SOLUTION OF AN INVERSE PROBLEM BY CORRECTION OF
TABULAR FUNCTION FOR MODELS OF NONLINEAR DYNAMIC
SYSTEMS
∗
I. A. Bogulavsky
State Institute of Aviation Systems, Moskow Physical - Technical Institute, Moscow, Russia
Keywords:
Polynomial approximation method, Uniformly small estimate errors, Mean-square optimal estimate.
Abstract:
In this paper, we present a solution to the problem of correction of parameterized tabular nominal functions
for the motion equations in a model of a nonlinear dynamical system using observations in discrete time.
The correction vector is determined by the mean of the multi-polynomial approximation algorithm (MPA-
algorithm) using observations of the noise functions of the components of the state vectors. The method
of correction of tabular functions is demonstrated by correcting 204 parameters in an example involving a
mathematical model of the motion of an F-16 aircraft.
1 INTRODUCTION
In this paper, an algorithm is presented for the a nu-
merical process of correction of a nominal mathemat-
ical model of a nonlinear dynamical system based on
experimental data. The problem of estimation of the
vectors of mathematical model parameters (the tra-
ditional problem of identification of the constant un-
knownparameters) has been considered in many prior
works (see, for example, Klein and Morelli, 2006,
Cappe et al., 2005, Gordon et al., 1993, Doucet et al.,
2000, Doucet et al., 2001, Ristic et al., 2004, Gosh
et al., 2008, Namdeo et Manohar, 2007, Cotter et al.,
2009, Boguslavskiy, 1996, Boguslavskiy, 2006, Bo-
guslavskiy, 2008 and Boguslavskiy, 2009).
The statement of the problem we consider here
differs from the traditional problem in that the nomi-
nal model –the model before the correction– contains
several nominal (previously obtained) tabular func-
tions of the components of the state vectors. The
problem of identifying the correction vectors is as fol-
lows: it is necessary to construct an algorithm to cor-
rect the tabular functions by processing the observa-
tion data . To do this it is necessary to identify param-
eters that are not constant and depend on a flowing
state vector of a dynamic system. These parameters
are hidden (sleeping); they do not influence the evo-
lution of system if the current state vector has not vis-
ited the corresponding areas of the phase space.
∗
This work has been supported by the Russian Founda-
tion for Basic Research.
This singularity distinguishes our problem from
traditional problems in which the evolution of the
state vector does not influence the constant unknown
parameters. The task differs from the task set in (Cot-
ter et al., 2009), where the Bayesian approach is used
to estimate a function of time that belongs to the
mathematical model of a dynamical system.
An example of a model with a tabular function
is the mathematical model of an aircraft with aero-
dynamic characteristics, i.e., the dimensionless coef-
ficients of aerodynamic forces and moments (Klein
and Morelli, 2006), given from tables of functions of
components of state vectors (e.g., angles of attack and
sliding ) and components of the vector of control(e.g.,
angles of deviation of steering surfaces).
Modern computational methods and wind tunnel
testing can provide, in many instances, comprehen-
sive data about the nominal aerodynamic characteris-
tics of the aircraft; these comprise the parameters of
the mathematical model.
” However, there are still several motivations for
identifying aircraft models from flight data, includ-
ing:
1. Verifying and interpreting theoretical predic-
tions and wind-tunnel test results (flight results can
also be used to help improve ground-based predictive
methods);
2. Obtaining more accurate and comprehensive
mathematical models of aircraft dynamics for use in
designing stability augmentation and flight control
systems;
134
Bogulavsky I. (2010).
SOLUTION OF AN INVERSE PROBLEM BY CORRECTION OF TABULAR FUNCTION FOR MODELS OF NONLINEAR DYNAMIC SYSTEMS.
In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 134-139
DOI: 10.5220/0002954901340139
Copyright
c
 SciTePress

3. Developing flight simulators, which require
an accurate representation of the aircraft in all flight
regimes (many aircraft motions and flight condi-
tions simply cannot be duplicated in the wind tunnel
or computed analytically with sufficient accuracy or
computational efficiency);
4. Expanding the flight envelope for new aircraft,
which can include quantifying stability and predict-
ing or controlling the impact of aircraft modifications,
configuration changes, or special flight conditions;
5. Verifying aircraft specification compliance”
(Klein and Morelli, 2006).
The nominal parameters for the problem of iden-
tifying actual aerodynamic characteristics are values
that correspond to knots of one-dimensional or two-
dimensional tables.
The correction vector of nominal (rated) parame-
ters, defined by an algorithm handling the streams of
digital information from the aircraft transmitters, has
a very high dimension that is on the order of several
tens or hundreds.
It should be noted that at NASA, projects based on
the theory and practice of identification of aircraft by
means of test flights are widely applied. An applica-
tion of (Klein and Morelli, 2006) in the internet soft-
ware package SIDPAC is published in the MATLAB
M-files language (systems identification programs for
aircraft), representing an implementation of the nu-
merous algorithms recommended by NASA for iden-
tification problems.
The most common method of identification is the
known nonlinear method of least squares, where the
sum of the squares of the discrepancies, i.e., the dif-
ferences between the actual measurements and their
rated analogues, obtained by numerical integration
of the system’s equations of motion is computed for
some realization of a vector of unknown parameters.
The outcome of a successful identification accepts the
vector of parameters, supplying a global minimum to
the mentioned sum of squares of the discrepancies.
It is necessary to note that this criterion is statis-
tically justified only for linear problems of identifica-
tion, problems in which the measurements are linear
in the unknown vector of parameters.
Significant computing difficulties arise when im-
plementing a nonlinearmethodof least squares to cor-
rect the nominal parameters of an aircraft according to
its test flights. The difficulties arise due to the large
dimension of the correction vector and due to the ex-
istence of numerous relative minima for the sum of
the squares of the discrepancies as functions of the
correction vector, and also because of the use of vari-
ants of Newton’s method, which requires a sequence
of local linearizations to define the stationary points
of the function.
The authors of the monograph (Klein and Morelli,
2006) presented a detailed exposition and analysis of
known algorithms for the identification of parame-
ters of the dynamic systems in chapters of[1]:[4 - 8].
However, only a regression method can be used for
a practical investigation. The regression method pre-
sented in (Klein and Morelli, 2006) solves this prob-
lem subject to the following restrictions:
1. All components of the state vector are mea-
sured.
2. The algorithm builds a vector of estimates for
the vector of derivatives
˙
dx at the moments of mea-
surement,
3. The vector functions on the right-hand side of
the equations of motion linearly depend on the esti-
mated vectors.
4. Prohibition of mathematical modeling without
the use of a Monte-Carlo method to analyze the the-
oretical observability of the components of the iden-
tified parameter vector if the laws of control are set
beforehand by test flights of the aircraft and informa-
tion about random errors of its transmitters.
In the monograph (Cappe et al., 2005), the prob-
lem of estimating the parameters is considered within
the limits of the common problem of smoothing;
this consists of the problem of constructing ap-
proximate conditional expectations for elements in
a non-observable sequence if these elements influ-
ence observable elements by means of a given sta-
tistical mechanism. Various approaches to solving
the smoothing problem by means of expectation-
maximization (EM) methods are stated and investi-
gated. However, the maximization operation requires
the definition of a point global (but non-local!) ex-
tremum, that is generally not guaranteedby numerical
methods.
In the last ten years, a significant number of stud-
ies (Gordon et al., 1993; Doucet et al., 2000; Doucet
et al., 2001; Ristic et al., 2004; Gosh et al., 2008;
Namdeo et Manohar, 2007), were published that rep-
resented the basic solution of the problem as the def-
inition of the conditional expectation of a vector of
parameters for a mathematical model of a nonlinear
dynamical system. By means of multiple applications
of Bayes’ formula to the state vectors and with nu-
merical quadratures, it is easy to determine the recur-
rence equations for the probability density function
(pdf) of state vectors of the dynamical system. The
actual solution of these recurrence equations, how-
ever, is not feasible because of the unwieldy dimen-
sions of the integrals involved (Namdeo et Manohar,
2007). Therefore, several alternative strategies have
been developed. One set of such alternatives consists
SOLUTION OF AN INVERSE PROBLEM BY CORRECTION OF TABULAR FUNCTION FOR MODELS OF
NONLINEAR DYNAMIC SYSTEMS
135

of developing suboptimal filtering, such as that based
on linearization or transformations, and the other con-
sists of methods that employ Monte Carlo simulation
strategies (e.g., estimation with the sequential impor-
tance sampling particle filter ) toapproximatelyevalu-
ate the multidimensional integrals in a recursive man-
ner.
This direction is perceptive, but it is bulky and not
suitable for the solution of practical applied (instead
of model!) problems of nonlinear identification.
The MPA algorithm (Boguslavskiy, 1996; Bo-
guslavskiy, 2006; Boguslavskiy, 2008; Boguslavskiy,
2009) makes use of this paper for the correction of
tabular functions. The MPA algorithm is a new re-
cursive algorithm that asymptotically and accurately
solves the nonlinear problem of construction of the
conditional expectation vectors of the vector of pa-
rameters for mathematical models of nonlinear dy-
namical systems, including random errors and pertur-
bations with given distributions. Therefore, the MPA
algorithm approximately solves the problem of quasi-
optimal mean-squares estimation.
The MPA algorithm has none of the disadvantages
of the NACA algorithm noted above, and in principle
it differs from all of the abovementioned algorithms;
a theoretical proof of its accuracy is presented in (Bo-
guslavskiy, 1996; Boguslavskiy, 2006; Boguslavskiy,
2008; Boguslavskiy, 2009).
2 A STATEMENT OF THE
PROBLEM OF CORRECTION
OF FUNCTIONS DETERMINED
IN THE FORM OF TABLES
Let x ∈ R
m
be a moving state vector of the mathemat-
ical model of the dynamical system and the compo-
nents of the vector z be a subset of the components
of x : z ∈ x,z ∈ R
r
,r < m. The vector z is an argu-
ment of the tabulated functions. We suppose that the
vectors z belong to the interior boundary of a paral-
lelepiped Ω ∈ R
r
under all realizable conditions of the
dynamical system – a realizable vector of functions of
t : u(t). Choose a Cartesian coordinate system for R
r
with axes parallel to the edges of the parallelepiped Ω.
The parallelepiped Ω is covered by nodes numbered
s
r
, where s is given as an integer.
Here s
r
is the number of nodes on which the table
determines the tabulated functions. We denote each
node by z(i
1
,...,i
r
),1 ≤ i
1
,...i
r
≤ s, where the inte-
gers i
1
,...i
r
are the indices of the coordinates of this
node. The nodes z(i
1
,...,i
r
) are vertices .of the small
parallelepipeds belonging to Ω ∈ R
r
.
We suppose that a presentation ( a description) of
the nominal mathematical model contains K tabular
functions ϑ
j
(z(i
1
,...,i
r
)), j = 1,...,K, which are given
on the specified nodes. The values ϑ
j
(z(i
1
,...,i
r
)) are
the nominal values of the tabular function, which, ac-
cording to the preceding experiments or in theory, all
represent the vertices of the small parallelepipeds.
Each tabular function ϑ
j
(z(i
1
,...,i
r
)) Is the skele-
ton of a continuous function f
j
(z), j = 1,..., K as fol-
lows : 1) on the vertices of the small parallelepiped
the values of the continuous function coincide with
the valuesof the tabular function; 2) on other points of
the parallelepiped, the values of the continuous func-
tion are linear combinations of the values of the tabu-
lar function at the vertices, such that these values are
multilinear functions of the values of the tabular func-
tion.
Let L
1
,...,L
r
be the lengths of the edges of any
of the small parallelepipeds. Then in the coordinates
tt
1
(α
1
),...,tt
r
(α
r
) of the 2
r
vertices, we can write the
formulatt
k
(α)=tt
k
0
+L
k
α
k
,k = 1...,r, where α
1
...,α
r
are independently determined with the values 0, 1.The
values α
1
= 0,..., α
r
= 0 correspond to the mini-
mal values of all coordinates for a given small par-
allelepiped.
Let z
1
,...,z
r
be the components of the vector
xx, ϕ
0
(z
k
)=L
−1
k
(tt
k
(1) − z
k
),ϕ
1
(z
k
)=L
−1
k
(z
k
−
tt
k
(0)),k = 1,...,r be linear functions of these compo-
nents ϕ
0
(z
k
)=0ifz
k
= tt
k
0
,ϕ
0
(z
k
)=1ifz
k
= tt
k
0
+L
k
,
ϕ
1
(z
k
)=0ifz
k
= tt
k
0
+ L
k
,ϕ
1
(z
k
)=1ifz
k
= tt
k
0
.
Then the components of the multilinear function
take the form
f
j
(z
1
,...,z
r
)=
∑
α
1
,···,α
r
=0,1
ϕ
α
1
(z
1
) ···ϕ
α
r
(z
r
)
ϑ(tt
1
0
+ L
1
α
1
,...,tt
r
0
+ L
r
α
r
), (2.1)
where the sum is over all binary values of an as-
pect α
1
···α
r
and over all 2
r
of the items.
The vector f
j
(z
1
,...,z
r
) is a nominal representa-
tion of the continuous functions defined by the tabular
functions ϑ
j
(z(i
1
,...,i
r
)).
The corrections of the K tabular functions
ϑ
j
(z(i
1
,...,i
r
)), j = 1,..., K replace the values ϑ
j
(...)
with values ϑ
j
(...)+θ
j
(...), where the correction
terms θ
j
(...) are the results of processing the new
data.
We shall designate by f
j
(z
1
,...,z
r
,θ
j
) the func-
tions obtained from f
j
(z
1
,...,z
r
) by substituting)
functions ϑ
j
(...)+θ
j
(...) for ϑ
j
(...) in (2.1)
The new data are the results of observationsvalues
y
1
,...,y
N
where y
k
= H
k
(x
k
)+ξ
k
, y
k
,x
k
,k = 1,..., N
are the results of observations of the vectors x at dis-
crete instants, H
k
(...) are the given functions, and ξ
k
are random errors with a given distribution.
ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics
136

Using the MPA algorithm, the correction vector
is quasi-optimal in the mean square estimation of the
variations of the components of the nominal vectors
ϑ
j
(z(i
1
,...,i
r
)), j = 1,...,K. The experiment responds
to these variations with new data about the dynamical
system. The vector of the estimates has K × s
r
com-
ponents
ˆ
θ
j
(i
1
,...,i
r
), j = 1,...,K,1 ≤ i
1
,...,i
r
≤ s.
The MPA algorithm is the Bayesian estimator of
the parameters (Boguslavskiy, 2006). A priori in-
formation, which is necessary for the training and
adjustment of the MPA algorithm by means of the
Monte Carlo method, includes the segment lengths
of expected scattering of values θ
j
(...) of the cor-
rections of the nominal tabular functions, expressed
in terms of components of these functions. There-
fore, the sum ϑ
j
(...)+θ
j
(...) replaces random values
ϑ
j
(...)(1+ ρ
j
ε(i
1
,...,i
r
)), where 0 < ρ < 1 and ran-
dom the values ε(...) are uniformly distributed on the
segment [−1,1].
If the nominal tabular function is smooth with re-
spect to the components of the vector xx, i.e., it varies
smoothly with changes in this component, then it is
appropriate to require this smoothness to be preserved
after the correction. The requirement is satisfied if
the correction includes the increments. of the nominal
values of the tabular function, This increments are the
tabular functions obtained by varying a given compo-
nent in the transfer from the given node to the subse-
quent node. These increments are approximately the
derivatives of the tabular functions with respect to the
given component.
If by a movement of the dynamical system the
moving vector z is found inside or on the bound-
ary of any small parallelepiped, then the equations of
the model use domains of the nominal tabular func-
tions ϑ
j
(z(i
1
,...,i
r
)) and the corresponding contin-
uous functions for which the points z(i
1
,...,i
r
) be-
long to this small parallelepiped Therefore the cor-
rection of values ϑ
j
(z(i
1
,...,i
r
)) is impossible if by a
given u(t) the current state vector x does not visit at
any instant the small parallelepiped with the vectors
z(i
1
,...,i
r
). The significant influence of the area cir-
cumscribed by the controlu(t) on the correction result
essentially distinguishes the problem of identification
of the vector correction of tabular functions from the
traditional problem of the identification of the param-
eters.
We illustrate the definitions on an example mod-
eling the motion of an aircraft. The continuous func-
tions correspondingto the tabular functions are piece-
wise linear approximations for the dimensionless co-
efficients of the aerodynamic forces and moments as
the functions of the angles of attack.
We present a formalized statement of the problem
of the correction of the tabular function, which is the
problem of quasi-optimal identification of the varia-
tions of this function. The variations are estimated
using data for the observed motion of the system. The
equations of the motion model are written in the form
dx/dt = F(x, f
1
(z
1
,...,z
r
,θ
1
),..., f
K
(z
1
,...,z
r
,θ
K
),u,t).
(2.2)
The equation of the observation is written in the form
y
k
= H
k
(x
k
)+ξ
k
, (2.3)
where k = 1,..., N.
Further, we designate by Y
N
the vector whose
components are y
1
,...,y
N
. In verifying the quality
of the correction after realization of the estimations
ˆ
θ
j
(i
1
,...,i
r
), j = 1..., K,1 ≤ i
1
,...,i
r
≤ s, we can use
the sum of quadrates of differences, which is
∑
k=1,...,N
(y
k
− ˆy
k
)
2
,
where the values ˆy
k
are computed from (2.2), (2.3) af-
ter substituting the values
ˆ
θ
j
(i
1
,...,i
r
),i = 1,...,K,1 ≤
i
1
,...,i
r
≤ s for the values θ
j
(i
1
,...,i
r
),i = 1,...,K,1 ≤
i
1
,...,i
r
≤ s
3 IDENTIFICATION OF SOME
PARAMETERS OF F-16
AIRCRAFT
1. The Driving Equations
It follows from monograph (Klein and Morelli, 2006)
that the driving equations of an F-16 plane stabilize
with respect to the magnitude of the airspeed velocity
V and roll angle rotation φ (
˙
V =
˙
φ = 0) and can be
approximated in the following form:
˙
α = q+(gcosυcosφ + ¯qSC
Z
/M)/V,
˙
β = −r+(gcosυsinφ + ¯qSC
Y
/M)/V,
p = 0,
˙q = 160c
7
r+ c
6
r
2
+ ¯qS¯cc
7
C
m
,
˙r = −(c
2
r+ 160c
9
)q+ ¯qSbc
9
C
n
,
where the pitch angle rotation υ and the yaw angle
rotation φ  are constants (within a small maneu-
vering time), the angle of attack is α, the sideslip
angle is β, p,q,r are the body-axis components
of the aircraft’s angular velocity, ¯q is the dynamic
pressure, S is the wing reference area, b is the
wing span, ¯c is the mean aerodynamic chord of the
wing, M,c
2
,c
6
,c
7
,c
9
are constants (see (Klein and
Morelli, 2006), C
Y
,C
Z
,C
m
,C
n
are non-dimensional
SOLUTION OF AN INVERSE PROBLEM BY CORRECTION OF TABULAR FUNCTION FOR MODELS OF
NONLINEAR DYNAMIC SYSTEMS
137

coefficients that are directly proportional to the aero-
dynamic forces and moments
C
Y
= −0.02β + 0.086(δ
r
/30)+(b/2V)C
Y
r
(α)r,
C
Z
= C
Z
0
(α)(1− (βπ/180)
2
) − 0.19(δ
s
/25)+(¯c/2V)C
Z
q
(α)q,
C
m
= C
m
0
(α,δ
s
)+(¯c/2V)C
m
q
(α)q+ 0.1C
Z
,
C
n
=C
n
0
(α,β)+C
n
δ
r
(α,β)(δ
r
/30)+(b/2V)C
n
r
(α)r−0.1( ¯c/b)C
Y
,
δ
s
is the stabilizer deflection,δ
r
is the rudder de-
flection, and α,β,δ
s
,δ
r
are in degrees.
Further, the functions C
Y
r
(α),C
Z
0
(α),C
Z
q
(α),
C
m
0
(α,δ
s
),C
m
q
(α),C
n
0
(α,β),C
n
δ
r
(α,β),C
n
r
(α) are
nominal functions. They are defined from the
vertices of the parallelepiped Ω ∈ R
4
by s = 8.
The functions accept ratings that are evaluated
using experiments in a wind tunnel at a finite num-
ber of reference nodes, covering the domains Ω.
The vectors of the nominal experimental data are
ϑ
1
(...),...,ϑ
8
(...)) in the equations (2.1). For the
functions C
Z
0
(α),C
Y
r
(α),C
Z
q
(α),C
n
r
(α),C
m
q
(α) the
vectors ϑ
1
(...),...,ϑ
5
(...) have dimensions 12 × 1;
accordingly, for the functions C
m
0
(α,δ
s
),C
n
0
(α,β)
the vectors ϑ
6
(...),ϑ
7
(...) have dimensions 12 × 5
and 12 × 7. Furthermore, we do not correct the
function C
n
δ
r
(α,β). Therefore, the number of
nominal parameters defining these functions equals
12× (5+ 5+ 7)=204.
The software package SIDPAC contains
a file F16 AERO SETUP Generates
aerodynamic data tables with ten one- and
two-dimensional tables of nominal experimental
data.
We emphasize that the nominal functions men-
tioned above are nonlinear functions of the argu-
ments.
2. Parametric Model of Aerodynamic Parameters of
the Subjects of Identification
We suppose that for all functions exceptC
δ
r
(α,β), the
nominal experimental data differ from the true data
by some random error vectors, which are designated
θ
1
,...,θ
7
.
We considerthe most complicated problem for the
MPA algorithm, where at each of the points of the ta-
ble, the actual parameter differs from the nominal pa-
rameter by a random magnitude subject to the a priori
limits θ
i
.
After collecting the measurements of the parame-
ters of the perturbed driving of the aircraft, the MPA
identification algorithm should estimate the 204 com-
ponents of the vector of random errors, generating the
vector of differences between the actual and nominal
parameters.
Let A
i
and B
i
(i = 1,..., 204) be the i-th compo-
nents of the nominal and actual (perturbed ) vectors
of the aerodynamic parameters corresponding to the
204 actual parameters subject to identification.
We suppose a fair parametrical model:
B
i
= A
i
+ Δ
i
,
The vector Δ is the vector of perturbations of nom-
inal parameters, i.e., the vector of errors of the aero-
dynamic parameters, and its component estimates are
subject to our identification. For the structure of these
components, we give the formula
Δ
i
= A
i
ρ
i
ε
i
,0 < ρ
i
< 1,−1 ≤ ε
i
≤ 1.
The positive number ρ
i
defines the maximum
magnitude that the ratio of the randomvariable of per-
turbations Δ
i
and nominal parameter A
i
is allowed to
attain under the conditions of our identification algo-
rithm. Each ε
i
is a random number that is uniformly
and independently distributed.
3. Transients of Characteristics of the Nominal and
Perturbed Movements
We suppose as abovethat the transients in the reduced
driving equation of the aircraft F-16 are α,β,q,r over
20 sec., if at t = 0 α = β = 0.3rad.,q= r = 10deg/sec
and magnitudes δ
s
,δ
r
are constant and equal to 10deg.
We shall discuss the precision of the estimate un-
der the following assumptions: during the 20 sec. pe-
riod, the current magnitudes α,β,q,r are measured at
intervals of 0.05 sec (N = 1600). We suppose that
the random errors of measurement are discrete white
noise, which is limited by the product of the true mea-
sured magnitudes on the magnitude of the set ε.We
shall suppose that the MPA algorithm supplies the
magnitudes

Δ
i
,i = 1,...,204, which are the estimates
of the magnitudes Δ
i
,i = 1,...,204. To characterize
the relative precision of the identification of the ran-
dom parameters Δ
i
we define ratios ε
i
=(

Δ
i
− Δ
i
)/Δ
i
We must emphasize that the state vector of the air-
craft corresponding to the modeled transient does not
visit all the reference points in which the nominal ex-
perimental data are set. Therefore, for some values of
i, the magnitude ε
i
has an order of 1 or more. The cor-
responding values ϑ
i
are not observable for the mod-
eled transient, and also cannot be corrected by means
of the MPA algorithm.
Table 1 presents a histogram of ε
i
.
Table 1.
2 ≥|ε
i
|≥11≥|ε
i
|≥0.50.5 ≥|ε
i
|≥0.25
37 44 37
0.25 ≥|ε
i
|≥0.10.1 ≥|ε
i
|≥0.05 0.05 ≥|ε
i
|
29 17 17
The practical purpose of identificationis to correct
the nominal experiment data, andthe outcome is to re-
place the nominal aerodynamic parameters with new
ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics
138

parameters. If the errors of identification are small,
the driving characteristics of the aircraft, obtained by
numerical integration after correction,should be close
to the perturbed driving characteristics of the aircraft,
as discovered early in real flight or by means of mod-
eling. In a real flight situation, the errors of the nomi-
nal experimental data can only be estimated only over
time and by observing the perturbed driving charac-
teristics.
In Table 2 , we present the ratios of the differ-
ence between the characteristics of the corrected and
perturbed movements and the difference between the
characteristics of the nominal and perturbed move-
ments as functions of discrete time with increments
of 1 sec.
In Table 2, for example, the expres-
sion δ
c,p
n,p
α designates the difference ratio
(α(corr) − α(perturb))/(α(nomin) − α(perturb)).
The labels δ
c,p
n,p
β, δ
c,p
n,p
q, δ
c,p
n,p
r are defined similarly.
These ratios show how quickly the MPA algo-
rithm reduces the difference between the corrected
and perturbed movements compared to the difference
between the nominal and perturbed movements.
Table 2.
sec δ
c,p
n,p
α δ
c,p
n,p
β δ
c,p
n,p
q δ
c,p
n,p
r
0 -0.086 -0.050 0.215 -0.040
1 -0.094 -0.064 0.048 -0.112
2 -0.112 -0.067 0.012 -0.329
3 0.110 -0.054 -0.011 -0.024
4 -0.082 -0.050 -0.022 -0.130
5 -0.019 -0.046 -0.040 -0.083
6 -0.022 -0.042 -0.168 -0.114
7 -0.022 -0.039 -0.003 -0.126
8 -0.022 -0.036 -0.0105 -0.141
9 -0.022 -0.034 -0.029 -0.154
10 -0.022 -0.032 0.001 -0.079
11 -0.022 -0.031 -0.009 0.001
12 -0.022 -0.030 -0.007 0.108
13 -0.022 -0.028 -0.008 0.233
14 -0.022 -0.026 -0.008 0.297
15 -0.022 -0.025 -0.009 0.345
16 -0.022 -0.024 -0.009 0.0382
17 -0.022 -0.022 -0.009 0.434
18 -0.022 -0.021 -0.009 0.486
19 -0.022 -0.020 -0.009 0.599
20 -0.022 -0.019 -0.010 0.722
It follows from Table 2 that the corrected charac-
teristics become close to, and often coincide with, the
perturbed drivingcharacteristics (to within 2 digits af-
ter the decimal point), in the absence of observational
errors.
4 CONCLUSIONS
The data presented in this work show that a multi-
polynomial approximation algorithm can forma com-
putational basis for creating an effective solution for
inverse problems, thus identifying the parameters of a
nonlinear dynamical system, including the system of
aerodynamic parameters of an aircraft.
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SOLUTION OF AN INVERSE PROBLEM BY CORRECTION OF TABULAR FUNCTION FOR MODELS OF
NONLINEAR DYNAMIC SYSTEMS
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