 0 5 10 15 20 25 30 35 -15 -10 -5 0 5 10 phi(deg) 0 5 10 15 20 25 30 35 -15 -10 -5 0 5 10 15 beta(deg) time (sec) Figure 6.11 Testing data from a lateral maneuver (Euler roll and sideslip angle) 0 5 10 15 20 25 30 35 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 ay(g) time (sec) Figure 6.12 Testing data from a lateral maneuver (acceleration along the y axis)
- 85. Nonlinear model identification of airplanes using noisy flight data 85 The applied trim values during the longitudinal simulations can be seen in Table 6.3. In both cases, it was supposed that most of the necessary signals are available from a sensor. However, in some practical cases, not all of the signals can be obtained from a sensor. In these cases, the fusion of the on board sensors has to be done. The estimation of the missing signals is based on the available data sequences. Table 6.3 Applied trim values during the longitudinal maneuver State Trim value V 45 m/s α 0.13435 rad β -0.013502 rad p , q , r 0 rad/s ϕ ,ψ 0 rad θ 0.12104 rad ex , ey 0 m H 1000 m 0 5 10 15 20 25 30 35 -8 -6 -4 -2 0 2 4 6 alpha(deg) 0 5 10 15 20 25 30 35 -15 -10 -5 0 5 10 15 20 25 q(dps) time (sec) Figure 6.13 Teaching data from a longitudinal maneuver (angle of attack and pitch rates)
- 86. Nonlinear model identification of airplanes using noisy flight data 86 0 5 10 15 20 25 30 35 -10 -5 0 5 10 V(m/s) 0 5 10 15 20 25 30 35 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 az(g) time (sec) Figure 6.14 Teaching data from a longitudinal maneuver (velocity and acceleration along the z axis) 0 5 10 15 20 25 30 35 -6 -4 -2 0 2 4 6 el(deg) 0 5 10 15 20 25 30 35 -30 -20 -10 0 10 20 the(deg) time (sec) Figure 6.15 Teaching data from a longitudinal maneuver (elevator and pitch angles)
- 87. Nonlinear model identification of airplanes using noisy flight data 87 0 5 10 15 20 25 30 -8 -6 -4 -2 0 2 4 6 alpha(deg) 0 5 10 15 20 25 30 -15 -10 -5 0 5 10 15 20 25 q(dps) time (sec) Figure 6.16 Testing data from a longitudinal maneuver (angle of attack and pitch rates) 0 5 10 15 20 25 30 -10 -5 0 5 10 V(m/s) 0 5 10 15 20 25 30 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 az(g) time (sec) Figure 6.17 Testing data from a longitudinal maneuver (velocity and acceleration along the z axis)
- 88. Nonlinear model identification of airplanes using noisy flight data 88 0 5 10 15 20 25 30 -6 -4 -2 0 2 4 6 el(deg) 0 5 10 15 20 25 30 -30 -20 -10 0 10 20 the(deg) time (sec) Figure 6.18 Testing data from a longitudinal maneuver (elevator and pitch angles) 6.3 The applied noise model In this subsection, the applied noise model is presented based mainly on noise model can be found in [1]. Taking into account a standard instrumentation the measurement error has the following form: ( ) (1 ) ( ) ( ) 1,2,...z i y i b v i i Nλ= + + + = (6.5) where ( )z i is the ith measured output from the sensor, ( )y i is the true value of the output, ( )v i is a random measurement noise, λ is a constant scale factor error parameter and b is a constant bias error parameter. The noise model can be seen on Figure 6.19. Figure 6.19 Instrumentation error model
- 89. Nonlinear model identification of airplanes using noisy flight data 89 Note that in reality not all of the sensors have both a scale factor and a bias error factor. The simulated sensor signals then have the following form: ( ) (1 ) ( ) ( ), ( ) (1 ) ( ) ( ), ( ) (1 ) ( ) ( ), ( ) (1 ) ( ) ( ), ( ) (1 ) ( ) ( ), ( ) (1 ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) x x y y E V v v E E E E E xE x a a yE y a a V i V i b v i i i b v i i i b v i i i b v i i i b v i i i b v i a i a i b v i a i a i b v β β β α α α φ φ φ θ θ θ ψ ψ ψ λ β λ β α λ α φ λ φ θ λ θ ψ λ ψ = + + + = + + + = + + + = + + + = + + + = + + + = + + = + + ( ), ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ), z zzE z a a E p p E q q E r r i a i a i b v i p i p i b v i q i q i b v i r i r i b v i = + + = + + = + + = + + (6.6) where the elements with the subscript E are the simulated signals with additional measurement noises. The constant scale and the bias error factors were set after real flight data. As was mentioned above, simulated data sequences can be seen on Figure 6.5 - 6.18.
- 90. Nonlinear model identification of airplanes using noisy flight data 90 6.4 Prefiltering the signals In this subsection, results of applied prefiltering methods for the signals that will be used during identification are shown. Since the applied methods were presented in Chapter 3, only one example is given here. Global smoothing in the frequency domain was used to eliminate the effects of the noises. The cut-off frequency was 0.9Hz. Figure 6.20 shows the noisy sideslip data sequence from a simulated lateral maneuver of the Beaver aircraft. Smoothed time series can be seen on Figure 6.21. 0 5 10 15 20 25 30 35 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Figure 6.20 Simulated noisy time series of the sideslip angle 0 5 10 15 20 25 30 35 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Figure 6.21 Smoothed time series of the sideslip angle
- 91. Nonlinear model identification of airplanes using noisy flight data 91 7 Parameter estimation using time domain ML and OE methods In this chapter identification of the lateral and longitudinal aerodynamic parameters are done applying noisy simulated data sequences and time domain ML and OE methods. In both cases, identification is done by using the teaching part of the gathered simulated data. As was mentioned in the previous subsection, prefiltering and smoothing of the signals were done before identification. Teaching and testing part of the simulated lateral data sequences can be seen on Figure 6.5-12. Teaching and testing part of the applied longitudinal simulated data sequences are depicted on Figure 6.13-18. During the identification of the coefficients related to the lateral state variables, the ones that related to the longitudinal coefficients are held on a constant trim value; similarly, in the case of the identification of the longitudinal state variable coefficients. Finally, testing part is used to validate the identification. Identification of the lateral aerodynamic force coefficients Firstly, lateral non-dimensional aerodynamic parameters are estimated which means the coefficients of the second, fourth and sixth row from 6.4 ( , ,Y l nC C C ). In the case of output-error method, all of the coefficients are estimated at the same time. Note that these algorithms depend highly on the initial parameter values. Several experiments were made using different starting values; most of them diverged. Hence, results of the equation-error and linear regression methods were used as a starting value. These linear regression methods were in the focus of an earlier work [10] that was made for a student science conference. Modified Newton-Raphson method was used to minimize the cost function in (2.42). The estimated parameter covariance matrix was computed from (2.50) as the Cramer-Rao lower bound 1 0 ˆ( ) ( ).Cov Mθ θ− ≥ Note that the residuals are assumed to be white. The same model structure is used during identification as in the first example of Chapter 4 (4.1-4.3). Time domain output-error model fit to the testing part of the measured data is shown in Figure 7.1. It can be seen that a quite good prediction is obtained by using the estimated parameters. The solution took 37 Newton-Raphson iterations. Parameter estimation results are summarized in Table 7.1-7.3.
- 92. Nonlinear model identification of airplanes using noisy flight data 92 0 5 10 15 20 25 30 35 -20 -10 0 10 beta(deg) 0 5 10 15 20 25 30 35 -20 0 20 p(dps) 0 5 10 15 20 25 30 35 -10 0 10 r(dps) 0 5 10 15 20 25 30 35 -20 -10 0 10 phi(deg) 0 5 10 15 20 25 30 35 -0.5 0 0.5 time (sec) ay(g) Figure 7.1 Time domain lateral output-error model (red) fit measured data (blue) The last columns of the tables contain the real values of the coefficients. Second columns comprise the estimated values applying equation-error method and linear regression technique. These estimated values were used as a starting value for the iterative maximum likelihood method. Finally, third columns contain the results obtained by using maximum likelihood and output-error method in the time domain. Table 7.1 Parameter estimation results for YC Parameter Equation-error method ML OE methods Real value constant -0.0136 -0.0016 -0,002226 β -0.7497 -0.7185 -0,7678 pb 2V - - -0,1240 rb 2V 0.3662 0.4614 0,3666 δa - - -0,02956 δr 0.2035 0.2094 0,1158 δ αr - - 0,5238 β V b 2 ɺ - - -0,1600
- 93. Nonlinear model identification of airplanes using noisy flight data 93 Table 7.2 Parameter estimation results for nC Parameter Equation-error method Output-error in time domain Real value constant 0.0013 -0.0028 -0,003117 β 0.0106 0.0110 0,006719 pb 2V -0.1639 -0.1398 -0,1585 rb 2V -0.0875 -0.1087 -0,1112 δa -0.0075 -0.0018 -0,003872 δr -0.0799 -0.0809 -0,08265 δr - - 0,1595 β3 - - 0,1373 It can be seen that there are little differences between real and estimated values. However, the results of the applied linear method can be partly used also in the nonlinear model. Table 7.3 Parameter estimation results for lC Parameter Equation-error method Output-error in time domain Real value constant -0.0036 0.0043 0,0005910 β -0.0616 -0.0566 -0,06180 pb 2V -0.5045 -0.5013 -0,5045 rb 2V 0.1722 0.2000 0,1695 δa -0.1122 -0.1111 -0,09917 δr 0.0073 0.0144 0,006934 αδ α - - -0,08269
- 94. Nonlinear model identification of airplanes using noisy flight data 94 Identification of the longitudinal aerodynamic force coefficients Secondly, non-dimensional longitudinal aerodynamic parameters are estimated. The used model structure came from [1]. Similarly to the first example in chapter 3 a linear perturbation model is used. In (7.1-2), values with subscript o denote the measured trim values of the maneuver. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 sin cos ( ) , 2 sin cos ( ) , , 2 ∆ = − + ∆ + + − ∆ − ∆ − ∆ + ∆ = − + ∆ + + + + − ∆ − ∆ − + ∆ = + ∆ + + + ɺ ɺ ɺ ɺ ɺ ɺ ɺ V q V q V q D D D D e V L L L L e m m m m e q y q S V qc V C C C C m V V T g b m q S V qc C C C C mV V V g T q b V mV q Sc V qc q C C C C b I V V α δ α δ α δ α α δ α γ θ α α α α δ γ α θ α α δ ,= qθ (7.1) In order to use these equations for output-error parameter estimation, it has to be rewritten in the state-space form (7.2). Equation (7.4) describes the connection between (6.4) and the coefficients can be found in (7.1). Nominal parameter values are computed by using the parameter estimation results of ,X ZC C and mC [10] and equation (7.4). Detailed description of the calculation of the nominal parameters can be found in Appendix B. Note that in [10] the parameters were estimated by applying linear regression method. In the case of longitudinal parameter estimation, only the first 16 seconds of the gathered longitudinal data were used in the identification. 0 0 1 1 0 1 2 0 0 0 0 3 3 0 3 2 0 4 4 4 2 1 3 4 sin ( cos ) cos sin sin ( cos ) 1 0 0 1 1 0 0 − + − − − + + = + ɺ ɺ ɺ ɺ V q V q V q D D D L L L m m m D V L m q T k C k C g k C k g m VV T g k C k C g k C k m V qq k C k C k C k k C b k C b k C b b α α α δ δ δ α θ α γ γ α γ αα γ θθ , 1 eδ (7.2a)
- 95. Nonlinear model identification of airplanes using noisy flight data 95 0 0 0 0 1 2 3 4 5 0 0 1 0 0 0 0 1 0 0 . 0 0 1 0 0 0 0 1 , , , , 2 = = − = = − = = − y V V q q q S q S q Sc q Sc k k k k k m V mV I mg α α θ θ (7.3b) cos sin , cos sin , = − + = − − L Z X D X Z C C C C C C α α α α (7.4) The solution takes 99 Newton-Raphson iterations. The given results can be seen on Figure 7.2. Estimated parameter values can be seen in Table 7.4. 0 2 4 6 8 10 12 14 16 -500 0 500 V(mps) 0 2 4 6 8 10 12 14 16 -10 -5 0 5 10 alpha(deg) 0 2 4 6 8 10 12 14 16 -20 0 20 40 q(dps) 0 2 4 6 8 10 12 14 16 -40 -20 0 20 theta(deg) time (sec) Figure 7.2 Time domain longitudinal output-error model (red) fit measured data (blue) Table 7.4 Parameter estimation results for D LC ,C and mC Parameter DC LC mC constant -0.0155 -0.0037 -0.0003 V 0.1888 1.5689 0.0297 α 0.2927 5.4237 -1.4073 q 14.1835 -10.0255 -40.1514 eδ 0.4029 -0.8197 -2.6314
- 96. Nonlinear model identification of airplanes using noisy flight data 96 8 Parameter estimation using frequency domain ML and OE methods Similarly to the previous section identification of the lateral aerodynamic parameters is done. In this case, frequency domain ML and OE methods are used to determine the unknown parameters. Identification is done by using the teaching part of the gathered simulated data. Before identification, filtering and smoothing methods were applied to the simulated data sequences. Teaching and testing part of the simulated lateral data sequences can be seen in Figure 6.5-12. During the identification of the coefficients of the lateral model, the parameters used only in the longitudinal model are held on a constant trim value. Identification of the lateral aerodynamic force coefficients Estimation of the same lateral non-dimensional aerodynamic parameters as in the previous chapter is done by applying frequency domain output-error method. Results from equation-error linear regression [1] done in frequency domain were used as initial parameters values. All relevant time series were transformed into frequency domain using the high-accuracy finite Fourier transformation described in subsection 3.4 with the frequency vector [ ]0.10 0.12 0.14 ... 1.5 T f Hz= . All significant spectral components caught by applying this frequency range [1]. The same linear dynamic model (4.3) was used to estimate the non-dimensional aerodynamic coefficients. The cost gradient and the information matrix were computed using (2.123) and (2.124), respectively. It can be seen on Figure 8.1 that predicted outputs match quite well to the testing part of the measured data. Note that in the case of frequency domain methods, constant parameters cannot be estimated because biases and trends were removed before Fourier transformation. The constant parameters and other biases must be estimated in time domain. Furthermore, parameter estimation results are also written in Table 8.1-8.3.
- 97. Nonlinear model identification of airplanes using noisy flight data 97 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 abs(beta) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 abs(p) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 abs(r) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 abs(phi) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 frequency (Hz) abs(ay) Figure 8.1 Output-error model (red) fit measured data (blue) in the frequency domain Table 8.1 Parameter estimation results for YC Parameter Equation-error method Output-error in freq. domain Real value constant - - -0,002226 β -0.7335 -0.7166 -0,7678 pb 2V - - -0,1240 rb 2V 0.3453 0.3761 0,3666 δa - - -0,02956 δr 0.1897 0.1502 0,1158 δ αr - - 0,5238 β V b 2 ɺ - - -0,1600
- 98. Nonlinear model identification of airplanes using noisy flight data 98 Table 8.2 Parameter estimation results for nC Parameter Equation-error method Output-error in freq. domain Real value constant - - -0,003117 β -0.0594 0.0131 0,006719 pb 2V -0.4980 -0.1202 -0,1585 rb 2V 0.1730 -0.1259 -0,1112 δa -0.1112 0.0043 -0,003872 δr 0.0074 -0.0864 -0,08265 δr - - 0,1595 β3 - - 0,1373 The structure of the tables is similar to the ones can be found in the previous chapter. Real values of the parameters can be found in the last column of the tables. Second and third columns contain the results of the least squares equation-error and the maximum likelihood output-error in frequency domain, respectively. Table 8.3 Parameter estimation results for lC Parameter Equation-error method Output-error in freq. domain Real value constant - - 0,0005910 β 0.0094 -0.0572 -0,06180 pb 2V -0.1490 -0.4942 -0,5045 rb 2V -0.1171 0.1632 0,1695 δa -0.0016 -0.1047 -0,09917 δr -0.0840 0.0017 0,006934 αδ α - - -0,08269
- 99. Nonlinear model identification of airplanes using noisy flight data 99 9 Description of the applied software components and Simulink models The applied software components are presented in this chapter. As was mentioned above the non-public SIDPAC toolbox was used to perform identification. It is assumed that the user of my software has the password to obtain his/her own copy of SIDPAC. To put SIDPAC in the MATLAB path is task of the user. During the work several functions were added to the toolbox in order to make an interface between the toolbox and the simulated data sequences of the Beaver aircraft. Firstly, a brief description about the usage of the functions and scripts are given. Then a detailed description of each function and script is presented. Perform the virtual test flights In order to perform some virtual test flights and gather data sequences one has to use the following sequence of scripts (in the case of generating lateral teaching part of the data).: >>SetPath >>mode = ModeSet(‘Lat’); >>VirtualTestFlight; For generating the testing part of the lateral data sequence, the switch in the corresponding Simulink model has to be changed. The switch block can be found in the Beaver_LatSim_For_MSC_Thesis.mdl which has to be set to the Input Run 2 block. Similarly in the case of longitudinal simulation. Perform the identification of the non-dimensional aerodynamic parameters Identification of the non-dimensional aerodynamic parameters of the DHC-2 Beaver aircraft can be done by using three scripts (one script for each case). After gathering the necessary data sequences, identification of the lateral and longitudinal coefficients is done in time domain while in frequency domain only the lateral coefficients are identified. In the case of lateral non-dimensional aerodynamic coefficients in time domain: >> TD_OE_Beaver_Lat_ID
- 100. Nonlinear model identification of airplanes using noisy flight data 100 In the case of lateral non-dimensional aerodynamic coefficients in frequency domain: >> FD_OE_Beaver_Lat_ID In the case of longitudinal non-dimensional aerodynamic coefficients in time domain: >> TD_OE_Beaver_Long_ID Description of the elaborated scripts and applied Simulink models Simulink models The applied Simulink models can be seen on Figure 9.1 and 9.2. Observe the input selection switches in the models. In both cases, Input Run 1 is used to generate the teaching data sequences while Input Run 2 is applied to produce the testing part of the data. Furthermore under the mask named Non-linear Beaver, there is the non-linear model of the Beaver which is part of the FDC Toolbox [7]. Concerning the file structure, the following set-up is used MSc_Thesis – Contains the main scripts that perform the identification methods MSc_ThesisVirtual_Test_Flight. – Contains the scripts and Simulink models that perform the virtual test flights, it also contains data necessary to the simulations and functions that perform data saving. In order to execute simulations, SetPath.m has to be run first. The library also contains the FDC Toolbox (public). The applied SIDPAC functions smoo – Optimal global Fourier smoothing. lesq – Least squares linear regression fint – High-accuracy finite Fourier integral for arbitrary frequencies oe – Output-error parameter estimation in time domain fdoe - Output-error parameter estimation in frequency domain r_colores – Parameter covariance for colored residuals from linear regression.
- 101. Figure 9.1 Simulink model to perform a lateral test maneuver Figure 9.2 Simulink model to perform a longitudinal test maneuver
- 102. Matlab scripts TD_OE_Beaver_Lat_ID - Performs the identification of the non- dimensional aerodynamic parameters of the DHC-2 Beaver aircraft using output error method in the time domain Usage: TD_OE_BEaver_Lat_ID Calls: lesq r_colores oe beaver_params smoo FD_OE_Beaver_Lat_ID - Performs the identification of the non- dimensional aerodynamic parameters of the DHC-2 Beaver aircraft using output error method in the frequency domain Usage: FD_OE_BEaver_Lat_ID Calls: fint lesq zep fdoe beaver_params smoo TD_OE_Beaver_Long_ID - Performs the identification of the non- dimensional aerodynamic parameters of the DHC-2 Beaver aircraft using output error method in the time domain Usage: TD_OE_BEaver_Lat_ID Calls: lesq r_colores oe beaver_params smoo
- 103. Nonlinear model identification of airplanes using noisy flight data 103 beaver2fdata - Converts the simulated data sequences using the nonlinear model of the DHC-2 Beaver to the standard fdata structure of the SIDPAC. Usage: beaver2fdata Calls: None LatSave1 - Saves the gathered lateral data sequences to a LatData_Teach named file Usage: LatSave1 Calls: none LatSave2 - Saves the gathered lateral data sequences to a LatData_Test named file Usage: LatSave2 Calls: none ModeSet - Sets the mode flag Usage: mode = ModeSet(mode) where mode can be 'Lat' or 'Long' that stands for lateral and longitudinal movement Calls: none VirtualTestFlight - Simulation execution script applying the nonlinear model of the DHC-2 Beaver aircraft Usage: VirtualTestFlight Calls: sim tnlatss_beaver
- 104. Nonlinear model identification of airplanes using noisy flight data 104 fnlatss_beaver - Frequency-domain lateral state-space model file using non-dimensional parameters. Usage: [Y,X,A,B,C,D] = fnlatss(p,U,w,x0,c); Input: p = parameter vector. U = matrix of input vectors in the frequency domain. w = frequency vector, rad/sec. x0 = initial state vector. c = vector of constants. Output: Y = model output vector or matrix. X = model state vector or matrix. A,B,C,D = system matrices. tnlatss_beaver - Time-domain lateral state-space model of the Beaver aircraft file using non-dimensional parameters. Usage: [y,x,A,B,C,D] = tnlatss(p,u,t,x0,c); Input: p = parameter vector. u = input vector or matrix. t = time vector. x0 = initial state vector. c = vector of constants. Output: y = model output vector or matrix time history. x = model state vector or matrix time history. A,B,C,D = system matrices.
- 105. Nonlinear model identification of airplanes using noisy flight data 105 beaver_params - Assembles constants for dimensional state-space models. Usage: css = beaver_params; Input: fdata = flight test data array in standard configuration. Output: css = vector of constants for a dimensional state-space model. plot4ident_lat and plot4ident_long - Generates the figures that can be found in the subsection 6.2. tnlonss_beaver - Time-domain longitudinal state-space model of the Beaver aircraft file using non-dimensional parameters. Usage: [y,x,A,B,C,D] = tnlonss(p,u,t,x0,c); Input: p = parameter vector. u = input vector or matrix. t = time vector. x0 = initial state vector. c = vector of constants. Output: y = model output vector or matrix time history. x = model state vector or matrix time history. A,B,C,D = system matrices. long_inic – Initialize the identification method of the longitudinal model and computes the nominal parameter values. Usage: p0 = long_inic(xinco, fdata, selevator, sV, salpha, sq,t); Input: xinco = trim value of the states. fdata = standard SIDPAC data structure. selevator = smoothed data sequence of the elevator. sV = smoothed data sequence of the absolute velocity. salpha = smoothed data sequence of alpha. sq = smoothed data sequence of q
- 106. Nonlinear model identification of airplanes using noisy flight data 106 t = time vector Output: p0 = nominal parameter vector.
- 107. Nonlinear model identification of airplanes using noisy flight data 107 10 Summary and concluding results In this MSc thesis I reviewed and analyzed several parameter estimation methods such as the time-domain and frequency domain Maximum Likelihood (ML) methods. Particular focus was placed on the output-error method. The parameter estimations are based on noisy flight data sequences, so some of the essential smoothing and filtering methods were presented, in order to decrease the blast of the measurement noises. A high accuracy approximation of the finite Fourier transformation was also described to execute an adequate transformation of the measured data sequences between time-domain and frequency domain. In Chapter 4 and 5, identification ability of time-domain and frequency domain parameter estimation methods were demonstrated using the SIDPAC Toolbox in MATLAB environment. The identification methods studied in the preceding chapters were applied to reconstruct the dynamic model of a known aircraft from flight data. Simulated data sequences were determined by using the nonlinear model of the de Havilland DHC-2 Beaver commercial aircraft in MATLAB/Simulink environment. These data sequences were corrupted artificially by noises and by other kind of disturbances. The aircraft sensor noise model found in [1] was used. All of the parameters of the Beaver aircraft were known; thus, it was possible to validate the applied parameter estimation and smoothing methods. Several functions were elaborated to SIDPAC Toolbox in order to perform the identification of the parameters of the nonlinear dynamic model of the Beaver. To sum up, it could be seen that the estimated parameters obtained by applying time and frequency domain Maximum Likelihood and output-error methods were equal or very close to the real parameter values. These results are convenient for controller design techniques. Hence, the methods will be able to use in the case of any other kind of flight data, also for an unknown aircraft. Note that in reality there will be more disturbing factors in the parameter estimation; for instance important signals are not measurable, hence state estimation and sensor fusion is needed. Future research goal may be the identification of a real aircraft based on real flight data. In order to use the algorithms tested here, it is necessary to develop the fusion of GPS, IMU and magnetometer data and solve the state estimation problem. The methods presented in this work solved the first part of this large program and documented the applicability of the parameter estimation methods.
- 108. Nonlinear model identification of airplanes using noisy flight data 108 11 Appendix A. State equations of an aircraft State equations of an aircraft can be obtained considering rigid body equations of motions, completed by kinematic and dynamic equations including aerodynamic and engine effects. Equations showed here came from [1], [4] and [8]. Hence, state equations can be written in a compact form: Force equation in body frame: sin , cos sin , cos cos . = − + − + = − + + = − + + ɺ ɺ ɺ T X Y Z FqS u rv qw C g m m qS v pw ru C g m qS w qu pv C g m θ θ φ θ φ Force equation in wind-axes frame: cos cos (cos cos sin cos sin cos sin sin cos cos ), sin tan ( cos sin ) cos cos (cos cos cos sin sin ), cos sin cos cos sin cos sin ( cos sin = − + + + − = − + − + − + + = + − + + ɺ ɺ ɺ W W T D T L Y FqS V C g m m FqS C q p r mV mV g V qS q C p r mV V g V α β φ θ α β φ θ β θ α β α α β α α β β φ θ α θ α β β α α β φ θ β α θ cos sin cos cos ).− + TF g m α α φ θ Moment equations: 2 2 ( ) , ( ) ( ) , ( ) . − − = − + − = − − − + Ω − − = − + − Ω ɺ ɺ ɺ ɺ ɺ z yxz xz l x x x x px z xz m p y y y y y x pxz xz n p z z z z z I II IqSb p r C qr qp I I I I II I IqSc q C pr p r r I I I I I I II IqSb r p C pq qr q I I I I I
- 109. Nonlinear model identification of airplanes using noisy flight data 109 Acceleration sensor equations: 1 ( ), 1 ( ), 1 ( ), = + = = x X T y Y z Z a qSC F m a qSC m a qSC m where cos sin , cos sin , = − + = − − L Z X D X Z C C C C C C α α α α and cos sin , cos sin . = − + = − − W W D D Y Y Y D C C C C C C β β β β here the notations are the same as that is used in subsection 6.1. Furthermore, , , ,x y z pI I I I and xzI are inertia while , ,S m c and b (wing area, mass, mean chord line and wingspan) are physical parameters of the aircraft. Exact values of these parameters for the Beaver aircraft came from [7].
- 110. Nonlinear model identification of airplanes using noisy flight data 110 B. Computing nominal parameter values for longitudinal model identification In this part, calculation of the nominal parameter values for the Maximum Likelihood output-error identification of the longitudinal non-dimensional aerodynamic parameters is described. According to [1]: 0 0 , , = ∆ + = ∆ + X X X D Z Z C C C C C C cos sin , cos sin , = − + = − − L Z X D X Z C C C C C C α α α α It can be obtained by combining these equations: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 cos sin , ( )cos( ) ( )sin( ) ( )(cos cos sin sin ) ( )(sin cos cos sin ) ... cos sin cos sin ( sin ∆ ∆ = − − = − ∆ + + ∆ − ∆ + + ∆ = − ∆ + ∆ − ∆ − ∆ + ∆ + ∆ ≈ − − − ∆ − ∆ + − D X Z D X X Z Z X X Z Z X Z X Z X Z C C C C C C C C C C C C C C C C C C α α α α α α α α α α α α α α α α α α α α α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 cos ) , cos sin , ( )cos( ) ( )sin( ) ( )(cos cos sin sin ) ( )(sin cos cos sin ) ... cos sin cos sin ( ∆ ∆ ∆ = − + = − ∆ + + ∆ + ∆ + + ∆ = − ∆ + ∆ − ∆ + ∆ + ∆ + ∆ ≈ − + − ∆ + ∆ + L Z X L Z Z X X Z Z X X Z X Z X Z C C C C C C C C C C C C C C C C C α α α α α α α α α α α α α α α α α α α α α α 00 0sin cos ) .+ ∆XCα α α Then substituting (6.4) can be taken into consideration. Note that nonlinear terms are dropped. We have to use also the results of the engine model of the Beaver aircraft [7]. 1 2 3 , 1 2 , . = + = = p dpt p dpt X X X Z P dpt C C V C C dpt C C dpt where P is the engine power and dpt is the non-dimensional propeller slipstream. Derivative of dpt according to V is then calculated to obtain the form that contains 0 ∆V V .
- 111. Nonlinear model identification of airplanes using noisy flight data 111 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos sin , cos ( ) sin ( ) ( sin cos ) ( cos sin ( sin cos )) ( cos sin = − − ∆ ≈ − ∆ + ∆ − ∆ + + ∆ + + − ∆ ≈ − − + − ∆ + − − + dpt dpt q q e q D D X Z D X Z X X X Z Z Z e X Z Z X Z X C C C C q C C dpt V q C dp c C C c C C C t C C V C CC C C α α δ α α α α α α α α α αα δ α α α α α α α α 0 0 0 0 2 3 00 ) ( sin ) (( cos sin )( 3) ) , 1 2 ∆ + − ∆ ∆ + − − − q e e D Dq dpt dpt DV C C X Z C Z Z e c C C C q P CC V V VV δ δ α δ α α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos sin , cos ( ) sin ( ) ( sin cos ) ( cos sin ( sin cos )) ( cos sin = − + ∆ ≈ − ∆ + + ∆ + ∆ + ∆ + + + ∆ ≈ − + + + ∆ + − + + dpt dpt q e q q L L Z X L Z X Z Z Z Z e X X Z X Z X C Z X C C C q C C dpt V q C dp c C C C c C C C C C t C C V C C α δ α α α α α α α α α α α α α α α α α δ α α α 0 0 0 0 2 3 00 ) ( cos ) (( cos sin )( 3) ) , 1 2 ∆ + − ∆ ∆ + − + − q e e L Lq dpt dpt LV C C C Z X X Z e c C C C q P CC V V VV δ δ α δ α α
- 112. Nonlinear model identification of airplanes using noisy flight data 112 C. Lateral parameter identification results both TD and FD issues in common tables Parameter estimation results for YC Parameter EE time domain ML OE freq. dom. ML OE Real value constant -0.0136 -0.0016 - -0,002226 β -0.7497 -0.7185 -0.7166 -0,7678 pb 2V - - - -0,1240 rb 2V 0.3662 0.4614 0.3761 0,3666 δa - - - -0,02956 δr 0.2035 0.2094 0.1502 0,1158 δ αr - - - 0,5238 β V b 2 ɺ - - - -0,1600 Parameter estimation results for nC Parameter EE time domain ML OE freq. dom. ML OE Real value constant 0.0013 -0.0028 - -0,003117 β 0.0106 0.0110 0.0131 0,006719 pb 2V -0.1639 -0.1398 -0.1202 -0,1585 rb 2V -0.0875 -0.1087 -0.1259 -0,1112 δa -0.0075 -0.0018 0.0043 -0,003872 δr -0.0799 -0.0809 -0.0864 -0,08265 δr - - - 0,1595 β3 - - - 0,1373
- 113. Nonlinear model identification of airplanes using noisy flight data 113 Parameter estimation results for lC Parameter EE time domain ML OE freq. dom. ML OE Real value constant -0.0036 0.0043 - 0,0005910 β -0.0616 -0.0566 -0.0572 -0,06180 pb 2V -0.5045 -0.5013 -0.4942 -0,5045 rb 2V 0.1722 0.2000 0.1632 0,1695 δa -0.1122 -0.1111 -0.1047 -0,09917 δr 0.0073 0.0144 0.0017 0,006934 αδ α - - - -0,08269
- 114. Nonlinear model identification of airplanes using noisy flight data 114 D. List of figures and tables Figure 2.1 Block diagram of filter-error method Figure 2.2 Block diagram of output-error method Figure 2.3 Block diagram of Equation-error method Figure 2.4 Approximation of colored noise with band-limited white noise Figure 3.1 Pitch rate measured time series Figure 3.2 Pitch rate measured time series with endpoints discontinuities removed Figure 3.3 Smoothed pitch rate measured time series Figure 4.1 Measured input and output data sequences for lateral maneuver run1 Figure 4.2 Measured input and output data sequences for lateral maneuver run2 Figure 4.3 Output-error model fit to measured perturbation outputs about trim values for lateral maneuver run1 Figure 4.4 Model prediction for lateral maneuver run2 using re-estimated biases and estimated parameters applying data sequence run1 Figure 4.5 Output-error model fit to measured data using lateral maneuver run1 Figure 4.6 Output-error residuals for lateral maneuver, run1 Figure 4.7 Measured outputs for a longitudinal maneuver Figure 4.8 Output-error residuals for longitudinal maneuver Figure 5.1 Model fit to measured output in the frequency domain Figure 5.2 Measured input-output data sequences for a longitudinal maneuver, run1 Figure 5.3 Measured input-output data sequences for a longitudinal maneuver, run2 Figure 5.4 LOES model fit to measured data Picture 4.1 De Havilland DHC-6 Twin Otter Figure 6.1 De Havilland DHC-2 Beaver Figure 6.2 The ABC coordinate frame Figure 6.3 The Wind-axis coordinate frame Figure 6.4 The actuators Figure 6.5 Teaching data from a lateral maneuver (aileron, rudder) Figure 6.6 Teaching data from a lateral maneuver (roll and yaw rate) Figure 6.7 Teaching data from a lateral maneuver (Euler roll and sideslip angle) Figure 6.8 Teaching data from a lateral maneuver (acceleration along the y axis) Figure 6.9 Testing data from a lateral maneuver (aileron, rudder)
- 115. Nonlinear model identification of airplanes using noisy flight data 115 Figure 6.10 Testing data from a lateral maneuver (roll and yaw rate) Figure 6.11 Testing data from a lateral maneuver (Euler roll and sideslip angle) Figure 6.12 Testing data from a lateral maneuver (acceleration along the y axis) Figure 6.13 Teaching data from a longitudinal maneuver (angle of attack and pitch rate) Figure 6.14 Teaching data from a longitudinal maneuver (velocity and acceleration along the z axis) Figure 6.15 Teaching data from a longitudinal maneuver (elevator and pitch angle) Figure 6.16 Testing data from a longitudinal maneuver (angle of attack and pitch rate) Figure 6.17 Testing data from a longitudinal maneuver (velocity and acceleration along the z axis) Figure 6.18 Testing data from a longitudinal maneuver (elevator and pitch angle) Figure 6.19 Instrumentation error model Figure 6.20 Simulated noisy time series of the sideslip angle Figure 6.21 Smoothed time series of the sideslip angle Figure 7.1 Time domain lateral output-error model (red) fit measured data (blue) Figure 7.2 Time domain longitudinal output-error model (red) fit measured data (blue) Figure 8.1 Output-error model (red) fit measured data (blue) in the frequency domain Figure 9.1 Simulink model to perform a lateral test maneuver Figure 9.2 Simulink model to perform a longitudinal test maneuver Table 4.1 Parameter estimation results for lateral maneuver using data sequence run1 Table 4.2 Parameter estimation results for lateral maneuver using data sequence run1 Table 4.3 Parameter estimation results for longitudinal maneuver with corrected error bounds Table 5.1 Parameter estimation results for lateral maneuver using data sequence run1 Table 5.2 Tu-144L longitudinal short-period LOES model parameters Table 6.1 Description of the elements of the state and input vector Table 6.2 Applied trim values during the lateral maneuver Table 6.3 Applied trim values during the longitudinal maneuver Table 7.1 Parameter estimation results for YC Table 7.2 Parameter estimation results for nC Table 7.3 Parameter estimation results for lC
- 116. Nonlinear model identification of airplanes using noisy flight data 116 Table 7.4 Parameter estimation results for D LC ,C and mC Table 8.1 Parameter estimation results for YC Table 8.2 Parameter estimation results for nC Table 8.3 Parameter estimation results for lC
- 117. Nonlinear model identification of airplanes using noisy flight data 117 E. References [1] Vladislav Klein and Eugene A. Morelli, Aircraft System Identification: Theory and Practice, 1st ed.: AIAA, 2006 [2] Micheal Verhaegen and Vincent Verdult, Filtering and System Identification: A Least Squares Approach, Cambridge, 2007 [3] Micheal Verhaegen, Vincent Verdult and Niek Bergboer, Filtering and System Identification: An Introduction to using Matlab Software, TU Delft, 2007 [4] Brian L. Stevens, Frank L. Lewis, Aircraft Control and Simulation, John Wiley and Sons, 1992 [5] Hungarian Academy of Sciences, Computer and Automation Research Institute, Systems and Control Laboratory, Flight data sequences measured on UltraStick-25 [6] Bauer Péter, Háromüzemmódú kibıvített Kalman szőrık repülıgép orientációjának becslésére, 2010 Repüléstudományi közlemények [7] Marc Rauw, FDC 1.2 A Simulink Toolbox for Flight Dynamics and Control Analysis, 2001, [8] Béla Lantos, Lırinc Márton, Nonlinear Control of Vehicles and Robots, Springer, 2011 [10] Gerlai Tibor, Vízer Dániel, Repülıgép nemlineáris dinamikus modelljének identifikációja szenzorfúzióval, BME-IIT, TDK dolgozat, II. díj, 2010
- 118. Nonlinear model identification of airplanes using noisy flight data 118 [11] Vízer Dániel, Nonlinear dynamic model identification for aircraft applying time and frequency domain identification techinques, BME-IIT, TDK dolgozat, III. díj, 2011 [12] FöldesiTamás, Repülıgépek robusztus irányítási módszerei, Diplomamunka, BME-IIT, 2009