. , , ,

,,,

,

.

, 1, - , , , , (Machmoud, 1977; Jamshidi, 1983). , , , , , , , , . , . - , , , , . 4.1 () . , (supremal coordinator), , , -, .. , , , .. . , , , , (March and Simon, 1958). Mesarovic . (1970) () . (Schoeffler and Lasdon, 1966; Benveniste et al., 1976; Smith and Sage, 1973; Geoffrion, 1970; Schoeffler, 1971; Pearson, 1971; Cohen and Jolland, 1976; Sandell et al., 1978; Singh,1980; Jamshidi, 1983; Huang and Shao, 1994a,b). Mahmoud (1977).

, . 4.6.

, :

1.           , .

2.           , .

3.           ( ).

4.           , , .

( ) , , , , .

1. . . , .

2. . . , 4.2 (Singh and Titli, 1978). , , ( ) , ( ), ( ) ( ).

3. . ; , , ( ) , . , 4.1, . , . 4.2 , , .

, . , , . Jamshidi (1983).

, , , . 4.3. , : () , () , ( ).

4.2 . 4.3 . 4.4, interaction prediction . 4.5 , . . . 6. 4.6 .


4.2. .

 

, , , , . . , , , : (feasible) (dual-feasible). , , ( ).

4.2.1 .

:

(4.2.1)

(4.2.2)

x , u , y . :

(4.2.3)

(4.2.4)

xi, ui, yi i- , . () . , yi, i=1,2 . y1 y2 wi, i=1,2:

(4.2.5)

:

i:

(4.2.6)

(4.2.7)

:

(4.2.8)

:

(4.2.9)

(4.2.10)

wi, yi, . , , . , x, u y, . , , . , .


             .

 

(4.2.1)-(4.2.2). . i- yi, zi. , .. . , zi x, u y. , , , , . , , , .. yi zi (Mesarovic ., 1969; Schoeffler, 1971).

, . , . , , , . :

(4.2.11)

( ), y-z. z, :

(4.2.12)

(4.2.13)

:

(4.2.14)

S0:

(4.2.15)

(4.2.11)-(4.2.13) , :

1:

(4.2.16)

(4.2.17)

2:

(4.2.18)

(4.2.19)

, :

(4.2.20)

, , , .. (4.2.16) (4.2.18) , . 4.4 . 4.4 4.5.

, , , .

4.3 .

. , .

:

(4.3.1)

u (n x l) (m x l). , N si, i=1,,N, i- :

, (4.3.2)

x, u, xi, ui n, m, ni, mi, , gi i- , :

(4.3.3)

(4.3.4)

u1,,uN, ,

(4.3.5)

(4.3.1) :

(4.3.6)

(4.3.1) N (4.3.2), (4.3.6) gi(x,t) (4.3.2), :

(4.3.7)

(4.3.8)

(4.3.9)

zi ( ) N . , :

(4.3.10)

(4.3.11)

(4.3.12)

(4.3.13)

, , . .


4.3.1

:

(4.3.14)

:

(4.3.15)

(k x l), :

(4.3.16)

N-1 , Gij ni x nj. N , (4.3.15)-(4.3.16) :

(4.3.17)

Qi ni x ni, Ri Vi mi x mi ki x ki ,

(4.3.18)

(4.3.17) . , , , . Mesarvic . (1970), linear-quadratic Pearson (1971) Singh (1980) Jamshidi (1983).

N zi(t), , (4.3.15)-(4.3.16), . SG. . SG N , a=(a1,,aN) si(a), i=1,,N. , SG si(a) (Sandell ., 1978) , a*, Si(a*), i=1,,N, SG. , , , (Geoffrion, 1970). 4.6 . , i- i ( ), () .

, , , ..:

(4.3.19)

ei l- , dl, , :

(4.3.20)

zi(*) (4.3.20) i, a(t) , (4.3.16). , , . .

(4.3.21)

(4.3.15), L(*) :

(4.3.22)

k . , ( ) , (4.3.16) ai, i=1,,k. , , (Geoffrion, 1971a, b; Singh, 1980) ,

(4.3.23)

, J (4.3.17) (4.3.15)-(4.3.16) q(a) (4.3.21) a. , , =*, :

(4.3.24)

, , , Li . Li, (4.3.24) (4.3.15) a*, . q(a*) (4.3.21). , , q(a*) , , . , q(a) :

(4.3.25)

, f . a (4.3.19) 4.6. ( ), dl (4.3.19) l- el(t). , :

(4.3.26)

(4.3.27)

d0=e0. e(t) , s. .

4.1. .

1. , Li, a=a*, , . . ( 4.3.2, ).

2. , (4.3.26)-(4.3.27), a*(t) (4.3.19).

(4.3.28)

, . .

. , Pearson (1971), Singh (1980) Jamshidi (1983), . (Beck, 1974; Singh, 1975). 4.6, 6. 4.6.

4.3.1. 12- Pearson (1971) 4.7 :

(4.3.29)

:

(4.3.30)

:

(4.3.31)

( ).

: , 4.7 ( ) (4.3.29) , ( . 4.7):

(4.3.32)

ei, i=1,,6 . :

(4.3.33)

Ki(t) ni x ni . , Davison Maki 1973 Jamshidi 1980, (4.3.33). - , (4.3.19), (4.3.26)-(4.3.27), (Hewlett-Packard, 1979) . =0.1, (Pearson, 1971; Singh, 1980). 1 , 4.8, (4.3.29), Singh (1980). .

4.3.2. .

(4.3.34)

() x1 () ( ) 2 () u1 .

(4.3.35)

Q=diag(2,4,2,4) R=diag(2,2), , (4.3.35) (4.3.34) x(0)=(11 -11)T.

: (4.3.34)-(4.3.35), , (4.3.33) - =0.1. 15 , 4.9. 4.10.

4.3.2. .

, , , - , Takahara (1965), . , N ,

(4.3.36)

zi:

(4.3.37)

ui(t), (4.3.36)-(4.3.37), :

(4.3.38)

ai(t), pi(t), (4.3.37) (4.3.36) , .. i- :

(4.3.39)

:

(4.3.40)

(4.3.41)

(4.3.42)

(4.3.43)

ai(t) zi(t) , ai(t) zi(t), , . , . , ui(t) (4.3.43):

(4.3.44)

(4.3.40)-(4.3.42), :

(4.3.45)

(4.3.46)

() , , (4.3.33) . , . :

(4.3.47)

gi(t) , ni. (4.3.47) (4.3.46) (4.3.45) , (4.3.47) xi(t), :

(4.3.48)

(4.3.49)

Ki(tf) gi(tf) (4.3.41) (4.3.47).

(4.3.50)

(4.3.51)

() . . , , , ni(ni+1)/2 xi(0). , Ki(t), gi(t) (4.3.49) zi(t) xi(0). 4.4, .

. :

(4.3.52)

ai(t) zi(t) :

(4.3.53)

(4.3.54)

..:

(4.3.55)

(l+1) :

(4.3.56)

:

4.2 :

1. N (4.3.48) (4.3.50) Ki(t), i=1,2,N. ai(t) zi(t).

2. l- (4.3.49), (4.3.50). gi(t), i=1,2,,N.

3.

(4.3.57)

xi(t), i=1,2,,N.

4. 2 3 (4.3.56) :

5. , :

(4.3.58)

6. . , l=l+1 2.

, , , , N- 1 xi(0), , (4.3.56). , zi(t) , , , .

, Tokahara (1965), , . Titli (1972) (Singh, 1980) Cohen . (1974), . Smith Sage (1973) , 6. , , 4.4, 4.5. , .

4.3.3.

(4.3.59)

x(0)=(-1,0.1,1.0,-0.5)T, Q=daig(2,1,1,2), R=diag(1,2) . tf =1.

: , 4.2. Davison Maki (1973), -. (Newhouse,1962), :

(4.3.60)

(4.3.49) , 4.2 3, -

(4.3.61)

[a11(t),a12(t),z11(t),z12(t)] [a21(t),a22(t),z21(t),z22(t)]T (4.3.56), (4.3.58) . , 4.11. Ci =(1 1) . (4.3.59) , , i(t), i=1,2,3,4; yj(t) uj(t), j=1,2. , , 4.12. . , .

.

4.3.4.

u*.

: tf=2, =0.1 , Q1=Q2=I4, R1=R2=1. , i=1,2 0 , . , 4.13. . x0 .

4.3.1. 4.3.1 (4.3.59):

x(0)=(-1,0.1,1.0,-0.5)T , Q =diag(2,1,1,2), R=diag(1,2) . LSSPAK tf=2.

: , , RICRKUT LSSPAK/PC, . INTRPRD LSSPAK/PC . . , ; Enter, .

DOS GRAPHICS , , shift-PrtScr.

Optimization via the interaction prediction method.

Initial time (to): 0

Final time (tf): 2

Step size (Dt): .1

Total no. of 2nd level iterations = 6

Error tolerance for multi-level iterations - .00001

Order of overall large scale system = 4

Order of overall control vector (r) = 2

Number of subsystems in large scale system = 2

Matrix Subsystem state orders-n sub i 0.200D+01 0.200D+01

Matrix Subsystem input orders-r sub i 0.100D+01 0.100D+01

Polynomial approximation for the Ricatti matrices to be used.

Matrix Ricatti coefficients for SS# 1

0.453D+01 -.259D+01 0.794D+01 -762D+01 O.186D+01
0.978D-01 -.793D-01 0.252D+00 .233D+00 0.571D-01
0.490D+00 0.759D-02 -.109D+00 0.975D-01 -.531D-01

Matrix Ricatti coefficients for SS# 2

0.112D+01 -.815D+01 0.361D+01 0.455D+01 0.105D+01
-0.149D+00 -.322D-01 0.697D-01 .284D-01 0.183D-01
0.815D+00 0.642D-01 -.295D+00 0.305D+00 -.138D+00

System Matrix A

0.200D+01 0.100D+00 0.100D-01 0.000D+00
0.200D+00 0.100D+01 0.100D+00 0.500D+00
0.500D-01 0.150D+00 0.100D+01 0.500D-01
0.000D+00 -0.200D+00 0.250D+00 -0.120D+01

Matrix Input Matrix B

0.100D+01 O.OOOD+00
0.100D+00 O.OOOD+00
O.OOOD+00 O.250D+O0

Matrix Input Cost Function R

0.100D+01 O.OOOD+OO
0.000D+O0 0.200D+01

Matrix Lagrange Multiplier Initial Values

0.100D+01
O.IOOD+Ol
0.100D+01
0.100D+01

Matrix Initial conditions vector xO

-.100D+01
0.100D+00
0.100D+01
-.500D+00

Subsystem no. 1 at 2nd level iteration no. 1

Subsystem no. 2 at 2nd level iteration no. 1

At second level iteration no. 1 interaction error = 0.347D+00

Subsystem no. 1 at 2nd level iteration no. 2

Subsystem no. 2 at 2nd level iteration no. 2

At second level iteration no. 2 interaction error = 0.771D - 03

Subsystem no. 1 at 2nd level iteration no. 3

Subsystem no. 2 at 2nd level iteration no. 3

At second level iteration no. 3 interaction error = 0.507D - 03

Subsystem no. 1 at 2nd level iteration no. 4

Subsystem no. 2 at 2nd level iteration no. 4

At second level iteration no. 4 interaction error = 0.323D - 04

Subsystem no. 1 at 2nd level iteration no. 5

Subsystem no. 2 at 2nd level iteration no. 5

At second level iteration no. 5 interaction error = 0.310D - 05

4.14, 4.15.

.


4.3.3

(4.3.15) (4.3.17) , , Si (4.3.17) , . , Li (4.1.24) (4.3.15). i- :

(4.3.62)

i-

(4.3.63)

(4.3.64)

, . . .

4.3.5. :

(4.3.65)

(u1,u2), , (4.3.65),

(4.3.66)

.

: (4.3.65) (4.3.66) , .

(4.3.67)

(4.3.68)

(4.3.69)

:

(4.3.70)

. , z1 (4.3.70). .

1,

1,

2,

2,


4.3.3.. 1.

Bauman (1968)

(4.3.71)

:

(4.3.72)

(4.3.73)

. (4.3.72) (4.3.73) :

ki(t) i- . :

, .


4.3.3.. 2.

 

Singh (1980) , , , z , .. z :

G :

G , . :

, p2 2 2, , . , (4.3.40) (4.3.51), , () .

ki(t) gi(t) . , .

(4.3.74)

, p ( ) . A, B, Q R -, (4.3.74) N , z, V=G-1.

. , 1, - , , , ,

 

 

 

! , , , .
. , :