Стохастические модели и оценки. Лабораторный практикум по курсу ''Теория оптимального управления

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  • H

    : .. , ..

    2001

  • 62-50 (076) 32.977

    C81

    - . , ..

    --

    C 81 . /.: .. , .. . :, 2001. 42 c.

    - . - . - .

    , -.

    .

    62-50 (076) 32.977

    c. , 2001

  • 1 4

    2 7

    3 8

    4 334.1 ,

    () . . . . . . . 334.2 . . . . . . . . . . . . . . . . . . . 344.3 . . . . . . . . . . 344.4

    . . . . . . . . . . . . . . 384.5 . . . . . . . 39

    5 40

    6 41

    3

  • 1

    - . .

    1) , . - () - . - , - . , - , , , -. H , , () , , , . - () .

    2) . , , - . - , , , . -

    4

  • N 1 2 3 4 5 6 7 8 9 10 11 12O 1.7 2 2.3 2.7 3 3.3 3.7 4 4.3 4.7 5 5.3

    . .

    1: .

    . , - .

    3) 7 14 12 , (). , , . , , , , .

    4) : 2 , - 12-. 3, 4, 5 6 12-. 4, 7, 8 9 12-. 5, 10, 11 12 12.

    O N 1.

    , - , . , - . : - ; -

    5

  • ; - - ; - .

    5) H - . (3, 4, 5) , - ( ).

    6

  • 2

    2 . - 12- . , ( ), , - 2 '' ( 12- ). - -. .

    A 1 2 5 7 9 10 11 15 18 19 20 23B 1 3 5 6 8 10 12 16 18 19 22 23C 1 4 5 9 10 13 14 17 18 19 21 23

    2: .

    , '', -, . 2-5 .

    7

  • 3

    : - - . () -. .

    1. t - OX , x(t), (-) , x(t0) = x0, - (t) c 2w. x(t) - , . - ti, i = 1, 2, . . . R(ti) , .

    - :

    (a) 2w;(b) R(ti);(c) 2w/R(ti);(d)

    t R(t) R(t) = 0.

    : , ( ).

    8

  • 2. , 1 - , x(t) - ut, u (t).

    3. x(t0) t OX , vx(t), () x x(t) c - qx = 2w. , 1, .

    (a) x(t)

    x(ti+1) = x(ti) + Gdwd(t),

    Gd ; wd(t) .

    (b) , - ( ) . - SI (- ), .

    (c) H - , , . , - x(t) vx(t) t.

    (d) - , . - -

    9

  • . , , - , -. - . ?

    4. , 3 , vx(t), x(t) OX (-) a (t)c q = 2w.

    5. 3, (a) (b), - :

    () , - x y;

    () OX , - 3; OY , x - y, qx = qy = 2w;

    () (x, y) - D = (x2 + y2)1/2 = arctg(y/x), x = D cos , y = D sin . D - 2D 2.

    :(a) ex ey -

    zx = x + ex zy = y + ey ;

    (b) ,

    10

  • ;(c) ,

    3, (d); 1 x y, ;

    (d) LDLT , L -, D ;

    (e) , L1, - ( L D);

    (f) , 3, (d); 2 - x y, () ;

    (g) (c) (f) 5. - 2 ( ), - - (x, y) (vx, vy) .

    6.

    x(ti+1) = (1)2i+1x(ti)

    11

  • i = 0, 1, . . ., x(t0) P0. - , x(ti+1) = x(ti).

    - R(ti), i = 1, 2, . . ..

    (a) .(b) , : P0 > 0 0 < R(ti) < i 1,

    P (t+i ) i; R(ti) - i, ; R(ti) = 0 i, .

    (c) .

    7. x -

    x(ti+1) = x(ti) + w(ti),

    x(t0) - P0; = const, w(ti) Q. H

    z(ti) = x(ti) + v(ti),

    v(ti) R. x(t0) .

    (a) .

    12

  • (b) , : ( ) - Q; ( ) [0, R]; K [0, 1]; K [0, 1] - P

    P

    R=

    [(1K)2 + K2 ][1 (1K)22]

    = Q/R /.(c) P K

    K K.(d) - , -

    P /R K 2 = 1/2 : 0.1; 0.2; 0.5; 1.0; 2.0; 3.0; 4.0; 5.0. , K = K, K .

    (e) - P Q = 0 : 0 2 < 1 2 > 1. , ( ) - - , 2 > 1 ( ) , - P = (2 1)R/2.

    (f) - , . - (. 3).

    K - -.

    13

  • 1 = 0.78 Q = 0.39 R = 0.5 P0 = 1.02 = 0.78 Q = 0.78 R = 0.5 P0 = 1.03 = 0.78 Q = 0.39 R = 1.0 P0 = 1.04 = 0.78 Q = 0.78 R = 1.0 P0 = 1.05 = 0.78 Q = 0.39 R = 5.0 P0 = 1.06 = 1.0 Q = 25.0 R = 32.0 P0 = 1.0

    3: 7.

    (g) () (f). - .

    8. (. 1) y(t), - 1/(s + a), - y(t) , - - T :

    E{y(t)y(t + )} = 5e| |/T .

    - -y(t) x(t)1s+a

    . 1: 8.

    z(ti) = x(ti) + v(ti),

    v(ti)

    E{v(ti)2} = 2.

    .

    14

  • (a) , .

    (b) - - y(t) Q , . - ? t, T a - , ?

    (c) - ( ) .

    1. - , - . -, Q = 10T .

    2. Q , - x(t) . ,

    Q =10T

    1 + aT.

    (d) , : (1) , 1; (2) , 2; (3) . .

    (e) , - ( ) - ( 1 2 ) , - .

    (f)

    15

  • ( 1 2 ), ( ) . 1 2 -? () ?

    (g) -.

    1. (e), , - .

    2. - . , . .

    9. (. . 2), - . 1 1 1, , . - 0 . 0 -. 1, , u(t) :

    E{u(t)} = 0; E{u(t1)u(t2)} = Q(t2 t1); Q = 2b2 .

    (a) 1, , -. ,

    16

  • e

    e

    jR2

    C1 C2u(t) e0(t)R1

    . 2: 9. R1 = R2 = 1; C1 = C2 = 1F .

    , - .

    (b) , , - 0 v(ti) - {v(ti)v(tj)} = Rij; R = 0.2b2.

    10. y(t) -

    y(t) + y(t) = 0,

    y(0) y(0) -

    E{y(0)} = 0, E{y(0)} = 0,

    E{y(0)2} = 4, E{y(0)2} = 2, E{y(0)y(0)} = 1. z(ti)

    z(ti) = y(ti) + v(ti),

    v(ti) , y(0) y(0),

    E{v(ti)} = 0, E{v(ti)2} = 1.(a) y(ti).

    ?

    17

  • (b) - ( ), , 2 . - . ? - , . - , ?

    (c) , (a) (b).

    11.

    x(ti+1) = x(ti) + wd(ti),

    () wd() -

    E{wd(ti)} = 0, E{w2d(ti)} =1

    2, E{wd(ti)wd(tj)} = 0 (i 6= j).

    E{x(t1)} = 1, E{x2(t1)} = 2. t1 t2

    z(ti) = x(ti) + v(ti) (i = 1, 2),

    v()

    E{v(ti)} = 0, E{v2(ti)} = 14, E{v(ti)v(tj)} = 0 (i 6= j).

    (a) , z(t1) = z1 z(t2) = z2, x(ti ) x(t+i ) ;

    18

  • P (ti ) P (t+i ) K(ti) t1 t2. , P1(ti ) t1 t2?

    (b) , , x() z() - .

    (c) , .

    12.

    x(ti) = 0.7x(ti1) + wd(ti1), i = 1, 2, . . . ,

    x(t0 = 0) = x0, wd() , :

    E{wd(ti)} = b = 0.2, E{[wd(ti) b]2} = 0.01.

    ti :

    z1(ti) = 2x(ti) + v1(ti), z2(ti) = sin x(ti) + v2(ti),

    v1() v2() , :

    E{v1(ti)} = 0, E{v21(ti)} = 1,

    E{v2(ti)} = 0, E{v22(ti)} = cos2 ti.(a) , ti1,

    ti1, x(t+i1) P (t+i1). - , .. x(ti ) P (ti ). - . x(t+i1) = 4 P (t+i1) = 1, x(ti1) P (ti )?

    19

  • (b) ?

    (c) , ti

    z1(ti) = z1i = 3; z2(ti) = z2i = 1.

    , x(t+i ) P (t+i ) z1i z2i , , -

    z(ti) =

    z1(ti)

    z2(ti)

    .

    , , - .

    (d) . (c) R(ti) . - () , . (c), - () ()? , , ?

    (e) , .

    13. -

    h(s)

    hc(s)=

    0.3(s + 0.01)

    s2 + 0.006s + 0.003,

    h hc . - hc hc0 hc(t)

    20

  • :hc(t) = hc0 + hc(t).

    hc0 -:

    =3000 , =22500 2. :

    E{hc(t)} = 0; E{hc(t)hc(t + )} = Nc(); Nc = 36 2,

    hc(t) . - , - . ,

    hm(t) = h(t) + m(t),

    m(t) :

    E{m(t)} = 0; E{m(t)m(t + )} = Nm(t); Nm = 81 2.

    (a) , - ( ) h(t). - . , - .

    (b) , , - 81 2.

    (c) . (a) (b) . . (a) - MATLAB , - . . (b) . .

    21

  • 14. - - - . ,

    x(ti+1) = x(ti) + ut + wd(ti),

    u , t = ti+1ti = const;wd() E{w2d(ti)} = Qd = const; x(t0) - x0 P0. -, ti -

    z(ti) = x(ti) + v(ti),

    v() E{v2(ti)} = R = const. , x(t0),wd() v() .

    t = 1 , , - () 10- - . ti = 0, 1, . . . , 9( ti = 10), , , . : ?

    (a) P0 = Qd = R = (30 )2 .

    (b) , , , 75 , -

    22

  • , - - . , .

    (c) (b) P0 = (90)2, Qd = R = (30)2.(d) (b) Qd = (90)2, P0 = R = (30)2.(e) (b) R = (90)2, P0 = Qd = (30)2.(f) -

    . , - ( ?).

    15. s(t) - () , - w1(t). -- n(t), - w2(t) , . 3.

    -

    -

    HHH - s(t)

    n(t)

    z(ti)w1(t)

    w2(t)

    -6

    . 3: 15

    ()

    F0(s) =1

    s + w0,

    Fn(s) =1

    s + wn.

    23

  • :

    E{w1(t)} = 0, E{w1(t)w1(t + )} = 2(t),

    E{w2(t)} = 0, E{w2(t)w2(t + )} = 1(t). w1(t) w2(t) , - :

    E{s(t0)} = 0, E{n(t0)} = 0, E{s2(t0)} = 1, E{n2(t0)} = 12.

    (a) , z(ti).

    (b) , P (t+i ) - ti1. .

    (c) , - , : w1(t) 2 1 ( 3); n(t) 1/2 1 ( 2); w0 = wn.

    16. ,

    x(t) = w(t),

    w(t)

    E{w(t)} = 0, E{w(t)w(t + )} = 4().

    t = 0 x(0) -

    E{x(0)} = 10, E{[x(0) 10]2} = 25.24

  • z(t) = x(t) + v(t)

    c v(t), w(t), -

    E{v(t)} = 0, E{v(t)v(t + )} = 16().(a) -

    t. - t . , .

    (b) - . ( z).

    (c) , . (a) (b) , , , - . . (a) MATLAB .

    17. (. 4), w1(t), w2(