Symbolical Combinatory Models of Algorithms for
Processing Flight Information at the Flight Test Stage
of Aerospace Objects
Genady Burov, Riga Technical University
Abstract – A new class of algorithms for identification of
dynamic parameters of aircraft and their onboard equipment
that is suitable for processing low-dynamic transients recorded
during aircraft flight test state is described. Traditional
computing algorithms are not adapted for such situations when it
is necessary to find the inverses of ill-conditioned matrices of
systems of identification equations. Therefore, more efficient
algorithms have been developed on the basis of their symbolical
combinatory models (SC models). They have allowed to derive
symbolical descriptions of algorithms, which are suitable for
realization of parallel modes of calculations necessary for
increasing the performance of algorithms in real-time mode of
test flights. Such performance is necessary to ensure indication of
occurrence of emergency situations in the functioning of aircraft
onboard equipment.
Keywords – Symbolical combinatory model, parallel
algorithm, analog transfer function, difference equation
I. INTRODUCTION
The problem of indication of occurrence of emergency
situations in aircraft flight modes is very important for
ensuring flight safety. This problem is especially important
during aircraft flight test stage when flight modes that are
outside the normal flight restrictions are used [17]. Therefore,
it is necessary to create software for aircraft onboard
computers that monitors the correctness of functioning of
aircraft control systems [18]. In this case, processing of basic
flight information is necessary to perform directly on onboard
computers working in real time. It allows to reduce material
and time expenses at flight test stages of new aircraft designs.
This demands creation of special software for onboard
computers, adapted for processing of low-dynamical
transients. Such problem of indication arises at the stage of
normal aircraft operation for ensuring flight safety [17], [19],
as well. It requires development of special high-performance
computing algorithms adapted for processing of slowly
changing transients. However, unlike the stage of normal
operation, at the stage of aircraft flight tests it is necessary to
process considerably larger amounts of flight information,
which demands increase in speed of processing. It is possible
to achieve only by application of parallel modes of
calculation. In [5], [9], [10], it is noted that in this case, there
are difficulties related to coordination of parallel architecture
of onboard processors with the parallel architecture of
computing algorithms, but currently formalized mathematical
methods for such coordination are not developed. In given
paper, the possibility of application of symbolical combinatory
models (SC models), suitable for realization of parallel
algorithm of identification, is examined. Creation of parallel
algorithms of identification was earlier considered in [6] –
[10].
Results of studies have shown that such algorithms can be
created on the basis of their symbolical descriptions when
results of transient measurements are used as conditional
addresses in SC model based descriptions of algorithm. It
allows to apply methods of optimization of such symbolical
descriptions and to reduce the complexity of computing
algorithm simultaneously with increase in processing
accuracy. In this paper, the questions of optimization of
parallel algorithms for processing of low- dynamical transients
with increased accuracy are considered. Practical suitability of
SC models has been proved by a numerical experiment of
finding the inverse of 20th order Hilbert’s matrix used in a task
of polynomial approximations of experimental data.
Previously it was believed that above 10th order this problem
is impossible to solve even with application of modern
supercomputers, since such matrices are considered to be
standards of ill-conditionality. The 20th order inverse matrix
was calculated with 100% accuracy due to application of SC
models for optimization of the algorithm. This paper describes
some variants of algorithms for indication of dangerous
emergency situations in flight modes that based on solving
systems of difference equations, which are equivalent to
differential equations that contain information about dynamic
properties of identified objects.
II. PROBLEM STATEMENT
For creating a system of indication of emergency flight
situations, it is necessary to use the information about dynamic
characteristics of the identified object. With this purpose, it
would be desirable to find an estimation of coefficients of
differential equation describing the object, which is usually
expressed in the form of its transfer function:
m
A( p)
W ( p) 

Q( p)
(p a )
i
i 1
n
(p b )
j
j 1
n

Cj
 (p b )
j 1
j
(1)
However, finding their estimations is complicated because
of weak variability of processed transients and also because of
approximation errors resulting from the discrete character of
their processing that requires using a discrete approximation
of the operator (1):
m
D( z )  Fz (T )  W ( p )  FИНТ ( z , T ) 

i
z ( k i )
i 1
n


i
z n
y( z)
(2)
x( z )
i 1
The operator (2) is the basis for formation of system of
difference equations
X   Y   y
(3)
from which the estimations of coefficients of D(z ) are
calculated. In test modes of onboard control systems, there is
an opportunity to create a system of difference equations of
lower order for determining the coefficients of characteristic
polynomial (2):
Y ( N n ) 
( n)
y
(N )
n
;
Y  r L  
Ci q i ( L  r 1)
(4)
i 1

( n)
 Y 1  y (n)
(5)
A feature that is characteristic for almost all aircraft is weak
variability of transients from which the matrix of system (5) is
formed. Therefore, it almost always is ill-conditioned and the
algorithm for its inversion is numerically unstable, which
leads to large errors in solution of (5). Sharp transients are
undesirable from the point of view of flight safety. Therefore,
they are damped by special devices. For example, angular
fluctuations of aircraft around its centre of mass are reduced
by special dampers.
The small degree of variability of signal y (t ) is a
prominent feature of almost all aerospace objects. Therefore,
application of special algorithms, capable to work in
conditions of low conditionality of systems of difference
equations, is required.
Nominal values of coefficients of operator (1) are usually
obtained at the design stage and written down in the object’s
technical passport. The coefficients of (1) have clear physical
interpretation and are best suited for diagnosis of physical
condition of object. At the same time, the coefficients of
operator (2) do not possess this quality; they are abstract
numbers unsuitable for practical use and demand application
of special algorithms for their decoding. Meanwhile, in main
works on identification [22], [23], operation of decoding is
completely ignored. Moreover, they recommend introduction
of additional members into difference equations, erroneously
believing that it will make also possible the determination of
characteristics of present noise in addition to parameters of the
object. In [1], it was mathematically proved that such
approach completely distorts the results of identification and
increases their abstract character. Therefore, the statement that
such stochastic models of identification can be tools for “the
practical user”, to put it mildly, are not proved at all. The
problem of decoding the coefficients of difference equations is
complicated by essentially nonlinear character of relation
between operators (1) and (2). It is a problem belonging to the
class of mathematically incorrect inverse problems and the
complexity of its solution is aggravated by numerical
instability of operation of inversion of matrix Y (5).
Therefore, to create of software for indication of emergency
situations, it is offered to use methods of decoding that are
based on application of algorithms of Fourier decomposition,
described in [2], [3]. Development of algorithms of direct and
inverse mapping between operators (1) and (2) were
considered in [24] – [26] These results can be used for
calculation of vector of coefficients of (2) for normal
operational modes, to which the vector (5) found from
experimental data of a real flight mode will be compared. For
smoothing the mistakes discrete approximation at transition
from operator (1) to operator (2), in the mathematical
description of calculated vector of nominal mode, the operator
of interpolating filter is introduced, which depends on analog
and discrete argument [24]. During decoding, it is necessary to
separate coefficients of this operator from the coefficients of
the operator of object.
Using the mathematical relation between operators (1) and
(2), vector of solution of the system describing a normal
condition of identified object can be calculated beforehand
and compared with the one calculated from the system (1)
made from measurements of real transients measured during
flight modes at the test stage. On the basis of such method,
conclusions about condition of object and absence of
emergency situations in its functioning can be made.
III. ALGORITHM FOR CALCULATING ESTIMATIONS OF
DYNAMIC PARAMETERS FOR INDICATION OF EMERGENCY
SITUATIONS IN TEST FLIGHT MODES
However, the problem of creating numerically stable
algorithms for processing of low-dynamical transients
remains. Algorithms for matrix inversion based on various
modifications of method of Gaussian elimination are not
adapted for finding reliable estimations of (5) in conditions of
ill-conditionality.
These algorithms use a recurrent procedure in which the
results found in the previous step are divided by small
numbers that are comparable with round-off noise. The
method of choosing largest pivot elements is not effective as
they all are small numbers, so there is nothing to choose. It is
impossible to predict such occurrence of singular situations.
Therefore, traditional algorithms are unsuitable for processing
transients in the problem of identification of characteristics of
aircraft and its onboard equipment.
Development of new algorithms, in which these drawbacks
would be eliminated, is required. First of all, it is necessary to
avoid the recurrent procedure with division by small noiseaffected numbers.
For this purpose, it is offered to find the inverse of (5) in the
form of adjugate matrix, the elements of which are expressed
as algebraic cofactors (minors with signs) for elements of the
matrix of transient Y .
Let’s denote the adjugate matrix as:
Y~11 Y~21
~
~
~ Y12 Y22
Y 
 
~
~
Y1n Y2 n
~
 Yn1 
~ 
 Yn 2 
 
~ 
 Ynn 
(6)
~
Here, Ai j  Y j i /  , because the matrix is transposed.
~
Elements Y j i are formed in the form of cofactors (minors with
signs) for corresponding elements of Y . It is known that
elements of inverse matrix Y 1 can be found by division of
~
elements of adjugate matrix Y (8) by the determinant of the
initial matrix; it follows from identity
Y 1
 Y  E   det Y

(7)
~ 1
Y 1  Y 

(8)
so the vector of solution of system (5) can be represented by
expression:
~ 1
y

 Y 
(9)
Let’s notice that cofactors of row elements are located in
~
H in corresponding columns, that is, operation of transposing
is made. Here   det H is the determinant of the initial
matrix H . Like all minors in (8), it is found using formula:
det H 
(1)   (a 
1
1
 a2 2 ,  an n )
(10)
P
Here the sum is distributed over all possible permutations
P  (1 , 2 , n ) of elements 1, 2, …., n, not containing
repetitions of elements, hence, (13) contains n! addends and
  0 , if permutation is even, and   1 , if permutation is
odd.
However, with the use of expression (13) in algorithm for
inversion of matrix (8), the amount of calculations increases
because it is necessary to calculate the minors. Therefore, for
increasing the performance of algorithm, capable to work in
conditions of rapid flight modes, it is offered to use mode of
parallel calculations. Proceeding from this, the algorithm
should have parallel architecture adapted for realization in a
parallel onboard computer [5] – [10]. In this case, it is
necessary to coordinate the architectures of computing
algorithms and onboard computers. The formalized
mathematical methods of such coordination were considered
in [5], [7] – [10] and these results are applied to calculation of
minors of the matrix Y . The specified problems are solved
with the use of address symbolical combinatory models (SC
models) of used algorithms.
IV. FORMATION OF ADDRESS SC MODELS OF IDENTIFICATION
ALGORITHMS
Processing large amounts of information during the test
flight demands application of high-performance computers.
However, their capacities cannot be fully utilized by
traditional computing algorithms, which are based on the
principle of consecutive execution of computing operations.
Therefore, it is necessary to solve these problems using a more
integrated approach. High performance can be achieved by
using computers working on parallel principles, but then also
the computing algorithms must be parallelized. Solving this
problem was considered in [8] – [10], and the solution was
found in development of new information technologies that
use symbolical combinatory models of execution of
computing operations.
As the basic tool for enabling parallel operation in both
computers and algorithms, graph structures were used that
allowed to solve ill-conditioned equation systems. Graph
structures possess the property of fragmentation and
consequently they can be used for parallelization of computing
algorithms, which can then be used to improve the
performance of aircraft onboard computers using software
methods [2] – [4]. Performance in this case can vary over a
wide range and it is determined by the number of used
processors. For their commutation and work synchronization
the same switching addressing schemes that are used in graphs
can be used.
For the central problem of algorithms – solving illconditioned systems of equations formed from results of
measurements of flight processes, a graph structure that is
created on the basis of symbolical combinatory models is used
[4], [5].
The advantage of the new information technologies based
on graph structures is that many different approaches can be
used for optimization of computing operations. They are
chosen depending on the character of the problem being
solved. The main principles of such optimization, however,
can be specified. They are reduced to application of methods
of specification of multisets located in graph sections that
allow to allocate their carriers using the SC operators [5], [10],
[12]. Their use is facilitated by the fact that problem is solved
in symbolical space of integers. It allows to realize the
specification using SC operators of address switching [4], [9],
[11], [12]. The system of switching is formed in a separate
working information space and it works in mode of
monitoring with the information space of objects processed in
arithmetic registers. Operations of switching are realized on
the basis of ordered index sequences. It allows to avoid the use
of enumeration operations, which are usually used in
combinatorial algorithms, and to use determined and
minimized algorithms.
In [3], methods for formation of such operators have been
developed. Their properties are suitable for carrying out
operations of optimization of computing algorithms. It was
found that operations on ordered numerical sequences, which
are generated by these operators, can be replaced with
operations on arguments of these SC operators [4].
From here follows that the initial software should be created
in the field of arguments of such operators. And then, using
only formal mathematical methods, hierarchical principles of
formation of symbolical combinatory models must be applied
to obtain the software in completed form.
The possibility of using graph structures for realization of
parallel calculations follows from their properties of
regularity, recursivity, decomposition, and hierarchy. It allows
to apply new information technologies to design universal
algorithms for processing of flight information in both the
systems of onboard measurements and the ground control
systems used in different stages of flight tests.
Let's represent the symbolical combinatory model in the
form of address lexicographic combinatorial configurations. In
this model, we use combinatorial operators developed in [3] –
[5].
The symbolical combinatory computing model we shall
present in the form of address lexicographic combinatorial
configurations, formed in the for of graph (4):
 
Gr r, L : q~1(n)  Dpv * Kcm1 * q~1  
 Dpv * Kcm2  * q~2  
 Dpv * Kcmk  * q~k  ;
k
m
i
n
(11)
i 1
Branches of graph are affected by operator Dpv
( n) T
i 1
( ni )
k

i  {Kc(v ) :[(i  1), (i  2),..., n)]}
i 1
k  n  (v  1)
(14)
The expression (11) describes the graph in the form of a
branching tree. Therefore, the SC model of algorithm for
determining the elements of the inverse has the form of a
decomposition of independent fragments as graph branches.
It proves that the graph has a parallel architecture, which
can be used for representation of the algorithm for solving
equation systems in a parallel form.
The properties of the operator Dpv have been investigated
and the proof is found that it possesses filtering properties in
relation to multiset, allocating the carrier. This property can be
used for optimization of calculation process with simultaneous
increase in its accuracy. Components of Ims containing the
coordinates of elements of allocated submatrix are used as
arguments of operator Dpv . Therefore, all operations related
to inversion of matrix Y, first of all, the re-addressing
operation, can be executed in index symbolical space.
V. OPTIMIZATION OF SC MODELS OF COMPUTING ALGORITHMS
The graph SC model will consist of addresses of submatrix
for which the minor according to (13) is found. As is it
apparent from (4), these addresses determine the degrees of
discrete poles in the expression of result of transient
measurement. Therefore, indices of addresses simultaneously
determine the degrees of discrete poles in (4). Values of
numbers of rows and columns, specified in an index matrix
( n)
 q1 , q2 ,, qn 
(12)
which forms products of elements located in graph branches.
The resulting expressions are summed in the whole set of
branches:
Here the designation of associative matrix is used in the
form of coordinates Y designated by vectors of row and
column numbers, forming its index matrix grid. It determines
the ordering of sampling of transient values from Y .
Therefore, it is possible to use shorter designation of SC
model for calculation of minors:
V ( r , L )  Dpv (r , L) * q ( n)  C


qi
grid of the allocated submatrix ims(r
V ( r , L )  Dpv * Gr( r , L ) * q (n)
q
k
( n) 

(13)
Here the notation of direct product of vectors of values of
discrete poles and the coefficients of decomposition,
describing the set of partial components of transient, is used.
The sets in its sections are formed in view of the sets in
previous sections using the residual principle and described by
symbolical formula:
( n)
L
( n)
) , are the
( n)
argument of the operator Dvp(r  L ) forming the graph
SC model [4], [5]. It can designate the addresses of a sample
of results of transient measurements and an array of degrees of
poles that is allocated by the operator Arang over the
elements of the set of discrete poles:
Dvp (Im s) * q~ ( n)  q~ ( n) *Arang Dvp (r

( n)
L
( n)
)

(15)
As the base position vector, relative to which the
permutations of other vector are allocated in sections of graph,
designated by the operator Perm * , both the vector of rows
and vector of columns can be chosen. It follows from the
known principle that the determinant of a matrix can be found
by decomposition of matrix by elements of rows or by
elements of columns:
( n)
( n)
q~ ( n) * Arang Dvp (r  L ) 


( n)
( n) 

 q~ ( n) * Arang Perm * r   L ) 



( n)
(n) 

 q~ ( n) * Arang Perm * L   r )




(16)
If the components of permutations in the base vector, for
example, the columns, contain a pole q i with the same degree
L0 it can be separated as a constant multiplier of SC model:
Dvp (Perm * r
( n)

)  (L
( n)
KC( i ) * (1.m), i 1.n, if (m  n); i 1.m, if (m  n) (22)
~ ( n)
 L0 
*q 

( n)
( n) 

 q~( n) * Arang Pern * r    L   qi * Arang( L0 ) (17)



Realization of operator of permutation can be made with the
help of graph SC model, but only even permutations are
located in its branches, unlike the same SC model at the
calculation of minors in adjugate matrix (8) using expression
(10) when the operator Dpv in which the operations of
multiplication of elements in graph branches and summation
of these products is used. At allocation of degrees over
q~ ( n) the property of conjugacy of operation of element
allocation of one set over elements of other set
~
b (m) *Arang(a~ (m) ) is used:
~
~
b (m) * Arang(a~ (m) )  a~ (m) * Arang(b (m) )
 
n
Dpv * 1.m  KC(n) * (1.m) ; if (m  n)
(19)
Dpv*Graf ( ) * q~ *
(m)
(20)
KC (n) * (1.m)  R(m, n)  
m
 C
i1
i 2 qi 2
 
 1.m
( m)
m!
(24)
n !(n  m) !
(25)
This expression shows that operator of allocation of set of
powers (24) operates relative to any component of (23). The
received result is placed into graph SC model to which the
operator Dpv is applied. Using the principle of equivalent
representation, an ordered component of dimension n can be
chosen from (25), which we shall designate as a set q~ ( n) .
From here follows that SC model (25) possesses property of
parallelism and is suitable for realization on parallel
processors [5] – [7].
The second multiplier in (25) designates the operation of
allocation of permutation formed from elements of coordinate
( m)
vectors ims(r  L ) . Therefore, in view of the property
(23), it is possible to write:
q~ *Arang Perm * r   L )  


 q~ *Arang r   Perm * L  




 


( n)
( n)
( n)
k
( n)
( n)
k
n
(21)
i2
Here m is the number of partial components of dynamic
process, the weight coefficients C i 2 of which show the degree
of its variability. As they do not depend on coordinates
( m)
; d
Graf ( ) *KC(n) * (1.m)*
( n)
n
(d )
Therefore, the algorithm for formation of (23) is applied
directly to components of (21):
( m)
For optimization of this SC model, we shall introduce in it a
local address SC model denoting the operation of
multiplication of results of transient measurements which we
shall present in the form of lexicographic power of numerical
interval:
(23)
The result of influence of operator KC(n) * (1.m) , as
mentioned above, can be represented as a vector from
components of ordered numerical series:
( m)
( m)
*ArangGraf ( ) * ims (r   L ) 


Then in view of the dual role carried out by graph SC
model, the algorithm of calculation of minor in symbolical
form can be represented as:
( m)
( m)
*ArangGraf ( ) * ims (r   L ) 


As a result of summation of expressions (26), in the
branches there is filtration of components of local address SC
model and the result of filtration can be written down as [3],
[12]:
(18)
Using property (18), it is possible to write:
( n) 
q~ ( m) *Arang 
(Perm * r )  


( n)
( m)
~
 (Perm* q ) *Arang(r )
In [3], [6], [8], it has been proved that local SC model (26)
can be expressed in a canonical form as decomposition by
vectors formed by operators:
ims(r  L ) , they are accompanying parameters in SC
models and can be extracted out of the expression of
fragments of local model as constant multipliers.
q~k ( n)  KC (n) * (1.m)
(26)
Having chosen as the base position vector, the vector of
(n )
rows r  , we obtain SC model of formation of minor from


elements of vector of columns L
(n )
:
 
Dpv * q~k ( n) *Arang r
( m)

( n) 

 Perm * L

( n)  
 
(27)
( m)
If ims(r  L ) has coordinate vectors with break of
regularity of ordering of numbers of rows
 

r ( n)   1. p  1. p  p ;
1
2
0


are
1
placed
accordingly.
For

q1
q1 P1
( n)
( n)
)  (Perm * L
( n)

) * q~ ( n) 

( n)
) * q~ ( n)   Fg ( L ) * q~ ( n) 
 

(34)
( n)
If r  m . m  (n  1) then the common multiplier is
allocated:

 q~



Dvp Perm * 0. n  1  m[n] * q~ (n) 
( n)

* Arang(m )  Fg * q~ (n)
[ n]

(35)
The conjugate property of decomposition is observed:
1 
 q n 


P1 
 qn 

1
 Fg (r

(28)
KC( p1 ) * (1.n) , there exist an matrix-original [13], [14]:
 1
q
A 1

 P1
q1

Dvp (Perm * r

Lm
the SC model (11) of formation of minor for the matrix being
the original (32) will represent a graph consisting of two
sections in components of which the direct combinative vector
and
its
difference
complement
KC( p1 ) * (1.n)
1.n/KC( p ) * (1.n)
For finding the elements of adjugate matrix (6),
permutations for both vectors of rows and columns are used.
Two vectors formed by operators of permutation are
accordingly allocated:
(29)
Dvp (Perm * r
( n)
) * q~ ( n) * Arang (r

( n)
L
( n)
) 

( n)  
( n) 

 q~ ( n) * ArangDvp * r   q~ ( n) * Arang L  (36)

 




It is obvious that the minor of a submatrix with
ims 1. p1  L has expression:
Dpv * q1 ; q 2 ;  q S *
*Arang Graf ( 1  p 1 ; 2  n  p 1 ) * L

 Fg * q1 ; q 2 ;  q S 
( n) 

( n)
) * q~ ( n) * Arang (r

( n)
L
( n)
) 

( n)  
( n) 

 q~ ( n) * ArangDvp * L   q~ ( n) * Arang r  (37)

 




(30)
Here the strip of the allocated submatrix (35) has regular
order of following of powers 1. p1 , therefore, its minor is
defined by the operator Fg * q~ (n) and in the field of originals
has the expression:
Fg * q1 ; q 2 ;  q S   (q1  q1 )  (q1  q 2 )  (q1  q S )
 (q 2  q3 )  (q 2  q 4 )  (q 2  q S ) (q S 1  q S )
Dvp (Perm * L
The SC model possesses recursive properties that allow to
apply the methods of reduction of algorithm complexity. For
this purpose, we shall represent the product in graph branches
in the form of ordered numerical sequence R (m, n) [4], [5]:
Perm * KC ( ) *1. n*Arng (Z )


m
R(m, n) 
1
Z  Perm * Part ( ) * m
(38)
(31)
It has the following property:


 
On the basis of Im s  0. n  1   0. n  1 in the SC model,
the difference operators Fg [13,14,15] can be introduced:
Fg * q~ ( n) 
q
i
qj
(32)
i, j ( i  j )
( n)

 Fg ( L

 (Perm * L
( n)

( n)
~ ( n)
)
* q 

 
) * q~ ( n)   q~ ( n) * Arang r ( n)

 
 
 
r (m) * 1. n  KC(m) * 1. n  (0 . m 1)

(40)
Using the above described properties, we have:
Dvp(Perm* r
(33)
(39)
On their basis, lexicographic forms for Kronecker products
of vectors can be generated:
r (m) * 1. n  R(m, n)
The vectors consisting from components of (30), with the
help of the operator (21), can be allocated from structure of
SC model in a Kronecker product:
Dvp r
G(m, n)  (0. m  1)  R(m, n)
 Dpv(Perm* r
( n)
( n)
) * G(m, n) 

 
) * KC(m) * 1. n
(41)
Dvp (Perm * r
( n)

) L
( n) 
Y
 * G(m, n) 

( n)

Dpv (Perm * r )

(n)
~
 q * Arang 



* KC (m) * 1. n


 
  P  u
 u  Dvp* Q
 h  Dvp* P
( nm)
( mn)
T
h
(43)
( nn)
i
i
i
i
( nn)
G(n, m)  (0. n)  R(n, m) :
 u  Dvp1. n   G(n, m) * Q ;  h 
 DvpG(n, m)  1.n* P
i
i
i
(45)
i
Then the expression (18) can be written down in the
following form:
 Sum*Dvp(arg ) *Q
(46)
 Dvp(arg ) * P
arg  1. n  G(n, m) ; arg  G(n, m)   1. n (47)

Dvp* Q
( nm)
P
( mn)
1
2
T
1
2
Let's find the argument set for Dpv , acting on the product.
Q
( nm)

 M (mm)  W (mn) With the help of the vector
ims i  G(n, m) i  G(n, m) , we shall allocate a submatrix.
S n  m  M (mm) Using (4), we find:
i

arg Dpv * S i  W   G(n, m) i   G(n, m) T

 
  G(n, m)   1. n

M
(48)
W
The result of the influence of the operator Dpv we shall
write down as:

Arg Dpv * M (mm)
 Dpv * Q
( nm)

 M (mm)  W (mn) (49)

d TZ w  
G(n, m)   G(n, m) 


T
M
(50)

 
w  Dpv1.n  G(n, m) *W
d T  Dpv 1.n   G(n, m) * Q
Z  i j Dpv G(n, m) i   G(n, m)
T



j  M
( mm)



 Dpv *W (G(n, m)) 


( kk )
 Dpv *W (G(n, m) ) (52)
The left and right multipliers of (52) are generated from
diagonal matrixes of powers of discrete poles which are
formed with the help of the operator Fg (31), and the
middle matrix is generated from weight coefficients of
transient Ci . Therefore, the description of algorithm for
inversion of matrix Y (4) has the form of:
(44)
The coordinates of submatrices are used as arguments of the
operator Dpv .
As them, we use the vectors made from the components of
ordered numerical sequences [4]:

 Dpv (G(n, m)   G(n, m) * Y 
 Dpv (G(n, m)   G(n, m) * M
(42)
Using the property of decomposition of the operator Dpv ,
we shall find:
Dvp* Q

1
Y
1





 Diag  ( FG )  Dpv * M ( k k )  Diag  ( FG ) (53)
From here follows that algorithm (53) can be generated as a
decomposition the fragments of which can be processed
independently, and it is suitable for parallel processing of
transients at the stage of aircraft flight tests. The structure of
algorithm can be changed with the help of arguments of
combinatory operators. It allows to coordinate the architecture
of algorithm with the parallel architecture of computer using
software methods.
VI. CONCLUSIONS
A method for symbolical description of computing
algorithms for solving the problem of identification of
dynamic characteristics of aircraft its onboard equipment is
developed. It has allowed to derive, for the first time, an
analytical expressions for the vector of solution of systems of
difference equations of identification.
Use of descriptions of computing algorithms in the form of
SC models has allowed to establish the fact that SC models
possess filtering properties that allow to minimize the
complexity of computing algorithm. The developed method of
mapping of symbolical descriptions of computing algorithms
into the area of arithmetic operations allows to create, on the
formalized mathematical basis, the software for onboard
computers which are carrying out functions of control and
diagnosis of the condition of aircraft onboard equipment and
indication of occurrence of emergency situations at
performance of test flights.
The specified problems remain actual for modes of normal
aircraft operation, as well. On the basis of SC models,
algorithms for processing of flight information in modes of
parallel calculations, which will allow to realize such
processing in real time of test flight, can be created. Use of SC
models allows to choose the required architecture of
computing algorithm and to coordinate it with the parallel
architecture of onboard parallel processors.
Developed SC models can be used as means of designing of
computing algorithms and software that are efficient in
conditions of deficiency of dynamism of processed signals.
(51)
Using the expression (11), we find the SC model for the
inverse matrix of the dynamic process:
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Centre of Riga Technical University.
G. Burovs. Lidojumu informācijas apstrādes algoritmu simboliskie kombinatoriskie modeļi aerokosmisko objektu izmēģinājumu lidojumu etapam
Rakstā apskatīti lidaparātu un to aprīkojuma dinamisko parametru identifikācijas algoritmi, kas piemēroti testa lidojumu laikā reģistrēto pārejas procesu ar zemu
dinamismu apstrādei. Šādos apstākļos tradicionālie identifikācijas algoritmi nevar tikt izmantoti, jo invertējot slikti nosacītās signālu matricas rodas nepieļaujami
lielas kļūdas. Tādēļ ir izstrādāti jauni identifikācijas algoritmi, kas pamatojas uz simboliskiem kombinatoriskiem modeļiem (SK modeļiem). Tie ļāva iegūt
simboliskus aprakstus algoritmiem, kas ir piemēroti paralēlai skaitļošanas realizācijai. Šī pieeja ļauj radīt ātrdarbīgus algoritmus lidojumu informācijas apstrādei
pašā lidaparātā testa lidojuma reālā laika režīmā. Tas ļauj realizēt sistēmas lidaparāta aprīkojuma funkcionēšanas ārkārtas situāciju indikācijai.
G. Burov. G.Burovs. Lidojumu informācijas apstrādes algoritmu simboliskie kombinatoriskie modeļi aerokosmisko objektu izmēģinājumu lidojumu
etapam
Algorithms for identification of dynamic parameters of aircraft and their onboard equipment that are adapted for processing low-dynamical transients recorded in
flight modes during aircraft flight tests are examined. In these conditions, traditional identification algorithms cannot work because during inversion of illconditioned matrices of signals inadmissibly large errors occur. Therefore, new algorithms of identification have been developed on the basis of symbolical
combinatory models (SC models). They have allowed to derive symbolical descriptions of algorithms, suitable for realization of parallel modes of calculation.
This approach will allow to create high-performance algorithms for processing of flight information directly on onboard aircraft computers in real time of test
flight mode. It will allow to realize systems of indication of occurrence of emergency situations in functioning of aircraft onboard equipment.
Г. Буров. Символьные комбинаторные модели алгоритмов обработки полетной информации на этапе летных испытаний аэрокосмических
объектов
Рассматриваются алгоритмы идентификации динамических параметров летательных аппаратов (ЛА) и их бортового оборудования, приспособленных
для обработки слабо динамичных переходных процессов, регистрируемых в полетных режимах на этапе летных испытаний ЛА. В этих условиях
традиционные алгоритмы идентификации теряют свою работоспособность, поскольку при обращении плохо обусловленных матриц сигналов
возникают недопустимо большие погрешности. Поэтому были разработаны новые алгоритмы идентификации на основе символьных комбинаторных
моделей (СК- моделей). Они позволили получить символьные описания алгоритмов, пригодных для реализации параллельных режимов вычислений.
Этот подход позволит создавать быстродействующие алгоритмы обработки полетной информации непосредственно на борту самолетов в темпе
реального времени испытательного полетного режима. Это позволит реализовать системы индикации возникновения нештатных ситуаций в
функционировании бортового оборудования ЛА.