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Submit to Transactions of JSASS
Parameter Estimation of Polytopic Models for a Linear ParameterVarying Aircraft System
Atsushi FUJIMORI�
and Lennart LJUNG†
�
Department of Mechanical Engineering, Shizuoka University3-5-1 Johoku, Hamamatsu 432-8561, JapanPhone & Fax: +81-53-478-1064, Email: tmafuji@ipc.shizuoka.ac.jp
† Department of Electrical Engineering, Linkoping UniversitySE-581 83, Linkoping, Sweden
Abstract
This paper presents a parameter estimation method of continuous-time polytopic models for
a linear parameter varying (LPV) aircraft system. The prediction error method for linear time
invariant (LTI) models is modified for polytopic models. The modified prediction error method
is applied to the parameter estimation of an LPV aircraft system whose varying parameter is
the flight velocity and estimation parameters are the stability and control derivatives (SCDs). In
an identification simulation, the SCDs of the initial polytopic model are adjusted so as to fit the
response of the model to the data obtained from the original LPV aircraft system. Moreover,
the polytopic model is more suitable for expressing the behavior of the LPV system than the
LTI model from the viewpoints of the time and the frequency responses.
Key Words: Polytopic model, System identification, Prediction error method, Aircraft
model
Mailing Address : Atsushi Fujimori, Dr.
Department of Mechanical Engineering, Shizuoka University
3-5-1 Johoku, Hamamatsu 432-8561, Japan
Phone & Fax : +81-53-478-1064
E-mail : tmafuji@ipc.shizuoka.ac.jp
1. Introduction
Linearized equations of aircraft are regarded as linear time invariant (LTI) systems if the al-
titude and the flight velocity are constant, but linear parameter varying (LPV) systems if they are
varying. Recently, a number of flight control designs in which the aircraft is treated as an LPV
system have been proposed by gain scheduling techniques1���
2�. In those gain scheduling de-
signs, the LPV system is expressed or approximated by a polytopic model which is constructed
by a linear combination of multiple LTI models at the vertices of the operating region1�. Then,
the constraints in the gain scheduling control design are expressed by linear matrix inequalities
(LMIs)3�. A gain scheduling controller is obtained by solving the LMIs numerically. Unfortu-
nately, in general it is not always possible to exactly transform an LPV system into a polytopic
model. It depends on the structure of the LPV system4�. One of the simplest ways for con-
structing a polytopic model is that multiple operating points are first chosen on the range of the
varying parameters, an LTI model is obtained at each operating point, and a polytopic model
is then constructed by interpolating between the operating points1�. However, the polytopic
model may contain an error in the interpolated region, although it depends on the interpolating
function. The error is one of causes of conservative controllers. Therefore, it is worthwhile to
obtain a polytopic model which is suitable for the original LPV system.
The aim of this paper is to obtain a desirable polytopic model for an LPV aircraft system.
Since an polytopic model is constructed by interpolating between the operating points as men-
tioned above, it may not suitably express the behavior of the original LPV system over the
entire operating region. In this paper, a desirable polytopic model is found by a system iden-
tification technique5�. The system identification technique used in this paper is the prediction
error method based on the quadratic criterion error in the time domain5�. This paper modifies
the standard prediction error method for LTI models to the parameter estimation of polytopic
models. The modified prediction error method is applied to the longitudinal linearized equation
of aircraft where the flight velocity is the varying parameter. The estimated polytopic model is
evaluated by the time response and the ν-gap metric, which is a criterion associated with the
1
model uncertainty6�.
2. Polytopic Model and Objective of Parameter Estimation
Let us consider a continuous-time LPV system given by
dx � t �dt � Ac � τ � ξ � τ ��� x � t ��� Bc � τ � ξ � τ ��� v � t �
y � t � � Cc � τ � ξ � τ ��� x � t ��� Dc � τ � ξ � τ ��� v � t ��(1)
where x � t � , v � t � and y � t � are respectively the nx-dimensional state, nv-dimensional input and
ny-dimensional output vectors. τ � t � is a measurable varying parameter with respect to time t,
but the argument t is usually suppressed. ξ � τ � is the p-dimensional estimation parameter vector
which is varying with respect to τ . In particular, when the matrices of Eq. (1) are written as the
following polytopic form:
Ac � τ � ξ � τ ��� �r
∑i 1
wi � τ � Ai � τi � ξi ��� Bc � τ � ξ � τ ��� �r
∑i 1
wi � τ � Bi � τi � ξi �
Cc � τ � ξ � τ ��� �r
∑i 1
wi � τ � Ci � τi � ξi ��� Dc � τ � ξ � τ � � �r
∑i 1
wi � τ � Di � τi � ξi ���(2)
Eq. (1) is called a polytopic model, where wi � τ ��� i � 1 ��������� r � are the weighting functions
satisfying the following relations.
wi � τ ��� 0 � � i (3a)r
∑i 1
wi � τ � � 1 (3b)
τi is a frozen varying parameter and is called the i-th operating point. Ai � τi � ξi � , Bi � τi � ξi � ,Ci � τi � ξi � and Di � τi � ξi � are constant matrices with including a constant estimation parameter
vector ξi at the i-th operating point. The set of � Ai � Bi � Ci � Di � is called the i-th local LTI model
in this paper. The polytopic form (2) is constructed by interpolating between r operating points
with the weighting functions satisfying Eqs. (3a) and (3b). When r � 1, w1 � τ � � 1; that is, the
polytopic model becomes an LTI model. This paper considers the case where r � 2. Ai � τi � ξi �preserves the same structure as Ac � τ � ξ � with respect to the estimation parameters. The same
2
holds for Bi � τi � ξi � , Ci � τi � ξi � and Di � τi � ξi � . The element of ξi is denoted as
ξi
��
�ξ1
�i ����� ξp
�i � T � (4)
The polytopic model is one of blended multiple models7� �
8�
in which the varying parameter
depends on the input and/or the state. In the polytopic model considered in this paper, the
varying parameter τ is independent of these, but depends on time t.
As pointed out in Section 1, in general it is not always possible to exactly transform an LPV
system into a polytopic model. It depends on the structure of the LPV system. One of methods
for constructing a polytopic model is that r frozen τi � i � 1 ����� ��� r � ; that is, the operating points
are chosen on the range of τ . A local LTI model is obtained at each operating point. A polytopic
model is then constructed by interpolating between the operating points. However, a model error
may be included in the interpolated region.
There are options for constructing the polytopic form (2) associated with the operating
points and the weighting functions. In the former, when the number of the operating points
is increased, the model error is decreased9�, but the polytopic model is complicated. In the lat-
ter, there are a number of candidates for the weighting functions satisfying Eqs. (3a) and (3b).
One of the simplest weight functions is a triangular function whose center is at the operating
point as shown in Fig. 1. Other weighting functions are introduced in Ref. 10). This paper does
not discuss the selection of the weighting functions, furthermore. Anyway, it is assumed that the
number of the operating points is determined and the weighting functions w i � τ � � i � 1 ��������� r �are given in advance.
The objective of this paper is summarized as follows; using the input- and the output-data
measured from the original LPV system � v � t �� y � t ��� and the varying parameter τ � t � , estimate p
times r parameters ξl�i � l � 1 ������� � p � i � 1 ������� � r � in the polytopic form (2) so that the output
of the model denoted by y � t � is fitted to the output-data y � t � as close as possible. To avoid
over-parameterized estimation11�, the size of ξ is restricted as
p ����� nx � nv � ny ��� nvny � Dc � 0 �
nx � nv � ny � � Dc � 0 �(5)
3
where ny is the number of the output elements which are independent in the sense of the linear
operator. A simple method for estimating the parameters is that the parameter estimation is done
at each local LTI model by using data which are obtained at each operating point. However,
when τ � t � is varied with time t, there is no guarantee that the response of the constructed
polytopic model is fitted to that of the LPV system, especially in the intermediate region because
the interpolation of Eq. (2) may be an approximated expression of the original LPV system.
For this situation, the model error contained in the polytopic model should be made as small
as possible. This is a motivation why p times r estimation parameters in the polytopic form
(2) have to be estimated at the same time. A similar parameter estimation for blended multiple
models in the discrete-time is discussed in Ref. 12). This paper shows the estimation of which
parameters are related to physical systems in the continuous-time.
3. Prediction Error Method for Polytopic Model
This section shows the estimation computation in which the prediction error method for LTI
models is modified for the case of polytopic models. Compared to the case of LTI models,
there are two novelties in the case of polytopic models: the first is that the number of the
estimation parameters is proportional to the number of chosen operating points. The second is
an assumption on the discretization of the predictor and the gradient. In fact, both of them are
caused to increase the computational burden on the parameter estimation.
3.1. Predictor
A predictor of a polytopic model Eq. (1) with Eq. (2) is given by
dx � t � ξ �dt � Ac � τ � ξ � x � t � ξ ��� Bc � τ � ξ � v � t � � Kc � τ � ξ � � y � t ��� y � t � ξ ���
y � t � ξ � � Cc � τ � ξ � x � t � ξ � � Dc � τ � ξ � v � t ���(6)
4
where Kc � τ � ξ � is a filter gain which is given so that Ac � τ � ξ � � Kc � τ � ξ � Cc � τ � ξ � becomes a
stable matrix. Using Eq. (2) and giving Kc � τ � ξ � by the following polytopic form:
Kc � τ � ξ � �r
∑i 1
wi � τ � Ki � τi � ξi � � (7)
Eq. (6) is then written as
dx � t � ξ �dt � Fc � τ � ξ � x � t � ξ ��� Gc � τ � ξ � z � t �
y � t � ξ � � Cc � τ � ξ � x � t � ξ � � Hc � τ � ξ � z � t ���(8)
where
z � t ���
�yT � t � vT � t � � T
Fc � τ � ξ ���
r
∑i 1
wi � τ ��Ai � τi � ξi ��� Ki � τi � ξi �
r
∑j 1
w j � τ � C j � τ j � ξ j ���Gc � τ � ξ �
��
�r
∑i 1
wi � τ � Ki � τi � ξi �r
∑i 1
wi � τ ��Bi � τi � ξi � � Ki � τi � ξi �
r
∑j 1
w j � τ � D j � τ j � ξ j �����Hc � τ � ξ �
��
�0
r
∑i 1
wi � τ � Di � τi � ξi ���3.2. Discretization
When the data z � t � used for estimation are sampled by a constant time period T , a discrete
representation of Eq. (8) is derived under an assumption that τ � t � is frozen for each sampling
interval;
τ � t � � τk � � const � � � kT � t � kT � T � � (9)
Then, Fc � τ � ξ � , Gc � τ � ξ � , Cc � τ � ξ � and Hc � τ � ξ � are constant for a fixed ξ . Applying the zero
order hold discretization to Eq. (8), the following piecewise discrete-time predictor is obtained.
x � kT � T � ξ � � F � τk � ξ � x � kT � ξ � � G � τk � ξ � z � kT �
y � kT � ξ � � C � τk � ξ � x � kT � ξ � � H � τk � ξ � z � kT �(10)
� kT � t � kT � T ���
5
where
F � τk � ξ ��� eFc � τk
�ξ�T
G � τk � ξ ���
� T
0eFc � τk
�ξ�sdsGc � τk � ξ �
C � τk � ξ ��� Cc � τk � ξ ��� H � τk � ξ �
�� Hc � τk � ξ � �
For convenience, hereafter, the notation ‘kT ’ in x, y and z is replaced by ‘t’ and τk is written as
τt in the discrete-time representation.
3.3. Quadratic function and Gauss-Newton method
Let N be the number of the sampled data. The data set ZN is defined as
ZN��
�zT � 1 � ����� zT � N � � � (11)
The function to be minimized is given by the following a quadratic function
JN � ξ � ZN ���
1N
N
∑t 1
12
e2 � t � ξ ��� (12)
where e � t � ξ � is the prediction error vector defined as
e � t � ξ ��� y � t � � y � t � ξ � � (13)
Then, the estimated parameter is obtained as
ξ � argminξ
JN � ξ � ZN � � (14)
Since the estimation parameter ξ is implicitly included in the prediction error e � t � ξ � , JN can not
be explicitly expressed with respect to ξ . The minimization of JN is then done numerically by an
iterative calculation. In this paper, the Gauss-Newton method is used to search a ξ minimizing
JN . For convenience, let the estimation parameter vector of the polytopic model be collectively
denoted as
�
�ξ T
1 ����� ξ Tr � T � (15)
6
Let the superscript � k � be the k-th iteration. The estimation parameter vector is updated by
ξ � k � 1�
� ξ � k �� µ � k � �
H � k �N � � 1J
�N � ξ � k � � ZN � (16)
where
J�N � ξ � k � � ZN � � �
1N
N
∑t 1
ψ � t � ξ � k � � e � t � ξ � k � � (17)
H � k �N �
1N
N
∑t 1
ψ � t � ξ � k � � ψT � t � ξ � k � � (18)
ψ � t � ξ � k � ���
∂ y � t � ξ � k � �∂ ξ
(19)
H � k �N
is called the Hessian and is usually invertible. µ � k � is the step size.
3.4. Gradient
Differentiating the piecewise discrete-time predictor (10) with respect to ξl�i, the gradient
of the prediction error ψ � t � ξ � is obtained as the following piecewise discrete-time state-space
representation.
∂ x � t � 1 � ξ �∂ξl
�i � F � τt � ξ � ∂ x � t � ξ �
∂ξl�i
��∂F � τt � ξ �
∂ξl�i
∂G � τt � ξ �∂ξl
�i
� ��x � t � ξ �z � t � ξ �
��
ψ � t � ξ � � C � τt � ξ � ∂ x � t � ξ �∂ξl
�i
��∂C � τt � ξ �
∂ξl�i
∂H � τt � ξ �∂ξl
�i
� ��x � t � ξ �z � t � ξ �
�� (20)
The derivatives of F � τt � ξ � , G � τt � ξ � , C � τt � ξ � and H � τt � ξ � are numerically obtained as
∂F � τt � ξ �∂ξl
�i �
F � τt � ξ � δl�i ��� F � τt � ξ � δl
�i �
2δl�i
(21)
where δl�i is a small positive value.
Before closing this section, some comments on the estimation computation will be given.
The procedure of the computation described in this paper is essentially the same as that for LTI
models except the following two points: one is that the number of the estimation parameters
is proportional to the number of the operating points. Another is that the discretization for
7
the predictor and the gradient calculation; that is, Eqs. (10) and (20) has to be done at every
sampling because they are piecewise discrete-time LTI representations.
To estimate the parameters accurately, all local LTI models have to be excited by the input
v � t � and the varying parameter τ � t � . In particular, τ � t � should be varied over the entire range.
This means that all the weighting functions wi � τ � � i � 1 ��������� r � may be not constant during
measuring the data. Otherwise, the Hessian H � k �N
in Eq. (16) may be singular, so the iterative
calculation is stopped. In this situation, some local LTI models are not needed in the polytopic
form (2). Let us consider a simple scenario that w1 � τ � is given by a constant. If w1 � τ � � 0,
the 1st local LTI model � A1 � B1 � C1 � D1 � is not needed because of no contribution to Eq. (2). If
w1 � τ � � 1, the rest of local LTI models are not needed. Moreover, if w1 � τ � is a constant in the
range of 0 � w1 � τ � � 1, Eq. (3b) is written as
r
∑i 2
wi
1 � w1 � 1 � (22)
Replacing wi by wi
�� wi � � 1 � w1 � , Eqs. (3a) and (3b) are satisfied. Then, the polytopic model
is constructed by r � 1 local LTI models.
4. Identification Simulation of LPV Aircraft System
This section presents an identification simulation of an LPV aircraft system. The estimated
parameters in the linearized aircraft equations are the stability and control derivatives (SCDs)
which express linear contributions of the perturbed velocities and the angular rates to the aero-
dynamic forces or moments. The SCDs are varied according to the flight conditions; that is,
the flight velocity and the altitude. This section demonstrates the identification simulation of an
LPV aircraft system when the flight condition is varying.
An LPV aircraft system for the longitudinal motion and a polytopic model are first shown
in the following subsection. The estimation of the SCDs in the LTI and the polytopic models
is shown in the next. The estimated models are evaluated by the time response and the ν-gap
metric6�
which is one of measures of model error.
8
4.1. LPV aircraft system for longitudinal motion
In steady flight, the dynamics of aircraft can be generally divided into two parts; the lon-
gitudinal and the lateral motions. This paper considers the identification of the longitudinal
motion which is regarded as an LPV system. The longitudinal motion in the continuous-time is
expressed as the following linearized equations:
dudt
� Xuu � Xαα � gcosΘ0θ � 0
� Zuu � Vdαdt
� Zαα � � V � Zq � q � gsinΘ0θ � Zδeδe
� Muu � Mαdαdt
� Mαα � dqdt
� Mqq � Mδeδe
dθdt � q
(23)
u is the x-axis velocity, α the angle of attack, θ the pitch angle, q the pitch rate and δe the
elevator angle. The notations used in Eq. (23) are based on the symbols which have been
usually used in flight dynamics13�. The variables denoted by small letters mean the perturbed
values. Θ0 is the pitch angle in the steady-state. g is the acceleration of gravity. In Eq. (23),
there are nine stability derivatives; Xu, Xα , Zu, Zα , Zq, Mu, Mα , Mα and Mq, and two control
derivatives; Zδeand Mδe
, which are varied with the flight velocity V and the altitude H. Since
V is more considerably influenced on the characteristics of Eq. (23) rather than H, the varying
parameter considered in this paper is τ � V and its range is given by
V1 � V � V2 � V1 � V2 � � (24)
Defining x � t � , y � t � and v � t � as
x � t ���
������
�u
θ
α
q
������
� � y � t ���
���
� u
θ
α
���
�� v � t �
�� δe � (25)
a continuous-time LPV system (1) for the longitudinal motion of aircraft is then written as
dx � t �dt � Ac � V � ξ � V ��� x � t � � Bc � V � ξ � V ��� v � t �
y � t � � Ccxc � t ��� Dcv � t � �(26)
9
where matrices in Eq. (26) are given by
Ac � E� 1c Qc � Bc � E
� 1c Rc
Ec
��
������
�1 0 0 0
0 1 0 0
0 0 V 0
0 0 � Mα 1
������
� � Qc
��
������
�Xu � gcosΘ0 Xα 0
0 0 0 1
Zu � gsinΘ0 Zα V � Zq
� Mu 0 Mα Mq
������
� �
Rc
��
������
�0
0
Zδe
Mδe
������
� � Cc �
���
� 1 0 0 0
0 1 0 0
0 0 1 0
���
�� Dc �
���
� 0
0
0
���
� �(27)
Collecting the SCDs, the estimation parameter vector in the LPV system is given by
ξ � V � ��Xu Xα Zu Zα Zq Mu Mα Mα Mq Zδe
Mδe � T � (28)
The elements of y � t � are selected to avoid the over-parameterized estimation11�.
A polytopic model of the longitudinal motion of aircraft is constructed as follows: two
operating points are chosen at the edges of Eq. (24); that is, V � V1 � V2 and two local LTI
models are obtained. Using linear interpolation as shown in Fig. 1, Ac and Bc of the polytopic
model are then constructed as
Ac � V � ξ � �2
∑i 1
wi � V � Ai � Vi � ξi ��� Bc � V � ξ � �2
∑i 1
wi � V � Bi � Vi � ξi � (29)
where
w1 � V ���
V2 � V
V2 � V1� w2 � V �
��
V � V1
V2 � V1(30)
The number of the estimation parameters is then 11 � 2 � 22.
4.2. Data for parameter estimation
In the identification simulation, the flight velocity was changed in the range of Eq. (24).
As an example, it was considered a situation that the flight velocity V � t � in the continuous-time
was constantly accelerated as
V � t � � V1 � avt (31)
10
where av was the acceleration. The input was given by a random binary signal. Using the
flight velocity and the random input, the output-data were generated by the LPV aircraft system
(26) which was converted to the discrete-time representation at each sampling as shown in
discretization of the predictor in Section 3.2. The SCDs of the LPV aircraft system at each
sampling were obtained by an analytical method based on the quasi-steady aerodynamic theory.
They were given by13�
Xu �ρV S2m
� Cxu � 2CL tanΘ0 ��� Zu �ρVS2m
� Czu � 2CL �
Mu �ρVSc2Iyy
Cmu � Xα �ρV 2S
2mCxα
Zα �ρV 2S
2mCzα � Mα �
ρV 2Sc2Iyy
Cmα � Mα �ρVSc2
4IyyCmα
Zq �ρV Sc
4mCzq � Mq �
ρV Sc2
4IyyCmq
Zδe �ρV 2S
2mCzδe
� Mδe �ρV 2Sc
2mCmδe
�
(32)
where m was the mass of aircraft, S the main wing area, c the main wing chord, and b the
main wing span. CL was the lift coefficient. Cxu, Cmα , etc. were the non-dimensional stability
and control derivatives and were obtained from the structural parameters of the aircraft13�. The
numerical values were referred from Ref. 14). The change of the altitude of aircraft due to the
response was taken into consideration in the calculation of the SCDs.
The number of the data was N � 100. The sampling time was given by T � 0 � 5 sec. The
acceleration of the flight velocity was given by av � 2 m/s2. The initial state was given by
xc � 0 � � 0. Using the flight velocity and the data explained above, The estimation of the SCDs
was done in the cases of the LTI and the polytopic models.
4.3. Parameter estimation results
(1) LTI model case Table 1 shows the initial and the estimated SCDs in the case of LTI
model, where the initial SCDs were given by Eq. (32) where the flight velocity was V � 110
m/s. The estimated SCDs were changed from their initial values. Figure 2 shows the outputs
of the initial and the estimated LTI models. The solid- and the dashed-dotted-lines mean the
11
output-data and the outputs of the models, respectively. The responses of both LTI models were
not well fitted to the data.
The root mean square (RMS) of the prediction error for each output channel is used to
evaluate the outputs of the models quantitatively. The RMS indicates the averaged amplitude
of the prediction error. Table 2 shows the RMS of the prediction error for each output channel
of the initial and the estimated LTI models, where eu, eα and eθ are the prediction errors of
x-axis velocity, the angle of attack and the pitch angle, respectively. Although the estimated LTI
model showed smaller values of the RMS than the initial LTI model, it was not enough to be
acceptable.
The ν-gap metric is one of criteria measuring the model error in the frequency domain.
It had been introduced in robust control theories associated with the stability margin6�. Let
Pl pv � s � V � be the transfer function of the LPV system where the varying parameter is V . Let
Plti � s � be that of the initial or the estimated LTI model. The ν-gap metric between Pl pv � s � V �and Plti � s � is defined as
δν � Pl pv � Plti ��� sup
ωκ � Pl pv � jω � V ��� Plti � jω ��� (33)
where
κ � X � Y ��� σ
�� I � YY
� � 1 � 2 � Y � X � � I � XX� � 1 � 2 �
where σ� � � means the maximum singular value. The range is δν � �
0 � 1� . A large δν means
that the model error is large. Figure 3 shows the plots of δν � Pl pv � Plti � , where the solid- and the
dashed-dotted-lines indicate that Plti � s � is the estimated and the initial LTI models, respectively.
Since the initial SCDs were given at V � 110 m/s, δν � Pl pv � Plti � whose Plti � s � was the initial
LTI model was zero at V � 110 m/s. However, it was increased when V was shifted from
V � 110 m/s. On the other hand, the minimum of δν � Pl pv � Plti � whose Plti � s � was the estimated
LTI model was moved to V � 93 m/s. Similar to the initial LTI model, it was increased in other
flight condition. It was seen that the LTI model was not enough to express the characteristics of
the original LPV system in the time and the frequency domains.
12
(2) Polytopic model case The result of estimation of the polytopic model is shown in
next. Table 3 shows the initial and the estimated SCDs of the polytopic models, where the
initial SCDs were given as the values at V � V1, numbered by #1, and V2, numbered by #2,
by using Eq. (32). The a polytopic model using the initial SCDs has been often used in gain
scheduling control design3���
9�. The estimated SCDs were moved from the initial values to the
inside of the range (24). Figure 4 shows the outputs of the initial and the estimated polytopic
models. The response of the estimated polytopic model was better fitted to the output-data than
that of the initial polytopic model. It was also seen in the RMS as shown in Table 4.
Letting Ppoly � s � V � be the transfer function of the initial or the estimated polytopic model,
Fig. 5 shows the plots of δν � Pl pv � Ppoly � , where the solid- and the dashed-dotted-lines indicate
that Ppoly � s � V � is the estimated and the initial LTI models, respectively. δν � Pl pv � Ppoly � whose
Ppoly � s � V � was the estimated polytopic model was smaller than that whose Ppoly � s � V � was the
initial polytopic model except near both edges of the flight region.
Summarizing the identification simulation which has been shown so far, the polytopic model
was more suitable for expressing the behavior of the LPV aircraft system than the LTI model
from the viewpoints of the time and the frequency responses. Applying the prediction error
method to the polytopic model, the parameters of the polytopic model were adjusted so as to fit
the response of the model to that of the original LPV system and make the model error small
over the entire flight region.
(3) Response for other input-data and varying parameter The estimated polytopic
models was evaluated by using other input-data and the varying parameter. Table 5 shows the
RMS of the prediction error in which different random binary signals were used as the input-
data and the varying parameter; that is, the flight velocity V � t � was given by the following three
types: increase, decrease and sinusoidal as shown in Fig. 6. The RMS of the prediction error
was not constant for each trial since the input-data were given by different random binary signals
at each trial. Therefore, the values of the RMS in Table 5 are the ten-trial-average for each type
of V � t � . Compared to the prediction error by using the estimation input-data, the errors were
13
increased by using different input-data and different types of V � t � . In particular, the change of
the varying parameter to “Decreased” and “Sinusoidal” was influenced on the prediction error.
Table 6 shows the reduction ratio of the RMS of the prediction error between the estimated
and the initial polytopic models for each output channel. The reduction ratio for the x-axis
velocity, for example, is defined as
rd � eu ���
RMS � eu � estimated
RMS � eu � initial� 100 � % � � (34)
“Original” means the reduction ratio when the estimation input-data was used. Except the RMS
of eu in the sinusoidal case, the prediction error of the estimated polytopic model was reduced
compared to the initial model. This means that the parameter estimation of polytopic models is
effective in improving the quality of the model.
5. Concluding Remarks
This paper has the presented a parameter estimation method of continuous-time polytopic
models for an LPV aircraft system. The prediction error method for LTI models was modi-
fied for polytopic models. The modified prediction error method was applied to the parame-
ter estimation of an LPV aircraft system whose varying parameter was the flight velocity and
estimation parameters were the stability and control derivatives (SCDs). In an identification
simulation, the SCDs of the initial polytopic model were adjusted so as to fit the response of
the model to the data obtained from the LPV aircraft system. The polytopic model was more
suitable for expressing the behavior of the LPV system than the LTI model from the viewpoints
of the time response (prediction error) and the frequency response (ν-gap metric).
A polytopic model is used as an approximated representation for nonlinear systems in which
the reference trajectory is given in advance9���
15�. The presented technique is applicable for
adjusting the models of such nonlinear systems. In this paper, the weighting functions were
assumed to be given in advance. Including the parameters of the weighting functions into the
estimation parameters, the quality of the estimated polytopic model will be improved. This is
14
also a future subject of research.
References
1) Apkarian, P. Gahinet, P. and Becker, G.: Self-Scheduled � ∞ Control of Linear Parameter-
varying Systems: a Design Example, Automatica, 31 (1995), pp. 1251-1261.
2) Apkarian, P. and Adams, R. J.: Advanced Gain-Scheduling Techniques for Uncertain
Systems, Advances in Linear Matrix Inequality Methods in Control, SIAM (2000), pp.
209-248.
3) Boyd, S., Ghaoui, L. E., Feron, E. and Balakrishnan, V.: Linear Matrix Inequalities in
System and Control Theory, SIAM, Vol. 15, 1994.
4) Fujimori, A.: Descriptor Polytopic Model of Aircraft and Gain Scheduling State Feed-
back Control, Transactions on the Japan Society for Aeronautical and Space Sciences, 47
(2004), pp. 138-145.
5) Ljung, L.: System Identification - Theory for the User, 2nd edn, Prentice Hall, Upper
Saddle River, NJ., 1999.
6) Vinnicombe, G.: Uncertainty and Feedback ( � ∞ Loop-Shaping and the ν-Gap Metric),
Imperial College Press, 2001.
7) Leith, D. J., and Leithead, W. E.: Analytic Framework for Blended Multiple Model Sys-
tems Using Linear Local Models, Int. J. Control, 75 (1999), pp. 605-619.
8) Shorten, R.,Murray-Smith, R. Bjorgan, R. and Gollee, H.: On the Interpretation of Local
Models in Blended Multiple Model Structures, Int. J. Control, 75 (1999), pp. 620-628.
9) Fujimori, A., Terui, F. and Nikiforuk, P. N.: Flight Control Designs Using ν-Gap Met-
ric and Local Multi-Objective Gain-Scheduling, AIAA Paper, 2003-5414-CP, Guidance,
Navigation and Control Conference, 2003, pp. 1729-1745.
15
10) Boukhris, A.,Mourot, G. and Ragot, J.: Non-Linear Dynamic System Identification: a
Multi-Model Approach, Int. J. Control, 75 (1999), pp. 591-604.
11) McKelvey, T. and Helmersson, A.: System Identification Using an Over-Parameterized
Model Class - Improving the Optimization Algorithm, Proc. 35th IEEE Conference on
Decision and Control, 1997, pp. 2984-2989
12) Verdult, V.,Ljung, L. and Verhaegen, M.: Identification of Composite Local Linear State-
Space Models Using a Projected Gradient Search, Int. J. Control, 75 (2002), pp. 1385-
1398.
13) Schmidt, L. V.: Introduction to Aircraft Flight Dynamics, AIAA, Reston, 1998.
14) Isozaki, K., Masuda, K., Taniuchi, A. and Watari, M.: Flight test Evaluation of Variable
Stability Airplane, K.H.I. Technical Review, 75 (1980), pp. 50-58 (in Japanese).
15) Fujimori, A., Gunnarsson, S. and Norrlof, M.: A Gain Scheduling Control of Nonlinear
Systems Along a Reference Trajectory, Proc. of 16th IFAC World Congress.
16
List of Figures
1 Triangular interpolative function. . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Comparison between output-data and outputs of initial and estimated LTI models. 19
3 ν-gap metric between Pl pv � s � V � and Plti � s � in flight region, 50 � V � t � � 150 m/s. 20
4 Comparison between output-data and outputs of initial and estimated polytopic
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 ν-gap metric between Pl pv � s � V � and Ppoly � s � V � in flight region, 50 � V � t � � 150
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Three types of flight velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
List of Tables
1 Initial and estimated SCDs in case of LTI model. . . . . . . . . . . . . . . . . 23
2 RMS of prediction error for each output channel of initial and estimated LTI
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Initial and estimated SCDs in case of polytopic model. . . . . . . . . . . . . . 23
4 RMS of prediction error for each output channel of initial and estimated poly-
topic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Ten-trial-average of RMS of prediction error of estimated polytopic model for
different input and varying parameter. . . . . . . . . . . . . . . . . . . . . . . 24
6 Reduction ratio of RMS of prediction error between estimated and initial poly-
topic models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
17
τ i τ i+1τi -10
1wi-1 wi +1i
τ... ...
w
Fig. 1. Triangular interpolative function.
18
0 25 50-20
-10
0
10
u [m/s]
t [s]
Data vs Initial LTI model
0 25 50-20
0
20
θ [deg]
t [s]
0 25 50-5
0
5
α [deg]
t [s]
data
initial model
(a) Initial LTI model.
0 25 50-20
-10
0
10
u [m/s]
t [s]
Data vs Estimated LTI model
0 25 50-20
0
20
θ [deg]
t [s]
0 25 50-5
0
5
α [deg]
t [s]
data
estimated model
(b) Estimated LTI model.
Fig. 2. Comparison between output-data and outputs of initial and estimated LTI mod-els.
19
50 75 100 125 1500
0.2
0.4
0.6
0.8
δ v
ν-gap metric bwtween Plpv and Plti
V [m/s]
initial Plti
estimated Plti
Fig. 3. ν-gap metric between Pl pv � s � V � and Plti � s � in flight region, 50 � V � t � � 150 m/s.
20
0 25 50-20
-10
0
10
u [m/s]
t [s]
Data vs Initial polytopic model
0 25 50-20
0
20
θ [deg]
t [s]
0 25 50-5
0
5
α [deg]
t [s]
data
initial model
(a) Initial polytopic model.
0 25 50-20
-10
0
10
u [m/s]
t [s]
Data vs Estimated polytopic model
0 25 50-20
0
20
θ [deg]
t [s]
0 25 50-5
0
5
α [deg]
t [s]
data
estimated model
(b) Estimated polytopic model.
Fig. 4. Comparison between output-data and outputs of initial and estimated polytopicmodels.
21
50 75 100 125 1500
0.1
0.2
0.3
0.4
δ v
ν-gap metric bwtween Plpv and Ppoly
V [m/s]
initial Ppoly
estimated Ppoly
Fig. 5. ν-gap metric between Pl pv � s � V � and Ppoly � s � V � in flight region, 50 � V � t � � 150m/s.
0 25 500
50
100
150
200
V [m/s]
t [s]
Increased
0 25 500
50
100
150
200
V [m/s]
t [s]
Decreased
0 25 500
50
100
150
200
V [m/s]
t [s]
Sinusoidal
Fig. 6. Three types of flight velocity.
22
Table 1. Initial and estimated SCDs in case of LTI model.
Model Xu Xα Zu Zα Zq
Initial -0.0237 7.8829 -0.2423 -88.877 -1.6869
Estimated -0.0452 4.4446 -0.2469 -85.799 -19.583
Mu Mα Mq Mα ZδeMδe
0 -5.2892 -1.0613 -0.3274 -5.9923 -3.7699
-0.0743 -32.227 27.776 -35.420 -21.065 -9.6394
Table 2. RMS of prediction error for each output channel of initial and estimated LTImodels.
Model eu m/s eθ deg eα deg
Initial 1.7100 1.7706 0.9801
Estimated 1.1029 1.3528 0.9319
Table 3. Initial and estimated SCDs in case of polytopic model.
Model Xu Xα Zu Zα Zq
Initial #1 -0.0108 1.6287 -0.1101 -18.363 -0.7668
Initial #2 -0.0324 14.658 -0.3303 -165.27 -2.3003
Estimated #1 -0.0148 0.9756 -0.1201 -17.392 -0.9337
Estimated #2 -0.0298 12.609 -0.3065 -161.54 -1.5464
Mu Mα Mq Mα ZδeMδe
0 -1.0928 -0.4824 -0.1488 -1.2381 -77.890
0 -9.8353 -1.4472 -0.4465 -11.143 -7.0101
0.0047 0.2142 -3.0443 2.2265 -1.0250 -0.5593
0.0013 -7.9295 -1.8485 0.1167 -11.374 -5.9544
23
Table 4. RMS of prediction error for each output channel of initial and estimated poly-topic models.
Model eu m/s eθ deg eα deg
Initial 0.4115 0.6847 0.4228
Estimated 0.0108 0.1255 0.1086
Table 5. Ten-trial-average of RMS of prediction error of estimated polytopic model fordifferent input and varying parameter.
Type of V � t � eu m/s eθ deg eα deg
Increased 0.1442 0.2276 0.1618
Decreased 0.3457 0.5398 0.2971
Sinusoidal 1.1852 0.9987 0.5009
Table 6. Reduction ratio of RMS of prediction error between estimated and initial poly-topic models.
Type of V � t � rd(eu) % rd(eθ ) % rd(eα ) %
Original 2.6315 18.329 25.697
Increased 49.019 36.827 33.171
Decreased 74.187 66.591 55.354
Sinusoidal 107.808 69.427 72.059
24