УКЛОНЕНИЕ ПОДВИЖНЫХ ОБЪЕКТОВ ОТ ОБНАРУЖЕНИЯ НА ПЛОСКОСТИ И В ПРОСТРАНСТВЕ

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  • 2 CONTROL SC I ENCES 3 2008

    517.977

    .. , .. , ..

    . .. , .

    - - ( ) -, , [15] . -- , --.

    - .

    - , - , - [2, 6].

    - - :

    - [6, 7], [2, 3, 8] [9];

    [3, 4] [5, 8] ;

    - [15] [7, 8];

    [2, 3, 5, 8, 10]; [2, 3, 5, 6].

    -- - [1118].

    - , -, - . , - , , - , (-) . - , , .

    - - -, -. ( - ) ( ) , - .

    - ,

    , -

    .

    ( ) ( -

    ) .

  • 3 3 2008

    ( ) .

    - [1, 6, 10, 19].

    1.

    1.1.

    P E, - , , - ( , - [20]). R ( - (R, 2)), - (. 1). - .

    X0Y, -

    P0, - - .

    = v u, w(0) = w0, (1)

    w = w(t) , P E t. , - ,u = (1; 0). v (-) E ||v || m < 1; - . ; 0 , |||| .

    - :

    G = ||w(t)|| . (2)

    (1), (2) [1, 4, 6]. - . , , * = ; cos = ( ). ABCD (. . 1).

    -:

    < y = x ctg; (3)

    l y = x ctg. (4)

    l arccos - , < arccos .

    , .

    (3) (4) , - - - . - = . .

    y = cosx , , (2); - . -

    . 1. () ()

    w

    mint

    maxv

    R ( )cossin

    -------------------------------

    Rsin

    -----------

  • 4 CONTROL SC I ENCES 3 2008

    (2), . - , - .

    , - ABCD, . .

    1.2.

    P, E1

    E2

    , .

    , R, , . - - -. , - . - r m R.

    () X0Y (. . 1, ), -

    P0 -, - t = 0, - - , - , - (R/1, 0), 1 .

    -

    = = E 0 -

    , .

    ().

    = vi u; i = 1, 2, (5)

    wi = w

    i(t) ,

    P Ei t; u, v

    i

    () - E

    i

    ||u || m 1, ||vi|| m

    i < 1. (6)

    - , - -. .

    - :

    G1 = ||w1(t)|| .

    -

    G2 = T s ,

    s T ||w2(s)|| = R, ||w2(T )|| = r.

    , -, . - .

    . - - ( 2), ( 1).

    .

    X s0sY s,

    P s - s , -

    (. 2). s - .

    - :

    = vi u; i = 1, 2;E1

    0E2

    0

    w i

    mint

    maxv1v2

    ,

    minu

    . 2.

    E2s

    w i

  • 5 3 2008

    = wi(s); i = 1, 2,

    ||w2(s)|| = R;

    G1s = ||w1(t)|| , G2s = T

    ;

    ||w2(T )|| = r.

    . 2, ,

    , : s > 0. -

    - .

    [15]. -

    . -

    . -

    X s0sY s. r -

    P s .

    P s , 0SXS, [21]:

    (1 )2 + 2( r Rcos) + R2 r2 = 0

    - , .

    , P s -

    , :

    E2: cos = 2, sign =

    = signs;

    : cos* = +

    + , sign* = signs;

    E1:

    = 1 + *, sign = signs, (7)

    = r/R; cos1 =

    1.

    - , - .

    (7) . 2 , - E

    1 -

    , - .

    -

    = || ||cos(| | |s |)

    = || ||(1 )1/2sin(| | |s |).

    ()

    T = R r.

    , - s .

    s -

    :

    , ,

    , | | |s | = |1 +

    + *(22)| |s | = 0;

    | | |s| > 0.

    -

    , = | |.

    . , - , .

    . - - :

    (5) (6); X0Y, , - u = (1; 0);

    ||w2(s)|| = R;

    = || ||cos(| | |s|) . (8)

    [15] , - , -. (8) :

    = || ||cos(| | |s |) , 0 m t m s,

    i1, i = 1, 2 E

    i

    X0Y. (. 3).

    P S , -

    wis

    mint s

    maxv1v2

    ,

    minu

    w1s

    w2s

    E2s

    22 2

    2 22

    E2s

    22* 22*

    22

    1 22( ) 1 2

    22( )

    12* 12*

    G1s* w1

    s 12*

    w1s 1

    2 12*

    1 222( ) 1 2

    2( )1

    w1s

    w2s

    12*

    12*

    G1s* w1

    s

    G1s* w1

    s 12* maxv1v2

    ,

    G1s* w1

    s 12* max11

    21

    ,

  • 6 CONTROL SC I ENCES 3 2008

    . E

    i, i = 1, 2 s

    is E 0.

    s.

    P s P s -

    Ps > 0. -

    P sB . -

    P sB P sB .

    ( 21 )

    , 11

    , B

    1s. PsE 0

    : E 0P s 1,

    E 0P s 2, Ps s, E 0P sx q s.

    (8) , 1 2 .. 3

    1 > 0,

    2 < 0.

    -

    P sE 0. A(. . 3).

    21

    > 0 (. . 3):

    = = |P sC | + |BC | =

    = cos(| | |2|) +

    1s,

    l 0 = ||w0|| - .

    P sB E 0 ,

    E 0P sB = AE 0 , . .,

    , 1, 2 qs, + 2 = q

    s.

    , E

    1 -

    X0Y = + 2 + qs. ,

    (. . 3), + 2 + q s

    , - E

    1 ,

    X0Y. , - E1

    . -

    E2.

    P sE 0 : P sE 0 = |qs 21|.

    sin|2|/

    2s = sin|q s

    21|/R, (9)

    tgqs = (l 0sinq0)/(l 0cosq0 s), (10)

    s

    (l 0cosq0 + 2s cos21 s)2 +

    + (l 0sinq0 + 2s sin

    21)2 = R2. (11)

    21

    (9)(11)

    E2 .

    = R

    w0 , - - .

    . 4 13, - . -

    E1s

    E2s

    E2s

    E1s 12* E1

    s

    G1s*

    E1s

    . 3.

    E2s

    E1s

    E1s=

    E2s=

    E1s

    E2s=

    =

    E1s

    11*

    G1* max

    11

    G1s*

    l0

    q0

    cos s( )2

    l0

    q0

    sin( )2

    + 12*

    E1s

    E1s

    12* 12* 11*

    11* 12*

    12*

    E2s

    E2s

    G1*

    21*

    max21

    G1*

    . 4.

  • 7 3 2008

    1 =

    2 = 0,4 = 0 ( 1),

    = 0,2 ( 2), = 0,4 ( 3). , -

    .

    -

    . E 0 -

    -

    , ||w1(t)||.

    .

    s -

    , P s,

    , .

    -

    2s E 0 -

    R P s (. 5).

    P s E 0

    , .

    2s . PsE 0 -

    .

    -

    . P sE 0

    , .

    .

    -

    . ,

    |P s | = |P s |, |E 0 | = |E 0 | = 1s,

    |E 0 | = |E 0 | = 2s,

    s . - , - , - - .

    , - , - - .

    2.

    2.1.

    - P E.

    - 0XYZ,

    P 0; - ; 0Y - 0X ; 0Z -.

    = v u, w(0) = w0, (12)

    w = w(t) = [x(t), y(t), z(t)] , P E - t. , - , ; , , u = (1, 0, 0). v () E ||v || m < 1.

    -, - , :

    - ;

    - 0X , ;

    , , - x = kD,0 < k < 1, D , - .

    .

    mint

    . 5.

    E1s

    E2s

    E2s

    E2s

    E 2s

    E2sE 2

    s

    E 2s

    E2s

    E 2s

    E1s

    E 1s

    E 1s

    E 2s

    E2s

    E 2s

    E1s

    E 1s

    E2s

    E 2s

    w

  • 8 CONTROL SC I ENCES 3 2008

    -:

    = (x, y, z): (13)

    n2 = k2/(1 k2) [9].

    , w(t) ,t l 0.

    G, - 1, w(t) , 0 m t < , 0 .

    G .

    , x = h; 0 m h m D. - S(h) = S

    h. -

    , - . S

    h:

    + = 1;

    - . - (13) S

    h,

    h, , ah/b

    h = tg/tg.

    > , ah > b

    h, 0 < h < D.

    Ch -

    Sh:

    Ch = (x, y, z): + 1 = 0 ,

    0 m h m D.

    (w0) .

    CkD. w0, -

    - C

    kD, , -

    , v = (, 0, 0). -

    [9]. - S

    h -

    , - ., , -

    - .

    -. S

    h -

    , T w(T ) - C

    h -

    x(T ) l h. -

    Sh

    (12), G = x(T ) -

    + 1 = 0. (14)

    .. - , [9].

    - , - .

    x ,

    ( v) ,y , 0Y

    vYZ v Y0Z (. 6). -

    cos = , (15)

    ctg = . (16)

    - (16). w(T ) = [x(T ), y(T ), z(T )] -

    y2

    tg2

    -----------

    z2

    tg2

    ---------- x2+ 0< 0 x kD, ,

    x2

    D2

    -------

    y2

    n2D

    2tg

    2-------------------------

    z2

    n2D

    2tg

    2------------------------ 1+ + 0, 0.

    S(t, w0) W

    S

    w0

    G* = x(T *, w0) = h, - , .

    , - :

    ;

    Sh

    Sh,

    w0, ES(h)

    Sh, .

    w0 == [x(0), y(0), z(0)], [y(T *), z(T *)] (14), :

    = = . (18)

    (18) , E

    w0 w(T *).

    (21) , WS -

    S(t, w0)

    [18]. S

    h;

    , - S(h) .

    w0 = [x(0), y(0),z(0)] : E

    S(h)

    Sh

    [y(0), z(0)] E

    x

    x(0) > h x(T *) = h (1 2). -

    - :

    m > 0 , cos = S

    h: y = y(T ) = a

    hcosm, z = z(T ) = b

    hsinm,

    ah > b

    h.

    bh2

    ah2

    -----

    y T( )z T( )-------------

    ah2

    bh2

    ah2

    1 2

    x*

    x*

    h x 0( )

    2 1--------------------

    y T *( ) y 0( )

    1 2 y*cos---------------------------------------

    z T *( ) z 0( )

    1 2 y*sin--------------------------------------

    1 2

    y 0( )bhah-----z 0( )ctgm ahe

    2m,cos+=

    x 0( ) h[ ]2ctg2 y 0( ) ah mcos[ ]2 +=

    z 0( ) bh msin[ ]2,+

  • 10 CONTROL SC I ENCES 3 2008

    LS , -

    WS

    X0Y. w0 LS -

    , - ; , L

    S -

    w0 = (h, ah, 0) -

    Sh (,

    - S

    h).

    [9] , LS

    -.

    + = 1, cos = . (19)

    (19) , S

    h -

    .

    x(0) h = [ah y(0)]tg. (20)

    (20)

    (19) w0 = [h + ah(1 e2)tg, a

    he2, 0].

    (19) 0X x = h ++ b

    htg. L(h) , -

    Sh X0Y. . 7

    Sh 90

    X0Y LS. F -

    Sh, y

    F > a

    he2.

    HH1 S

    h -

    , L

    S.

    . 8 - - -

    S(t, w0). -

    WS S

    h

    ( ), 0 m h m kD.

    WS -

    , S

    h. (18)

    , LS , -

    X0Y.

    Sh;

    (ah= b

    h = r, = ) W

    S -

    . (15) (19)

    w0 = [x(0), y(0), z(0)] ES(h)

    Sh. -

    WS:

    [x(0) h r tg]2 = [y2(0) + z2(0)]tg2. (21)

    (21) (-, ), - (h + r tg, 0, 0).

    -

    . (13) -

    x 0( ) h[ ]2

    bh2tg

    2---------------------------

    y20( )

    ah2

    bh2

    -----------------

    . 7.

    . 8.

  • 11 3 2008

    -

    + + 1 < 0,

    kD m x m D, n2 = k2/(1 k2), 0 < k < 1.

    E - .

    w0 = [x(0), y(0), z(0)] , E, . {||v || = ,x= arccos,

    y = const} -

    E, -, E ( ) S(h) E

    S(h) [y(0), z(0)] -

    . h - E

    S(h) -

    [x(0) h]2ctg2 = [y y(0)]2 + [z z(0)]2. (22)

    h (22), - - , - E. [x(0), y(0), z(0)]; /2 , = arccos.

    , -, L

    S, -

    Sh,

    y(0) = z(0)ctgm + ahe2cosm,

    x(0) = h +

    + tg . (23)

    {h, y(0), z(0)}

    (23) w0 - x(0), E S(h), , 0X, ( m = m(h)).

    (w0) - [9]. , - ( ( = ;

    k = cos), (w0) ). :

    K = {(x, y, z): y2 + z2 x2tg < 0;

    0 m x m Dcos};

    = {(x, y, z): x2 + y2 +

    + z2 D2 < 0; Dcos m x m D}. -

    , , . .

    , m , cos = , -

    = [y2(0) + z2(0)]tg2. (24)

    -

    C: .

    . 9 - ,

    0X. P 0 ; AB - - . , - (24).

    , > , .

    P 0, , - K

    ,

    , = arccos. r = D sin. , , - W

    ,

    K (. 10). A, A

    1 C,

    ,

    x2

    D2

    -------

    y2

    n2D

    2tg

    ----------------------

    z2

    n2D

    2tg

    ---------------------

    bhah-----

    y 0( ) ah mcos[ ]2

    z 0( ) bh msin[ ]2+

    x 0( ) D ( )coscos

    --------------------------2

    D ( )coscos

    -------------------------- 0 0, ,

    . 9.

  • 12 CONTROL SC I ENCES 3 2008

    , . , W

    , .

    2.2.

    P, E1

    E2 , -

    , , - .

    R, . -. - - . - , - . - r < R.

    0XYZ, -

    P 0, - -. .

    - [9] = arcsin

    1,

    1 -

    -; (R/

    1, 0, 0).

    -

    = = E 0 -

    , . 0XYZ ,

    E 0 X0Y

    (x0, y0, 0). , -

    : () - ().

    = vi u,

    i = 1, 2, w1(0) = w

    2(0) = w0, w

    i = w

    i(t) -

    , P E

    i t, u v

    i

    () P Ei, .

    , - ; ||u || m 1; ||v

    i|| m

    i < 1, i = 1, 2.

    .

    -

    : G1 = ||w

    1(t)||

    .

    - ()

    : G2 = T s , s T -

    -, , - ||w

    2(s)|| = R; ||w

    2(T )|| = r.

    [16, 18]., - -

    , , , -

    P s s

    -

    E 0, (. . 3). . -

    , -

    E 0 P s

    . - - - , , -, .

    , ,

    . 10.

    E10

    E20

    w i

    mint

    maxv1v2

    ,

    minu

    w1s

    w2s

    w0s

    E1s

    E2s

  • 13 3 2008

    , , , - . 4.

    - - ( ) () - -.

    , , , - . -, - - , - , - () .

    - . - . - , - , - .

    , , -.

    1. .., .. . .:- , 1992.

    2. .., .. . .: . -, 1977. 334 c.

    3. .., .. . .: , 1978.

    4. .. . .: , 1989.

    5. .., .. - . .: - , 1987.

    6. .., .. -. // - . 1995. . 1, . 4. . 827862.

    7. .., .., .. // . 1987. 8. . 4655.

    8. .., .., .. - // . 1986. 10. . 3245.

    9. / .. ,.. , .. , .. // . 1992. 8. . 1122.

    10. .., .. - // . . .: - . 1984. . 7.

    11. .., .. // . 1984. 8. . 5262.

    12. Boyell R.L. Counterweapon Aiming for defense of a movingtarget // IEEE Trans. Aerospace Electron. Syst. 1980. Vol. 22. P. 402408.

    13. .., .. -- // : . 1991. . 23. . 3258.

    14. Shevchenko I.I. Successive Pursuit with a Bounded DetectionDomain // Journal of Opt. Theory and Appl. 1997. Vol. 95. N 1. P. 2548.

    15. / .. -, .. , .. , .. // . 2004. 10. . 312.

    16. .., .., .. - // . . 2005. 5. . 6265.

    17. .., .., .. - // -. 2005. 4. . 7581.

    18. .., .., .. - // . - . 2007. 5. . 611.

    19. .., .. - // . 1992. 6. . 3037.

    20. . . .: , 1967. 415 .

    21. .. . .: , 1960.

    (495) 334-91-81; e-mail: maslovep@ipu.ru

    .. .

    .., .., .. - / . - .. . .: - - , 2008. 244 .

    , - - - , : - , , , , - , , , , , - , .

    , -,- .

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