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(t), v(t), x1(0) x2(0)

    E{w1(t)w1(t + )} = (), E{w2(t)w2(t + )} = (),

    E{v(t)v(t + )} = 2(), E{x21(0)} = 1, E{x22(0)} = 2.(a) x3(t), -

    z(t). ( : - ?) H t .

    25

  • -dt

    -dt

    -

    -

    b

    a6

    -

    - -?w1(t)

    w2(t)

    x1(t)

    x2(t)

    x3(t)

    v(t)

    z(t)

    . 4: 17.

    (b) - . . ?

    18. - . r ( ) , , ., - ur u . :

    r(t) = r(t)2(t) Gr2(t)

    + ur(t),

    (t) = 2r(t)

    (t)r(t) +1

    r(t)u(t).

    () - , - x = (r, r, , ).

    (b) (-

    26

  • ) : r(t) = r0, r(t) = 0,(t) = wt, (t) = w, ur(t) = u(t) = 0, G = r30w2 t. - . () . , -.

    () , - = x3.

    z1(t) = x3(t) + v1(t),

    E{v1(t)} = 0, E{v1(t)v1(t + )} = R1().

    x1 = r. -

    z2(t) = x1(t) + v2(t),

    E{v2(t)} = 0, E{v2(t)v2(t + )} = R2().

    , . H- ? ? - - z1 z2?

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

    19. -

    27

  • / x(s)

    w(s)=

    s +

    s +

    , w(t) . H

    E{x(0)} = 0, E{x2(0)} = 1.

    H

    z(t) = x(t) + v(t)

    v(t)

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

    () , , t 0. - t . M

    M = AM + MAT MBM, M1

    M1 = ATM1 M1A + B.

    ( , t - M1(t)).

    (b) , - , .

    () . (b) -. . ()?

    20.

    x(t) = x(t) + w(t)

    28

  • z(t) = x(t) + v(t),

    E{w(t)} = 0, E{w(t)w(t + )} = Q(),E{v(t)} = 0, E{v(t)v(t + )} = R().

    Q = 1.() -

    P (t) , P (t = 0) = P0. P (t) lim P (t) t . R = 1, 2 4 P (t) - ( P0 = 1).

    (b) ,

    y = [1 P/R]y.

    V (, ),

    V (y, t) = yTP1(t)y,

    , . ?

    () z(ti), , .

    (d) , . (). P (t) . ().

    21.

    z(t) = at + n(t),

    29

  • a

    E{a} = 0, E{a2} = 1

    n(t)

    E{n(t)} = 0, E{n(t)n(t + )} = 2().

    (a) - a. ?

    (b) , z(ti), .

    (c) . (b). - ?

    22. , - F0(s) = 1/s - w1(t). - x2(t), - Fn(s) = 1/(s + a) w2(t) (. 5).

    -

    - -

    6

    - -1s

    1s+a

    @@

    w2(t)

    x1(t)

    x2(t)

    z(t) z(ti)w1(t)

    . 5: 22.

    w1 w2,

    E{w1(t)} = 0, E{w1(t)w1(t + )} = Q1(),30

  • E{w2(t)} = 0, E{w2(t)w2(t + )} = Q2(), ,

    E{x1(t0)} = 0, E{x2(t0)} = 0,

    E{x21(t0)} = 21, E{x22(t0)} = 22, E{x1(t0)x2(t0)} = 0,E{z2(t0)} = 2z , E{z(t0)x1(t0)} = 21.

    (a) x1. t . -,

    x1(s)

    z(s)=

    c1(s + a)

    s + ac1, c1 =

    Q1Q1 + Q2

    .

    (b) x1 x2. z1(ti), - .

    (c) , . - .

    (d) - , - .

    23. , - , , . - ,

    nn() = E{n(t)n(t + )} = Ne5| |

    31

  • , -

    yy(w) =5

    2 N(w

    2 + 16)

    (w2 + 4)(w2 + 25)

    (a) ?(b) , , -

    . , , ,

    ss(w) =5/12

    w2 + 4, nn(w) =

    7/12

    w2 + 16

    . - . - ?

    (c) - . . . .

    (d) . (c). . (b)?

    32

  • 4

    4.1 , ()

    : vx(t) = cx + x(t)

    x(t) = vx(t)

    vx(t) = x(t) = (t)x(t) =

    x(t)

    vx(t)

    ddtx(t) =

    0 1

    0 0

    x(t) +

    0

    1

    (t)

    x(t0) =

    x(t0)

    vx(t0) = cx

    :

    1. E{x(t0)} =

    x0

    cx

    = x0.

    2. E{[x(t0) x0][x(t0) x0]T} = P0 =

    p011 p012

    p021 p022

    (p021 = p

    012).

    (t):

    1. E{(t)} = 0.

    2. E{(t)(t)T} = 2(t t) = Q(t t), Q = 2.

    :

    d

    dtx(t) = Fx(t) + G(t) (1)

    F =

    0 1

    0 0

    , G =

    0

    1

    , Q =

    [2

    ].

    33

  • x0 = E{x(t0)} =

    x0

    cx

    ,

    p011 p012

    p021 p022

    .

    4.2

    z(ti) = Hx(ti) + v(ti), H =[

    1 0]. (2)

    v(t):

    1. E{v(ti)} = 0.

    2. E{v(ti)v(tj)T} = Riij.

    4.3

    1. (1):

    x(t) = (t t0)x(t0) +t

    t0

    (t )G()d() (3)

    2. (t):(t) = eFt - :

    d(t)

    dt= F (t)(t), (0) = I. (4)

    :F (t) = F = const

    G(t) = G = const

    (4) :

    s(s) (0) = F(s)(Is F )(s) = (0) = I(s) = (Is F )1 .

    :

    (s) =

    s 10 s

    1

    = 1s2

    s 1

    0 s

    34

  • :

    s 10 s

    1s2

    s 1

    0 s

    = 1s2

    s2 0

    0 s2

    =

    1 0

    0 1

    = I.

    (s) =

    1/s 1/s2

    0 1/s

    .

    (s) :

    (s) (t) =

    1 t

    0 1

    , (5)

    , -:

    1

    s 1 1

    s2 t

    , (t).

    3. (3) :t0, t1, t2, . . . , tk, tk+1, . . . , x(tk) - tk tk+1:

    x(tk+1) = (tk+1 tk)x(tk) +tk+1

    tk

    (tk+1 )Gd(). (6)

    d(tk) =

    tk+1

    tk

    (tk+1 )Gd().

    , d(tk), -:

    (a) E{d(tk)} = 0.(b)

    Qd(tk) = E{d(tk)d(tj)T} =

    =

    tk+1

    tk

    (tk+1 )GQGTT (tk+1 )d = Qd (7)

    35

  • (6) :

    x(tk+1) = x(tk) + d(tk). (8)

    , = (tk+1 tk) - . 2 , Qd d(tk) (7).

    . - = tk+1 tk = const, ( !) - (7).

    :

    =

    1

    0 1

    . (9)

    Qd =

    tk+1

    tk

    1 (tk+1 )0 1

    0

    1

    Q

    [0 1

    ]

    1 0

    (tk+1 ) 1

    d =

    = Q

    tk+1

    tk

    (tk+1 )2 (tk+1 )(tk+1 ) 1

    d = Q

    0

    2

    1

    d =

    = Q

    3/3 2/2

    2/2

    0

    = Q

    3/3 2/2

    2/2

    . (10)

    , (9) (10).

    . (8) :

    x(tk+1) = x(tk) + Gdd(tk), (11)

    Gd , , .

    36

  • Qd :

    Qd = GdQdGTd , (12)

    Qd ,

    Qd =

    d1. . .

    dn

    . (13)

    :

    Qd =

    q11 q12

    q12 q22

    =

    Gd

    1 0

    g21 1

    Qd

    d1 0

    0 d2

    GTd

    1 g12

    0 1

    ; Q =

    2

    q11 = 2

    3/3

    q12 = 2

    2/2

    q22 = 2

    ;

    q11 = d1, q12 = d1g21

    q22 = d2 + d1g221

    :d1 = q11 = (

    3/3)2

    g21 = q12/d1 = (3/(2))

    d2 = q22 d1g221 = 2 26 33 94 6 2 =

    2 [1

    3

    4] = 2

    4. (14)

    Gd =

    1 032 1

    , Qd =

    ( 3/3)2 0

    0 (/4)2

    .

    (11) , , : (9) (14).

    . (11) (tk) :

    (tk) =

    (1)(tk)

    (2)(tk)

    .

    37

  • , , {d1, d2, . . . , dn}, (13). (14).

    . (11), d(tk) . (12)

    Qd = GdGTd .

    , (13) (14). (14), (13):

    Qd =

    d1. . .

    dn

    =

    Qd

    d1

    . . .

    dn

    QTd

    d1

    . . .

    dn

    = QdQTd

    Qd = GdQdGTd = GdQdQ

    Td G

    Td = (GdQd)(GdQd)

    T = GdGTd

    Gd = GdQd

    :

    Gd =

    1 032 1

    3

    3 0

    0

    2

    =

    3

    3 034

    2

    .

    4.4

    :

    x(tk+1) = x(tk) + Gdwd(tk)

    z(tk) = Hx(tk) + v(tk).

    , Qd wd(tk). : I. , tk tk+1, k = 0, 1, . . ..

    38

  • 1. : x(tk+1) = x(t+k )

    : x(t+0 ) = x0.

    2. :

    P (tk+1) = P (t+k )

    T + GdQdGTd

    : P (t+0 ) = P0.

    II. , k = 0, 1, . . ..

    1. : x(t+k ) = x(tk ) + K(tk)[z(tk)Hx(tk )]

    2. :

    P (t+k ) = P (tk )K(tk)HP (tk )

    3. :

    K(tk) = P (tk )H[HP (t

    k )H

    T + R]1

    4.5

    , -, ( ).

    F, G, Q, R, ; 2.

    x0 =

    x0

    cx

    P0 =

    p011 p012

    p012 p022

    .

    , .

    39

  • 5

    , .. , .

    - - , .

    .

    (, , ), - , , .

    - ( ).

    - (, , , ..).

    TurboVision 2.0 .

    , - (, , ..), .

    .

    40

  • 6

    1. . . . .: , 1973.

    2. . -. .: , 1973.

    3. . ., . ., . . . .: , 1980.

    4. . . . .: , 1986.

    5. . . . . .: ,1971.

    6. . . .: , 1988.

    41

  • :

    ..

    - LATEX.

    . . 020640 22.10.97. 09.02.01. 6084/16. . . . . 2,63. .-. . 2,00. - Computer Modern. . 100 . .

    432027, , . , 32.

    , 432027, , . , 32.

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