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AYMAN AL-LAWAMA,ABDEL-RAHMAN - WSEAS TRANSACTIONS on COMMUNICATIONS AL-QAWASMI and OMAR AL-AYASRAH Optimum Control Using Signal Processing in Integrated Aerospace navigation systems AYMAN AL-LAWAMA* , ABDEL-RAHMAN AL-QAWASMI**and OMAR AL-AYASRAH* * Electrical Engineering Department Mutah University Karak– 61710 P.O. Box (7) JORDAN ay_lawama@yahoo.com http://www.mutah.edu.jo ** Communications and Electronics Department Philadelphia University P.O. Box: 1 Amman 19392 JORDAN 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 ISSN: 1109-2742 1081 Issue 11, Volume 7, November 2008 AYMAN AL-LAWAMA,ABDEL-RAHMAN - WSEAS TRANSACTIONS on COMMUNICATIONS AL-QAWASMI and OMAR AL-AYASRAH 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 characteristics 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 ) 2 dx d D / X0 D V x2 x 2 (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 2 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 dx DM (while condition (6) is implemented) has the 2 same dispersion as the result of DMs. dx( x0 , 0 ) Relations (8) show that while condition (6) is D 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 x VX (9) Therefore, on accomplishment of necessary x0 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 2 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 ( YY 1 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 2 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) z 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 dy1 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 dy1 An Index parameter of the minimal value of 2 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 2 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. ISSN: 1109-2742 1085 Issue 11, Volume 7, November 2008 AYMAN AL-LAWAMA,ABDEL-RAHMAN - WSEAS TRANSACTIONS on COMMUNICATIONS AL-QAWASMI and OMAR AL-AYASRAH Wi 12 10 W1 8 W1,W2,W3 W2 6 W3 4 2 0 0 1 2 3 4 5 6 7 8 9 10 U 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 » m 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) h2 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 0.8 0.6 0.4 W4(u) 0.2 W4,W5,W6, W4(m0) 0 1 2 3 4 5 6 7 W5(u) -0.2 -0.4 W6(m0) -0.6 -0.8 u,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 m 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. 2 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 - WSEAS TRANSACTIONS on COMMUNICATIONS AL-QAWASMI and OMAR AL-AYASRAH 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", NAU,2001. 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

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