PAw2


"
"
"
" g(t)
y(t) = g(t) " u(t)
t
= g()u(t - )d
0
t
= g(t - )u()d.
0
" G(s)
Y (s) = G(s) U(s)
U(s) Y (s)
G(s)
"
g(t) "! G(s)
G(s) = L [g(t)]
g(t) = L-1 [G(s)] .
"
(t) = Ax(t) + Bu(t)
"
y(t) = Cx(t) + Du(t)
" x(t)
" u(t)
" y(t)
"
t0
u(t) t e" t0
x(t), y(t) t e" t0
t
x(t) = exp(At)x(0)+ exp(A(t-))Bu()d
0
"
(t) = Ax(t), x(0)
x(t) = exp(At)x(0).
"
L [(t)] = L [Ax(t) + Bu(t)]
"
sX(s) - x(0)|x(0)=0 = AX(s) + BU(s)
"
sX(s) = AX(s) + BU(s),
"
(sI - A)X(s) = BU(s),
I
"
X(s) = (sI - A)-1BU(s),
"
Ś(t) = exp(At) = L-1 [Ś(s)]
Ś(s) = (sI - A)-1.
"
L [y(t)] = L [Cx(t) + Du(t)]
"
Y(s) = CX(s) + DU(s),
"
Y(s) = C(sI - A)-1B + D U(s).
"
G(s) = C(sI-A)-1B+D = CŚ(s)B+D
"
G(t) = L-1 [G(s)]
= L-1 C(sI - A)-1B + D
= CŚ(t)B + D
"
ea(t)
ia(t)
eb(t)
(t)
Ń(t)
J
b
"
(ia)
"(t) = k "ia(t),
k
"
Ł
"eb(t) = kb"Ń(t).
" k kb
"
d"ia(t) d"Ń(t)
Ra"ia(t)+La +kb = "ea(t)
dt dt
d2"Ń(t) d"Ń(t)
J = k"ia - b .
dt2 dt
"
"Ea(s) = Ra"Ia(s)+sLa"Ia(s)+skb"Ś(s),
s2J"Ś(s) = k"Ia(s) - sb"Ś(s).
"
k
"Ś(s) = "Ia(s).
s(b + Js)
"
"Ś(s) k
= .
"Ea(s) s [kkb + (Ra + Las)(b + Js)]
"
" La
"Ś(s) k0
=
"Ea(s) s(1 + T0s)
k
k0 =
kkb + bRa
JRa
T0 =
kkb + bRa
" T0
"
"
d"Ń(t))
Ł
"Ń(0), "Ń(0) = , "ia(0).
dt
t=0
T
Ł
x(t) = .
"Ń(t) "Ń(t) "ia(t)
"
(t) = Ax(t) + b"ea(t)
ł łł
0 1 0
ł ł
A = 0 -b/J k/J
0 -kb/La -Ra/La
ł łł
0
ł ł
b = 0 .
1/La
" La H" 0
T
Ł
x(t) =
Ż
"Ń(t) "Ń(t)
Ż
"  b
0 1
0
Ż
 = , b = .
kbk
k
b
0 - +
JRa
JRa J
"
"Ń(t)
La = 0

"Ń(t) = 1 0 0 x(t) ! C = 1 0 0
La = 0
"Ń(t) = 1 0 x(t) ! C = 1 0
Ż
"
"Ń(t)
=
Ł
"Ń(t)
"
kŃ kŃ
Ł
kŃ 0 0
La = 0 C =

0 kŃ 0
Ł
kŃ 0
La = 0 C =
0 kŃ
Ł
"
"
T T
x(t) = x1(t) x2(t) = c(t) %0ł(t)
r(t)
d(t)
c(t)
"
1
X1(s) = X2(s)
s
2
X2(s) = (U(s) + D(s))
4 + s
2
= [k(E(s) - ktX2(s)) + D(s)]
4 + s
2
= [k(R(s) - X1(s) - ktX2(s)) + D(s)] .
4 + s
"
sX1(s) = X2(s),
sX2(s) =
= -2kX1(s)-(2kkt+4)X2(s)+2kR(s)+2D(s).
"
1(t) = x2(t),
2(t) = -2kx1(t)-(2kkt+4)x2(t)+2kr(s)+2d(s)
0 1
(t) = x(t)
-2k -2 (2 + kkt)
0 0 r(t)
+
2k 2 d(t)
c(t) = 1 0 x(t).
"
n-1
Y (s) bisi
i=0
G(s) = = , an = 1, bn-1 = 0.

n
U(s) aisi
i=0
"
"
{Ac, bc, cc, dc}
ł łł ł łł
0 1 0 0 0
ł śł ł śł
0 0 1 0 0
ł śł ł śł
ł śł ł śł
Ac = , bc =
ł śł ł śł
ł ł ł ł
0 0 0 1 0
-a0 -a1 -a2 -an-1 1
cc = b0 b1 bn-2 bn-1 T , dc = 0.
"
{Ao, bo, co, do}
Ao = AT, bo = cT, co = bT, do = dc.
c c c
ł łł ł łł
0 0 0 -a0 b0
ł ł
1 0 0 -a1 śł b1 śł
ł śł ł śł
ł ł śł
Ao = 0 1 0 -a2 śł , bo =
ł śł ł śł
ł ł ł
0 bn-2 ł
0 0 1 -an-1 bn-1
T
co = 0 0 0 1 , do = 0.
"
{Ad, bd, cd, dd}
G(s)
n
ci
G(s) = ,
i=1
s - pi
i " {1, . . . , n}
pi
ci
n
ci
Y (s) = U(s).
i=1
s - pi
U(s)
Xi(s) =
s - pi
i
sXi(s) = piXi(s)+U(s), i " {1, . . . , n} ,
n
Y (s) = ciXi(s).
i=1
ł łł
p1 0 0
ł śł
0 p2 0
ł śł
Ad = = {pi}n
1
ł ł
0 0 pn
ł łł ł łł
1 c1
ł śł ł śł
1 c2
ł śł ł śł
bd = , cd = , dd = 0.
ł ł ł ł
1 cn
"
{As, bs, cs, ds}
G(s)
n-1
(s - zi) bn-1
G(s) = ,
i=1
(s - pi) s - pn
(s - zi)/(s - pi) i
i " {1, . . . , n - 1}
s - zi 1/s
= (s - zi)
s - pi 1 - pi/s
(s - zi)/(s - pi)
(s - zi)/(s - pi)
G(s)
"
1(t)
1(t) = p1x1(t) + u(t),
2(t)
2(t) = 1(t) - z1x1(t) + p2x2(t).
2(t) = (p1 - z1)x1(t) + p2x2(t) + u(t).
3(t)
3(t) = (p1 - z1)x1(t) + (p2 - z2)x2(t)+
+p3x3(t) + u(t).
n(t)
n-1
n(t) = (pi - zi)xi(t)+pnxn(t)+u(t).
i=1
y(t) = bn-1xn(t).
G(s)
(t) = Ax(t) + bu(t)
ł łł
p1 0 0 0
ł śł
p1 - z1 p2 0 0
ł śł
ł śł
= p1 - z1 p2 - z2 p3 0 x(t)
ł śł
ł ł
p1 - z1 p2 - z2 p3 - z3 pn
ł łł
1
ł śł
1
ł śł
ł śł
+ 1 u(t)
ł śł
ł ł
1
y(t) = 0 0 0 bn-1 x(t),
T
x(t) = x1(t) x2(t) xn(t)
A " Cnn
A
G(s)
A
A = {pi}n
i=1
i
A pi - zi
i = 1, . . . , n - 1
"
Y (s) 48 + 44s + 12s2 + s3
G(s) = =
U(s) 105 + 176s + 86s2 + 16s3 + s4
2 + s 4 + s 6 + s 1
= .
7 + s 5 + s 3 + s 1 + s
n = 4, (p1 = -7, z1 = -2)
(p2 = -5, z2 = -4)
(p3 = -3, z3 = -6), p4 = -1.
ł łł ł łł
-7 0 0 0 1
ł śł ł śł
-5 -5 0 0 1
ł śł
(t) = x(t)+ł 1 śł u(t)
ł ł ł ł
-5 -1 -3 0
-5 -1 3 -1 1
y(t) = 0 0 0 1 x(t).
" M1 " Rnn M2 " Rnn
P " Rnn
M2 = P-1M1P.
M1 = PM2P-1.
"
" M1 " Rnn M2 " Rnn
P " Rnn
det (M2 - In) = det (P-1M1P - In)
= det P-1(M1 - In)P
= det (M1 - In).
"
M1 = M2
det M1 = det M2
M1 = M2
" M =
[mij]n,n " Rnn
n
det M = i,
i=1
n n
M = mii = i,
i=1 i=1
i " M i " {1, . . . , n}
" A " Rnn
exp (At) A
"
M = x1 xn , M " Rnn
xi i "
{1, . . . , n} A " Rnn
i i " {1, . . . , n}
A
"
Rn
M " Rnn
"
A i = j

i = j i, j " {1, . . . , n}

" xi = 0 A

i
Axi = ixi.
i " {1, . . . , n}
AM = M
 " Rnn (Cnn)
A
ł łł
1 0 0
ł śł
0 2 0
ł śł
 = {i}n = .
i=1
ł ł
0 0 i
" M = n
 = M-1AM
A = MM-1
" A
 = {i}n
i=1
P = M
A
" exp(At)
"
Aiti
exp (At) = .
i=0
i!
"i
(MM-1)i = MiM-1.
A = MM-1
exp(At) = M exp(t)M-1,
exp(t) = e1t, . . . , ent
ł łł
e1t 0 0
ł śł
0 e2t 0
ł śł
= .
ł ł
0 0 ent
exp(At)
exp(t)
A
"
M-1
ł łł
T
y1
T
ł śł
y2
ł śł
M-1 = , yi " Cn, i " {1, . . . , n}
ł ł
T
y2
exp(At)
n
T
exp (At) = xiyi eit.
i=1
"
ńł
-1 3 -1
ł
(t) = x(t) + u(t)
-1 -5 1 .
ół
y(t) = 0 -1 x(t)
"
-1
Y (s)
1 0 -1 3 -1
= 0 -1 s -
0 1 -1 -5 1
U(s)
1
s + 5 3 -1
= 0 -1
-1 s + 1 1
(s + 2)(s + 4)
1
-s - 2
= 0 -1
s + 2
(s + 2)(s + 4)
-(s + 2)
=
(s + 2)(s + 4)
-1
= .
s + 4
"
-1 3
= {1, 2} = {-2, -4}
-1 -5
1 3 x1 0
1
1 = -2 :
-1 -3 x2 = 0
1
x1 3
1
! x1 = ;
x2 = -1
1
3 3 x1 0
2
1 = -4 :
-1 -1 x2 = 0
2
x1 1
2
! x2 = .
x2 = -1
2
3 1
M = x1 x2 =
-1 -1
1/2 1/2
M-1 = .
-1/2 -3/2
ńł
-1 3 -1
ł
ł
ż(t) = M-1 Mz(t) + M-1 u(t)
ł
ł
-1 -5 1
ł
-2 0 0 (!)
ł = z(t) + u(t)
ł
ł 0 -4 -1
ł
ół
y(t) = 0 -1 Mz(t) = 1 1 z(t).
"
ńł
1(t) = -x1(t) + 3x2(t) - u(t)
ł
2(t) = -x1(t) - 5x2(t) + u(t)
ół
y(t) = -x2(t)
ńł
ż1(t) = -2z1(t)
ł
ż2(t) = -4z2(t) - u(t)
ół
y(t) = z1(t) + z2(t)
"
{A, b, c}
M-1AM, M-1b, cM .
"
M-1b cM)
" b
c
A
"
e-2t
"
" n
(t) = Axx(t) + Bxux(t x(0)
yx(t) = Cxx(t) + Dxux(t)
ż(t) = Azz(t) + Bzuz(t z(0)
yz(t) = Czz(t) + Dzuz(t)
"
P " Rnn
Az = P-1AxP
Bz = P-1Bx
Cz = CxP
Dz = Dx.
"
ux(t) a" uz(t) a" u(t), x(0) = Pz(0).
"
Cz(sIn - Az)-1Bz + Dz =
CxP(sIn - P-1AxP)-1P-1Bx + Dx =
-1
CxP P-1(sIn - Ax)P P-1Bx+Dx =
Cx(sIn - Ax)-1Bx + Dx.
"
"
x(t) a" Pz(t), yx(t) a" yz(t) a" y(t).


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