�
"
"
"
"
"
"
"
"
r1 r2
r1Ń1(t) = r2Ń2(t) �1(t)/r1 = �2(t)/r2
Ń1(t) Ń2(t)
�1(t) �2(t)
N1 N2)
�1(t) �2(t)
N1Ń1(t) = N2Ń2(t) oraz = .
N1 N2
Ł Ł
�1(t) = Ń1(t) �2(t) = Ń2(t)
N1�1(t) = N2�2(t)
N1 : N2
u1(t)i1(t) =
u2(t)i2(t) u1(t) i1(t)
u2(t) i2(t)
N1i1(t) = N2i2(t) u1(t)/N1 = u2(t)/N2
ZL
N1 2
ZL1 = � ZL.
N2
�
JLŃ2(t) = �2(t)
�
(N1/N2)JL � Ń1(t) =
(N2/N1) � �1(t)
N1 2
�
JL � Ń1(t) = �1(t)
N2
JL
N1 2
JL1 = � JL.
N2
 p1(t)
q(t)
q(t) = k p1(t) - p2(t), p1(t) e" p2(t)
k
p2(t) = p2 p2
Ż Ż
p1(t) p1 p1(t) =
Ż
p1 + "p1(t)
Ż
"
"p1(t)
k p1 + "p1(t) - p2 H" k p1 - p2 � 1 + = q + "q(t)
Ż Ż Ż Ż Ż
2(p1 - p2)
Ż Ż
"
q = k p1 - p2 p1
Ż Ż Ż Ż
k
"q(t) = " � "p1(t)
2 p1 - p2
Ż Ż
"p1(t) p1(t)
"Q(s) k
= " .
"P1(s) 2 p1 - p2
Ż Ż
"
R = 2 p1 - p2/k
Ż Ż
 S
p(t)
q1(t) q2(t)
dh(t)
S � = q1(t) - q2(t)
dt
h(t) p(t)
p(t) = �gh(t)
� g
dp(t)/dt = (�g/S) � (q1(t) - q2(t))
t
�g
p(t) - p(t0) = � (q1(�) - q2(�))d�.
S
t0
C = S/(�g)
"q1(t)
"q2(t) q1(t) q2(t)
 q2(t)
p1(t) - p2(t)
q2(t) = k p1(t) - p2(t), p1(t) e" p2(t), "t
k
p1(t) -
p2(t) g�h(t)
h(t) �
"
g q2(t) = k g� �
h(t) h(t) v(t)
dv(t)/dt = q1(t) - q2(t)
v(t0) = v dv(t)/dt
Ż
dv(t) dv(h) dh(t)
= �
dt dh dt
dv(h)/dh
h dv(h)/dh = S
S h
dh(t) "
S � = q1(t) - k g� � h(t).
dt
Ż Ż
q1(t0) = q1 q2(t0) = q2 h(t0) = h q1 = q2 = k g�h a" q
Ż Ż Ż Ż Ż
Ż
q1(t) = q1 + "q1(t) q2(t) = q2 + "q2(t) h(t) = h + "h(t)
Ż Ż
Ż
(q, h)
Ż
d"h(t) "
Ż
S � = q + "q1(t) - k g� � h + "h(t)
Ż
dt
"h(t)
Ż
H" q + "q1(t) - k g�h � 1 + .
Ż
Ż
2h
im(t) -Im0 +Im0
e = f(im) e
e(t)
e(t) = kmim(t)
km = tan ą Lm
um(t)
dim(t)
Rmim(t) + Lm = um(t)
dt
Rm
Rm Lm
e(t) + 
(t) = um(t).
km km
 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)
"ea(t) = Ra"ia(t) + La + kb
dt dt
d2"Ń(t) d"Ń(t)
J = k"ia - b .
dt2 dt
La
d"Ń(t) d2"Ń(t)
+ T0 � = k0 � "ea(t)
dt dt2
k JRa
k0 = oraz T0 = .
kkb + bRa kkb + bRa
T0
k0
T0
q(t)
T1(t)
T2(t) T3(t)
T1(t) - T2(t)
q1(t) =
R1
R1
dT1(t) T1(t) - T2(t)
C1 = q(t) -
dt R1
C1
T2(t) - T3(t)
q2(t) =
R2
dT2(t) T1(t) - T2(t) T2(t) - T3(t)
C2 = -
dt R1 R2
 R2 C2
eA eB A
B
uf (t)
Ń(t)
J b
� Ł
JŃ(t) = �(t) - bŃ(t)
Ń(t) �(t)
Śf (t) ia(t)
�(t) = k1Śf (t)ia(t)
Ł
Rf if (t) + kf Śf (t) = uf (t)
Ł
Raia(t) + kaŚa(t) + eb(t) = ua(t)
Ł
Śa(t) eb(t)
�m(t)
Ł
eb(t) = kb�m(t)Ń(t) ua(t)
k1 kf ka kb
d"if (t)
Rf "if (t) + Lf � = "uf (t)
dt
Lf
If0 (If0, Ia0)
if (t) = If0 + "if (t)
ia(t) = Ia0 + "ia(t) Ia0
�(t) = k1Śf (t)ia(t)
"�(t) = kt(If0"ia(t) + Ia0"if (t))
kt
Ra"ia(t) + Lad"ia(t)/dt = -k2(&!0"if (t) + If0d"�(t)/dt)
 La
ia0) &!0
If0, Ia0) k2
x(t)
x = 0 y(t)
Ż
q(t) p1(t) p2(t)
m b
m S
� d D D > d)
x(t) y(t)
q(t) = q = 0 p1(t) = p1 = p2(t) = p2
Ż Ż Ż
q = f(x, "p)
(x = 0, "p =
0)
q(t) = k1x(t) - k2"p(t),
"f(x, "p) "f(x, "p)
k1 = oraz k2 = - .
"x ""p
(0,0) (0,0)
k1 k2
dt
S�dy(t) = qdt
"p(t)
k1 S�
"p(t) = x(t) - Ź(t).
k2 k2
"p(t)
f(t) m
S
f(t) = S"p(t) = (k1x(t) - S�Ź(t)).
k2
S
m�(t) = -bŹ(t) + (k1x(t) - S�Ź(t)).
k2
Ź(t) + T0�(t) = k0x(t)
Sk1 mk2
k0 = oraz T0 = .
S2� + bk2 S2� + bk2
m
 m1 f1(t)
m2
x1(t) x2(t)
k1 k2
b1 b2
m1 m2
m1ć1(t) = -b1�1(t) - b2(�1(t) - �2(t)) - k1x1(t) - k2(x1(t) - x2(t))
m2ć2(t) = -b2(�2(t) - �1(t)) - k2(x2(t) - x1(t)).
x1(0) �1(0) x2(0)
�2(0)
q1(t) q1(t) - q2(t)
L1q1(t) + R1q1(t) + R2(q1(t) - q2(t)) + + = 0
� Ł Ł Ł
C1 C2
q2(t) - q1(t)
L2q2(t) + R2(q2(t) - q1(t)) + = 0
� Ł Ł
C2
q1(0) q1(0) q2(0) q2(0) q1(t)
Ł Ł Ł
i1(t) q2(t) i2(t)
Ł
 m
u(t) y(t)
m
m�(t) = -b(Ź(t) - u(t)) - k(y(t) - u(t)).
Ł
 fi(t)
fo(t)
fi(t) - b�(t) - kx(t) = 0.
ui(t) a" fi(t)
uo(t) a" fo(t) i(t) a" �(t) R a" b C a" 1/k)
fi(t) - b1�(t) - k(x(t) - y(t)) = 0
-b2Ź(t) - k(y(t) - x(t)) = 0.
ui(t) a" fi(t)
uo(t) a" fo(t) ix(t) a" �(t) iy(t) a" Ź(t) R1 a" b1 R2 a" b2
C a" 1/k)
�i(t)
Ńi(t)
Ńo(t)
� Ł Ł
JŃo(t) = -k1(Ńo(t) - Ńi(t)) - b2Ńo(t) - b3Ńo(t)
Ł
0 = �i(t) - k1(Ńi(t) - Ńo(t)) - b1Ńi(t)
Ł
Ńi(0) Ńo(0) Ńo(0)
Ł Ł
ui(t) a" �i(t) i1(t) a" Ń1(t) i2(t) a" Ń2(t) R1 a" b1
R2 a" b2 R3 a" b3 C a" 1/k1 L a" J
Ł Ł
ii(t) a" �i(t) u1(t) a" Ń1(t) u2(t) a" Ń2(t) R1 a" 1/b1 R2 a" 1/b2 R3 a" 1/b3
L a" 1/k1 C a" J
k2
b2 xa(t)
b1(�i(t) - �o(t)) + k1(xi(t) - xo(t)) - b2(�o(t) - �a(t)) = 0
b (�o(t) - �a(t)) - k2xa(t) = 0.
2
�i
�l
Ń1 Ń2 Ń3)
N1 : N2 N3 : N4) J1 J2 J3)
 b1 b3)
� Ł
JeqŃ1(t) = �i(t) - beqŃ1(t) - �l eq(t)
N1 2 N1 2 N3 2
Jeq = J1 + J2 + J3
N2 N2 N4
N1 2 N3 2 N1 N3
beq = b1 + b3, �l eq = �l(t).
N2 N4 N2 N4
a b
A
 u(t) y(t) A C
x(t)
B
b a
x(t) = u(t) - y(t).
a + b a + b
(a + b)/(ak0) 1
y(t) H" (b/a)u(t)
 q1r q2r hr
q1(t) q2(t) h(t)
qd(t)
z(t)
z(t) = ah(t)/b q1(t) =
-c1z(t) c1 > 0
z(t)
dh(t)
A = q1(t) + qd(t) - q2(t)
dt
A q2(t)
= �gh(t)/Rh Rh
�h = ARh/(�g)
�g dh(t)
� h(t) + �h = q1(t) + qd(t)
Rh dt
C1 C2
Z
qd(t)
Ż
q h
Ż
 q(t) qd(t)
h1(t) h2(t)
q2(t)
dh1(t)
C1 = q1(t)
dt
dh2(t)
C2 = q(t) - q1(t) - q2(t)
dt
qd(t) = 0
h2(t) - h1(t)
= q1(t)
R1
h2(t)
= q2(t)
R2
 R1 R2
h2(t)
x(t)
a
x(t) = h2(t).
a + b
y(t)
y(t) = �k0 x(�)d� k0
Z
q(t) = -kzy(t) kz
�d = const
 er(t)
�r(t) er(t) = c1�r(t)
�o(t)
eo(t) �o(t)
eo(t) = c2�o(t) er(t) - eo(t)
ef (t) ef (t) =
ka(er(t) - eo(t)) if (t)
dif (t)
Rf if (t) + Lf = ef (t).
dt
eg(t)
eg(t) = kgif (t) kg
�d
dia(t)
Raia(t) + La = eg(t) - eb(t)
dt
ia(t) eb(t)
eb(t) = kb�o(t) �(t)
�(t) = ktia(t) J
b �d(t)
d�o(t)
J = �(t) - b�o(t) - �d(t).
dt
Ja 1 : N
Jl bl
Ł
ut(t) = ktŃa(t)
uŃ(t) = kŃŃ(t) "ur(t)
ur(t) "Ń(t) Ń(t)
q(t))
Ż
q h
Ż
qd(t)
h(t)
hr(t)
u(t)
i(t) = y(t)
 t iL(t)
uC(t) u(t)
x(t) = [ x1(t) x2(t) ]T = [ iL(t) uC(t) ]T
diL(t) duC(t) 1
u(t) = L + uC(t), iL(t) = C + uC(t)
dt dt R
diL(t) 1 1 duC(t) 1 1
= - uC(t) + u(t), = iL(t) - uC(t).
dt L L dt C RC
1
y(t) = iR(t) = uC(t).
R
0 -1/L 1/L
�(t) = Ax(t) + bu(t) = x(t) + u(t)
1/C -1/(RC) 0
y(t) = cT x(t) = [ 0 1/R ]x(t).
ea(t)
ia(t) Ra
La eb(t)
�(t)
Ń(t) J
b
"�(t) = k1Ś"ia(t) Ś
k1
Ł
"eb(t) "eb(t) = k2Ś"Ń(t) k2
Ł
"ea(t) = Ra"ia(t) + La"ia(t) + "eb(t)
� Ł
J"Ń(t) = "�(t) - b"Ń(t)
Ł Ł
Ra"ia(t) + La"ia(t) + k2Ś"Ń(t) = "ea(t)
� Ł
J"Ń(t) - k1Ś"ia(t) + b"Ń(t) = 0.
 ("Ń(0),
Ł
"Ń(0)) "ia(t) "Ń(t) t e"
0
Ł
"ia(0) "ea(t), "Ń(t)
"ia(t) t e" 0
Ł
x(t) = [ "Ń(t) "Ń(t) "ia(t) ]T
�(t) = Ax(t) + b"ea(t)
�ł łł �ł łł
0 1 0 0
�ł �ł �ł �ł
A = 0 -b/J k1Ś/J , b = 0 .
0 -k2Ś/La -Ra/La 1/La
Ł Ż
x(t) = [ "Ń(t) "Ń(t) ]T A
Ż
Ż
b
0 1
0
Ż Ż
A = , b = .
k1Ś
-k1k2Ś2 b
0 -
JRa
JRa J
"Ń(t) La = 0 "Ń(t) = [ 1 0 0 ]x(t)

La = 0 "Ń(t) = [ 1 0 ]x(t)
Ż
y(t) + a2�(t) + a1Ź(t) + a0y(t) = u(t)
u(t) y(t)
y(t)
x1(t) = y(t) x2(t) = �1(t) = Ź(t) x3(t) = �2(t) =
�(t) �3(t) = -a0x1(t) - a1x2(t) - a2x3(t) + u(t)
x(t) = [ x1(t) x2(t) x3(t) ]T
�ł łł �ł łł
0 1 0 0
�ł �ł �ł �ł
�(t) = 0 0 1 x(t) + 0 u(t)
-a0 -a1 -a2 1
y(t) = [ 1 0 0 ]x(t)
 m fx(t)
m0
l
m x
m0 x - y
T (t) V (t)
2 2
1 1
T (t) = m�2(t) + m0 (x(t) + l sin ą(t)) + (l cos ą(t)) =
2 2 t t
1 1
= m�2(t) + m0 (�(t) + lą(t) cos ą(t))2 + (lą(t) sin ą(t))2 ,
Ł Ł
2 2
V (t) = V0 + m0gl cos ą(t),
x(t) ą(t)
V0
 g L(t) =
T (t) - V (t)
1 1
L(t) = m�2(t) + m0 (�(t) + lą(t) cos ą(t))2 + (lą(t) sin ą(t))2
Ł Ł
2 2
-V0 - m0gl cos ą(t).
x(t) = x1(t) x2(t) x3(t) x4(t)
Ł
= ą(t) ą(t) x(t) �(t) , x(t) " R4.
"L "L
- = -b�(t) + fx(t)( )
t "� "x
"L "L
- = 0( )
t "ą "ą
Ł
b fx(t)
"L/"x = 0
"L/"� = m�(t) + m0(�(t) + lą(t) cos ą(t))
Ł
("L/"�)/ t = mć(t) + m0ć(t) + m0lą(t) cos ą(t) - m0lą2(t) sin ą(t)
� Ł
(m + m0)ć(t) = m0lą2(t) sin ą(t) - m0lą(t) cos ą(t) - b�(t) + fx(t).
Ł �
"L/"ą = m0gl sin ą(t) - m0l�(t)ą(t) sin ą(t)
Ł
"L/"ą = m0l2ą(t) + m0l�(t) cos ą(t)
Ł Ł
("L/"ą)/ t = m0lć(t) cos ą(t) - m0l�(t) ą(t) sin ą(t) + m0l2ą(t)
Ł Ł �
ć(t) cos ą(t) + lą(t) - g sin ą(t) = 0.
�
�(t) = f(x(t), fx(t)), x(0).
(m + m0 sin2 ą(t))ć(t) = m0lą2(t) sin ą(t)
Ł
-m0g sin ą(t) cos ą(t) - b�(t) + fx(t).
m0lx2(t) sin x1(t) m0g sin x1(t) cos x1(t)
2
�4(t) = -
m + m0 sin2 x1(t) m + m0 sin2 x1(t)
bx4(t) fx(t)
- +
m + m0 sin2 x1(t) m + m0 sin2 x1(t)
g(m + m0) sin x1(t) m0x2(t) sin x1(t) cos x1(t)
2
�2(t) = -
l(m + m0 sin2 x1(t)) m + m0 sin2 x1(t)
bx4(t) cos x1(t) cos x1(t)
+ - fx(t)
l(m + m0 sin2 x1(t)) l(m + m0 sin2 x1(t))
�1(t) = x2(t) �3(t) = x4(t)
"t ą(t)
ą(t) H" 0 ą(t) H" 0
Ł
sin ą(t) H" ą(t) cos ą(t) H" 1 ą2(t) sin ą(t) H" H" ą2(t)ą(t) H" 0
Ł Ł
g(m + m0) b 1
�2(t) = x1(t) + x4(t) - fx(t)
lm lm lm
m0g b 1
�4(t) = - x1(t) - x4(t) + fx(t).
m m m
�(t) = Ax(t) + bfx(t), x(0),
 A " R4�4 b " R4
�ł łł �ł łł
0 1 0 0 0
�ł śł �ł śł
g(m + m0)/(lm) 0 0 b/(lm) -1/(lm)
�ł śł �ł śł
A = , b = .
�ł �ł �ł �ł
0 0 0 1 0
-m0g/m 0 0 -b/m 1/m
m
m0 m
l
(m + m0)ć(t) + k�(t) - m0l(ą2(t) � sin ą(t) - ą(t) � cos ą(t)) = p(t)
Ł �
ć(t) � cos ą(t) + lą(t) + g � sin ą(t) = 0
�
x(t) x0(t) y0(t) p(t)
n(t)
b
g
 x(t0) �(t0) ą(t0) ą(t0)
Ł
Ł
x(t) = [ x(t) �(t) ą(t) ą(t) ]T .
�(t) = Ax(t) + bp(t)
ą(t)
ą(t)
Ł
�ł łł �ł łł
0 1 0 0 0
�ł śł �ł śł
0 -k/m m0g/m 0 1/m
�ł śł �ł śł
A = , b = .
�ł �ł �ł �ł
0 0 0 1 0
0 k/(lm) -g(1 + m0/m)/l 0 1/(lm)
m P
u(t)
"f(t) m y(t)
"x1(t)
x(t) = [ "x1(t) "�1(t) "x2(t) ]T
�ł łł �ł łł
0 1 0 0
�ł �ł �ł �ł
�(t) = -(k1 + k2 + k3)/m 0 k1/m x(t) + 1/m u(t)
k1/b1 0 -k1/b1 0
y(t) = [ 1 0 0 ]x(t).
x1(t) x2(t)
y(t) R2
R1 = R2 = R3 = 1 M&! C1 = C2 = 1 �F)
�1(t) -2 1 x1(t) 1 0 u1(t)
= +
�2(t) 1 -2 x2(t) 0 1 u2(t)
x1(t)
y(t) = [ 1 -1 ] .
x2(t)
x1(t) = uC(t) x2(t) = iL(t)
y(t) =
uC(t)
x1(t) = uC(t) x2(t) = i2(t) x3(t) = i1(t)
y(t) = uC(t)
u(t)
-1/(R1C) -1/C 1/(R1C)
�(t) = x(t) + u(t)
1/L -R2/L 0
y(t) = [ 1 0 ]x(t).
�ł łł �ł łł
0 1/C 0 0
�ł �ł �ł
�(t) = -1/L2 -R/L2 R/L2 �ł x(t) + 0 u(t)
0 R/L1 -R/L1 1/L1
y(t) = [ 1 0 0 ]x(t).