1 , ( 2 .2).
: 1, 1; 0.5, 1; 1, 0.5.
1 | 2 | 3 | 4 | 5 | |
1, 0 | |||||
0, 1 | |||||
1, 1 | |||||
0.5, 1 | |||||
1, 0.5 |
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4. .
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6. 1 . . .
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1 | 2 | 3 | 4 | 5 | 6 | |
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7. .
1. ( ).
2. ( , ).
3. ().
4. ( ).
5. .
6. ( , , ).
7. .
3 (4 )
: Excel. (2.51), (2.35) - (2.37) .
Excel, K(t) ..
.
zk(t) = Akzk(t) + Bkuk(t),
zk(0) = zk0, k = 1, K,
yk(t) = Ckzk(t), (3.1)
zk, uk,, yk - c, , ; Ak, Bk,, Ck ; t , k = 1, K .
.
ek(t) = pk(t) yk(t), k = fixe,
T
Jkl(T) = 1/2 ò{ek(t)Qklek(t) + uk(t)Rkluk(t)}dt à min, (3.4)
0
ek, pk - ; Qkl, Rkl ; l = 1, L; L ; .
. , , .
. , , .
|
|
, -. m. m = 1 m = 2 . , .
, m > 4, , , . m = 3 m = 4. m = 4 . . , m . m = 4 , .
.
K(t) + K(t)A + ATK(t) - K(t)BR-1BTK(t) + CTQC, (29)
K(T) = CTSC, (30)
(29)
.
K(t) = F(t, K(t)),(31)
F , (29).
[T] [ti] = [(ti) (ti - 1)], i = 1, I, . [ti] = [t] = const.
K((ti + 1) = K((ti) + (k1 + 2k2 + 2k3 + k4)/6, (32)
kr, r = 1, 4 , .
k1 = [t]* F(K(ti)), (33)
k2 = [t]* F(K(ti) + k1/2), (34)
k3 = [t]* F(K(ti) + k2/2), (35)
k4 = [t]* F(K(ti) + k3), (36)
m = 4 [t]m + 1.
. (33) (36) K(ti), (32).
.
.
z(t) = az(t0 + u(t),
y(t) = z(t),
T
J = ½*fe2(T) + ò[q (t) + ru2(t)]dt à min.
0
u(t) = r -1*(K(t)z(t) g(t)),
.
K(t) = 2aK(t) + r -1*K(t)2 q (37), K(T) = f,
.
g(t) = [a - r -1*K(t)]g(t) qp(t),
g(T) = fp(T),
.
z(t) = az(t) r 1*(K(t)z(t) g(t)), z(0) = 0.
f = 0, T = 1, a = 1, q = 1, r = 0.01, 0.1, 1. p(t) - . () K(t).
.
1. ( ).
2. ( , ).
3. ().
4. ( ).
5. .
6. ( , , ).
7. .
2. MatLab.
4 (2 )
: MatLab.
1 4 MatLab. Optimization Toolbox, SIMULINK.
1. Optimization Toolbox. linprog.
|
|
.
f = cx à min
Ax ≤ b
Aeq = beq
xl ≤ x ≤ x u,
x - ; f - ; c - ; A, Aeq ; b, beq - ; xl, x u - .
linprog linprog(f,A,b,Aeq,beq,lb,ub), lb, ub . , linprog(f,A,b,[],[], lb, ub).
F = 5 p 1 + 4 p 2 + 6 p 3 à max
p 1 + p2 + p3 £ 20
3 p 1 + 2 p 2 + 4 p 3 £ 42
3 p 1 + 2 p 2 £ 30
P = { p j}, p j ³ 0, j = 1, 3.
linprog
function x= lp13(f,A,b,lb);
f=[-5; -4; -6];
A=[1 -1 1; 3 2 4; 3 2 0];
b=[20; 42; 30];
lb=zeros(3,1);
[x]=linprog(f,A,b,[],[],lb);
, ; . , .
MatLab File-New -. . : Debug-Run. MatLab.
.