Optimum Control Using Signal Processing in Integrated Aerospace

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					                                                                          AYMAN AL-LAWAMA,ABDEL-RAHMAN -

   Optimum Control Using Signal Processing in Integrated Aerospace
                        navigation systems
                                 Electrical Engineering Department
                                         Mutah University
                                    Karak– 61710 P.O. Box (7)
                          ay_lawama@yahoo.com http://www.mutah.edu.jo
                              Communications and Electronics Department
                                      Philadelphia University
                                    P.O. Box: 1 Amman 19392
                     qawasmi@philadelphia.edu.jo http://www.philadelphia.edu.jo

Abstract:-The global – air – navigation telecommunication network is an integrated system which joins
telecommunication, navigation, and observation. This integrated system provides communication in real-time
scale between any two inverse points .Direct and indirect measurement ( DIM ) of basic navigational
parameters of aircraft movement ( position , velocity , acceleration ,….. etc ) should be done accurately and
precision . This paper provides general theoretical fundamentals of Optimum Control (OC) of processed signal
in integrated navigational systems to obtain relatively simple and practically convenient relations for
calculating maximum accuracy.
The well known maximum probability method is used to obtain Optimum Complex Evaluation (OCE) for the
processed results of Unequal Accuracy (UA) measurements .Two lemmas are considered for absolute concepts
and ratio errors in DIM. Features of the proposed method are illustrated using the simplest case of two
measurement (m=2) and then generalized using the method of induction for (m>2). The essential conditions of
existence theorem of optimum integration (OI) are proved for these problems. Theoretical regulations of OI are
illustrated using the results of experiment, numerical examples and graphics. Through the results of
investigations, conclusions are formulated and practical recommendations are developed for the use of OI.

Key-Words: - aerospace navigation systems, Optimum control, DIM measurements, complex evaluation.

1 Introduction                                                  movement (Position, velocity and acceleration),
The approach on air-lines of global aircraft liners as          airborne, ground-based and satellite navigational
Boing 777-200 LR and the practical realization of               systems. Later, such systems shortly named PVA
the concept of global control of air traffic                    systems. The accuracy can be increased using
(“seamless sky conception”) open principally new                methods of integrated systems (IS) [1-3].
capabilities for enhancing safety flights, quality of           Theoretically, this designates the application of
passenger's service and commercial efficiency of                optimum processing of UA results. When sensors
aircraft companies. Global aircraft liners with 300             (designed on different physical principles and have
passengers on broadside support distance with                   different errors of measurements) are used in
nonstop flights equals to 17000 km. The use of                  measuring, the problem of experimental processed
global Air-navigation Telecommunication Network                 results, control, diagnosing and other many cases
(ATN), which joins systems of telecommunications,               will be actual even in the theory of approximated
navigation and observations (CNS/ATM) allows                    calculations.
provide communication in real –time scale between                   In all these cases, it is important to have a
any two inverse points. All of this creates important           theoretical justification about how to get the best
conditions for global control of air traffic.                   general evaluation of measurable parameter due to
    One of main directions of CNS/ATM ATN                       results of measurements of inaccurate sensors.
application is to increase the measurement precision            Elements of this theory are developed by many
of basic navigational parameters of aircraft                    specialists in the area of mathematical statistics and

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                                                                          AYMAN AL-LAWAMA,ABDEL-RAHMAN -

theory of statistical solutions, where the considered          The work objective is to create general theoretical
problem is known as a problem of processing of                 fundamentals of OC of processed signal in
samples from nonhomogeneous -serial statistics [4].            integrated navigational systems to get practically
However, obtained results, as a rule, are oriented on          convenient and relatively simple relations for
the prove of asymptotic characteristic of derivable            complex and Optimum Evaluation (OE) using
evaluations- property to give more accurate                    criterion of maximum accuracy; and to show
evaluations by increasing the number of samples i.e.           asymptotic convergence to true values of
(sensors). Practically, we usually have to work with           evaluations derivable as a result of OI.
relatively small number of sensors –from 3 to 10,                   The following well-known information is
which give unpractical results.                                presumed:
    The concept of integration of nonhomogeneous                    1- The mathematical model of results taken from
results of PVA measurements in aviation is not new.            Indirect Measurement (IM) Yi i-th of real- value
The simplest example of integration of two systems             system X0 of navigational parameter at instant time t
is the complex «GPS/INS integration» [1-3]. An                 of measurements can be defined as
example of more complicated situation is the                    Yi (t ) X 0 (t ) i (t ) ,                         (1)
measurement of flight altitude of aircraft depending           where m (i =1,2,….. ,m) is the total number of
on the results of measurement of its altitude using            systems forming the complex integrated system and
six to seven different PVA systems such as airborne            ξ(t) is the random absolute error of Direct
sensor of barometrical altitude, airborne radio                Measurements (DMs) (See below lemmas 1, 2).
altimeter, earth radio navigation system "Omega",                   2- The real value of measurable parameter in (1)
satellite Global Position System (GPS), Primary                is a deterministic and constant value, where the
Surveillance Radar (PSR), Secondary surveillance               error is presented as a Gaussian stationary signal
radar (SSR) and Aeronautical Mobile Satellite                  with known numerical characteristics.
Service (AMSS).                                                     3- The mathematical expectation of error
    As shown by works [4-6], the use of redundant               M i (t ) 0,                                      (2)
number of measurements (essentially redundant                  and dispersion
volume of signals) allows successfully to create                D i (t )        2
                                                                               i ,i 1, m .                       (3)
high-precision structure - redundant info-
                                                                    In this paper we need to:
measurement from relatively inaccurate systems.
                                                                    1- create an OC method to process m signals (1)
    All well-known methods of integration can be               (optimum integration method of UA systems) using
conditionally divided into three classes:                      criterion of maximum probability,
    -Majority processing method of redundant                        2- find function of optimum integration using
inaccurate measurements;                                       criterion of maximum probability.
    -Methods using averaging value without                     Z m f ( Y 1, Y m ),                               (4)
considering the different precisions from sensors;
                                                                with arguments presented by results of IMs (1).
    -Methods of substituting results of measurement
                                                                    3- determinate the necessary conditions of non-
using more-accurate sensors by the results of                  bias, opulence and asymptotic effectiveness of
measurement using less-accurate sensors.                       evaluation (4),
    Each class has its advantages and disadvantages.                4- evaluate the comparative effectiveness of OE
Selection of suitable integration method in every              application, advantages and disadvantages of
specific case without theoretical justification                proposed optimum integration method and to
presents laboriousness and insufficiently explored                  5- bring out convenient practical applications
problem.                                                       relative to simple calculated recurrent relations for
    Brief examination prehistory of PVA integration            OE (4) and its numerical characteristic.
shows actuality, theoretical value and practical
concernment of solution of optimum-controlled
problems of processed signals in integrated                    3 Program and procedure of analysis
navigational systems.                                          First, two lemmas about properties of absolute error
                                                               under DIMs are proved. Second, the well-known
                                                               method of maximum probability (4), in convenient
2 Work          objective       and      Problem               way, is used to get OEs. The proposed method can
definition                                                     be illustrated in simple case when m=2. Later on,
                                                               this method is used for main set of convenient

        ISSN: 1109-2742                                 1082                  Issue 11, Volume 7, November 2008
                                                                                  AYMAN AL-LAWAMA,ABDEL-RAHMAN -
         WSEAS TRANSACTIONS on COMMUNICATIONS                                     AL-QAWASMI and OMAR AL-AYASRAH

recurrent relations. Results of this method, by                       The second Lemma is about necessary conditions
induction, will be generalized into a complex                     of using mathematical model of the IMs result in the
problematical situation for m>2. For these cases,                 form (1).
theories about essential conditions of OI are proved.             If the variation coefficient (9) is
Calculated theoretical relations are introduced for               Vx      1                                      (10)
various indices of effectiveness of OI. Theoretical               and at the same time, the following necessary
situation of OI is illustrated using numerical and                conditions are satisfied
graphical examples, depending on the results of
                                                                  1. M X        X0                               (11)
analysis,       conclusions        and        practical
recommendations are formulated to use OI.                         2. M                   0           0                                                                                 (12)
    In order to theoretically substantiate the fairness                                                          2
                                                                  3. D            DX                             x                                                                     (13)
of used mathematical model of IMs result in the
                                                                                                         2                                          2
form (1), two lemmas are considered to prove the                         dy ( x0 ,           0   )                   dy ( x0 ,          0   )
properties of random absolute and relative errors of              4.                                                                                                    1              (14)
results of DMs and the necessary conditions for
                                                                              dx                                          d
using the mathematical model (1).                                 where M[.] and D[.] are the mathematical
    We remind that in theory of approximate                       expectation and dispersion respectively. The result
calculations traditionally the term "error" is used,              of IMs Y(X, Δ), as function of two variables using
whereas in measurement theory more often the term                 (6) approximately described by three terms of
"mistake" is used. In future these terms are used as              Taylor's series at point(x0, Δ0):
convertible terms, preference return to standard                                                                             dx( x0 ,           0   )
terms of mathematical statistics.                                 Y(X , )            Y ( x0 ,                0   )                                      (X                  x0 )
    The first Lemma is about random properties of                                                                                dx
absolute and relative errors of DMs results.                           dx( x0 ,      0   )
                                                                                             (                   0   )                                                             (15)
If the true value of parameter X0 is measured with                         d
random absolute error of DMs                                   then and only then IMs result can be approximately
      X X0                                          (5)        presented in the form of sum of the DMs result and
and the random result of DM X has Gaussian                     the absolute error of measurements
distribution with numerical characteristics                     Y(X, ) X                                       (16)
M X        X 0,D X        2                                       This can be proved using the mathematical
                          x,                         (6)
                                                               expectation and dispersion of IMs results that can be
then the absolute and relative errors of DMs results
                                                               determined using standard ensemble averaging
also have Gaussian distribution with numerical
                                                               method of multiple realizations. Thus
                             2                                M Y( X, )        M Y ( x0 , 0 )
 M          0   0, D         x,                      (7)
                                                                         dx( x0 ,        0       )                               dx( x0 ,                   0   )
M     / X0     M           0,                                     M                                  (X              x0 )                                           (              0   )
                                                                             dx                                                      d
D    / X0     D       V x2          x
                                                    (8)                                  dx( x0 ,                    0   )     dx( x0 ,                 0   )
                                X   0                             x0      M                                                                                                 x0         (17)
                                                                                             dx                                    d
   Relations (7) denote that the absence of so-called
"Systematic error of DMs" is accomplished due to                  and
the absence of bias of evaluation of DMs (6).                                                                                dx( x0 ,           0   )
   Relations (7) show that the absolute error (5) of              D Y(X , )                      D x0                                                               DX
DM (while condition (6) is implemented) has the
same dispersion as the result of DMs.                                   dx( x0 ,         0   )
   Relations (8) show that while condition (6) is                                                        D
achieved, the mathematical expectation of relative
                                                                                                                 2                                                  2
error δ equals to zero, but its dispersion equals to the                      dx( x0 ,                       )                 dx( x0 ,                     )                      2
                                                                                                         0                                          0
square of variation coefficient X.                                DX                                                                                                               x   (18)
                                                                                  dx                                               d
VX                                                  (9)              Therefore, on accomplishment of necessary
                                                                  conditions (6)-(10) the IMs result represents non-
                                                                  bias evaluation, which has Gaussian distribution
                                                                  with parameters

         ISSN: 1109-2742                                   1083                              Issue 11, Volume 7, November 2008
                                                                                                    AYMAN AL-LAWAMA,ABDEL-RAHMAN -
            WSEAS TRANSACTIONS on COMMUNICATIONS                                                    AL-QAWASMI and OMAR AL-AYASRAH

M Y (X , )                    x0                                       (19)               D. Result of i-th IM in compliance with
                                                                                     statement Lemma 1 can be represented in the form
D Y (X , )                        x                                    (20)          Y i (t ) X 0 (t ) i (t ) , i = 1, 2,           (27)
and can be represented in the form (16), which                                       where i is the Absolute error of i-th DMs result.
required to be proved.
                                                                                        E. Numerical characteristics of i-th error are
   If the DMs are unbiased (7) and the systematical
                                                                                     defined by
error entire them is absented (12), then the IMs
using formula (12) give unbiased evaluation.                                          M i 0, D i D Yi , i=1, 2                       (28)
   Derivatives in (14) characterize transmission                                        F. Numerical characteristics of i-th IMs result in
coefficients of converted DMs results into IMs                                       (27) of initial point Yio are determined by
taking into account their absolute errors.                                            M Yio 0, D Yio 0, i 1,2                        (29)
   Condition (14) essentially represents the                                            G. Numerical characteristics of complex
condition of such normalization of transmission                                      evaluation Z,2) of IMs at initial point Y1o, Y2o are
coefficients of converters, which maintain adequacy
                                                                                     determined by
of DIMs
    Since                                                                             M Z Y1o , Y2 o     0, D Z Y1o , Y2 o   0.       (30)
dx( x0 ,     0   )           dx( x0 ,       0   )                                       H. The Condition of normalization of
                                                    ,                  (21)          transmission coefficients of channel transducers is:
    dx                           d                                                    2               2
from condition (14) followed that                                                           dz
                                                                                                              gi    1,                           (31)
                                                                                     i 1   dyi 0     i 1
  dx( x0 ,           0   )                 dx( x0 ,     0   )                                   dz
2                                     1,                        1/ 2   (22)          where            is the mathematical expectation of i-th
      dx                                       d                                               dy i 0
   If in DMs the absolute precision is achieved and                                  derived at initial point Yio.
represented by references, then the DMs result of                                       I. The condition of OCE Z (Y1 , Y2 )
corresponds to the real value                                                        dD Z
 X X 0, x 0 2
                                               (23)                                                 0, i 1,2                                     (32)
                                                                                      dg i
and the transmission coefficient of transducers in
                                                                                     where D[z] is the dispersion of complex evaluation
IMs is
                                                                                         J. The condition of minimum achievement D[z]
dx( x 0 ,    0   )                                                                   at point Zopt
                             1.                                        (24)
    dx                                                                               d 2D Z
                                                                                                     0, i 1,2.                                   (33)
                                                                                       dg i2
                                                                                     ,then and only then the complex evaluation
4 Analysis of optimum complex                                                                             2
evaluation of indirect measurements                                                  Z (Y1 ,Y 2 )              g iopt Yi                         (34)
Considering the case m =2, the founded results will                                                   i 1
be generalized into the case m >2. This case can be                                  will be optimum using criterion of maximum
analyzed using theorem 1, which contains necessary                                   accuracy, unbiasedness and effectiveness. Optimum
conditions of unbiasedness and effectiveness of                                      coefficients can be determined by relations
OCE in UA of IMs.                                                                                     D(Y2 )                     1
    Theorem 1 states that if in UA of IMs the                                        g1opt                                                   , (35)
                                                                                                   D(Y1 ) D(Y2 )           1 D(Y1 ) / D(Y2 )
following necessary conditions are accomplished:
    A. Measuring transducers of all channels that                                                      D(Y1 )                   1
                                                                                     g 2opt                                                  , (36)
have linear Gaussian characteristics transformation                                                D(Y1 ) D(Y2 )           1 D(Y2 ) / D(Y1 )
of DMs results into IMs results.
                                                                                     and the minimum value of dispersion of OCE (34)
    B. Numerical characteristics of initial real value                                                                          2
of parameter X00 is defined by the relation                                          D min Z opt          g iopt D Yi      1/         1 / D Yi   (37)
 M X 00 0, D X 00 0                               (25)                                                                          i 1

    C. Numerical characteristics of current real value                                  To prove that we use condition A and
X0 are defined by the relations                                                      represent Z (Y1 , Y2 ) in the form of expansion in
 M X0      X0,D X 0 0                             (26)                               Maclaurain series, in which we keep only the linear
                                                                                     terms as:

            ISSN: 1109-2742                                                   1084                         Issue 11, Volume 7, November 2008
                                                                                                AYMAN AL-LAWAMA,ABDEL-RAHMAN -
            WSEAS TRANSACTIONS on COMMUNICATIONS                                                AL-QAWASMI and OMAR AL-AYASRAH

                                           df                                         2
 Z ( Y1, Y 2)      f ( Y1, Y2 )                (Y1         Y10 )
                                           dy1                                      g2    D (g)
                                                                                                     2( D1       D2 )   0,                 (46)

      df                                                                               The condition of minimum achievement (I) is
          (Y2     Y20 ) .                                           (38)
                                                                                   carried out and the founded value gopt gives the
                                                                                   minimum of (42) that equals to
   To prove the unbiasedness of evaluation (34) we
define the mathematical expectation (38) and                                                         D1 D2
                                                                                   D2 min ( g opt )           g opt D1 (1 g opt ) D2 , (47)
considering the conditions B-F for corresponding                                                    D1 D2
mathematical expectations, which yields:                                           hence, the evaluation is effective
                                df            df                                       The received result allows entering three obvious
 M Z (Y1 , Y2 )            M        (Y1 Y10 )     (Y2               Y20 )          parameters of efficiency of OI of PVA systems:
                                dy1           dy1                                      An index parameter of maximum value of
          df          df                                                           efficiency of OI
      M                   X0,                                       (39)                  max(D1 , D2 )        1              D1
          dy1         dy1                                                          W1                                   1        ,         (48)
    If the condition of normalization G (31) is                                             D2 min            g opt           D2
accomplished                                                                       where D1>D2
 M Z (Y1 , Y2 ) X 0 ,                          (40)                                    An Index parameter of average value of
                                                                                   efficiency of OI when instead of an OE
, then the evaluation (38) represents an unbiased
evaluation.                                                                        Z 2 opt g optY 1 (1 g opt )Y2          (49)
   To prove the effectiveness of evaluation (38) we                                an arithmetic-mean value of results of two
will define its dispersion and considering at the                                  measurements is used
same time conditions B-F for corresponding                                          X 20 (Y1 Y2 ) / 2                     (50)
dispersions, we will get
                                                                                           D20        ( D1 D2 ) 2           (1 D1 / D2 )
                                                        df                          W2                                                   (51)
 D Z (Y1 , Y2 )         D f (Y 1, Y2 )                D     (Y1 Y10 )                      D2 min        4 D1 D2              4 D1 / D2
                                                                                       An Index parameter of the minimal value of
       df                            2
                                              df                                   efficiency (Minorant) of OI when instead of
 D         (Y2        Y20 )                                D(Yi )                  evaluation (49) the measured value on an output of
       dy1                           i 1     dy i 0                                that sensor with a smaller error is used
                                                                                          min(D1 , D2 )            D1
       g i2 D(Yi ).                                                  (41)          W3                        1        , D1<D2              (52)
 i 1                                                                                        D2 min                 D2
     Considering (41) in condition (31) and choosing                                  Let's designate dimensionless relation D1/D2
 g1 = g, we get                                                                    through U, U.         [1, ) . We shall express
D z ( ) g 2D 1 (1 g ) 2 D 2
      g                                          (42)                              parameters of efficiency of OI as function of U:
     The optimum value of g at which condition H                                                         (1 U ) 2                    1
 (32) is satisfied when the minimum value of                                       W1     1 U , W2                , W3         1       .   (53)
 dispersion is reached:                                                                                    4U                        U
                                                                                      Let's establish interrelations between these
  g opt arg min z (g )
                 g [0,1]   D                    (43)
                                                                                   parameters. As U=W1-1:
    A standard method can be used to find an                                       W1 W2 /(W2 1) ,W 2 W12 / 4(W1 1) ,
extremum of one-variable function. First, we find                                  W3 W1 /(W1 1) .                                  (54)
the first derivative of function (42) on g and second,
we equate the result of derivation to zero. The                                        On figure 1 graphics of functions (53) depending
optimized value will be                                                            on U are shown. By increasing U from 1 (a case
 2 g opt D1 2(1 g opt ) D2 0 .                    (44)                             equally accurate measurements) and up to          (the
                                                                                   case of integration with reference system at which
       Solving equation (44), we get                                               D2 = 0), a proportional increasing in W1 is observed.
                                              1                                    It designates that the use of a complex evaluation
 g opt     D2 /( D1           D2 )                                   (45)
                                           1 D1 / D2                               (49) gives the maximal gain on the accuracy,
       As the second derivative of function (42) is                                proportional to the relation of dispersions.

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                                                                      AYMAN AL-LAWAMA,ABDEL-RAHMAN -

                 10           W1


                         0     1    2       3      4          5   6       7       8       9       10

                                         Figure1: Graphics of function 49
    The use of OE (49) comparing to the mean-                   For definition of an arithmetic-mean error of
arithmetic value (50) gives an increasing in                measurement of airplanes altitude by radio locators
effectiveness parameter in number of times (curve           range we shall consider, that between the range of
2), in how many times the square of arithmetic-             operation R and height Н of airplanes flight is a
mean value of dispersions of measurements exceeds           relation
the square of compound value of these dispersions.
                                                             H R sin arctan( H / R 2 H 2 ) R sin (55)
The application of substitution method (Curve 3) is
more effective because there is more difference                 At R = 400 км, Н = 10 км, tg (10/400) ≈ 0.25, β
between dispersion measurements. When U=1                   ≈ 14º, sin14º ≈ 0.2419. Applying linear
(D1=D2) measurements are to be equally accurate             approximation (55) for evaluation root mean square
and formulas (53) result in known results                   errors of measurement of altitude of radiolocators
W1(1)=W3(1)=2, W2(1)=1. Thus, OI (Coarse) and               range               we           shall           get
                                                                              2     2
(Exact) measurement systems with the application               H2     0.2419 250         60.475 M.
of unbiased evaluation (49) and coefficients (45)                                      2
                                                                   Thus, DH1 ≈ 3657 м , DH2 ≈ 1406.25 м2 , U ≈
provide achievement of minimum value of general             2.6,     gopt ≈ 0.72226, 1-gopt ≈ 0.27774, D2min ≈
dispersion of measurements (47). The integration is         0.72226 x 1406.25 ≈ 0.27774 x 3657≈1015.82 m2,
more effective when the errors of complex system            σ2min ≈ 31.87 м, W1(2,6) ≈ 3.6, W2(2.6) ≈ 1.24615,
have more differences between each other.                   W3(2,6) ≈ 1.38461. From here follows, that in
    We will observe two characteristic examples of          comparison with an arithmetic-mean evaluation OI
OI of aerospace navigational systems.                       allows to reduce approximately by 25 % a
    In example1 we will observe what the OI of              dispersion of radiolocators measurements of flight
primary and secondary radiolocators gives, when             altitude ,that is especially actual at reduction of
considering the measurement of flight altitude of           ranges separation and use the concept         « free
airplanes as control objects.                               flight ».
    We will use the data of work [7]. Arithmetic                In Example2, we shall examine features of OI
mean error of measurement of distance R using               systems GPS and INS [1]. For GPS root-mean-
radiolocators range at flight altitudes H=(10-20) км        square error of Doppler, a measuring instrument of
equals to σR = (0,2-0,25) км. Arithmetic mean error         velocity σv2 ≈ 0.01 m/s. For INS root-mean-square
σн2 altitude transmission H by transponders of              error of an independent measuring instrument of
secondary radiolocators through readings of flight          velocity σv1 ≈ 0.5 m/s for one hour of flight. We
instruments equals to σн2 ≈ 37,5 м. We will perform         shall determine the efficiency of OI-GPS and INS
through this data the OI and we will evaluate its           on measurement of velocity.
effectiveness using formulas (53).                              Let's calculate the dimensionless parameter U
                                                            and weight factors (45) and evaluations (51):

            ISSN: 1109-2742                            1086              Issue 11, Volume 7, November 2008
                                                                                  AYMAN AL-LAWAMA,ABDEL-RAHMAN -
         WSEAS TRANSACTIONS on COMMUNICATIONS                                     AL-QAWASMI and OMAR AL-AYASRAH

    U = σv12/σv22 = 0.52/0,012 ≈ 2500, gopt =                    Thus one system asymptotically comes nearer to
σv22/(σv12+σv22) = 0.012/(0.52+0.012) = 0.0339984, 1-            reference so that its error approaches zero. It is a
gopt = 0.9360016.                                                limiting case of s rough-exact measurements when
    Let's determine values of D2min, σ2min and                   one system allows defining the approached value of
parameters of efficiency W1 - W3:                                parameter, and the second system allows estimating
    D2min ≈ gopt x σv1 ≈ 0.0339984 x 0.25 ≈ 0.0339984
x 0.25 ≈ 0.04936001.
                                                                 value with high accuracy.                        m0 [2, ] , is   a
    σ2min =0.009998 m/s. W1 = 1+U =                              number that can conveniently be used as integrated
1+2500=2501. W2 = (1+U) 2/4U = (1+2500)2/4 x                     characteristic of various UA complexes which
x2500 = 625.5. W3 = 1+1/U = 1+1/2500 = 1.034.                    allows comparing them among themselves.
    Let's calculate the contribution σv12 in general                 In turn, the size u can be considered as inverse
dispersion D20:                                                  function for (59), which is as a function of
    ΔD1 = gopt2 D1 ≈ (0.0339984)2 x 0.25 ≈ 0.073996.             characteristic number of a complex.
    Let's determine a relative error of a substitution                m0 1
method:                                                          u         ,                                                  (62)
    Δ = ΔD1/ D20 ≈ 0.073996/0.04936001 ≈                              m0 2
(0.0339975986) ≈ 0.04%.                                              Relations (59) and (62) form pair transformations
    Results of calculations allow doing an                       which completely describe complexes UA
unequivocal conclusion that in this case the most                measurements in which attitudes of dispersions of
effective is use of a method of substitution. It leads           the next measurements are identical.
relative root-mean-square to an error, smaller than                  We shall establish a relation and attitudes of the
0,2 %. In other words, results of measurements INS               entered integrated characteristics of complexes UA
are expedient for adjusting approximately in each                measurements with parameters of accuracy [8]. A
hour of flight by results of satellite measurements              parameter of accuracy hi i-th measurements Yi and
GPS which can be considered as reference.                        its dispersion D [Yi] are connected by relations:
    For complexes with UA systems it is useful to                hi   1 / 2 D Yi , D Yi                      1 / 2hi2 .       (63)
enter characteristic number m0 as follows. We shall
designate dispersions of measurements so that the                   In the classical theory of UA measurements enter
condition was satisfied                                          « a normal measure of accuracy »
 D Y 1 D Y2 .                                      (56)
                                                                          2         gi ( X 0        Yi ) 2
    Let's consider these dispersions as first two                             i 1
members of a decreasing arithmetic progression                   h                                                2D Z m ,    (64)
                                                                                      m 1
with a difference                                                   a weight coefficients gi is to be found from the
 d D Y1 D Y2 ,                                    (57)           condition
    Then from the condition                                            hi2
 D Y1 d (m0 1) 0,                                 (58)           gi        , i 1, m,                                         (65)
    It is possible to find such number m0 « the virtual             for evaluation of parameter that can be found in
channel » which possesses a zero dispersion of                   the form
measurement, that has a zero error:                                            m                m

          D Y1           2u 1                                    Z ( m)              g i Yi /         g i , but thus leaves open a
m0      1                     .                   (59)                         i 1              i 1
           d             u 1                                     question of a choosing h for a case when gi is
     We investigate limiting properties of function              unknown.
m 0 (u )                                                            To increase the accuracy of measurement, it is
lim m 0 ( u)         ,                            (60)           convenient to use the index
u   1
                                                                          max hi                min   Z opt (m)
lim m 0 ( u)     2                                (61)           W4                                                 .         (66)
u                                                                         min hi                       (Y1 )
   The singular case (60) corresponds to asymptotic                  The value (66) shows, what percentage makes
approach of d to zero that is approach UA                        minimal root mean square value of an OCE rather
measurements. The singular case (61) corresponds                 root mean square values of the worst measurement.
to asymptotic approach d to D (Y1). It shows                     It shows relative reduction of a field of the tolerance
approach of a complex from large number of UA                    of an OE in comparison with a field of the tolerance
measurements to a complex of only two systems.                   for the first measurement.

         ISSN: 1109-2742                                  1087                          Issue 11, Volume 7, November 2008
                                                                                                AYMAN AL-LAWAMA,ABDEL-RAHMAN -
                    WSEAS TRANSACTIONS on COMMUNICATIONS                                        AL-QAWASMI and OMAR AL-AYASRAH

                   1               1       1             m0 2
W4                         1                                   (67)
                   W1              W2    1 u            2 m0 3




                               1               2             3                     4                  5                   6         7

                  -0.4         W6(m0)



                                        Fig.2. Graphics of functions W4(u), W4(m0), W5(u), W6(m0)

    It is convenient to consider both of root signs in                                 We Use (71) for the proof of conditions of
(67) evidently to illustrate narrowing a field of the                              existence and uniqueness of the optimum decision at
tolerance. On fig.2 graphics W4 (u) W4 (m0) are                                    of UA measurements.
shown, negative branch W4 (u) is designated as W5                                      Theorem 2 states that if:
(u), negative branch W4 (m0) is designated as W5                                       A. Conditions A-J of the theorem 1 are carried
(m0).                                                                              out for all of UA measurements.
    It is possible to notice, that as function of                                      B. Range sequence of the known dispersions,
arguments u and m0 value W4 has the following                                      satisfying to a condition (70), it is possible to
limiting properties:                                                               approximate an arithmetic progression (71).
lim W4 (u ) 1 / 2 , lim W 4 ( u)                   0,                (68)              C.    The Characteristic number m0 of UA
u 1                                 u                                              measurements satisfies samples of results to an
lim W4 (m0 )                0, lim W4 (m0 ) 1 / 2 .                  (69)          inequality
m0    2                            m0
                                                                                               D0 (Y1 )
   Let's pass to consideration of a case m> 2. We                                  m0     1                     3,                        (73)
shall number dispersions by a way of their decrease.                                            d0
We shall receive range sequence D[Y1], D[Ym], for                                       That for all m which satisfy to an inequality
which the condition is valid                                                       3     m    m0                                 (74)
 D Yi D Y i 1 , i 1, m.                        (70)                                   There are optimum values of evaluation of a true
   We approximate decreasing sequence of                                           value of parameter
dispersions as an arithmetic progression on a                                      Z opt (Y1, Y m )       Z opt (Y1 , Ym 1 )
method of the least squares:
                                                                                          Dmin Z opt (Y1 , Ym 1 )
 D0 (i ) D0 (1) d 0 (i 1), i 1, m,             (71)
where D0(i) - is an OE of i-th member of the                                       Dmin Z opt (Y1 , Ym 1 )           D(Ym )
progression, received by optimization D0(1)and d0                                  Z opt (Y1 , Y m 1 ) Ym ,                               (75)
from a condition of minimization of the sum                                        that have the minimal values of dispersions
S D0 (1), d 0                      D0 (1) d 0 (i 1) D(Yi ) (72)
                                                                 2                  Dmin Z opt (Y1 , Ym
                            i 1                                                                   D(Ym )
                                                                                   Dmin    Z opt (Y1 , Ym 1 )        D(Ym )

                    ISSN: 1109-2742                                         1088                      Issue 11, Volume 7, November 2008
                                                                            AYMAN AL-LAWAMA,ABDEL-RAHMAN -
          WSEAS TRANSACTIONS on COMMUNICATIONS                              AL-QAWASMI and OMAR AL-AYASRAH

D min Z opt (Y1 , Ym 1 )                          (76)           enough to prove, that the minimal value of a
                                                                 dispersion of the OE, received in the previous step,
    To prove this theorem its clear that as the
arithmetic progression (71) is decreasing, for the
proof of validity of recurrent relations (75), (76) it is
      m                1       2          3           4           5    6      7      8                         9
      Ym             0,98 1,02           0,95       1,04        1,01 1,005  0,996  0,999                     1,000
      D(m)           0,04 0,035          0,03      0,025        0,02 0,015   0,01  0,005                     0,000
      V(m),%          20     18,7       17,32      15,81        14,1 12,25   10,0   7,07                     0,000
      Z(m)           0,98 0,998666 0,980000 0,998912 1,002046 1,002854 1,000889 1,000194                     1,000
      Dmin(m)        0,04 0,01867 0,01151 0.00788 0,005653 0,004105 0.002911 0,00184                         0,000
      V0(m),%        20,4 13,68          10,9       8,88         7,5 6,389   5,39  4,288                     0,000
      V0(m)/V(m) 1,02 0,7355 0,62933 0,56167 0,53191 0,52155 0,539                0,6065                     0,000
      W4,%           100 68,32           53,6       44,3       37,59 32,03   26,9  21,44                     0,000
      Em,%            -2       2          -5          4           1   0,5    -0,4   -0,1                      0,00
      E0,%            -2    -0,133     -1,999 -0,1088          0,204 0,285 0,0889 0,0194                     0,000
                                                          Table 1

    always there is less than dispersion of UA                   accordingly, in the second - the fourth lines. In the
measurement on a following step.                                 fifth - the seventh lines results OI by means of
    We shall show validity of this statement for first           recurrent parities (75), (76) are reflected,
three measurements. For this purpose it is necessary             accordingly: Zopt (m), Dmin (m), V0 (m). In the eighth
to prove validity of an inequality                               line the attitude of factors of a variation of OEs and
                        D0 (1)                                   measurements is given. In the ninth line the change
 D Zopt (Y1 ,Y2 )
  min                              D0 (2) D0 (3) . (77)          of a parameter of efficiency OI is shown. The tenth
                    D0 (1) D0 (2)
                                                                 and eleventh lines show, how the values of relative
    Let's consider that the approximated values of
                                                                 errors in this realization of experiment is changed.
dispersions of UA measurements are connected by a
                                                                     The analysis of experimental results shows, in
condition (71) then from (77) we shall receive
                                                                 the first, validity of the offered theoretical positions,
D0 (1) D 0 (1) d 0          D0 (1) 2d 0                          and in the second, high efficiency OI even at rather
    2 D0 (1) d 0 .                            (78)               low accuracy of primary measuring systems. The
   Removing the brackets and carrying out the                    third and fifth lines evidently show speed
elementary transformations of this inequality, we                asymptotic to convergence of the minimal
shall receive finally                                            dispersion to a zero with growth m.
 D0 (1) 2d 0          0, D02 (3)   0.            (79)
    From validity of an inequality (79) validity of              5 Conclusions
recurrent relations (75) and (76) follows at m=3 ,to                1. Optimum integration aerospace of navigation
use (45) in (47) and (49) at definition Zopt (Y1, Y3).           systems is an important and actual problem. Solving
Applying a method of an induction for m> 3 for all               it will promote an increase of accuracy of the
m, which satisfy to an inequality (74), we shall get             approached calculations, approximations of the
analogues of an inequality (79) for dispersions with             functions, scientifically use of redundant volumes of
these numbers as was shown.                                      the measuring information, optimum construction of
    We shall consider results of experimental digital            processing algorithms of UA measurements and the
imitating modeling of UA measurements. Optimum                   control and diagnosing of systems to construct exact
integration results are executed at the following                complexes from inexact systems.
modeling initial data:                                              2.      The Examined lemmas and the proved
X0=1.0, D0(1)= 0.04, d0= - 0.0025; m0 = [D0(1)/d0]               theorems about necessary conditions of OI results in
+ 1 = 9.                                                         UA measurements are a theoretical basis of
    In table 1 which contains eleven lines and ten               construction of exact complexes from inexact
columns, results of realization of UA measurements               systems through UA measurements.
and their OI are shown. The first line contains                     3. In-depth, the study of the simplest case of
number of measurement m. Realizations of UA the                  two UA measurements allows entering a number of
measurements, the smoothed values of their                       constructive offers to assess the capability of
dispersion and factors of a variation are displayed,

          ISSN: 1109-2742                                 1089                  Issue 11, Volume 7, November 2008
                                                                        AYMAN AL-LAWAMA,ABDEL-RAHMAN -

efficiency of optimum indirect of UA                              and Innovative Application of          Satellite
measurements. The Use of concept of characteristic                Navigation, NATO, Paris, July 1996.
number m0 (59) complexes of UA measurements                   [2] Greenspan, R. L., “GPS and Inertial
deserves special attention. This number is                        Integration,” in Global Positioning System:
unequivocally connected with all parameters of                    Theory and Applications II, edited by Bradford
efficiency of a complex and allows not only to                    W. Parkinson and James J. Spilker, Volume 164
compare successfully such complexes among                         of the Progress in Astronautics and aeronautics
themselves, but also to determine a degree of their               Series, AIAA, Washington, DC, 1996, pages
proximity to complexes of equal accuracy                          187-220.
measurements.                                                 [3] Copps, E., M., Geier, G. J., Fidler, W. C., and
    4. Conditions ABC, (73), (74) theorems 2,                     Grundy, P.A., “ Optimum Processing of GPS
existence of optimum decisions for the general case               Signals ”, Navigation: Journal of the Institute
2<m<m0 allow to construct recurrent relations (75),               of Navigation, Vol. 27, No 3, Fall 1980.
(76). They are useful not only for OC of processing
of the results of UA measurements, but also can find          [4] Ignatov V.A., Bogolyubov N.V, "Management
application in the decision of problems of the                    of information redundancy of systems for
measurement, the approached calculations, the                     diagnosing and controlling ", the collection of
control, diagnosing, evaluate of reliability and                  proceedings, Kiev, KIUCA, 1990, pages 3-13.
others. Obvious advantages of application (75), (76)          [5] Ignatov V.A., Markevich S.K, Dynamic logic-
consist in reduction of memory sizes of computers                 expectation method of studying special cases of
and in an opportunity of a choice of demanded                     flight. ", the collection of proceedings, Kiev,
number of systems from a condition of maintenance                 KIUCA, 1988, pages 3-14.
of the set accuracy of measurements.                          [6] Ignatov V.A. "Theory of the Information and
Acknowledgement                                                   Signal       transmission",      Radio     and
Authors is grateful to Prof. Ignatov V.A ,for his                 communication, 1991.
advices ,useful discussions of the results and                [7] Androsak A.I,Demanchuk V.C, Yoriev U.M.
inspiration in preparing this publication .                       "Merega         of       Avia-Communications",
References                                                    [8] Guter R.S., Ovcjinski B.V " Element of the
[1] Phillips, Richard E. and George T. Schmidt,                   numerical      Analyis     and    Mathematical
    “GPS/INS Integration”, AGARD Lecture                          processing of the results of experience",
    Series MSP LS 207 on System Implications                      FIZMATGIZ,1962

        ISSN: 1109-2742                                1090                 Issue 11, Volume 7, November 2008