Теория управления [часть 2] 5-87851-123-1

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1

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. . 98

: : ISBN 5-87851-123-1

2. .-

:

, 2000. - 128 .:

.

. .

,

2

Ы

Ы

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Ы

Ы

, -

550200 210100 -

.

. 681.5 32.96

ISBN 5-87851-123-1



 , 2000 . ., 2000

20-

2000

681.5 32.96 98

2

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. ,

;

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,

,

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,

-

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-

,

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. ,

,

,

,

, .

, ,

. ,

-

-

,

210100 . -

, ,

, .

, ,

. -

. ( )

.

, (8÷16) Ы

1.

Ы

Ч

-

. .

,

-

, 1.1.

. . . -

,

,

,

-

, .

,

, ,

. ,

-

1.2.

.

,

-

.

( (

)

.

, ,

, -

(

,

) .

,

)

-

. -

, .

)

( [18].

-

3

, (

,

)

( (

-

),

,

. 1.1).

: 1)

-

2)

-

-

3) ) . 1.1. -

-

ϕ = f(x)

: ;

-

:

(

const); )

.

1.2).

( T = const, = const); ( A = const, T = const); , : ( , . . T, A = const, T = const,

: A = f(x) : = f(x) -

( : ω0 = f(x)

-

A = const,

-

=

=

const). ,

. ,

-

I. , -

II.

;

.

. 1.2.

k,

x x

*

=γ (

(

ω0 = 2π/ ), -

γ, 0 ≤ γ ≤ 1)

w(t). , -

.

. 1.2) 1) 2) 3)

(

.

:

: ; ; (

. .

) (

,

) ,

. , ω0 = 2π/ . -

1) 2) 3)

-

, ,

. ,

-

, . . ,

. .

4

T

,

,

.

-

x[nT] = x[n] = x(t)

. ,

=

k

A x

xA-

= const ,

-

t = nT,

,

,

.

-

. -

. , .

-

,

, . [5]: 0 1,

(1.96)

zi < 1

.

. z

.

,

, (

lim y [n, σ ] = const

. 1.13).

(1.92)

n→∞

,

-

. σ

lim y [n, σ ]

n→∞

σ = 0.

σ=0 (1.91), (1.92) .

,

(1.90),

σ≠0-

. 1.13. ,

,

a0y[n,σ] + a1y[n−1,σ] + ... + amy[n−m,σ] = 0,

.

-

. -

, (1.93)

.

А

-

.

,

. n y [n, σ ] = ∑ C z , m

i =1

zi -

Z

, .

m-

-

i i

(

. 1.2).

: (1.94)

1. 2

20

, pπ (

a0+a1>0, a0−a1>0 a0+a1+a2>0, a0−a1+a2>0, a0−a2>0 a0+a1+a2+a3>0, a0−a1+a2−a3>0, a0(a0−a2)−a3(a3−a1)>0, 3(a0+a3)−a1−a3>0

m=1 m=2 m=3

. .

(−1, j0)

), pz = e jωT.

,

-

m m ≤ 3.

А

. . 1.15.

. mπ,

ω

mD(e jωT)

, 0

π/T

. 1.15 -

z

-

∆ arg D (e jωT) = mπ , 0 ≤ ω ≤ π/T.

D (e jωT)

.

(1.97)

z

w(

e jωT w=

D(z) = a0zm + a1zm-1 + ... + am-1z + am , z = e jωT. . 1.14

. 1.16)

[5] z− 1 z+ 1

,

z=

1+ w 1− w

.

(1.98)

m = 3.

. 1.16. . 1.14. А

z .

, , (−1, j0 ).

W(e ) jωT

(1.98)

1+ w a0  1 − w

(1.95)

m

1+ w + a1  1 − w

m− 1

+.. .+ a

m

= 0,

(1.99)

21

~a w m + ~a w m − 1 +...+ ~a 0 1

zi

m

Y(z,σ)

z−1

= 0.

Y(z,σ) = Y0 + Y1 z−1 + Y2 z−2 + Y3 z−3 + ... .

(1.100)

(1.95), w ( . 1.16).

,

wi (i = 1, 2, ..., -

(1.100) , m)

.

t = (n+σ)T. , -

Y(z,σ) Y0, Y1, Y2, ... .

.

. ,

.

.

, :

1.8.

(z) = z-

,

,

, -

,

. Z, Y(z) =

[9, 15, 17, 18]. Zσ−1{Y(z,σ)}.

(1.102)

=

zy[n,σ] = (1.41), ,

y[n, σ ] = ∑ R esY (z, σ ) z n − 1 k

i =1

z=z

,

=

(1.101)

5.74 z − 7.69 z + 4.12 z−1 3

G(z)=z/(z−1).

(z)G(z) =

3.71 z − 1.52 z − z 3

2

5.74 z − 13.43 z + 11.81 z − 5.12 z+ 1 4

3

3.71 z

5.74 − 13.43 z

−1

−1

2

− 1.52 z

+ 11.81 z

−2

−2

−z

−3

− 5.12 z

ResY (z, σ ) z n − 1 = lim (z− z )Y (z, σ ) z n − 1 ,

r

ResY (z, σ ) z n − 1 =

i

i

−3

+z

z

−4 −4

−4

=

=

1. 3

i

1 d r−1 lim [(z− z ) r Y (z, σ ) z n − 1 ]. i (r − 1) ! z → z dz r − 1

×

z

. 1.3, .

. 1.17 z→z

.

2

= 0.64z-1+1.25z-2+1.42z-3+1.34z-4+1.2z-5+1.11z-6+1.08z-7+... .

i

Y(z,σ); i = 1, 2, ..., k.

zi -

3.71z 2 −1.52 z−1

t=nT 0 1T

y[nT] 0 0.64

22

2T 3T 4T 5T 6T 7T . .

Т

1.25 1.42 1.34 1.20 1.11 1.08 ...

,

.

,

,

-

,

,

, [13]. .

,

-

. ( g (1.73)

. 1.10) f

x,

-

,

(1.76),

zX(z,σ) = Xg(z,σ) + Xf(z,σ) =

= G(z, σ ) −

. 1.17. g[nT]

y[nT]

.

(z) m≥l

y[n] ≡ 0, f[n] ≡ 0

y[n]=

(1.103)

n < 0.

(1.105)

y[n] ≡ 0

g[n] ≡ 0

σ) Z {F (s) W (s)}. {W (s) F (s)} − 1W(z, + W(z)

xg(∞,σ) xf(∞,σ) -

n→∞

.

2

z-

: . -

z→1

z− 1 z

X(z, σ ) = x (∞, σ ) + x (∞, σ ), (1.106) g

f

; . ,

-

,

.

(1.104)

n < 0.

2

(1.37):

(1.106)

{

σ=0

}  =

  z− 1 G(z)   z− 1 Z W2 (s) F (s) x( ∞) = lim  × + × lim   1 + W(z) z 1 + W(z)  z → 1 z z → 1  = x (∞) + x (∞). g

1.9.

Xf(z,σ) -

σ

t = nT.

0

(1.104)

1 + W(z)

:

m  1  l  ∑ b i g[n − i] − ∑ a i y[n − i] ; n=0, 1, 2, ... a i = 0 i =1 

G (z) + Z

x( ∞, σ ) = lim x[n, σ ]= lim

(1.42)

a0y[n]+a1y[n−1]+...+amy[n−m]=b0g[n]+b1g[n−1]+...+blg[n−l],

Xg(z,σ) -

W(z, σ )

(1.105)

f

 

(1.107)

23

g(t)

.

g(t) = g0 ×1(t)

f = 0. x ( ∞) = lim

z → 11 +

g

g

x[n] = c g[n] + c g ′[n] + 0

0

W(z)

c = i!

g T

z → 1 (z − 1) W(z)

g

1

i

.

2

(1.108) ,

-

di xg

dz i

(z)

z =1

i = 0, 1, 2, ..., k.

,

(1.109)

.

g t2 2!

c (k) g ′′[n]+... + k g [n], k! 2! ,

c

. x ( ∞ ) = lim

2

1

c0, c1, c2, ... . .

g(t) = g1×t

g(t) =

-

,

,

: c0 = c1 = ... = cr-1 = 0,

x g (∞) = lim

g 2T

z → 1 (z − 1)

2

.

sin[ωnT].

W(z)

r-

.

g(t) = gmsin(ωt)

2

ω g[nT] = gm

.

-

, ,

. W(z)

W (z) W(z) =

W1(z)

1

(z − 1)

z = 1,

z = 1.

xm = 

jωT )×gm , xg(e

x[nT] = xm sin[ωnT+ψ],

ψ(ω,σ) = arg 1.6,

jωT ). xg(e

,

r

, r=0

, r=1W(1)→ ∞.

-

.

. . . ,

,

xg(∞) = 0,

x g (∞) =

gkT

-

r

.

,

k

-

kr.

y2 =

1π ∫

π

0

(e j ω T ) S* (ω ) d ω 2

g

(1.111)

24

x2 = (ejωT)

jωT ) xg(e

1π ∫

π

(e j ω T ) S* (ω ) d ω , 2

xg

(1.112)

g

RCRC-

0

-

:

, .

-

;

S* (ω ) -

[3, 4].

x

-

g

, u

.

К

-

.

,

. -

,

,

.

[13]. .

-

, . ,

.

[15].

,

,

, ,

, ,

, . [9]: ,

. .

, , -

,

,

-

. ,

,

, . . w [n] = wa(t)|t=nT.

, .

-

. ,

, . . W (s) = L[w (t)].

[5, 15]. , u [13].

-

x

(1.64)

-

-

W (z) = u[n] = ∑ w [n − i]x[i], k

i=0

k

W (s) -

. ,

. Wk(z) = Z{wk[n]}.

, , . .

(1.114) .

(1.114) [5].

(1.115)

(1.113)

k

wk[n] -

z− 1 1 W (s) }, Z{ × z s W (s)

.

δ-

25

, ,

W (z) =

-

,

K

=

U (z) X(z)

b +b z 0

1

1+ a z

−1

U(z)

-k

k

−1

+... + a z

1

W (s) = sL[h (t)],

+... + b z

-k

=

B(z) , 1 + A(z)

(1.117)

k

X(z) - z-

. .

h (t) -

.

A(z) = 0,

.

,

(1.117)

W(z) =

b z 1

m− 1

+b z

m− 2

2

z

+... + b

m

m

u[n] + ∑ a u[n − i] = ∑ b x[n − i], k

(1.116)

,

k

i=1

, mT;

-

i=0

i

-

h[n]

:

h[m].

u[n]= ∑ b x[n − i] − ∑ a u[n − i]. k

,

k

i=0

z ,

-

i=1

i

(1.119) [3]. 2k

,

-

. 1.18. .

W (s).

,

,

, -

.

.

( :

u[n]= ∑ b f[n − i];

-

k

[9]:

1)

,

i=0

-

i

f[n]= x[n] − ∑ a f[n − i],

.

k

; 2)

-

,

(1.115)

.

(1.119)

i

T.

.

(1.118)

i

,

, ,

-

. . -

i=1

f[n] -

.

i

. 1.19)

(1.119)

26 x[n]

b0 b1

1 Z-1

x[n]

k Z-1

bk

b1

b0

bk-1 u[n]

bk

Z-1

Z-1

1

k

ak

a1 ak-1 ak

bk-1

Z-1

Z-1

k

1

ak-1

u[n]

a1

. 1.20. :

-

[9].

. 1.18.

.

u[n]

b0

b1

bk-1

W (z) = k K

bk

1+ z z o1

1+ z z

−1

1

x[n]

f[n]

Z-1

Z-1

1

k

1+α z +α z 1

2

0

−1

1

−2

1

1+ z z

−1

.

(1.120)

k

0

-

. 1.21, ), -

, (

. 1.21, ).

(1.120) -

,

k

i =1

.

. 1.20)

[3]. .

.

W (z) = ∑ W (z) K ki

. 1.19.

(

×.. .×

, ak

ak-1

×

β + β z −1 + β z −2

, (

a1

−1

(1.117)

(1.121) .

27

y[n + m] + ∑ a y[n + m − i] = ∑ b u[n + m − i], m

xi[n]

Z-1

ui[n]

Z0i

Zi

i=1

i

i=0

(1.123)

i

,

β2

)

m

-

. 1.22.

β1

xj[n]

Z-1

β0

Z-1

α1

uj[n]

α0 )

. 1.21. -

: ;

-

,

[3, 13]. . 1.22.

1.10.

,

,

-

u[n]

-

[9]. my[n], m− 1

+... + b Y(z) b 0 z + b1 z m . (z) = = U(z) 1 z m + a z m − 1 +... + a m

1

, . . l < m,

b0 = , ..., = bm-l-1 = 0. (1.122)

(1.122)

m

xi[n] -

-

(i= 1, 2, ..., m), :

x 1 [n+ 1] = x 2 [n] + h 1 u[n];   x 2 [n+ 1] = x 3 [n] + h 2 u[n];  L  (1.124)  x m − 1 [n+ 1] = x m [n] + h m − 1 u[n];  x m (n+ 1) = − a m x 1 [n] − a m − 1 x 2 [n] − ... − a 1 x m [n] + h m u[n];  y[n] = x 1 [n] + h 0 u[n]. 

hi (i=0, 1, 2, ..., m) (1.124) -

(1.123)

:

h0 = b0; h1 = b1 − a1h0; h2 = b2 − a1h1 − a2h0;

28

.................. h i = b i − ∑ a j h i− j , i

i = 0, 1, 2, ..., m.

j=1

(1.124)

(1.125)

-

:

X(n+ 1) = AX(n) + BU(n);   Y(n) = CX(n) + DU(n),

 x 1 (n)   x (n)   X(n) =  2  M   x (n)   m m ×1  0   0  A=  0  ...  − a m 

: ; A,B,C,D -

-

m-1

.

xi[n]

0

...

0 0

1 0

... ...

...

...

...

−a

m− 1

m− 2

 h 1 (n)     h 2 (n)  B=    M  h (n)  m ×1  m

. ,

0   0   0 

-

, -

... 

. A

;

,

B ,

... − a 

1 m × m

C ,

,

.

-

; . k (1.126),

A

.

r m×k,

, r×m.

;

, .

,

(1.126)

-

[9]

. ,

-

. ,

. 1.23.

. ,

. .

,

C = [1 0 0 ... 0 0 ]1 x m D = [h0]1 x 1 -

. 1.23. Z-1 -

;

1

−a

(1.126)

-

[zE − A]

(n) = Z−1{[zE − A]−1 z}.

(1.127) , .

,

29

X(n + 1) = ∑ n

i=0

(n − i) BU (i) +

(n + 1) X(0).

.

(1.128) ,

W(z) =

z− 1

z − 1.75 z+ 1.125

.

2

. (z) =

W(z) 1+ W(z)

=

-

z− 1

z − 0.75 z+ 0.125

)

,

2

y[n+2] − 0.75 y[n+1] + 0.125y[n] = u[n+1] − u[n]. (1.125)

(1.126)

 1  1   0 A= ; B=  ;  −0.25 −0.125 0.75

= [1 0]; D = [0]. ) . 1.24. -

 x [n+ 1]= x [n]+ u[n]; 1 2  x 2 [n+ 1]= −0.125 x1 [n]+ 0.75 x 2 [n] − 0.25 u[n];  y[n]= x [n].  1

-

(z1 = 0.5; z2 = 0.25) (z) =

z − 0.75 z+ 0.125 2

=

A

z− 0.25

+

B

z− 0.5

=

3

z− 0.25

−

. 1.24, .

: q [n+ 1] = 0.25q [n]+ 3 u[n]; 1  1  q 2 [n+ 1] = 0.5q 2 [n] − 2 u[n];  y[n] = q [n]+ q [n],  1 2

. z− 1

-

,

. 1.24, -

,

: ;

2

z− 0.5

.

0.25 0  A = ; n  0 0.5

3 B =  ; n −2

n=

[1 1];

Dn = [0].

. .

30

Ы

А

1

1.

. ?

2. . -

3.

.

-

. 4. . ? -

5. ? 6. ? .

7. 8. 9.

? ? .

-

. .

10. .

31

. . Ы

2.

Ы

,

Ч

. (

, ),

-

. .

2.1.

. ,

, ,

,

-

, ,

. ,

,

-

. 2.2.

-

, . ,

,

.

-

. , ,

. 2.2. -

, .

.

, .

,

. ,

, ,

,

.

. 2.1. -

),

. 2.1.

: ;

-

. 2.2, .

, (

(

)

)

. 2.2, ) ,

( 2.2,

. 2.2, ) ) − .

. ,

(

. 2.2, ) − .

-

( (

, ,

: ;

. 2.3.

(

. -

32

. 2.3. -

;

-

: ;

-

; -

2.3, ) − , .

(

. 2.3, )

, ,

( .

(

, . . 2.4.

.

. 2.3, ) -

. 2.3, )

−

(

-

. 2.4. -

; (

,

( , . ,

: ; -

. 2.4, ) . 2.3, ) − ( . 2.3, ) −

-

. 2.4

.

, . -

. . , .

,

,

33

. ,

-

-

,

, ,

.

.

. .

, ,

-

.

-

, [10].

.

1. ;

-

.

2.2.

2. ,

. “

“

”,

”, “

. .

.

”.

, “

”,

)

( .

“

( ”,

“

”,

)

)

, ,

,

. ( -

xi (i = 1, 2, ... , n)

.

g = g0 n-

f = f0

3. (

-

),

. .

-

:

. 1)

i = 1, 2, ... , n,

Fi -

(2.1)

, F1 = F2 = ... = Fn = 0

; 2) 3)

,

:

d xi = Fi (x1, x2,..., xn ) dt

. 4.

.

-

x1 = x2 = ... = xn = 0,

(2.2)

; .

. ,

.

.

. -

V

,

.

, ,

,

,

. . ,

, . .

-

, ,

.

.

V

.

-

V

,

,

, . V

,

.

34

1.

.

(2.1)

V

, V = V(x1, x2, ..., xn ),

(2.3)

, . .

x1 = x2 = ... = xn = 0, x1, x2, ..., xn .

, (2.1)

,

. 2.

(2.3)

, W

V

d V = n ∂ V × dxi . ∑ dt ∂ xi dt i=1

, , (2.1),

(

. 2.1)

-

(2.5) y = F(σ) , (

V,

(

V,

-

,

[2]:

-

nV(x1, x2, ..., xn ), W(x1, x2, ..., xn ), ,

(g = 0)

-

 dx n  i = ∑ a x + b y ij j i  dt j=1  y = F (σ ), 

( V,

-

W . [2]:

(2.1) W(x1, x2, ..., xn)

n,

V(x1, x2, ..., xn) -

V ,

,

,

i = 1,2,..., n;

σ = ∑ c x ; n

k =1

k

k

aij, bi, ck -

.

, .

. 2.5.

.

;

)(

(2.6)

W, x1 = x2 = ... = xn = 0.

(2.1)

)

2.5).

d V = W (x , x ,..., x ), 1 2 n dt

W,

-

.

,

),

-

.

(2.4)

dxi (i = 1, 2, ... , n) dt d V = n ∂ V F (x , x ,..., x ). ∑ n dt ∂ xi i 1 2 i=1

(2.2)

, ,

. (2.7)

. V

(2.7)

35

∂V n W= ∑ [ ∑ a ij x j + b i F(σ )] . i = 1∂ x i

. . 2.6)

n

(

(2.8)

j=1

ax12 + bx22 = 1.

. . L(x)

F(σ) σ

V = L(x) + ∫ F (σ ) d σ ,

-

(2.9)

0

L(x) = ∑ α i x 2i . i=1 n

, ,

. 2.6.

2.3. Ч

:

0≤ k-

F (σ )

σ

.

≤ k; F (0) = 0,

. .

. .

(2.10)

, (

.

,

dx1 dt dx 2 α, β, , b -

dt

= −(x 2 + α

y = F(x), ,

dV dt

= 2α x

dx 1

dt

1

y = F(x) arctg k ( . 2.7), . .

W0

(2.12)

y

+ 2β x

dx 2

2

x

=

dt

= −2(1 − ax12 − bx22)( αx12 + βx22) . bx22 )

, x

0 ≤ F(x) ≤ kx.

= −2αx1 (x1 − βx2 )(1 − ax12 − bx22) −2βx2 (x2 + αx1 )(1 − ax12 − bx22) =

2

(2.11) -

W (s). bx 22 ) ,

.

W=

-

,

2 V = αx1 + βx22 .

.

.

:

= −(x1 − β x 2 )(1 − ax12 − bx 22 ) ; x1 )(1 − ax12 −

k . 2.5)

2

ax1 +

bx22

НЭ

y = F( x )

x

)

) . 2.7.

) < 1.

arctg k

y

: ; )

36

. .

[2]:

U*(ω) −

ω≥0

q,

V*(ω) +

T

1

> 0,

1

=0

(2.16)

k

0

Re[(1+ jωq)W (jω)] +

W*(jω).

(2.13)

, [−1/k, j0]

,

k

kW (jω) .

q

1/q.

;

. .

-

[2]:

W*(jω), (−

.

1

W*(jω)

, j0),

.

k

, Im W (jω) → −∞ Re W (jω) → −∞

ω → 0,

.

ω → 0,

Im W (jω) < 0

. 2.8.

ω.

,

-

, W*(jω),

)

:

)

. 2.8.  U * (jω ) = Re W * (jω ) = Re W (jω ),   * * V (jω ) = Im W (j ω ) = ω T0 Im W (j ω ), 

T0 = 1 -

;

k

ω ≥ 0. ,

(2.13)

T

0

V*(ω) +

. 2.8, .

(2.14), q

, . .

, (2.12),

= Re W (jω) − ωq Im W (jω)] +

U*(ω) −

-

. 2.8,

(2.13)

1

:

-

(2.14)

.

Re[(1+ jωq)W (jω)] +

. .

-

1

>0

,

1

. . W*(jω),

k

. .

0≤

(2.15)

F (x) x

-

k

≤k

(−

1

)

-

k

W*(jω)

,

k

.

, .

, q,

, (2.15)

,

-

37

.

.

arctg k (

,

,

. 2.7, )

.

k > 0. -

, :

. k → ∞.

(2.13)

2.4.

. .

.

k ,

W (s) =

10

s(s + 1)

-

. .

, .

.

, , . .

. W

(j ω ) =

10

j ω (j ω + 1)

W (j ω ) = − *

=−

10

ω2 +1

ω +1 , 2

10

−j

ω

−j

ω (ω 2

+ 1)

+1

-

.

,

(

. 2.1)

, ω.

10 2

.

К 10

( ω

∞(

0

.

y = F(x),

2.9). ω

,

. 2.9.

a 1,

-

,

W*(jω)

.

(2.18)

(2.19)

.

k

ψ = ωt,

;

-

[0, −j0].

(2.17)

y 1 = a 1 sin(ψ + ψ 1),

ψ 1-

. 2.10),

a, . .

x = a sin ψ,

W*(jω) [−10, −j10] ,

,

38

-

x 2 /

,

a

y

q

q′ -

.

,

t x

y

. a1

ψ

ω

,

y

a

t

q

,

1

y

q′ .

(2.21), -

,

-

.

1

. 2.10. ω

a

-

[7, 17]. q(a, ω) q′(a, ω) . q(a)

,

q′(a)

a (2.17)

-

, q′(a) = 0.

. (2.21)

-

F(x)

,

s

jω (s = jω),

-

(2.18)

y 1 = b1F sinψ + a1F cosψ, ,

b1F, a1F , b 1F =

W (jω, a) = q + jq′ = A (ω, a) e jψ (ω, a),

(2.20) -

-

:

π

1

2π

∫0 F ( a sin ψ

) sin ψ d ψ ,

a 1F =

π

1

2π

∫0 F ( a sin ψ

(2.22)

A (ω, a) = mod W (jω, a) =

[q( a, ω )]2 + [q ′( a, ω )]2 ;

ψ (ω, a) = arg W (jω, A) = arctg[q′(a, ω)/q(a, ω)].

) cos ψ d ψ .

px = aω cos ψ,

(2.19) (2.18)

p = d/dt,

, . .

a 1 = a×A (ω, a); ψ 1 = ψ (ω, a).

y 1 = [q + q ′

p

ω

] x,

(2.21)

И

-

.

q = b1F/a, q′ = a1F/a.

. ,

ω0

-

a0.

39

(

W

(s) =

B(s) A(s)

=

. 2.5), b 0 s + b 1s m

a 0 s + a 1s n

m− 1 n− 1

+ ... + b m

+ ... + a n

Re D(jω, a) = X(ω, a); Im D(jω, a) = Y(ω, a),

(2.23)

X(ω, a) + jY(ω, a) = 0.

W (jω, a) = q(ω, a) + jq′(ω, a) = A (ω, a) e jψ (ω, a).

p

ω

]}x = 0.

X(ω 0 , a0 ) = 0; Y(ω , a ) = 0.  0 0

: (2.29)

(2.29)

(2.25)

,

x = a0 sin ω0t

X(ω 0 , a0 , k) = 0; Y(ω , a , k) = 0.  0 0

. ω = ω0

,

a = a0

A(p) + B(p)×[q(ω, a) + q ′(ω , a) λi+1 = −jω0.

k, .

p

ω

]=0

.

(2.26)

, -

-

[−1, j0].

.

-

, . . W (jω, a) = −1.

jω

D(jω, a) = A(jω) + B(jω)×[q(ω, a) + jq′(ω, a)].

(2.31) -

(2.27)

, W (jω, a) = W (jω)×W (jω, a).

D(jω, a) = 0, .

(2.30)

a0 = f(k), ω0 = f(k)

. -

.

p

, . .

:

,

А

(2.29)

k

,

λi = jω0 .

k -

,

.

,

-

,

-

{A(p) + B(p)×[q(ω, a) + q ′(ω , a)

(2.28)

,

(2.24)

(2.21),

ω0

a0

-

(2.28)

(2.32) -

(2.31)

W (jω) = −

1

.

mod W (jω)W (jω, a) = 1; arg W (jω)W (jω, a) = − (2k+1)π,

(2.33)

W ( a)

(2.33)

2.11).

L (ω) + L (ω, a) = 0; ψ (ω) + ψ (ω, a) = − (2k+1)π,

W (jω) ( ,

. (2.34)

ω4

(ω 0,α 0) α4

1 − W (a )

α5

Im

α1

ω0

ω=0 Re ω1

a0 + ∆a

.

a = a0 + ∆a

a = a0 − ∆a,

a0 − ∆a

.

∆a > 0 -

ω = ω0 -

.

,

ω0 ,

ψ (a),

,

-

− W −1 ( a) ,

,

L (ω) = − L (a); ψ (ω) = − (2k+1)π,

, a0−∆a. . 2.11 ,

-

,

a3 < a0
0 ( ) ( ψ (ω) , L (ω)≥−L (ω0,a0+∆a), L (ω)≥−L (ω0,a0−∆a).

a = a0

,

,

-

a01, a02

−L (ω02, a)

−L (ω01, a),

−1800.

a03

(2.37)

−L (ω03, a).

ψ (ω) ψ (ω)

ψ (ω)

−1800,

ω03

−

ω01

ω = ω03 ,

,

L (ω)≥−L (ω03,a03−∆a),

,

. 2.12 . ω01, ω02 ω03, ψ (ω)

−π

)

L (ω)≥−L (ω02,a02+∆a), −1800. a = a03 L (ω)≥−L (ω03,a03+∆a),

41

−1800.

a03, -

a01. .

, W (s) =

L, L ( )

− L (a )

-

k , s(T1s + 1)(T2 s+ 1)

k=200 c-1; T1=1.5 c; T2=0.015 c, (

Ψ,

,

0 03

01

0

02

02

. 2.4, ) .

=10 , b=2 . [7]

-1

q(a) =

01

,

03

µ = a/b.

-1

-180

( )

. 2.12.

1,

ω = ω01 a = a01 L (ω)≥−L (ω01,a01+∆a), −1800, 2, ψ (ω) a = a02

= ω02

. 2.12, ,

ψ (ω) L (ω)≥−L (ω01,a01−∆a), −1800. ,

4 cb 2 π b a2

q(µ ) =

2

,

a ≥ b, q′(a) = 0.

4c b 1− 2 πa a

q(a) = 1

:

2

k =

.

µ= ω ,

k

µ

4c = 6. 4 πb

2

-

a2 −1. b2

µ2 −1, ;

a b

. k

-

q( µ ) =

µ

1 2

µ 2 − 1 , q′(µ) = 0 -

µ

− L (µ ) = 20 lg

µ → 1,

.

,

(

+20 / . 2.13).

,

L,

L

, L = 0, µ = 1

− L (µ)

( ) -20

60 -40

+20

40

Ψ,

20

180

0

90

-20

0

0,1

0,1

µ1

1 µ2

10

ω0 1

-90

100

,

−1

-60

10

Ψ (

-180

100

,

−1

)

. 2.13.

W (s) =

ψ (ω).

. 2.13

−L (µ).

2

µ2 −1 µ >> 1 −L (µ) = 20 lg µ.

−L (µ) → ∞;

µ1 = 29

ω0 = 4.3 c-1 ψ (ω) −1800. µ2 = 1.08

k k s(T1s + 1)(T2 s+ 1)

L (ω), −L (µ)

ω0 = 4.3 c-1

µ1

42

L (ω) ,

µ2 .

a0 = b×µ1 = = 58 .

-

43

,

-

, k k W (s) = s(T1s + 1)(T2 s+ 1)

ψ (ω).

−L (µ).

ω0 = 4.3 c-1 ψ (ω) −1800. µ2 = 1.08 µ1

ω0 = 4.3 c

,

.

L (ω), −L (µ)

. 2.13

µ1 = 29

.

М

.

L (ω) ,

µ2

, .

,

“

”

-

-

, -

,  dy  d t = f (x, y);  dx  = y,  dt

2.5.

. [1]. ,

x, y f(x, y) -

, .

. (2.39)

,

, (n-1)

,

,

(2.39)

;

n-

.

, . .

a0 = b×µ1 = = 58 .

,

-

,

.

-1

n-

-

-

-

,

-

t: dy f (x, y) . = dx y

(2.40)

y = F(x)

(2.41)

. ,

. .

,

. ;

(x, y). -

y0)

.

(x0, -

. -

.

.

(2.40): 1)

f(x, y)

-

44

, ,

. 2.14.

-

,

.

,

,

-

; 2)

y>0

dx/dt>0

x

,

dx d2x + a1 + a 2x = 0 . dt dt 2

t . . y=0, f(x, y)≠0 (

; 3)

-

,

y=

),

dx/dt,

. (2.40)

,

dx  = y;  dt  dy  = − a1y− a 2 x ,  dt

, [2, 5, 10]. , i,

dy/dx=ci .

,

(2.40) f (x, y) = ci , y

dy x = − a1− a 2 . dx y

y = ϕ(x, ci ).

y = F(x)

i

,

t,

, (

, . 2.14). arctg

i

.

-

p2 + a1p + a2

(x, y). , = 0,

.

. 2.14 ,

.

y

-

c0 y0

1 1. a1=0, a2>0

c1

A

c2

2

x

Im Re

c3

3

y t

x -

x0

x

45

2. a12>4a2, a1>0, a2>0

.

y

x

Im

x

Re

.

t

, ,

-

,

-

. . 2.15, .

,

. , 1

2

.

3

3. a24a2, a10

x

Im

. 2.15.

y x

;

-

;

-

Re

t

:

-

-

, (

(

. 2.15, ),

. 2.15, ).

-

, .

. (

.

. 2.15, ).

, ,

46

.

. 1.

. ,

(

. 2.4, )

F(x) = csign(x).

,

(2.44) -

.

dy

.

dx

=−

1 T

. :

(x, y) )

(

. -

,

W

kT-

(s) =

s(T s + 1)

c0 -

.

x

y = dx/dt.

x

dt

+

dx(t) dt

+ kF (x) = 0 ,

(2.43)

dx

=−

T

−

=−

1 T

−

kc Ty

(2.46)

.

(2.47) .

k T

×

(2.42)

y

[2]

=−

1 T

+

kc Ty

,

x = −kcT lny − kc − Ty + c0,

(2.48)

(2.49) y

(2.43)

-

F (x)

y=

x < 0

dx

, 1

-

(2.45) dy

1 k  dy  d t = − y − F (x); T T  dx  = y. dt 

dy

,

.

,

2

x = 0. . 2.16, )

,

−kc.

y = F(x).

-

d x(t)

(2.45)

.

c0

,

T

y

x = kcT lny + kc − Ty + c0,

,

;

2

c sign(x)

[2]

-

k

T

×

x>0

dx

. 2.1),

k

.

dy

.

−

(

.

(

-

,

(2.44) .

= kc

. . 2.16, (x0, 0).

47 y

0  F(x) =  + c  −c 

А

y

у

kc x0

x

x

-b С

b

x = −b (

0

-kc

(

− b ≤ x ≤ + b, x > + b,

x < − b.

AB ,

CD).

. 2.17, ) CD

AB (

D

x = +b -

. 2.17, ). CD

AB −

(2.46), )

(2.48).

)

. 2.16. ;

( + c F(x) =   − c

+ c − c

F(x) = 

. 2.16, );

x > + b,

. 2.4, )

x < + b,

x < − b,

dx < 0. dt

x > − b,

x = −b (

y < 0, ABCD

(

dx

, . 2.16,

1 T

1

,

(2.50)

x+ c 0 ,

(2.51)

T

y

y

C

A

C

A

-

-

, . . .

-

-1/T.

x = +b, y>0 CD). ( . 2.16, ) -

−b

.

. yM

=−

y= −

,

-

(2.44) dy

− (2.48). . 2.16, )

ABCD (2.46),

,

-

dx > 0; dt

. 2.17, )

−b ≤ x ≤ +b,

.

-

2.

AB , .

(

:

-

(

,

−b

x

b

x0

x

b

x0

a; -

. 3.

(

. 2.4, )

B

-

D

B

D

Л )

. 2.17.

)

:

48

-

;

–

Y

. 2.17, (x0, 0). ,

. ( . 2.3, ) k x − b ≤ x ≤ + b,  F(x) =  + c x > + b,  −c x < − b. 

4.

0

AB

CD (

-

−b ≤ x ≤ +b,

.

,

dx

М

(x0, 0) x1, x2

=−

1 T

−

kk T

×

-

. x0i

x

.

-

0X, xi

(2.52)

.

y

. 2.17, (x0, 0)

-

. .

dy

X

x0

0X. 0X

(2.44)

x1

. 2.18.

. 2.17, ).

,

x2

xi = f(x0i)

. (

-

)

(2.53)

-

[10].

.

,

,

xi

, (

,

0X

y

-

. 2.18). x

1 2

А

x0i 1

x01

x02 2

x03

49

. 2.19.

(

450

. 2.19) ,

xi = x0i, , . .

0X

xi =

. f(x0i)

xi = x0i (

A

B)

.

, ,

,

.

,

, x0

, ,

. 2.19. . 2.19 A)

( ,

, , (x0, y0), ,

(

B)

.

,

.

-

50

.

xi = f(x0i)

xi = x0i (

A

B)

-

. ,

,

,

. ,

, x0

-

, ,

. 2.19. . 2.19 A)

(

, , (x0, y0), ,

,

(

B)

.

,

. . 2.20.

:

-c

2.6.

;

.

[10]:

,

; . , .

К

(

щ

W

-

kT-

. 2.20, , (s) =

k

s(Ts + 1)

,

; F(σ);

, -

.

. 2.20, ) ( . 2.20, ).

-c

-

, W (s) = (T s + 1), T -

.

.

, ,

W

. 2.20, , W (s) = W (s) ×W (s) . 2.20, , W (s) = W (s) + W

(s) =

k(T s + 1) s(Ts + 1)

( . 2.20, ) (g = 0) σ

(s).

.

(Tp2 + p)σ + k(T p + 1)F(σ) = 0, ,

σ = −(T p + 1)x,

(2.54) (2.54)

p=d/dt.

-

(2.55)

(2.56)

51

x T

d 2 x(t) dt

2

+

dx(t) dt

− kF (σ ) = 0 .

(2.57) -

x (2.57)

  Toc y + x = b   T y + x = − b  oc 

y = dx/dt

1 k  dy  dt = − y + F(σ ); T T  dx  = y, dt 

x= ±b

y=−

(x − b);

1 T

y=−

oc

1 T

(2.63)

(x + b).

oc

, ,

α = arctg  1  .  Toc 

(2.58)

(2.64)

. 2.21, (x0, 0).

dy dx

=−

1 T

+

k T

×

F (σ ) y

, ,

(2.61)

(2.59)

.

(2.62),

,

. (2.60),

-

,

σ≤ b

2.3, ),

(

T .

.

, ,

F(σ) = k σ = − k (T p + 1)x ,

, dy dx

=−

1 + kk T T

−

F(σ) = ±c, (2.46)

(2.48)

kk T

×

x

, (T = 0)

α = 900. .

(2.60)

, . T ;

y

y

y

C

A

A′

(2.59)

C

A

C′

: dy dx dy dx

=− =−

1 T 1 T

− +

kc Ty kc Ty

σ < −b

(T p + 1)x > +b;

σ > +b

(T p + 1)x < −b.

(2.61)

b

−b

x0

b

−b

x

x0

′

x

D′

(2.62)

B

B′

D B

D

Л

σ≤ b

a)

(2.56), :

)

. 2.21. -

: ;

,

-

52

, ,

-

.

.

T s

-

α

(2.63); 2.21,

(2.64). (

. . 2.4, )

(x0, 0).

. 2.22.

. , α′.

F(σ)

T

. 2.4, ,

,

,

, F-1(σ)

W

W (s) = 1 + k ,

(2.65)

σ = −(1 + k )x.

(2.66)

-

(s) (

. 2.23).

:   k oc x + x = b   k x + x = − b  oc 

x=

b

1+ k

x=−

. 2.23.

;

(2.67)

oc

b

1+ k

,

. oc

,

-

,

,

, -

К

. .

,

T = 2π/ω (

g(t) u(t) ( . 2.24). ( . 2.24, , ) g(t) u(t), ω g(t) . 2.24, ), . . x(t) = g(t) + u(t),

. F(σ) 1

(σ).

-

,

.

.

. 2.22

F-

ω

F1(g)

y = F(x) = F[g(t) + u(t)] = F1(g) + F2(u).

F(x) (2.68) , F2(u),

(2.69)

53

,

A, -

.

F1 [g(t)] ≈

ω 2π

t+ π /ω

.

∫ F [g(t) + u(t)]dt .

F1(g)

(2.70)

t −π /ω

-

,

k A. y (2.69)

ω

. g = const

(2.70)

u(t)

-

,

,

,

F2(u)

-

-

,

(

F2(u) . 2.24, )

-

.

. W(s) = k W (s).

ω

,

(2.72)

g(t) < A (

. 2.24, )

-

u(t), ,

. u(t)

.

2.7.

, . 2.24.

-

:

-

;

-

ω

; (2.70)

, T.

. , g(t)

-

. 2.24,

ω

F1(g) u(t)

±A.

. .

A. g(t),

-

-

, ,

-

,

[10]. , x0 ≠ 0

. 2.20, , k

=

C

A

.

(2.71) AB

C.

y0 = 0. (x0, 0) (

. 2.25), 1 -

54

, 2

D.

,

D

. -

, 1.

(kp→∞).

-

,

, 2

, . . -

, ,

-

,

,

. 2.20, ,

. ,

,

D,

1

2

-

(s) =

,

k

0. . 2.25, 2

AB ( (x02, 0) D

1

→ ∞ 1 + k p W (s) Woc (s)

p

,

,

x01 (

M0

C M0 x01 x0

x03

x02

. 2.25)

-

2

,

. , . Toc

M0 M1 0 M ′ M ′ ,

1

1

-

;

0

, D

. -

M′0 0

, x

,

.

-

.

M1′

.

, .

: ,

B

-

,

. 2.25.

. ,

,

,

, 2.26.

. .

-

,

M1

(2.73)

M1

y

2

1 . Toc s + 1

-

. A

Woc (s)

=

,

-

AB .

1

.

. (x03, 0)

=

,

,

D).

k p W (s)

lim

-

.

55

,

y

.

A

, .

-

,

x

. [2, 9, 10],

-

B

. :

. 2.26.

:

;

AB -

,

.

-

, .

. 2.27 .

y = F(x)

(2.74)

y = k×x,

(2.75) .

(2.75) y = k0 mx + k11 xo, xo -

. k0

. 2.27. -

, ,

.

(2.76)

k11

,

my = k0 mx = my ; σ 2 = k 2 σ 2 y

mx, my , my, σ 2 , σ 2 , σ 2 x

y

11

x

=σ

2

y

,

(2.77)

-

y

. (2.77)

,

,

-

2.8.

, ,

-

, ,

,

-

,

,

-

,

-

k

0

=

m

y

m

x

;

(2.78)

56

k

k11 F(x). k0

=±

11

σ

σ

;

y

(2.79)

,

k11

k

-

. m y = ∫ F (x)ω (x) dx ;

ω(x) -

y

1

=

k +k 11

12

2

(2.87)

.

:

∞

σ

.

:

x

-

2

-

k12

−∞ ∞

.

= ∫ F (x)ω (x) dx , −∞

-

,

(2.80)

2

, .

(2.81)

-

, .

k0

k1

. ,

-

, ,

.

-

. ,

,

(2.74)

(y − y ) 2 = k 2 m 2 + k 2 σ 2 − 2 k m m 0

x

12

x

.

(2.76)

0

x

y

−2k

12

(xo y ) + y2

[17].

(2.82)

Ы

. k0

k12 ,

А

2

1.

2 k 0 m 2x − 2 m x m y = 0;

(2.83)

2 k 12 σ x2 − 2(x o y ) = 0.

(2.84)

,

.

.

2.

, ?

3. 4.

. -

-

. .

.

5. k

k 12 =

(x o y )

σ x2

=

0

R xy (0) R x (0)

=

m

y

m

(2.85)

;

x

1 ∞ = ∫ (x − m x ) F(x)ω (x) dx .

σ x2

−∞

(2.86)

?

6. 7. 8. 9.

. ? ? ? ?

10.

,

k0 ,

?

?

57

13. Ч

. . .. . .-

14. ,

-

.:

, 1989. - 304 .

.:

, 1979. - 256 . /

15.

.

. .

.-

.:

-

, 1973. - 336 .

.

.

,

, , .

16.

: . . .

17.

. 2-

.-

. . . - .: . .

18.

.:

.

2-

/

, 1986. , 1989. - 752 . .-

.:

, 1963.

- 968 . 19. 20.

1.

. .,

. . MATLAB. -

. . . . 1980. - 412 .

.-

.: . - .:

, 1999. - 467

.:

. 2. .-

.:

3.

. ., . . , 1975. - 768 . . .

.-

.:

, 1976. -

576 . 4. 5.

. ., . - .: . .

6.

. .,

. . , 1987. - 320 .

. .

.. ., . - .:

. . 1978. - 609 . 8. . ., . .

.:

, 1981. - 304 . . .

.-

-

. . . . , 1973. - 507 . 11. . .,

-

, 1989. - 284 . . - .:

7.

9. 10.

.,

. ., .

,

-

,

, .-

. .,

.:

,

. ., .

.

, 1998. - 172 . , 1990. - 335 . . - .: . . . -

. ., -

,

.:

. .

2000. - 549 . 12.

. . , 1986. - 616 .

.

-

. -

.:

, 1974. - 576 . ,

58

2. . ........................................... 3 . . 1.

........ 4

1.1. 1.2. 1.3.

..................................... 4 .......... 5 ............................................. 9

1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5. 1.6. 1.7. 1.8. 1.9. 1.10.

................................. .......... ..................... .......... .............. ...

2.

30 34 41 46 49 60

. . . . . . . 68

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

. .

. . . . . . . 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 . . . . . . . . . . . . . . . . . . . . . . . . 78 . . . . . . 83 . . . . . . . . . . . . . . . . . . . . . . . 95 . . . . . . . . . . . . . . . . . . . . . . . 109 . . . . . . . . . . . . . 118 . 122

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

№ 020717 60 × 84 1/16

2.02.1998 12.10.2000 .

. . . 8,0 35 . ________________________________________________ . , . , 15