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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