A LOCAL LEARNING APPROACH TO REAL-TIME

PARAMETER ESTIMATION

Application to an Aircraft

Lilian Ronceray, Matthieu Jeanneau

Stability and Control Department, Airbus France, Toulouse, France

Daniel Alazard

Philippe Mouyon

epartement Commande des Syst

emes et Dynamique du Vol, ONERA, Toulouse, France

Sihem Tebbani

epartement Automatique,

Ecole Sup

erieure d’

Electricit

e, Gif-sur-Yvette, France

Keywords:

Local learning, radial-basis neural networks, real-time parameter estimation.

Abstract:

This paper proposes an approach based upon local learning techniques and real-time parameter estimation, to

tune an aircraft sideslip estimator using radial-basis neural networks, during a ﬂight test. After a presentation

of the context, we recall the local model approach to radial-basis networks. The application to the estimation

of the sideslip angle of an aircraft, is then described and the various results and analyses are detailled at the

end before suggesting some improvement directions.

1 INTRODUCTION

In the aeronautical ﬁeld, although most of the useful

parameters (like inertial data, airspeed) are calculated

directly using probes, it is often relevant to use esti-

mators to consolidate the information, thus increasing

the redundancy of the aircraft systems or to replace

the probes in order to save weight. It then becomes

critical to have these estimators tuned in the early days

of the ﬂight tests of a new aircraft, when it is only

known by an inaccurate numerical model.

The problem that is dealt with in this paper, is that

of retuning a speciﬁc estimator in real-time (or near

real-time) using ﬂight test data. We must then take

into account that during ﬂight tests, the aircraft ﬂies

around in small regions of the ﬂight domain, yielding

a strong locality constraint on the retuning.

The general idea will be to make a combined use

of local learning and real-time parameter estimation

techniques to tune the estimator, in a small neighbour-

ing of its input space, without degrading its perfor-

mance in other regions.

2 ABOUT NEURAL NETWORKS

In this section, some generalities about RBF networks

are recalled.

2.1 General Description

Such a network is composed by a set of N local es-

timators {

(x)}

i=1

, deﬁned in the neighbouring of

some points

{

}

i=1

in an input space

I (Murray-

Smith, 1994).

The resulting global estimation

ξ is the weighted

sum of the outputs of all local estimators, for a query

point x ∈ I (see equation 1).

The weighting ϕ

of the local estimation

(x) is a

function of

x−c

for all i = 1. . . N. ϕ

is then consid-

ered as a radial-basis function. The resulting output is

then:

ξ =

∑

i=1

(x) (

399

Ronceray L., Jeanneau M., Alazard D., Mouyon P. and Tebbani S. (2007).

A LOCAL LEARNING APPROACH TO REAL-TIME PARAMETER ESTIMATION - Application to an Aircraft.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 399-403

DOI: 10.5220/0001635103990403

 SciTePress

2.2 Local Parameters Computation

The local models

(x) can either be constants in the

neighbouring of the centers, or afﬁne models in the

inputs, or any nonlinear model.

From now on, we will show how the parameters of

the local models are computed in the particular case

where these models are linear in the inputs (Haykin,

1999):

(x) = x

∀x ∈

The problem to solve is set as follows: given a

set of K different points

{

∈ R

}

k=1

and a cor-

responding set of K reference values to estimate

{

∈ R

}

k=1

, ﬁnd a function

ξ such that :

ξ(x

) = ξ

, ∀k = 1. . . K (2)

The RBF technique consists in choosing a func-

tion

ξ that has the following form :

ξ(x) =

∑

i=1

∑

i=1

(x)

(3)

= ψ(x)

θ (4)

where N is the chosen number of local estimators, and

with ψ

= ϕ

With Ψ =

{

}

{

)

}

, the interpolation

condition (2) can be written as a linear system :

θ = ξ (5)

In order to ﬁnd a solution to equation (5) and as

Ψ is not a square matrix, we must verify that ΨΨ

nonsingular, which can be done using Michelli’s the-

orem (Michelli, 1986). A solution

θ satisfying the

interpolation condition (2), can then be found using

least square optimization :

∗



ΨΨ



−1

Ψξ (6)

2.3 Selection of the Centers

A simple solution is to make a regular gridding on the

normalized input space. As minimum and maximum

variations of the input parameters are known, the in-

put space can be normalized. The gridding is then

made on a unitary hypercube.

The issue is that we face the curse of dimension-

ality though the dimension of each network’s input

space is smaller than 5. However, some physical con-

siderations, depending on the considered application,

may help reducing the number of neurons by making

a “truncated” hypercube.

For instance, in our application, some parameters

have a dependency in Mach number and angle of at-

tack α. As the aircraft is not designed to ﬂy at both

high Mach and high α, we may remove the corre-

sponding part of the domain.

3 APPLICATION

The application we considered here is the estimation

of the sideslip angle β of a civilian aircraft, which is

the angle between the aircraft longitudinal axis and

the direction of ﬂight (Russell, 1996).

To do so, we have a formula for the estimation

of β, based on equation (7) that describes the air-

craft lateral force equation where we neglect the lon-

gitudinal coupling terms and equation (8) which is a

classical decomposition of the lateral force coefﬁcient

(Boifﬁer, 1998):

mg·Ny

= P

SCy−F

eng,y

(7)

Cy = Cy

β+ ∆Cy

tas



r+ Cy



+∆Cy

δr

+ ∆Cy

δp

(8)

where Ny

denotes the lateral load factor, P

the

dynamic pressure, F

eng,y

the projection of thrust on

the lateral axis, β the sideslip angle, p the rolle rate,

r the yaw rate, δp the ailerons deﬂection, δr the rud-

der deﬂection, Cy

⋆

the Cy gradient w.r.t. ⋆ (β, p or r),

∆Cy

⋆

the Cy effect due to ⋆ (δp, δr or β), l the mean

aerodynamic chord andV

tas

the true airspeed velocity.

An approximation of the aircraft sideslip can then

be deduced :

β = −

−

∆Cy

δp

−

∆Cy

δr

−δ

∆Cy

−

tas





The key points treated in the sequel are the deﬁ-

nition of the architecture and the initialisation of the

neural networks from a given set of simulated data,

and the in-ﬂight tuning of these networks.

3.1 Rbf Networks Applied to Sideslip

Estimation

We will start by noticing that the expression (9) is lin-

ear in ratios of aerodynamic coefﬁcients.

In order to ease the reader’s effort, the following

notations are introduced. Let M denote the number of

unknown ratios of aerodynamic coefﬁcients,

the

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

400

m-th unknown ratio with m = 1. . . M and y

its at-

tached auxiliary measurement. (9) will then be writ-

ten as:

β =

∑

m=1

(9)

The m-th unknown ratio is modelled by a RBF net-

work with N

linear local estimators

i,m

(x). Its input

space

is a subset of the ﬂight enveloppe variables.

The

depends on the following variables : α,

Mach number, P

, δr and δp, the last two being only

used respectively for ∆Cy

δr

and ∆Cy

δp

According to equation (3), estimated sideslip can

then be rewritten as :

β =

∑

m=1

∑

m=1

= ζ

Θ (10)

A linear expression of the estimated sideslip is

thus obtained, allowing the recursive least algorithm

(RLS) algorithm to be directly applied. For a com-

plete formulation of the RLS algorithm and the re-

lated criterion, one may refer to (Labarr

ere et al.,

1993).

3.2 The Process in Details

The process will be divided into three main parts.

Initialization: the optimal network structure must

be found : RBF, distance function, feature scaling

(transformation on the input space), centers location,

and smoothing parameter σ

(Atkeson et al., 1997).

A direct ofﬂine learning: where each network is

trained individually on a database of aerodynamic co-

efﬁcient values, computed by the numerical model.

A particular attention will be paid to the norm of the

local parameter vector, which is an indicator of the

generalization performance of the network.

An indirect online learning: where the learning

criterion is no longer the error of the networks out-

puts but the error between the estimated sideslip and

the true sideslip. All the networks are trained at the

same time and must achieve both local performance,

i.e. on the considered ﬂight point and global perfor-

mance, i.e. on the whole ﬂight domain.

3.3 Analysis

The structure of the neural networks is a key element

in the outcome of the process and requires an analysis

of the networks’ ofﬂine performance, with respect to

the various degrees of freedom available for the net-

works’ structure.

For clarity reasons, we will only show the perfor-

mance of the (∆Cy

δr

) network in the sequel.

Prior to the analysis, the input space of the net-

work is normalized.The effect of the smoothing pa-

rameter σ on the learning performance and the Eu-

clidean norm of the parameter vector, which gives an

idea of the network’s capacity to generalize, will be

studied. Both the Euclidean and Inﬁnity norm will be

used as distance functions and inverse multiquadrics

as RBF (f : x 7→

√

). We perform an ofﬂine learn-

ing for each value of σ and compute the relative error

on the whole training data :

−2

−1

1000

3000

kΘk

−2

−1

Relative error in %

Figure 1: Inﬂuence of σ using

−2

−1

120

160

kΘk

−2

−1

11.6

12.4

12.8

Relative error in %

Figure 2: Inﬂuence of σ using

∞

The plain line represents the norm of the parame-

ter vector and the dashed line the relative error.

For both norms, we can see a behaviour that can

be compared with the overﬁtting phenomenon with

multilayer perceptrons (Haykin, 1999; Dreyfus et al.,

2004). We could name this “over-covering”, as it

seems that local models are strongly interfering with

each other, hence over-compensating their interac-

tion. σ must then be chosen that reaches a compro-

A LOCAL LEARNING APPROACH TO REAL-TIME PARAMETER ESTIMATION - Application to an Aircraft

401

mise between a decent performance and a relatively

small norm for the parameter vector.

In terms of compared performance, the Inﬁn-

ity norm allows better generalization, as the norm

reaches lower values for a quite similar performance.

This can be justiﬁed by ﬁgure 3: the covering of 2D

normalized space is presented using gaussian kernels

with Euclidean (solid line circles) and Inﬁnity norm

(dotted lines squares).

Figure 3: Input space covering using

and

∞

3.4 Implementation

We then chose the structure of our networks : linear

local models, inverse multiquadrics as RBF

, optimal

smoothing parameter according to the previous analy-

sis, normalisation of the input space as feature scaling

and a regular gridding to locate the centers.

To test our method, we used ﬂight tests recordings

to be as close as possible to real conditions.

3.5 Results

The obtained results are presented in this section,

from learning on the pre-ﬂight test identiﬁcation data

to the online tuning during simulated ﬂight tests (in-

ﬂight recorded data is fed through the estimator).

About the ofﬂine part, as it is basic least squares

optimization, we will only say that the points were

generated randomly using an inaccurate numerical

model of the aircraft.

For the online adaptation, we will present results

in clean conﬁguration for two distinct ﬂight points

(FP1 and FP2) on a steady sideslip maneuver.

First, estimations without and then with the RLS

algorithm are presented. The dot-dashed line rep-

resents the real sideslip at the center of gravity,

the dashed line the estimated sideslip computed by

They are better suited for an implementation on an em-

bedded computer

the current method and the solid line the estimated

sideslip computed by our estimator (ﬁgures 4 and 5).

The aim is to compare the performance of the ex-

isting sideslip estimator which uses interpolated ap-

proximate values for the aerodynamic coefﬁcients.

0 50 100 150

−7

−6

−5

−4

−3

−2

−1

time (s)

Sideslip (deg)

BETA

LBEST

BETAESTF

Figure 4: Flight point 1 - Fixed estimator.

0 50 100 150

−7

−6

−5

−4

−3

−2

−1

time (s)

Sideslip (deg)

BETA

LBEST

BETAESTF

Figure 5: Flight point 1 - RLS estimator.

The results are quite satisfactory because better

estimation than the existing estimator is achieved.

The noise we can see on both ﬁgures comes from the

Ny sensor and the derivation p and r.

The issue is the generalization and is emphasized

by the following procedure. The estimator is tuned on

FP1 then on FP2. Local performance is achieved for

both ﬂight points as shown in ﬁgure 5 for FP1. The

estimator is then veriﬁed on FP1. We can see on ﬁg-

ure 6 that the original performance has been degraded.

The learning algorithm does not tune locally enough

and impacts the whole ﬂight domain.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

402

0 50 100 150

−7

−6

−5

−4

−3

−2

−1

time (s)

Sideslip (deg)

BETA

LBEST

BETAESTF

Figure 6: Generalization from FP2 to FP1.

4 CONCLUSION

Throughout this paper, we studied the application of

RBF-based neural networks on the estimation of an

aircraft sideslip and tried to ﬁnd a method to tune it in

real-time during a simulated ﬂight test.

Though such networks have interesting local

properties, some improvements are required on the

different steps of the process, mainly on the gener-

alization performance. Work is currently on-going

about using total least squares algorithm instead of

the classical least squares (Huffel and Vandewalle,

1991; Bj

orck, 1996) and their recursive form (Boley

and Sutherland, 1993). Other work directions will be

investigated :

• allowing directional forgetting in the RLS algo-

rithm (Kulhavy and K

arny, 1984)

• reducing numerical complexity

• adaptive ﬁltering on the estimator output to soften

input noise effects

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