�(t) + 5Ź(t) + 6y(t) = 0
 y(0+) = a Ź(0+) = b
s2Y (s)-
sy(0+) - Ź(0+) + 5(sY (s) - y(0+)) + 6Y (s) = 0,
5a + b + as 5a + b + as
Y (s) = = .
6 + 5s + s2 (2 + s)(3 + s)
Y (s)
Y (s)
3a + b 2a + b
Y (s) = - .
2 + s 3 + s
y(t) = L-1(Y (s)) = (3a + b) � e-2t - (2a + b) � e-3t, t e" 0.
L(s) L(s)
y(t) = � e-2t + � e-3t, t e" 0
M (s) s=-2 M (s) s=-3
L(s) dM(s)
Y (s) = M (s) =
M(s) ds
M (s) = 5 + 2s
t
Ź(t) + 4 e(t-�)y(�)d� + 3y(t) = 1, y(0+) = 1.
0
sY (s)-
y(0+) + 4 � L(et " y(t)) + 3Y (s) = 1/s
s - 1 1 1
Y (s) = = -
s(1 + s) 1 + s s(1 + s)
y(t) = e-t - (1 - e-t) = 2e-t - 1, t e" 0.
y(3)(t) + 4y(2)(t) + 4y(1)(t) + 3y(t) = u(t)
u(t)
y(0+) = 1 y(1)(0+) = 2 y(2)(0+) = 3
L(y(1)(t)) = sY (s) - y(0+)
L(y(2)(t)) = s2Y (s) - sy(0+) - y(1)(0+)
L(y(3)(t)) = s3Y (s) - s2y(0+) - sy(1)(0+) - y(2)(0+)
1 + 15s + 6s2 + s3
Y (s) = .
s(3 + 4s + 4s2 + s3)
1 0.80952 -4.28571 + 0.14286s
Y (s) = + - .
3s 3 + s 0.866032 + (0.5 + s)2
y(t) = 0.33333 + 0.80952e-3t + 5.03322e-5t �
sin(0.86603t - 0.02839), t e" 0
f(t)
1 + 3s
F (s) = .
1 + s + s2
g(t) = fŁ(t) G(s) = L(g(t)) = sF (s) -
f(0+)
s + 3s2
f(0+) = lim f(t) = lim sF (s) = lim = 3.
s"
t0+ s" 1 + s + s2
s + 3s2 -3 - 2s
G(s) = - 3 = .
1 + s + s2 1 + s + s2
-3s - 2s2
g(0+) = lim g(t) = lim sG(s) = lim = -2.
s"
t0+ s" 1 + s + s2
g(0+)
f(t) F (s)
"
1 + s + s2" 3/4 + (1/2 + s)2 f(t) = -1/ 3 �
=
"
e-t/2 � sin( 3t/2) + 3e-t/2 � cos(
"3t/2) t e" 0 f(0+) = 3 "
"
f(t) g(t) = - 3e-t/2 � sin( 3t/2) - 2e-t/2 � cos( 3t/2),
t e" 0 g(0+) = -2
�(t) + 3Ź(t) + 2y(t) = 0, y(0+) = a, Ź(0+) = b.
y(t) = (2a + b) � e-t - (a + b) � e-2t t e" 0
�(t) + 2Ź(t) + 5y(t) = 3 � 1(t), y(0+) = 0, Ź(0+) = 0.
y(t) = 0.6 - 0.3e-t � sin 2t - 0.6e-t � cos 2t t e" 0.
y(t)
u(t) = E � 1(t) C
+Q0
y(t) = (E - U0) � e-t/T t e" 0 U0 = Q0/C
T = RC
�(t) + 4Ź(t) + 3y(t) = u(t)
u(t) = 2 cos 3t t e" 0 y(t)
y(t) = -0.0667 cos 3t+0.1333 sin 3t+0.1667 e-3t-0.1e-t
t e" 0
ńł
Ź1
�ł - 2Ź2 + y1 = 1(t)
y1 + Ź2 - 2y2 = e-t � 1(t)
ół
y1(0+) = 1, y2(0+) = 0.
2 1 1 1 1
y1(t) = 1 - e-2t + e-t - et y2(t) = - e-2t - et
3 3 2 6 3
t e" 0
s
F (s) = , a = b.

(a + s)(b + s)
a�e-at-b�e-bt
f(t) = t e" 0
a-b
"
f(t) = 1(t - n).
n=1
e-s
F (s) = L(f(t)) =
s(1-e-s)
n = 0, 2, 4, . . .
1 dla n d" t < n + 1
f(t) =
-1 dla n + 1 d" t < n + 2.
1-e-s
F (s) = L(f(t)) =
s(1+e-s)
 gn(t) = L-1(Gn(s))
n
Gn(s) = (i + s)-1, n " N.
i=1
n (-1)i-1
gn(t) = aie-it t e" 0 ai = i " N
i=1
(i-1)!(n-i)!
n-1 n
Gn(s) = (2i + s) � (2i - 1 + s)-1, n " N.
i=1 i=1
n
gn(t) = L-1(Gn(s)) = aie-(2i-1)t, t e" 0
i=1
(2(i - 1) - 1)!! � (2(n - i) - 1)!!
ai =
(2(i - 1))!! � (2(n - i))!!
1 � 3 � 5 � � � (2k - 1) = (2k - 1)!! 2 � 4 � 6 � � � (2k) = (2k)!! k!! = 1
k d" 0
24 + 30s + 8s2
G(s) = .
24 + 26s + 9s2 + s3
G(s)
L(s) 24 + 30s + 8s2
G(s) = =
M(s) (2 + s)(3 + s)(4 + s)
L(s) L(s) L(s)
g(t) = � e-3t + � e-3t + � e-4t, t e" 0
M (s) s=-2 M (s) s=-3 M (s) s=-4
dM(s)
M (s) = = 26 + 18s + 3s2
ds
g(t) = -2e-2t - 6e-3t + 16e-4t, t e" 0.
L(s) L(s) L(s)
h(t) = + � e-2t + � e-3t+
M(s) s=0 sM (s) s=-2 sM (s) s=-3
L(s)
+ � e-4t, t e" 0.
sM (s) s=-4
h(t) = 1 + e-2t + 2e-3t - 4e-4t, t e" 0.
1
F (s) =
s(1 + T s)
F (s)
a b
F (s) = +
s 1 + T s
a = 1 b = -T
f(t) =
1 - e-t/T t e" 0
F (s)
L(s)
F (s) =
sM(s)
 L(s) = 1 M(s) = 1 + T s f(t)
L(s) L(s)
f(t) = + � e-t/T , t e" 0
M(s) s=0 sM (s) s=-1/T
dM(s)
M (s) = = T f(t) = 1 - e-t/T t e" 0
ds
F (s)
1 1
F (s) = L L-1 L-1
s 1 + T s
f(t)
1 1
f(t) = L-1 L-1 .
s 1 + T s
1 1 e-t/T
L-1 = 1(t) oraz L-1 = � 1(t)
s 1 + T s T
t
-1
f(t) = T � e-(t-� )/T d� = 1 - e-t/T t e" 0
0
t
1
F (s) = L L-1 d� .
1 + T s
0
1 e-t/T
L-1 = � 1(t).
1 + T s T
f(t) = -e�/T |t = 1 - e-t/T t e" 0
0
1 + 3s + s2
F (s) = .
(1 + s)3
 F (s)
a1 a2 a3
F (s) = + +
1 + s (1 + s)2 (1 + s)3
a1(1 + s)2 + a2(1 + s) + a3 = 1 + 3s + s2 a1 = 1
2a1 + a2 = 3 a1 + a2 + a3 = 1 a1 = 1 a2 = 1
a3 = -1
1 tn-1 � eąt
L-1 = , n e" 1, t e" 0
(-ą + s)n (n - 1)!
f(t) = (1 + t - 0.5t2) � e-t t e" 0
45 + 20s + 7s2 45 + 20s + 7s2
G(s) = = .
15 + 11s + 5s2 + s3 (3 + s)(22 + (1 + s)2)
6 5 + s
G(s) = + .
3 + s 22 + (1 + s)2
g(t) = 6e-3t + (cos 2t + 2 sin 2t) � e-t, t e" 0.
H(s)
G(s) 3 2 1 + s
H(s) = = - -
s s 3 + s 22 + (1 + s)2
h(t) = 3 - 2e-3t - e-t � cos 2t, t e" 0.
2 + 8s + 16s2
G(s) =
(2 + s)(1 + s)2
u(t)
 u(t) = u1(t) + u2(t) +
u3(t) u1(t) = 2 � 1(t - 1) u2(t) = -(t - 2) � 1(t - 2) u3(t) =
(t - 4) � 1(t - 4) u(t)
2e-s e-2s e-4s
U(s) = - + .
s s2 s2
G(s) � U(s)
2 + 8s + 16s2 2 + 8s + 16s2
L-1 oraz L-1
s(2 + s)(1 + s)2 s2(2 + s)(1 + s)2
2 + 8s + 16s2 L(s) a0 a11 a12 a2
= = + + +
s(2 + s)(1 + s)2 sM(s) s 1 + s (1 + s)2 2 + s
L(s)
1 d(2-1) L(s)
a0 = , a11 =
M(s) (2-1)!
ds(2-1) sM1(s) s=-1
s=0
L(s)
1 d(2-2) L(s)
a12 = , a2 =
(2-2)! sM1(s)
ds(2-2) sM1(s) s=-1
s=-2
M1(s) = 2 + s
2 + 8s + 16s2 1 24 10 25
= + - -
s(2 + s)(1 + s)2 s 1 + s (1 + s)2 2 + s
2 + 8s + 16s2
L-1 = 1 + (24 - 10t) � e-t - 25e-2t, t e" 0.
s(2 + s)(1 + s)2
2 + 8s + 16s2 3 25
L-1 = + t - (14 - 10t) � e-t + e-2t, t e" 0.
s2(2 + s)(1 + s)2 2 2
y(t) = (2 + (48 - 20(t - 1)) � e-(t-1) - 50e-2(t-1)) � 1(t - 1) -
-(1.5 + (t - 2) - (14 - 10(t - 2)) � e-(t-2) + 12.5e-2(t-2)) � 1(t - 2) +
+(1.5 + (t - 4) - (14 - 10(t - 4)) � e-(t-4) + 12.5e-2(t-4)) � 1(t - 4).
1
G(s) = .
s2(a + s)
t e-at-1
g(t) = + t e" 0
a a2
Y (s) 3 + s + (1 + s)2(1 + 3s + s2)
G(s) = =
U(s) (1 + s)2(1 + 3s + s2)
y(t) t = 0 t "
u(t)
limt0 y(t) = 1 limt" y(t) = 4
3+2s+s2
G(s) =
(1+s)3
7+9s+5s2+s3
G(s) =
(1+s)(2+s)
12+2s
G(s) =
5+2s+s2
 t e" 0
g(t) = (1 + t2) � e-t
d�(t)
g(t) = + 2�(t) + 2e-t - e-2t
dt
g(t) = (5 sin 2t + 2 cos 2t) � e-t
u1(t)
u2(t)
U2(s) 1
=
U1(s) 1 + (T1 + T2 + T12)s + T1T2s2
T1 = R1C1 T2 = R2C2 T12 = R1C2
39+26s+5s2
G(s) =
(1+s)(3+s)(4+s)
45+20s+7s2
G(s) =
15+11s+5s2+s3
36+12s+4s2+s3
G(s) =
((2+s)(3+s))2
2+s+s2
G(s) =
(-1+s)(2+s)(1+s2)
8+12s+16s2+24s3+36s4+48s5
G(s) =
((1+s)(2+s))3
4+8s+16s2+36s3
G(s) =
s2(1+s)(2+s)
 t e" 0
g(t) = 3e-t - 3e-3t + 5e-4t
g(t) = 6e-3t + (2 sin 2t + cos 2t) � e-t
g(t) = -(32 - 20t) � e-2t + (33 + 9t) � e-3t
g(t) = 0.6667et - 0.2667e-2t - 0.4472 cos(t - 0.4636)
g(t) = -(908 - 220t + 12t2) � e-t + (956 + 388t + 552t2) � e-2t
g(t) = 1 + 2t - 24e-t + 59e-2t
t e" 0
h(t) = 3.25 - 3e-t + e-3t - 1.25e-4t
h(t) = 3 - 2e-3t - e-t � cos 2t
h(t) = 1 + (11 - 10t) � e-2t - (12 + 3t) � e-3t
h(t) = -1 + 0.6667et + 0.1333e-2t - 0.4472 sin(t - 0.4636)
h(t) = 1 + (712 - 196t + 12t2) � e-t - (713 + 470t + 276t2) � e-2t
h(t) = 5.5 + t + t2 + 24e-t - 29.5e-2t
G(s)
u(t)
20+40s
G(s) = u(t) = t � 1(t)
s(10+7s+s2)
12+24s+36s2
G(s) = u(t) = (-1 + t2) � 1(t)
(2+s)2(3+s)
y(t) = -2.02 + 2.6t + t2 + 2.5e-2t - 0.48e-5t t e" 0
y(t) = 2.0556 + 1.3333t + t2 - (70.5 - 27t) � e-2t + 68.4444e-3t t e" 0
G(s) = Y (s)/
U(s) = 1/s y(t)
u(t) = sin t � 1(t)
y(t) = (1 - cos t) � 1(t)
s
 s
k2 1
H1(s) = h1, H2(s) = h2, G1(s) = k1, G2(s) = , G3(s) =
s 1 + T3s
1 1 1
G4(s) = , G5(s) = , G6(s) = .
1 + T4s 1 + T5s 1 + T6s
G(s) = Y (s)/U(s)
G5(s)
G3G4G5G6(G1 + G2)
G = .
1 + H1G4G5 + H2G3G4 + G3G4G5G6(G1 + G2)
k2 + k1s
G(s) = .
k2 + k1s + (h2(1 + T5s) + s(1 + T3s)(h1 + (1 + T4s)(1 + T5s)))
Y (s)/U(s)
P1(s) = -2 � s-1 � 13 = -26s-1
P2(s) = -2 � s-1 � (-160) � s-1 � 9 = 2880s-2
P3(s) = 3 � s-1 � 9 = 27s-1
P4(s) = 3 � s-1 � 77 � s-1 � 13 = 3003s-2
L1(s) = 108s-1
L2(s) = s-1 � (-160) � s-1 � 77 = -12320s-2
L3(s) = -114s-1.
L1(s) L3(s)
"(s) = 1 - (L1(s) + L2(s) + L3(s)) + L1(s)L3(s) = 1 + 6s-1 + 8s-2.
P1(s) P2(s) P3(s) P4(s)
"1(s) = 1 + 114s-1
"2(s) = 1
"3(s) = 1 - 108s-1
"4(s) = 1.
4
Y (s) 1 3 + s
= Pi(s)"i(s) = .
U(s) "(s) 8 + 6s + s2
i=1
 Y (s)/U(s)
Y (s)
= 1 + G2(s) + G1(s)G2(s).
U(s)
Y (s)/U(s)
Y (s) G1(s) + G3(s)
= .
U(s) 1 + G1(s)G2(s)
 Y (s)/U(s)
Y (s) G3(s) + G1(s)G2(s)
G(s) = = .
U(s) 1 + G1(s)G4(s) + G2(s)G5(s) - G3(s)G4(s)G5(s)
U2(s)/U1(s)
U2(s) sR1C1
= 1 - .
U1(s) (1 + sR1C1)(1 + sR2C2) + sR1C2
C(s)/R(s)
C(s) (2+s)2
=
R(s) s(1+s)(4+s)
Y (s)/U(s)
Y (s)
14+11s+3s2
=
U(s) 15+11s+5s2+s3
C(s)/R(s)
C(s) 4k0k1
= .
R(s) 4k1 - 6 + s + s2
F1(s) F1(s)F2(s)F4(s)
G1(s) = , G2(s) =
1 + F1(s)F2(s)F3(s)F4(s) 1 + F1(s)F2(s)F3(s)F4(s)
F1(s)F3(s)F4(s) F4(s)
G3(s) = , G4(s) = .
1 + F1(s)F2(s)F3(s)F4(s) 1 + F1(s)F2(s)F3(s)F4(s)
 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
Ż
p1(t) = 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
d"h(t) q
Ż
S � = "q1(t) - � "h(t)
Ż
dt 2h
T0s"H(s) = k0"Q1(s) - "H(s)
Ż Ż
T0 = 2hS/q k0 = 2h/q
Ż Ż
"H(s)/"Q1(s) = k0/(1 + T0s)
Ż
"q2(t) = q"h(t)/(2h) = "h(t)/k0
Ż
"Q2(s)/"H(s) = 1/k0 "Q2(s)/"Q1(s)
"Q2(s) "Q2(s) "H(s) 1
= � = .
"Q1(s) "H(s) "Q1(s) 1 + T0s
 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
E(s) km
=
Um(s) Rm(1 + �ms)
�m = Lm/Rm
 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
"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 JRa
k0 = oraz T0 = .
kkb + bRa kkb + bRa
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
R1 R2
C1 C2
R1 1
T1(s) = Q(s) + T2(s)
1 + R1C1s 1 + R1C1s
R1
1
R2
T2(s) = T1(s) + T3(s).
R1 R1
1 + + R1C2s 1 + + R1C2s
R2 R2
T1(s) = Gq(s)Q(s) + GT3(s)T3(s)
Gq(s) GT3(s)
T1(s) R1 + R2 + R1R2C2s
Gq(s) = =
Q(s) 1 + (R1C1 + R2C1 + R2C2)s + R1C1R2C2s2
T1(s) 1
GT3(s) = = .
T3(s) 1 + (R1C1 + R2C1 + R2C2)s + R1C1R2C2s2
 eA eB
A B
Eo(s)/Ei(s)
EA(s) EB(s)
R1Cs R3
EA(s) = Ei(s), EB(s) = Eo(s).
1 + R1Cs R2 + R3
EA(s) = EB(s)
Eo(s) R2 + R3 R1Cs
= � .
Ei(s) R3 1 + R1Cs
I(s) R1 R3
1
I(s) = (Ei(s) - Eo(s))
R1 + R3
B
1
EB(s) = Ei(s).
1 + R2Cs
 A EA(s) = Ei(s) - UR1(s)
UR1(s) R1
R1
UR1(s) = R1I(s) = (Ei(s) - E0(s)).
R1 + R3
R3 R1
EA(s) = Ei(s) + Eo(s).
R1 + R3 R1 + R3
EA(s) = EB(s)
R3 R1 1
Ei(s) + Eo(s) = Ei(s).
R1 + R3 R1 + R3 1 + R2Cs
R2R3
1 - Cs
Eo(s)
R1
= .
Ei(s) 1 + R2Cs
R1 = R3
Eo(s) 1 - R2Cs
= .
Ei(s) 1 + R2Cs
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
(Rf + sLf )"If (s) = "Uf (s)
Lf
If0 (If0, Ia0)
if (t) = If0 + "if (t)
ia(t) = Ia0 + "ia(t) Ia0
 �(t) = k1Śf (t)ia(t)
"T (s) = kt(If0"Ia(s) + Ia0"If (s))
kt
(Ra + sLa)"Ia(s) = -k2(&!0"If (s) + sIf0"Ś(s))
La
ia0) &!0
If0, Ia0) k2
ktIa0
s(b + Js)"Ś(s) = ktIf0"Ia(s) + � "Uf (s)
Rf + Lf s
-k2&!0 sk2If0
"Ia(s) = � "Uf (s) - � "Ś(s).
(Ra + Las)(Rf + Lf s) Ra + Las
"Ś(s) kt(Ia0(Ra + Las) - k2If0&!0)
= .
2
"Uf (s) s((b + Js)(Ra + Las) + k2ktIf0)(Rf + Lf s)
If0 = 0
"Ś(s) kv
=
"Uf (s) s(1 + T1s)(1 + T2s)
kt J Lf
kv = Ia0, T1 = , T2 = .
bRf b Rf
"Ś(s) ka ktIa0
= , ka = .
"Uf (s) s2(1 + T2s) JRf
 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 =
Ż Ż
p2(t) = p2 q = f(x, "p)
Ż
(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
Y (s) k0
=
X(s) s(1 + T0s)
Sk1 mk2
k0 = oraz T0 = .
S2� + bk2 S2� + bk2
m
Y (s)/X(s) H" k0/s
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
x1(0) �1(0) x2(0) �2(0)
X2(s) k2 + b2s
= .
F1(s) (k1 + k2 + (b1 + b2)s + m1s2)(k2 + b2s + m2s2) - (k2 - b2s)2
m
u(t) y(t)
m
Y (s)/U(s)
m�(t) = -b(Ź(t) - u(t)) - k(y(t) - u(t)).
Ł
(y(0), Ź(0))
Y (s) k + bs
= .
U(s) k + bs + ms2
fi(t)
fo(t)
fi(t) - b�(t) - kx(t) = 0
Fo(s) 1
=
Fi(s) 1 + T0s
T0 = b/k
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.
Fo(s) k0 b2 b1b2
= , k0 = , T0 = .
Fi(s) 1 + T0s b1 + b2 k(b1 + b2)
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)
Śo(s) 1 J b1 + b2
= , � = , ś = " .
Śi(s) 1 + 2ś�s + �2s2 k1
2 k1J
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
Xo(s)/Xi(s)
Xi(s) Xo(s)
xi(t) xo(t)
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
Xo(s) k1k2 + (k1b2 + k2b1)s + b1b2s2
= .
Xi(s) k1k2 + (k1b2 + k2b1 + k2b2)s + b1b2s2
k1 + b1s b2s b2s
G1(s) = , G2(s) = , G3(s) = .
k1 + (b1 + b2)s) k2 + b2s k1 + (b1 + b2)s
�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
Eo(s)/Ei(s)
Eo(s) Z3(s)Z4(s)
a) = -
Ei(s) Z1(s)Z2(s) + Z1(s)Z3(s) + Z2(s)Z3(s)
Eo(s) Z2(s)Z3(s) + Z2(s)Z4(s) + Z3(s)Z4(s)
b) = - .
Ei(s) Z1(s)Z4(s)
 a b
A
Y (s)/U(s)
u(t) y(t) A C
x(t)
B
b a
x(t) = u(t) - y(t).
a + b a + b
b
Y (s)
a
=
a+b
U(s)
1 + s
ak0
k0
(a + b)/(ak0) 1
Y (s)/U(s) H" b/a
 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(s) Qd(s)
q(t) qd(t) H1(s)
H2(s)
h1(t)) h2(t)) Q2(s)
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(s) (1 + R1C1s)R2
Gp(s) = = .
Q(s) 1 + (R1C1 + R2C1 + R2C2)s + R1C1R2C2s2
Qd(s)
H2(s)/Qd(s)
h2(t)
x(t)
a
x(t) = h2(t).
a + b
y(t)
Y (s) = X(s) � k0/s X(s) Y (s)
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
kakg kt 1
Gf (s) = , Ga(s) = , Gb(s) =
Rf (1 + Tf s) Ra(1 + Tas) b(1 + Tbs)
Tf = Lf /Rf Ta = La/Ra Tb = J/b
 Ja 1 : N
Jl bl
Ł
ut(t) = ktŃa(t)
uŃ(t) = kŃŃ(t) "ur(t)
ur(t) "Ń(t) Ń(t)
R2 k RaJeq
k0 = , km = , Tm =
R1 kkb + Rabeq kkb + Rabeq
Jeq = Ja + Jl/N2 beq = bl/N2 k kb
q(t))
q
Ż
Ż
h qd(t)
h(t)
hr(t)
Xv(s) ki kds
Gc(s) = = kc + +
H(s) s 1 + �s
a1c2d1 kaa2c1
kc = , ki =
b2(c1 + c2)(d1 + d2) b2(c1 + c2)
b1c2d2� b
kd = , � =
b2(c1 + c2)(d1 + d2) k
ka
Xa(s) ka
=
X(s) s
Q(s)
= -cv
Xv(s)
H(s) kp
= .
Q(s) 1 + �ps
kp �p
 Gp(s) = kp/s kp > 0
kf > 0
k T
e(t)
k T
kp
kp
s
G(s) = = .
kpkf
s + kpkf
1 +
s
k
G(s) =
1 + T s
k T
k = 1/kf T = 1/(kpkf )
r(t) 1(t)
k
c(t) = L-1(G(s)R(s)) = L-1 = k(1 - e-t/T ) � 1(t)
s(1 + T s)
kp
r(t) t � 1(t)
k
c(t) = L-1(G(s)R(s)) = L-1 = k((t - T ) + T e-t/T ) � 1(t).
s2(1 + T s)
e(t) = r(t) - c(t)
1 k 1 - k + sT
E(s) = - = .
s s(1 + T s) s(1 + T s)
e(") = lims0(sE(s)) = 1 - k
k = 1
e(t) = t - k((t - T ) + T e-t/T ) � 1(t) = ((1 - k)t + kT - kT e-t/T ) � 1(t).
k = 1 e(t)

k = 1
e(") e(t)
1 1 T
E(s) = - =
s2 s2(1 + T s) s(1 + T s)
e(") = lims0(sE(s)) = T
Gp(s) = kp/s kp > 0
kf > 0
G(j�)
�3dB kf
�3dB = 10 rad � s-1 � = 0.1 � �3dB |G(j�)|dB H" 20 dB
kp kf
k 1 1
G(s) = , k = , T = .
1 + T s kf kpkf
k k
G(j�) = G(s)|s=j� = = " � e-j arctan(�T ).
2
1 + j�T
1 + �2T
 �3dB = 1/T �3dB =
kpkf
� = 0.1 � �3dB |G(j�)| H" k
kf = 1/k H" 0.1 kp = �3dB/kf H" 100
r(t) = 0, "t
d(t) = �(t) c(t)
C(s) 1
= .
D(s) (1 + k) + s
D(s) = L(�(t)) = 1
1
C(s) =
(1 + k) + s
c(t) = e-(1+k)t � 1(t) k
kp Tp
 kp = 9 Tp = 10 s
� = 0.1 rad�s-1
-Ą/4
kp H" 2.414 Tp H" 34.14 s
m = 1000 kg
u(t) = 200�1(t) N
b = 50 N � s � m-1 vmax
v(0) = 0 m � s-1
u(t)
vmax = 4 m�s-1
59.9 s
1 - s
G(s) = .
1 + 0.1s
1 s - 1
a) , b) .
s - 1 s + 1
1 1 s - 1
a) = " � e-Ą+arctan �, b) = 1 � eĄ-2 arctan �.
s - 1 s + 1
1 + �2
k kt
�% = 20%
T� = 1 s
Ts2% Ts5%
C(s) k
G(s) = =
R(s) k + (1 + kkt)s + s2
1
G(s) =
1 + 2ś�s + �2s2
1 1
�2 = oraz 2ś� = + kt.
k k
�% T�
0 < ś < 1
śĄ Ą
�% = � � 100% = exp - � 100%, T� = � �
1 - ś2 1 - ś2
| ln �| 1 - ś2
ś = oraz � = � T�.
Ą
Ą2 + ln2 �
ś = 0.456 � =
0.283 s k = 1/�2 = 12.46 kt = 2ś� - 1/k = 0.178
Ts2% = 2.359 s Ts5% = 1.488 s
Ts2% H" 4�/ś = 2.485 s Ts5% H" 3�/ś = 1.86 s
 k
kt
k kt
Ts2% d" 0.25s
C(s) 2k
G(s) = = .
R(s) 2k + (1 + 2kt)s + s2
1 1 kt
�2 = oraz 2ś� = + .
2k 2k k
ś = 1 G(s) = 1/(1 + �s)2
G(s) t
h(t) = L-1 = 1 - 1 + e-t/T , t e" 0.
s T
Ts"
ą" (1 + Ts"/�)e-Ts"/� = "
" = 0.02 " = 0.05
Ts2% H" 5.834 � � Ts5% H" 4.744 � �
� = 0.04285 s k =
272.284 kt = 22.8347
ś = 1
Ts"
Ts2% H"
4�/ś Ts5% H" 3�/ś
ś 1)
Ts2% H" 4�/ś
k = 128.0 kt = 15.5
 d(t)
k kt d(t) = 1(t)
|e(")| d" 0.005
Ts2% d" 0.5 s �%
C(s) k
Grc(s) = =
R(s) k + (3 + kkt)s + s2
E(s) -1
Gde(s) = = .
D(s) k + (3 + kkt)s + s2
D(s) = 1/s |e(")| = 1/k k > 0
k e" kmin = 200
�2 =
1/k 2ś� = 3/k + kt Ts2% H" 4�/ś
� = ś/8
k = kmin = 200 ś2 = 64/k ś = 0.566
�% = 11.59 %
kt kt = 2ś� -3/k
kt = 0.065 � = 0.0707 s
Ts2% = 0.415 s
2
�n
G(s) =
2
�n + 2ś�ns + s2
� d" �max = 0.2
Ts2% d" Ts2%max = 0.5 s
s1,2
s1,2 = -ś�n ą j�n 1 - ś2
0 < ś d" 1
1 - ś2
tan ą =
ś
1 - ś2
ą = arctan = arccos ś.
ś
 � = e-Ą/ tan ą
Ą
ą d" ąmax = arctan - .
ln �max
� Ts2%
� d" �max
Ts2% d" Ts2%max � d" �max
Ts2% d" Ts2%max
� �max
Ts2% H" 4/(ś�n)
4
-ś�n d" �max = - .
Ts2%max
Ts2%
Ts2%max
� Ts2%
ąmax = 62.9ć% �max =
-8 (s-1) s1,2 =
-10 ą j12
ś = 0.6402 �n = 15.621 (rad � s-1) ą = 50.2ć%
�% = 7.3% Ts2% = 0.384 s
C
1
G(s) = .
1 + 2śs + s2
ś
G(s)
�n = 1 rad � s-1
2
�n 1
G(j�) = = .
2 2
�n - �2 + j2ś�n�
� �
1 - + j2ś
�n �n
� = �n G(j�n) = 1/(j2ś)
|G(j�)|
ś
1
ś = .
2|G(j�)| �=1 rad�s-1
ś �
ś
 T = 2 s T0 = 0.3 s
k
Tv
Mr = 1.4
�3dB = 15 rad � s-1 �
T� Ts5%
C(s) k
G(s) = = .
R(s) k + (T0 + kTv)s + T0T s2
T0T Tv T0
� = oraz ś = + .
k 2� 2k�
1 1
2
Mr = , 0 d" ś d" "
4ś2(1 - ś2)
2
1 1
ś = 1 - 1 - , Mr e" 1
2
2 Mr
Mr ś
� �3dB
ś
1 - 2ś2 + (1 - 2ś2)2 + 1
� = .
�3dB
ś = 0.3874 � =
0.0924 s
k = T0T/�2 = 70.2969 Tv = 2ś� - �2/T = 0.0673 s
�% =
26.7% T� = 0.315 s T = 0.734 s Ts5% = 0.712 s
s5%
1 + ��s
G(s) = .
1 + 2ś�s + �2s2
ś 0 < ś < 1
�
�
�
Ż
�(ś, �) = �(ś, �) � e-śT�(ś,�)
�(ś, �) = �2 - 2�ś + 1
"
� 1-ś2
Ą + arctan
�ś-1
Ż
T�(ś, �) = .
1 - ś2
�(ś, �) �
Ż
"�(ś, �) � - ś "T�(ś, �)
Ż
= - ś�(ś, �) � � e-śT�(ś,�).
"� �(ś, �) "�
Ż
T�(ś, �) �
Ż
"T�(ś, �) 1
= - .
"� �2(ś, �)
"�(ś, �) �
Ż
= � e-śT�(ś,�).
"� �(ś, �)
� �(ś, �)
� = 0
"2�(ś, �)
Ż
= e-śT�(ś,0) > 0.
"�2 �=0
ś
�(ś, �)
� �(ś, �)|�=0
ś = 0.25
k
r(t) = t � 1(t)
C(s) k
T (s) = =
R(s) k + s + s2
"
1 1
ś = = " oraz �n = k.
2�n 2 k
ś = 0.25
k = 4
C(s) = T (s)/s2
2ś�n + s
E(s) = R(s) - C(s) =
2
s(�n + 2ś�ns + s2)
e(") = 2ś/�n = 0.25
C(s) k(1 + T s)
T (s) = = .
R(s) k + s + s2
1 - kT + s
E(s) = R(s) - C(s) = .
s(k + s + s2)
e(") = 1/k - T
T = 1/k
 T (s) = (4 + s)/(4 + s + s2)
kp
Gp(s) = , kp = 4, Tp = 0.15 s.
1 + Tps
1
Gm(s) = , Tm = 0.01 s
1 + Tms
Gc(s) = kc
kc �% c(t)
r(t)
�% = 20% c(") e(t) =
r(t) - c(t) kc
G(s)
C(s) Gc(s)Gp(s)
G(s) = = .
R(s) 1 + Gc(s)Gp(s)Gm(s)
1 + ��s
G(s) = G(0) �
1 + 2ś�s + �2s2
kckp Tm(1 + kckp)
G(0) = G(s)|s=0 = , � =
1 + kckp Tp
Tm + Tp TmTp
ś = , � = .
1 + kckp
2 TmTp(1 + kckp)
0 < ś < 1
�
kc Tm Tp
ą � �2 - ą2
� = exp - Ą + arctan � 1 + �2�2 - 2ł
ł - 1
�2 - ą2
Tm + Tp Tm Tm + Tp
ą = , � = oraz ł =
Tp 2Tp
2 TmTp
� = 1 + kckp
kc
ą = 2.06559 � = 0.258199 ł = 0.533333
�
�
� = 0.2 � = 3.590042
kc = 2.9721
e(") = 1 - G(0) =
1/(1 + kckp) = 0.07759
 k
k > 2.25
k = 2.25 k < 2.25
1
Gp(s) =
(5 + s)(0.2 + s)
k
k
 k �%
�% = 25%
k
k > 5.76
�% = 25% k =
40.476
h(") = 0.976
r(t) = 10 � 1(t)
r(t)
ś = 1
kv
ą2%)
k
C(s)
k
T (s) = =
R(s) k+kkvs+s2
c(") = 10
"
kv = 2/ k k > 0
k = 0.9454 kv = 2.0569
k0 k1 k2
�% H" 10%
Ts2% H" 0.5 s
k0 = k1 = 22.222 k2 = 2
�% = 9.48% Ts2% = 0.446 s
kt = 0)
�% = 50% Ts2% = 2 s
k kt
Ć Ć
T T = 0.254 s T
k = 12.478 kt = 0.0414
T = 0.272 s
�% = 26.4% Ts2% =
0.88 s
k0 = 8 T = 0.3 s
Ti k
�% = 15%
�3dB = 15 rad � s-1
Ti = 0.1859 s k = 0.3393
4
G(s) = .
4 + 4śs + s2
ś
"
ś = 1/ 2 H" 0.707
G0(s)
ś
�1 �2 �3
kv
G0(s) =
s(1 + T s)
1
kv = �1 oraz T = .
�2
�2
�1
kv �2 �3
2
�3 = �1�2 = oraz ś = = = .
T 2 2�3 2�1
 T = 5 s T0 = 0.5 s
k1 k2
Mr = 1.25 �3dB = 8 rad � s-1
�% T� Ts2%
k1 = 90.5905 k2 = 25.9207 �% = 20.79%
Ż
T� = 0.584 s Ts2% = 1.495 s Ts2% = 1.388 s
T0 = 4 s kc Ti
Mr = 1.3
�3dB = 3 rad � s-1 �%
T� Ts5%
� = 2ś
"
� + 1 + �2
1 - ą
ś = oraz � =
2ą �3dB
1
ą = 1 - oraz � = 1 + 2ś2.
2
Mr
 kc = 2śT0/� = 7.9618
Ti = 2ś� = 0.8885 s
Ż
�% = 22.28% T� = 1.51 s Ts5% = 3.30 s Ts5% =
2.90 s
Gp(s)
Gm(s) c(t)
0.6 1
Gp(s) = oraz Gm(s) = .
1 + 0.35s 1 + 0.03s
kc Gc(s) = kc
�% = 15%
r(t)
�
e(t) = r(t) - c(t) e(") = 0.05
kc
�%
kc = 13.2419
�% = 15%
e(") = 0.11179 e(") = 0.05 kc =
31.6667
�% = 52.8%
8
G(s) = .
10 + 9s + 4.5s2 + s3
�% = 10.6% T� = 2.328 s
Tr = 1.056 s Ts2% = 3.267 s
Mr = 1.031 �r = 1.071 rad�s-1
�3dB = 2.13 rad � s-1
G(s)
�% Ts2%
�% T�
8
G(s) = .
(2.5 + s)(4 + 2s + s2)
�1 = 0.4 s
ś0 = 0.5
�n = 2 rad � s-1 �2 = 1/(ś0�n) = 1 s
2
k�n k 1
Gr(s) = = , � =
2
�n + 2ś�ns + s2 1 + 2ś�s + �2s2 �n
k = G(s)|s=0 = 0.8
�% = 10.6%
ś = 0.581
Gr(s) T� = 3.861� Ts2% = 5.905� Tr = 1.8095� Mr = 1.057
�r = 0.569/� �3dB = 1.173/�
� Ts2% � = 3.267/1.905 s = 0.5533 s
2.6134
Gra(s) = .
3.2668 + 2.1013s + s2
T� = 2.136 s Tr = 1.001 s Mr =
1.057 �r = 1.029 rad � s-1 �3dB = 2.12 rad � s-1
 � T� � = 2.328/3.861 s =
0.6029 s
2.2007
Grb(s) =
2.7508 + 1.9282s + s2
Ts2% = 3.56 s Tr = 1.091 s Mr = 1.057 �r =
0.946 rad�s-1 �3dB = 1.944 rad�s-1
G(s)
G(s)
3.2
Grc(s) = .
4 + 2s + s2
�% = 16.3%
T� = 1.811 s Tr = 0.819 s Ts2% = 4.04 s Mr = 1.155 �r = 1.413 rad �
s-1 �3dB = 2.54 rad � s-1
h(t)
 �n =
0.2 rad � s-1 ś H" 0.1
� H" 0.05 rad � s-1 -40dB/dek
�
k 1 + j
0.05
G(j�) = .
2
� �
(j�)2 1 + 2 � 0.1 j + j
0.2 0.2
|G(j0.01)|dB = 80 dB k = 1
1 + 20s
G(s) = .
s2(1 + s + 25s2)
6
Gp(s) = .
(1 + 4s)(1 + s)(1 + 0.125s)
Gc(s)
 Mr = 1.3
� T� Ts5%
�r �3dB
C(s) Gc(s)Gp(s)
Grc(s) = =
R(s) 1 + Gc(s)Gp(s)
r(t)
Gw(s)
n
k0
Gp(s) =
n
(1 + Tjs)
j=1
1
GP I(s) = kc 1 +
Tis
1
GP ID(s) = kc 1 + (1 + Tds).
Tis
Grc(s)
Gw(s)
n e" 2
n
k0
�
Gp(s) = , TŁ = Tj
(1 + T1s)(1 + TŁs)
j=2
T1
n e" 3
n
k0
�
Gp(s) = , TŁ = Tj
(1 + T1s)(1 + T2s)(1 + TŁs)
j=3
T1 T2
T1 T1 T2 TŁ
P I : Ti = T1
P ID : Ti = T1, Td = T2.
kc
Mr
�
Gp(s) Gc(s)
�
Gc(s)Gp(s) k0kc
�
Grc(s) = = .
�
k0kc + T1s + T1TŁs2
1 + Gc(s)Gp(s)
�
Gw(s) Grc(s)
T1
� = 2śTŁ oraz kc = .
4ś2k0TŁ
ś = 0.42487
T1 = 4 s
TŁ = 1.125 s k0 = 6
T1 =
4 s T2 = 1 s TŁ = 0.125 s k0 = 6
kc = 0.8207 Ti = 4 s � = 0.95596 s �% = 22.89% T� =
Ż
3.318 s Ts5% d" Ts5% = 6.964 s �r = 0.836 rad � s-1 �3dB =
1.413 rad � s-1
kc = 7.3863 Ti = 4 s Td = 1 s � = 0.1062 s �% =
Ż
22.89% T� = 0.369 s Ts5% d" Ts5% = 0.774 s �r = 7.525 rad � s-1
�3dB = 12.719 rad � s-1
Ts5% = 0.771 s) �% = 27.4% T� = 3.22 s Ts5% = 7.18 s
Mr = 1.417 �r = 0.927 rad � s-1 �3dB = 1.52 rad � s-1
kc
kc
� - �
Ż
kc = kc 1 -
�
Ż
Ż
Mr - Mr
kc = kc 1 -
Ż
Mr
kc
Ż
� Mr � Mr
Ż
kc
 kc
kc kc = 0.85kc =
0.6976
� = 22.96% T� = 3.54 s Ts5% = 7.33 s Mr = 1.30 �r = 0.815 rad � s-1
�3dB = 1.365 rad � s-1
Grc(s)
Gw(s) Gp(s)
r(t)
Gru(s) = U(s)/R(s)
u(t)
Gc(s)
Gru(s) =
1 + Gc(s)Gp(s)
Gru(s) H" Gw(s)/Gp(s)
kp
�
Gp(s) = , T e" TŁ
(1 + T s)(1 + TŁs)
kp T
TŁ
kc
Gc(s) = kc + .
Tis
Ti
�
Ti = T G0(s) =
�
Gc(s)Gp(s)
T
�
G0(s)
kc
� T� Ts"
Mr �r �3dB
8
Gp(s) =
(1 + 2s)(1 + 0.04s)(1 + 0.008s)
�% = 15%
T� Ts2%
Mr �r �3dB
�
G0(s)
kc(1 + Tis) kp
� �
G0(s) = Gc(s)Gp(s) = � .
Tis (1 + T s)(1 + TŁs)
kckp
�
G0(s) = .
T s(1 + TŁs)
�
G0(s) kckp
�
G(s) = = .
�
kckp + T s + T TŁs2
1 + G0(s)
1 T
ś = " oraz � = "Ł
2 ą1 ą1
kckpTŁ
ą1 = .
T
�
T� Ts" Mr �r �3dB) kc
0 < ś < 1
T
kc > .
4kpTŁ
� = e-Ąą2, T� = 2Ąą2TŁ
"
2 ą1
Ż
Ts" d" Ts" = 2TŁ ln
" ą2
"
ą3
Mr = 2ą1ą2, �r =
TŁ
2
ą3 + ą2 + ą3
1
�3dB =
TŁ
1 1
ą2 = " oraz ą3 = ą1 - .
4ą1 - 1 2
"
ś < 1/ 2
T
kc > .
2kpTŁ
�
kc
T (Ą2 + ln2 �)
kc = .
4kckpTŁ ln2 �
8
�
Gp(s) = .
(1 + 2s)(1 + 0.048s)
0.04 s 0.008 s) TŁ = 0.048 s
kc = 4.8727
Ti = 2 s
T� = 0.182 s Ts2% = 0.390 s Mr = 1s-1
�3dB = 25.24 rad � s-1
�% = 18.26 T� = 0.173 s Ts2% = 0.389 s
Mr = 1.191 �r = 15.96 rad � s-1 �3dB = 27.9 rad � s-1
0 < ś < 1)
e-T0s
G(s) = .
1 + T s
G(s)
T T0
G(s)
G(s) T
1
T = .
�3dB
arg G(j�) = - arctan(�T ) - �T0.
arg G(j�3dB) = - arctan 1-T0/T = -Ą/4 - T0/T
T0
Ą
T0 = -T arg G(j�3dB) + .
4
12
Gp(s) = .
(1 + 6s)(1 + 2s)(1 + 0.1s)(1 + 0.012s)
kc
Ti Td
�% H"
20% Ts2%
Mr �r
12
�
Gp(s) = .
(1 + 6s)(1 + 2s)(1 + 0.112s)
kc = 5.3686 Ti =
6 s Td = 2 s
Ż
�% = 20% T� = 0.361 s Ts2% = 0.850 s Ts2% = 0.902 s
Mr = 1.232 �r = 7.484 rad � s-1
�% = 23.39% T� = 0.348 s Ts2% = 0.829 s
Mr = 1.31 �r = 8.26 rad�s-1
18
Gp(s) = .
(1 + 5s)(1 + 1.5s)(1 + 0.08s)(1 + 0.03s)
kc Ti
Mr H" 1.5
�% T� Ts2%
�r
kc = 0.3388 Ti =
5 s �% =
Ż
30.12% T� = 3.864 s Ts2% = 12.54 s Ts2% = 12.81 s Mr = 1.5
�r = 0.751 rad � s-1
�% = 34.60% T� = 3.80 s Ts2% = 12.55 s Mr = 1.65 �r = 0.799 rad �
s-1
kc
�%
Mr
kc = 0.9kc =
0.3049 �% = 31.8%
T� = 4.03 s Ts2% = 13.07 s Mr = 1.55 �r = 0.747 rad � s-1
s s
�c 1 + 1 +
�1 �2
G(s) = .
3
s
s 1 +
�3
e-T0s
G(s) = .
1 + T s
� = 2.5 rad � s-1 |G(j�)| = 0.5
arg G(j�) = -Ą rad T T0
T = 0.693 s T0 = 0.838 s
G(s)
G(s) = G0(s) � e-0.1s
40(1 + s)
G0(s) = .
s(0.2 + s)(64 + 2s + s2)
G0(s)
1
Gp(s) =
(1 + 0.2s)(1 + 0.6s)(1 + 1.4s + 4s2)
�% = 29.26% T� =
7.58 s Ts2% = 22.69 s Ts5% = 16.52 s
Mp = 1.47 �r = 0.428 rad � s-1
�3dB = 0.684 rad � s-1
Mr �r
Mr �3dB
Gp(s)
Gp(s)
Mr
�r
1
Gra(s) =
1 + 1.4612s + 4.0033s2
�% = 29.17% T� = 6.75 s Ts2% = 21.63 s Ts5% = 15.65 s Mr = 1.47
�r = 0.428 rad � s-1 �3dB = 0.702 rad � s-1
Mr
�3dB
1
Grb(s) =
1 + 1.499s + 4.21345s2
�% = 29.17% T� = 6.93 s Ts2% = 22.19 s Ts5% = 16.06 s Mr = 1.47
�r = 0.417 rad � s-1 �3dB = 0.684 rad � s-1
Gp(s)
1
Grc(s) =
1 + 1.4s + 4s2
�% = 30.92% T� = 6.69 s Ts2% = 21.96 s Ts5% = 15.76 s Mr = 1.52
�r = 0.434 rad � s-1 �3dB = 0.709 rad � s-1
 G(s) = e-T0s
T0
1 - s
2
Ga(s) = .
T0
1 + s
2
G(s) - Ga(s)
-Ą/4
� d" 2.6247/T0
 RH"
s
RH"
G = sup |G(s)| = sup |G(j�)|.
"
Re(s)>0 �"
 G(s) " RH"
s
RH" �" RL" RL"
jR
L" H"
W (s) = 48 + 28s - 56s2 - 35s3 + 7s4 + 7s5 + s6 = 0
s6 1 7 -56 48
s5 7 -35 28
s4 12 -60 48
s3 48 0 -120 0
s2 -30 48
s1 -43.2 0
s0 48 .
s
P (s)
P (s) = 48 - 60s2 + 12s4
dP (s)/ds = -120s + 48s3
P (s)
 s
W (s)
W (s) = 2 + 5s + 9s2 + 10s3 + 3s4 + 3s5.
s
s5 3 10 5
s4 3 9 2
s3 1 3
s2 � 0 2
s1 -2
�
s0 2 .
s
�
�)
W (s) = 0
W (s) = 3 + 4s + 3s2 + 2s3 + s4.
 W (s) = a0 + a1s + a2s2 + � � � + an-1sn-1 + sn
ai i = 0, 1, . . . , n - 1 an = 1
s
Hn " Rn�n
�ł łł
an-1 an-3 an-5 an-7 � � � � � � 0
�ł śł
1 an-2 an-4 an-6 � � � � � � � � �
�ł śł
�ł śł
0 an-1 an-3 an-5 � � � � � � � � �
�ł śł
�ł śł
0 1 an-2 an-4 � � � � � � � � �
�ł śł
Hn = .
�ł śł
�ł śł
0 0 an-1 an-3 � � � � � �
�ł śł
�ł śł
�ł �ł
� � � � � � � � � � � � � � � � � �
0 0 0 0 � � � � � � a0
W (s) deg W (s) = n
s "i > 0 i = 1, . . . , n
"i Hn "i = 0

i = 1, . . . , n W (s) m
s m
"1 "2 "n
"0, , , � � �
"0 "1 "n-1
"0 = an = 1
�ł łł
2 4 0 0
�ł śł
1 3 3 0
�ł śł
H4 = .
�ł �ł
0 2 4 0
0 1 3 3
"1 = 2 "2 = 2 "3 = -4 "4 = -12
"3 "4
"1 "2 "3 "4 2 2 -4 -12
"0, , , , = 1, , , ,
"0 "1 "2 "3 1 2 2 -8
k0
G0(s) = , k0 > 0.
(-1 + s)(3 + s)(4 + s)
k0 = 20
G0(j�)
� �
arg G0(j�) = -180ć% + arctan � - arctan - arctan .
3 4
arg G0(j�)|�=0 = -180ć% arg G0(j�)|�" = -270ć%
� " O+(0)
7�
arg G0(j�) = -180ć% + arctan � - arctan
12 - �2
�(5 - �2)
= -180ć% + arctan
12 + 6�2
arg G0(j�)|�"(0,"5) > -180ć%
 -k0/kmin -k0/kmax
(-1, j0)
k0 < kmin N = 0
kmin < k0 < kmax N = -1
k0 > kmax N = 1
1 1
kmin = = 12 oraz kmax = = 42
|G0(j�)| k0=1,�=0 |G0(j�)| k0=1,�="5
N
(-1, j0) �
-" +"
 P = 1)
N = -1)
Z = N +P = -1+1 = 0) N =
0
Z = N +P = 0+1 = 1
N = 1 Z = N + P = 1 + 1 = 2
k0 = 20 kmin < k0 < kmax
kmax k0
+ -
Mg = 20 log = 6.44 dB oraz Mg = 20 log = 4.44 dB.
k0 kmin
+
Mg
k0
-
Mg
k(1 + T0s)
G0(s) =
(-1 + T1s)(1 + T2s)(1 + T3s)
k = 10 T0 = 0.05 s T1 = 0.1 s T2 = 0.02 s T3 = 0.25 s
k(1 + j�T0)
G0(j�) = = U(�) + jV (�)
(-1 + j�T1)(1 + j�T2)(1 + j�T3)
-10 - 3.05 � 10-1 � �2 - 2.5 � 10-4 � �4
U(�) =
1 + 7.29 � 10-2 � �2 + 6.54 � 10-4 � �4 + 2.5 � 10-7 � �6
�(1.2 - 6 � 10-3 � �2)
V (�) = .
1 + 7.29 � 10-2 � �2 + 6.54 � 10-4 � �4 + 2.5 � 10-7 � �6
� = 0 : U(�) = -10, V (�) = 0
0 d" � < " : U(�) < 0
� " : U(�) 0, V (�) 0
"
1.2
� = �pc = = 200 rad � s-1 : U(�) = -1.85185, V (�) = 0
0.006
0 < � < �pc : V (�) > 0
�pc < � < " : V (�) < 0.
Z =
N + P Z
N
(-1, j0)
� -"
+" P
P = 1 N = -1
Z = 0
k
G0(s) = , k > 0.
s2(3 + s)
 s =
0 C
CI : s = �ej�, � > 0, 0ć% d" � d" 90ć%
CII : s = j�, � < � < " � > 0
CIII : s = ąj"
CIV : s = j�, -" < � < -�, � > 0
CI : s = �ej�, � > 0, -90ć% d" � < 0ć%.
G0 : C C s G0(s) � 0+
s " CI
k
G0(s)|s"CI =
�2ej2�(3 + �ej�)
� > 0
k
G0(s)|s=�e ,�>0,�=0ć% H" > 1
j�
3�2
k
|G0(s)||s=�e ,�>0,�=90ć% H" > 1
j�
3�2
�
arg G0(s)|s=�e ,�>0,�=90ć% = -180ć% - arctan .
j�
3
ć%
s = �ej90 � > 0 arg G0(s) < -180ć%
s s " CII
arg G0(s)|s"CII = -180ć% -arctan(�/3) s " CII lims" |G0(s)|
= 0 s " CI *"CII *"CIII
� > 0
s " CIV *" CI
(-1, j0)
N = 2
P = 0
G0(s) s Int C
C
Z = N + P = 2
k(2 + s)
G0(s) = , k > 0.
s(-1 + s)
k
s = 0
C
G0 : C C s G0(s) � 0+
s " CI
k(2 + �ej�)
G0(s)|s"CI =
�ej�(-1 + �ej�)
� > 0
2k
G0(s)|s=�e ,�>0,�=0ć% H" -
j�
�
2k
|G0(s)||s=�e ,�>0,�=90ć% H"
j�
�
�
arg G0(s)|s=�e ,�>0,�=90ć% = -270ć% + arctan + arctan �.
j�
2
s " CII
ńł
"
3�
�ł
�ł -270ć% + arctan dla � < 2 rad � s-1
�ł 2-�2
"
arg G0(s)|s"CII = -180ć% dla � = 2 rad � s-1
�ł
"
�ł
3�
ół
-90ć% + arctan dla � > 2 rad � s-1.
2-�2
s " CIII lims" |G0(s)| = 0
s " CI
"
� = �pc = 2 rad � s-1
� = �pc G0(j�pc) = -k
s " CI *" CII *" CIII � > 0
s "
CIV *" CI
k < 1 k > 1
s = 1
P = 1
k < 1
(-1, j0)
N = 1
Z = N + P = 2
 P = 1
N = 1
Z = N + P = 2 N = -1 Z = N + P = 0
k > 1
(-1, j0)
N = -1
Z = N + P = 0
k(1 + s)(3 + s)
G0(s) = , k > 0.
s(2 + s)(4 + s2)
G0(s)
s = 0
s = ąj2 C
CI : s = �ej�, � > 0, 0ć% d" � d" 90ć%
CII : s = j�, � < � < 2 - �, � > 0
CII : s = j2 + �ej�, � > 0, -90ć% d" � d" 90ć%
CII : s = j�, 2 + � < � < ", � > 0
CIII : s = ąj"
CIV : s = j�, -" < � < -2 - �, � > 0
CIV : s = -j2 + �ej�, � > 0, -90ć% d" � d" 90ć%
CIV : s = j�, -2 + � < � < -�, � > 0
CI : s = �ej�, � > 0, -90ć% d" � < 0ć%.
G0 : C C s G0(s) � 0+
s " CI
k(1 + �ej�)(3 + �ej�)
G0(s)|s"CI =
�ej�(2 + �ej�)(4 + �2ej2�)
� > 0
3k
G0(s)|s=�e ,�>0,�=0ć% H" > 1
j�
8�
3k
|G0(s)||s=�e ,�>0,�=90ć% H" > 1
j�
8�
arg G0(s)|s=�e ,�>0,�=90ć% = -90ć% + arctan � > -90ć%.
j�
s " CI
s " CII
k(1 + j�)(3 + j�)
G0(s)|s"CII =
j�(2 + j�)(4 - �2)
� �
arg G0(s)|s"CII = -90ć% + arctan � + arctan - arctan < �0
3 2
�0 = -90ć% + arctan 2 + arctan(2/3) - arctan 1 = -37.875ć%
s " CII
k(1 + j2 + �ej�)(3 + j2 + �ej�)
G0(s)|s"CII =
(j2 + �ej�)(2 + j2 + �ej�)(4 + (j2 + �ej�)2)
k 65
|G0(s)||s=j2+�e ,�>0,�=ą90ć% H" > 1
j�
16� 2
arg G0(s)|s=j2+�e ,�>0,�=-90ć%
j�
2-� 2-�
= -90ć% + arctan(2 - �) + arctan - arctan < �0
3 2
arg G0(s)|s=j2+�e ,�>0,�=90ć%
j�
2+� 2+�
= -90ć% + arctan(2 + �) + arctan - arctan - 180ć%
3 2
> -180ć% + �0.
s " CIII lims" |G0(s)| = 0
s " CI *" CII *" CII *" CII *" CIII s "
CIV *" CIV *" CIV *" CI � > 0
 (-1, j0) N =
2 G0(s)
P = 0
Z = N + P = 2
1
Gp(s) = , T > 0, T0 > 0
T0s(1 + T s)
1
Gc(s) = kc 1 +
Tis
kc Ti
Mp = 50ć%
k + kTis + s2 + T s3 = 0 k = kc/(TiT0)
kc > 0 Ti > T Ti > T
Ti
Ti = ąT ą > 1
G0(j�) = Gc(j�)Gp(j�)
arg G0(j�) = -180ć% + arctan(�ąT ) - arctan(�T )
�T (ą - 1)
= -180ć% + arctan .
2
1 + ą�2T
�max � arg G0(j�)
�
" �max = �max(ą) =
1/(T ą)
ą - 1
arg G0(j�)|�=�max(ą) = -180ć% + arctan " .
2 ą
 ą
Ti kc = kc(ą)
�max(ą)
�gc
|G0(j�gc)| = 1 �max(ą) ą
Mp
ą - 1
Mp = Mp(ą) = arctan " .
2 ą
T0
"
kc(ą) = T0�max(ą) = .
T ą
Mp
ą(Mp)
ą(Mp) = 1 + 2 tan2 Mp + (1 + 2 tan2 Mp)2 - 1.
 kc(ą) Ti = ąT ą > 1
G(p) p = sT
1 + ąp
G(p) = " " .
1 + ąp + ą ąp2 + ą ąp3
Mp)
Mp 30ć% 40ć% 50ć% 60ć% 70ć%
ą
�%
T�/T
Ts2%/T
Ts5%/T
�gcT
Mr
�rT
Mp = 50ć%
ą = 7.5486
kc = 0.364T0/T Ti = 7.549T
k = 5 T = 0.2 s
T0
T0max k
T
 G0(j�)
T0 = 0
k
G0(j�) = " � e-j(Ą/2+arctan(�T )).
2
� 1 + �2T
�gc
|G0(j�gc)| = 1
"
2
1+4k2T -1
2
�gc = .
T
arg G0(j�gc)
"
2
1 + 4k2T - 1
arg G0(j�gc) = -90ć% - arctan .
2
T0
arg G0(j�gc) + Ą
T0max =
�gc
"
2
Ą 1+4k2T -1
- arctan
2 2
T0max = .
"
2
1+4k2T -1
2
k = 5 T = 0.2 s
T0max = 0.230 s
T0 < 0.230 s
kp
Gp(s) = , kp = 600 [ � V-1 � s-1], Tp = 0.120 s,
s(1 + Tps)
-1
ks = 0.06 [V � ]
-1
kr = ks = 0.06 [V � ]
1 1 + Tds
Gc(s) = kc 1 + � , TD Td.
Tis 1 + TDs
r(t) rv(t)
ev(t) uv(t)
c(t) cv(t)
kc Ti Td TD
Mp = 40ć% Ts5% d" 0.05 s
rv(t)
cv(t)
kckpks(1 + Tis)(1 + Tds)
G0(s) = .
Tis2(1 + Tps)(1 + TDs)
1 + T ąs
(s) = " "
2 3
1 + T ąs + T ą ąs2 + T ą ąs3
ą Mp
ą(Mp) = 1 + 2 tan2 Mp + (1 + 2 tan2 Mp)2 - 1
T
Mp = 40ć%
ą = 4.5989 Ts5% = 11.311T T = Ts5%/11.311 =
0.00442 s
1 + T ąs
0(s) = " .
ą ą(T s)2(1 + T s)
1
kc = " , Ti = ąT, Td = Tp oraz TD = T.
kpksT ą
kc = 2.9302 Ti =
0.02033 s Td = 0.120 s TD = 0.00442 s
Tp
kc = " , Ti = Tp, Td = ąT oraz TD = T.
2
kpksT ą ą
kc = 17.297
Ti = 0.120 s Td = 0.02033 s TD = 0.00442 s
Ti > Td
 Gp(s) Gc(s) Gs(s)
1 2 - s 1
Gp(s) = , Gc(s) = oraz Gs(s) = .
(2 - s)(3 + s) 1 + s 1 + s
c(t) r(t) u(t)
m(t) d(t)
n(t)
Y1(s) Y2(s) Y3(s)
Y1(s) = R(s) - M(s) = R(s) - Gs(s)Y3(s)
Y2(s) = D(s) + U(s) = D(s) + Gc(s)Y1(s)
Y3(s) = N(s) + C(s) = N(s) + Gp(s)Y2(s).
�ł łł �ł łł �ł łł
1 0 Gs(s) Y1(s) R(s)
�ł �ł �ł �ł �ł �ł
-Gc(s) 1 0 Y2(s) = D(s) .
0 -Gp(s) 1 Y3(s) N(s)
 (R(s), D(s), N(s))
(Y1(s), Y2(s), Y3(s))
1 + Gp(s)Gc(s)Gs(s)
�ł łł �ł łł-1 �ł łł
Y1(s) 1 0 Gs(s) R(s)
�ł �ł �ł �ł �ł �ł
Y2(s) = -Gc(s) 1 0 D(s)
Y3(s) 0 -Gp(s) 1 N(s)
�ł łł �ł łł
Gry1(s) Gdy1(s) Gny1(s) R(s)
�ł �ł �ł �ł
= Gry2(s) Gdy2(s) Gny2(s) D(s)
Gry3(s) Gdy3(s) Gny3(s) N(s)
1 -Gp(s)Gs(s) -Gs(s) R(s)
1
= Gc(s) 1 -Gc(s)Gs(s) D(s) .
1 + Gp(s)Gc(s)Gs(s)
Gp(s)Gc(s) Gp(s) 1 N(s)
Gp(s) Gc(s) Gs(s)
s
s
1 + Gp(s)Gc(s)Gs(s)
s
Gp(s)Gc(s)Gs(s)|s" = -1.

1 + Gp(s)Gc(s)Gs(s)
 Gp(s)Gc(s)Gs(s)
4 + 7s + 5s2 + s3
1 + Gp(s)Gc(s)Gs(s) =
(1 + s)2(3 + s)
(3.20557 + s)(0.6654572 + (0.897215 + s)2)
= .
(1 + s)2(3 + s)
Gc(s)Gp(s)
p = 2 z = 2
(1 + s)2(3 + s) 1 + s
Gry1(s) = , Gdy1(s) =
4 + 7s + 5s2 + s3 (2 - s)(4 + 7s + 5s2 + s3)
-(1 + s)(3 + s) (2 - s)(1 + s)(3 + s)
Gny1(s) = , Gry2(s) =
4 + 7s + 5s2 + s3 4 + 7s + 5s2 + s3
(1 + s)2(3 + s) (-2 + s)(3 + s)
Gdy2(s) = , Gny2(s) =
4 + 7s + 5s2 + s3 4 + 7s + 5s2 + s3
1 + s (1 + s)2
Gry3(s) = , Gdy3(s) =
4 + 7s + 5s2 + s3 (2 - s)(4 + 7s + 5s2 + s3)
(1 + s)2(3 + s)
Gny3(s) = .
4 + 7s + 5s2 + s3
Gdy1(s) " RH" Gdy3(s) " RH"
p = 2
R(s)
N(s) C(s)
C(s) 1 + s
Grc(s) = = Gry2(s)Gp(s) =
R(s) 4 + 7s + 5s2 + s3
C(s) -1
Gnc(s) = = Gny2(s)Gp(s) = .
N(s) 4 + 7s + 5s2 + s3
100 + 225s + 186s2 + 74s3 + 14s4 + s5 = 0.
-3
s = p - 3
p
-3
130 + 77s + 26s2 + 6s3 + s5 = 0
-36 - 13s2 + 9s3 + 26s4 + 10s5 + 3s6 + s7 = 0
50 + 25s + 12s2 + 6s3 + 2s4 + s5 = 0
-12 - 4s - 9s2 - 3s3 + 3s4 + s5 = 0
s2 + 3s3 + 4s4 + 4s5 + 3s6 + s7 = 0
 s
1 + 3s + 4s2 + 4s3 + 3s4 + s5 = 0
W (s) = 2 + 4s + s2 + s3 + s4.
H4
"1 = 1 "2 = -3 "3 = -14 "4 = -28
W (s)
s
1
Gp(s) = , T1 = 2 s, T2 = 8 s
(1 + T1s)(1 + T2s)
1
Gc(s) = kc 1 + .
Tis
(kc, Ti)
T1T2 kc 8 kc
kc > 0 oraz Ti > � = � .
T1 + T2 1 + kc 5 1 + kc
1
Gp(s) = , T1 = 2 s, T2 = 8 s
s(1 + T1s)(1 + T2s)
 Gc(s)
kc Ti
T1 + T2 5 (T1 + T2)2 50
0 < kc < = oraz Ti > = .
T1T2 8 T1 + T2 - kcT1T2 5 - 8kc
k
G0(s) = .
s(1 + T s)2
k, T
H3
3 2
"1 = 2/T "2 = 2/T - k/T "3 =
5 4
2k/T - k2/T T > 0 k > 0
kT < 2
Ż
k k = k1k2k3 k1, k2, k3 > 0
T1 T2 T3
T1 T2 T1 T3 T2 T3
Ż
k = + + + + + + 2.
T2 T1 T3 T1 T3 T2
Ż
T1 T2 T3 k
Ż
kmin = 8 T1 = T2 = T3
k
G0 = , k > 0.
(1 + s)3
k
H3
"1 = 3 "2 = 8 - k
"3 = (1 + k)(8 - k) k <
8
kp
Gp(s) = , kp > 0, T > 0, n e" 3
(1 + T s)n
kc > 0
n
1 + tan2 Ą 2
1
n
kc < = .
kp
kp cosn Ą
n
kp
Gp(s) = , T1, T2 > 0
s(1 + T1s)(1 + T2s)
Gc(s) = 1 + Tds
Td
kp
T1T2 T1T2 1
Td e" > - .
T1 + T2 T1 + T2 kp
kc kp Td Tp
kc
Td > Tp
kckp
1
Gp(s) =
(1 + s)(1 + 2s)
Gc(s) = 1/s
 G0(s) = Gp(s)Gc(s)
Mg = 3.52 dB Mp = 11.52ć%
"
G0(s) �gc = 0.5716 rad�s-1 �pc = 1/ 2 rad�s-1
(a + s)(b + s)
G0(s) = k �
s2(c + s)
a b c a = ą1
b = ą3 c = ą2 (a, b, c)
k
P
G0(s)
N
(-1, j0) �
-" +" Z
(1, 3, 2) : P = 0, dla k < 0 mamy N = 1, zatem Z = 1,
dla k > 0 mamy N = 0, zatem Z = 0;
(-1, 3, 2) : P = 0, dla k < 0 mamy N = 2, zatem Z = 2,
dla k > 0 mamy N = 1, zatem Z = 1;
(1, -3, 2) : P = 0, dla k < -0.5 mamy N = 2, zatem Z = 2,
dla - 0.5 < k < 0 mamy N = 0, zatem Z = 0
(�pc = 1 rad � s-1),
dla k > 0 mamy N = 1, zatem Z = 1;
(-1, -3, 2) : P = 0, dla k < -2.75 mamy N = 3, zatem Z = 3,
dla - 2.75 < k < 0 mamy N = 1, zatem Z = 1,
dla k > 0 mamy N = 2, zatem Z = 2;
(1, 3, -2) : P = 1, dla k < 0 mamy N = 0, zatem Z = 1,
dla 0 < k < 2.75 mamy N = 1, zatem Z = 2,
dla k > 2.75 mamy N = -1, zatem Z = 0
"
(�pc = 11 rad � s-1);
(1, -3, -2) : P = 1, dla k < 0 mamy N = 1, zatem Z = 2,
dla k > 0 mamy N = 0, zatem Z = 1;
(-1, 3, -2) : P = 1, dla k < 0 mamy N = 1, zatem Z = 2,
dla 0 < k < 0.5 mamy N = 2, zatem Z = 3,
dla k > 0.5 mamy N = 0, zatem Z = 1;
(-1, -3, -2) : P = 1, dla k < 0 mamy N = 2, zatem Z = 3,
dla k > 0 mamy N = 1, zatem Z = 2.
24 + 10s + s2
G0(s) = k � , k = 1.
15 - 8s + s2
P = 2 N = -2
k
Ż
k = 0.8 < 1
-
Mg = 1.938 dB
-
G0(s) �pc = 4.3589 rad � s-1
60
G0(s) = .
(-2 + s)(5 + s)2
G0(s)
+ -
Mg = 3.5212 dB Mg = 1.5836 dB Mp = 4ć%
+ -
�pc = 2.236 rad � s-1 �pc = 0 rad � s-1 �gc = 1.11 rad � s-1
(2 + s)(1 + s)2
Gp(s) = .
(3 + s)2(4 + s)(6 + s)(1 - s + s2)
k = 160 k = 800
G0(s) = k �Gp(s)
k = 160
+ -
Mg = 6.737 dB Mg =
7.972 dB Mp = 22.9ć%
+ -
�pc = 5.449 rad � s-1 �pc = 1.316 rad � s-1 �gc = 3.292 rad � s-1
k k = 160
k = 800
(2 + s)2(4 + s)
G0(s) = k � , k = 10.
(-3 + s)(-1 + s)(1 + s)(3 + s)
-
Mg = 7.475 dB Mp =
46.28ć% G0(s)
-
�pc = 4.472 rad � s-1 �gc = 10.1695 rad � s-1
4 + s
Gp(s) =
(-3 + s)(-1 + s)(2 + s)
Gp(s)
(1 + s)(2 + s)(3 + s)
Gp(s) =
(-1 + s)(-2 + s)(-3 + s)(-4 + s)
k > 10.6878
G0(s) = k �Gp(s)
k
- -
k = 30 Mg = 17.28 dB Mp = 59.26ć% �pc = 0.9094 rad � s-1
�gc = 29.732 rad � s-1) k = 5)
k k = 30
k = 5
kp
Gp(s) = , kp = 15, T1 = 1 s, T2 = 2 s
(1 + T1s)(1 + T2s)
Gc(s) = 1/(Tis) Ti = 8 s
Ti
Mg = 10 dB Mp
 Re G0
(j�pc) = -1.25 < -1 Im G0(j�pc) = 0 G0(s) = Gc(s)Gp(s)
"
"
�pc = 1/ T1T2 = 1/ 2 rad �
s-1
kp(T1 + T2)
Ti = 10Mg/20 � = 31.623 s.
2 2
2 2
(1 + �pcT1 )(1 + �pcT2 )
Mp = 34.26ć% Mp = 90ć%-arctan(�gcT1)-
arctan(�gcT2) �gc = 0.3615 rad � s-1
e-T0s
G0(s) = k � , k = 6, T = 0.15 s, n = 3
(1 + T s)n
T0
Ą - n � arctan k2/n - 1
T0max = � T = 0.01754 s.
k2/n - 1
 Gp(s)
Gc1(s) Gc2(s)
1 -3 + s
a) Gp(s) = , Gc1(s) = , Gc2(s) = 1
-3 + s 2 + s
s -(2 + s) 4 + s
b) Gp(s) = , Gc1(s) = , Gc2(s) =
2 + s 1 + s 3 + s
1 -2 + s 1 + 0.5s
c) Gp(s) = , Gc1(s) = , Gc2(s) =
s(-2 + s)(3 + s) 1 + s 1 + 2s
5 + s 2 + s 2 - s
d) Gp(s) = , Gc1(s) = , Gc2(s) =
1 + s 3 + s 2 + s
1 -1 + s
e) Gp(s) = , Gc1(s) = , Gc2(s) = 1.
-1 + s2 1 + s
Gc(s) Gp(s)
4 + s s
Gc(s) = oraz Gp(s) = .
s 2 + s
 kc kp = 1
kc = 1 Mg
Mp
kc
Mg = 6 dB Mp
Ą
Mg(n) = -20n � log10 cos [dB].
n
Mp = 180ć% limn" Mg(n) = 0
1
kc =
2 cosn Ą
n
1
Mp = Ą - n � arctan - 1, 3 d" n d" 7.
22/n cos2(Ą/n)
n e" 8
Mp
4ś4 + 1 - 2ś2
2ś
Mp = arctan , gdzie �gc = .
��gc �
Mp e" 0
cos Mp � tan Mp
ś = .
2
 m(t)
kc
k
Mg = 10 dB |e(")|
|e(")| d" 0.2
kp = 2 T0 = 0.005 s T1 = 0.05s T2 = 0.01s
k = 0
-kp(1 + T1s)e-sT0
Gd1e(s)|k=0 =
1 + kckpe-sT0 + (T1 + T2)s + T1T2s2
1 + (T1 + T2)s + T1T2s2
Gd2e(s)|k=0 = - .
1 + kckpe-sT0 + (T1 + T2)s + s2T1T2s2
kp 1
|ed1(")||k=0 = oraz |ed2(")||k=0 = .
1 + kckp 1 + kckp
kckpe-sT0
G0(s)|k=0 = .
(1 + T1s)(1 + T2s)
�pc0 = 146.7 rad � s-1
G0(s)|k=0
arg G0(s)|s=j�pc0,k=0 = -180ć% oraz |G0(s)||s=j�pc0,k=0 = 0.15211kc.
k = 0 kc = 10-Mg/20/0.15211 = 2.079
Mg
|ed1(")||kc=2.079,k=0 = 0.3878
|ed2(")||kc=2.079,k=0 = 0.1939
k = 0
k > 0
Ż
-kpk(1 + T1s)e-sT0
Gc (s) =
d1e
Ż Ż Ż
1 + kckpke-sT0 + (T1 + T2)s + s2T1T2s2
Ż Ż
-(1 + (T1 + T2s) + T1T2s2)
Gc (s) =
d2e
Ż Ż Ż
1 + kckpke-sT0 + (T1 + T2)s + T1T2s2
Ż Ż
k = 1/(1 + k) T1 = T1/(1 + k)
kp 1 + k
|ed1(")| = oraz |ed2(")| = .
1 + k + kckp 1 + k + kckp
Ż
kckpke-sT0
G0(s) = .
Ż
(1 + T1s)(1 + T2s)
�pc
G0(s) arg G0(j�pc) = -180ć% k
kc Mg
10-Mg/20
2 2
2 Ż 2
kc = � (1 + �pcT1 )(1 + �pcT2 ).
Ż
kpk
 k
�gc
G0(s) |G0(j�gc)| = 1 Mp
k kc |ed1(")| |ed2(")| �gc [rad � s-1] Mp �pc [rad � s-1]
54.23ć%
64.26ć%
68.33ć%
70.26ć%
k
T1
Ż
T1
k
�gc
k = 1
 d(t)
d(t)
r(t)
C(s)/R(s) = 1/(1 +
TcTps2) Tc Tp
 C(s)/D(s) = (k + Tcs)(1 +
kTps + TcTps2)
Tc Tp k
d(t) = 1(t)
0 < k d" 0.01
G(s)
pi
i = 1, . . . , n zi i = 1, . . . , m
m
(s - zi)
i=1
G(s) = k0 n , k0 = 0, m < n.

(s - pi)
i=1
G(s) k0 m (s - zi)
i=1
G0(s) = = .
n
1 - G(s) (s - pi) - k0 m (s - zi)
i=1 i=1
G(s)|s=0 = 1
m n
k0 (-zi) = (-pi).
i=1 i=1
kv = lim(sG0(s)).
s0
m
k0s (s - zi)
i=1
kv = lim
n
s0
(s - pi) - k0 m (s - zi)
i=1 i=1
d
k0 m (s - zi) + k0sds m (s - zi)
i=1 i=1
= lim .
n
d d
s0
(s - pi) - k0 ds m (s - zi)
i=1 i=1
ds
n n n
d
lim (s - pi) = lim (s - pi) � (s - pi)-1
s0 ds s0
i=1 i=1 i=1
n n
= - (-pi) � p-1
i
i=1 i=1
n m m
d
-1
lim (s - zi) = - (-zi) � zi .
s0 ds
i=1 i=1 i=1
k0 m (-zi)
i=1
kv =
m -1 n n
k0 m (-zi) � zi - (-pi) � p-1
i=1 i=1 i=1 i=1 i
m n
1 1 1
= - .
kv i=1 zi i=1 pi
10 ki -0.5
Gp(s) = , Gc(s) = kc + , Gd(s) =
0.1 + s s 0.1 + s
kc ki
e(t) = r(t) - c(t) = er(t) + ed(t)
er(t)
ed(t)
er(") e(")
kc ki er(") = lims0(sR(s)(1 - Grc(s)))
C(s) Gc(s)Gp(s)
Grc(s) = = .
R(s) 1 + Gc(s)Gp(s)
er(")
sR(s)
er(") = lim .
s0 1 + Gc(s)Gp(s)
Gc(s)Gp(s) = 10(ki + kcs)/(s(0.1 +
s))
R(s) = 1/s
R(s) = 1/s2 er(") =
1/kv kv = lims0(sGc(s)Gp(s)) = 100ki
ed(") = - lims0(sD(s)Gdc(s)) Gdc(s)
C(s) Gd(s) 0.5s
Gdc(s) = = = -
D(s) 1 + Gc(s)Gp(s) 10ki + (10kc + 0.1)s + s2
0.5s2
ed(") = lim D(s) .
s0
10ki + (10kc + 0.1)s + s2
ed(") = 0.05/ki
Gp1(s) Gp2(s)
Gc(s)
3 1 2.5(0.1 + s)(4.2 + s)
Gp1(s) = , Gp2(s) = , Gc(s) = .
4 + s s(3 + s) s
eri i = 0, . . . , 4
edi i = 0, . . . , 4
er(")
r(t) = r0(t) = 2 + 0.4t + 0.25t2 t e" 0 ed(")
d(t) = d0(t) = 1.5 + 0.3t t e" 0
ed(") d(t) = r0(t)
E(s) 1
Gre(s) = =
R(s) 1 + Gc(s)Gp1(s)Gp2(s)
12s2 + 7s3 + s4
= .
3.15 + 32.25s + 19.5s2 + 7s3 + s4
E(s) -Gp2(s)
Gde(s) = =
D(s) 1 + Gc(s)Gp1(s)Gp2(s)
-4s - s2
= .
3.15 + 32.25s + 19.5s2 + 7s3 + s4
Ge(s)
"
1 di
Ge(s) = eisi, ei = Ge(s) , i = 0, 1, . . . .
i! dsi s=0
i=0
ei i = 0, 1, . . .
r(t) = 1(t)
" "
er(") = lim sR(s) eisi = lim eisi.
s0 s0
i=0 i=0
e0 = 0
Ge(s)
Ge(s)
s
n
bisi
i=0
Ge(s) = , a0 = 0.

n
aisi
i=0
n " n
aisi � eisi = bisi
i=0 i=0 i=0
s
Ge(s)
ńł
b0
e0 = dla i = 0,
�ł
a0
�ł
i
1
ei = bi - ajei-j dla i = 1, . . . , n,
j=1
a0
�ł
ół
1
-a0 n ajei-j dla i > n.
j=1
Gre(s) = 0 + 0s + 3.8095s2 - 36.78s3 + 353.292s3 + � � �
Gde(s) = 0 - 1.2698s + 12.6833s2 - 121.9919s3 + 1173.2706s4 + � � � .
r(t) = r0(t) d(t) = d0(t)
er(") ed(") er(") = 3.8095 � 2 �
0.25 = 1.905 ed(") = -1.2698 � 0.3 = -0.381
ed1 |ed(t)| d(t) = r0(t)
G0(s) = Gc(s)Gp1(s)Gp2(s)
1 - Gre(s)
G0(s) = .
Gre(s)
kv
ka
kv = lim(sG0(s)) oraz ka = lim(s2G0(s)).
s0 s0
0 gdy er0 = 0

kv =
e-1 gdy er0 = 0
r1
0 gdy er0 = 0 lub er1 = 0

ka =
e-1 gdy er0 = 0 oraz er1 = 0.
r2
kv = " ka = 0.2625
k0 = 1.5 s-1 T1 =
0.2 s T2 = 0.07 s
 kc
k
d(t)
kc k
|e(")| = 0.1 Mg = 12 dB
2 2
-T1 - 2T2 + T1 + 4kck0T1T2 � 10Mg/20
1
kc = oraz k = .
|e(")| 2T2
kc = 10 k = 1.31095
Mp = 43.7ć%
k = 0)
Mg = 12 dB
kc = 3.2296
|e(")| = 0.3096
Mp = 38.4ć%)
|e(")| = 0.1) kc = 10
Mg = 2.2 dB Mp = 6.5ć%
G(s)
m
(s - zi)
i=1
G(s) = k0 � , G(s)|s=0 = 1, m < n.
n
(s - pi)
i=1
ka
ka = lims0(s2G0(s)) G0(s)
G(s) pi i =
1, . . . , n zi i = 1, . . . , m G(s)
ka
n m m n
2 1 1 1 1 1 1
= - - + oraz = - .
2 2
ka kv i=1 p2 zi kv i=1 zi i=1 pi
i
i=1
R(s) C(s)
C(s) b0 + b1s + � � � + bn-1sn-1
Grc(s) = = .
R(s) a0 + a1s + � � � + ansn
E(s) = R(s) - C(s)
E(s)
Gre(s) = = 1 - Grc(s)
R(s)
(a0 - b0) + (a1 - b1)s + � � � + (an-1 - bn-1)sn-1 + ansn
= .
a0 + a1s + � � � + ansn
a0 - b0
e(t)|t" = .
a0
a0 = b0
 a0 = b0
a1 - b1
e(t)|t" = .
a0
a0 = b0 a1 = b1
G0(s) l0 + l1s + l2s2
G(s) = =
1 + G0(s)) 72 + 54s + 13s2 + s3
G0(s)
l0 l1 l2
t � 1(t)
t2/2 � 1(t)
G(s)
l0 = 72 l1 = 54 l2 = 13

l0 = 72 46.8 < l1 < 61.2 l2
l0 = 72 l1 = 54 9.4 < l2 < 16.6
 kt = 0) Ts2% = 0.981 s
� = 0.6
k kt
Mp = 50ć%
|e(")| = 0.002
a
2
a H" -2 ln(0.02 1 - ś0)/Ts2% ś0 =
0.1605 a = 8 s-1 k
k = 1/|e(")| = 500
kt kt = 2śt�t - 8/(5k) = 0.01591 s
"
�t = 1/ 5k = 0.02 s śt = 0.4777
Ts2%
Ts2% �
ś0 � = 0.6 Ts2% =
23.09� a a = 2�23.09�
ś0/Ts2% = 7.555 s-1
a = 8 s-1
Mp = 49.63ć%)
1
Gp(s) =
(1 + s)2(2 + s)
k
k
|e(")| < 0.01
|e(")|min = 0.1
Mg = 6 dB
|e(")|min = 0.182
kp 12
Gp(s) = =
1 + Tps 1 + 0.4s
Gc(s) = 1/(Tis)
Ti
� =
0.2
Ti Gc(s)
�
Mp
 � Ti = 4ś2kpTp
ś � = 0.2 ś =
0.45595 Ti = 3.9915 s
e(") = Ti/kp = 0.33262 s Mp =
48.148ć%
e(") = 0.1 Ti = kpe(") = 1.2 s
� = 0.44434
Mp = 28.02ć%
Gp(s)
Gc(s)
5 2
Gp(s) = oraz Gc(s) = 3 + + 2s.
(1 + s)2(2 + s) s
ei i = 0, . . . , 4
r(t) = 3 + 0.5t t e" 0
E(s)
Gre(s) = = e0 + e1s + e2s2 + e3s3 + e4s4 + � � �
R(s)
= 0 + 0.2s + 0.16s2 - 0.172s3 + 0.0724s4 + � � � .
r(t)
dr(t)
e(") = lim e1 � = 0.1.
t"
dt
r(t) = c1t�1(t)
d(t) = d � 1(t)
 e(") = er(") + ed(")
c1 d
Gt(s) = 0
er(")
Gt(s) =
kt kt > 0
Gt(s) = sTt/(1+sT0)
ed(") d(t)
e(") = er(") + ed(") = T2c1/(kck1) + kdd/kc
er(") = T2c1/(kck1) + ktT2c1
er(") = T2c1/(kck1)
Gp(s) Gc(s)
e-Tms
kp 1
Gp(s) = oraz Gc(s) =
(1 + Tps)2 Tis
 kp = 30 Tp = 0.15 s
Tm = 0
Mg = 12 dB
Tmmax Tm
Tm
Tm = Tmmax
Mg = 12 dB
�
G0(j�)
Tm = 0
�
�pc = 1/Tp |G0(j�pc)| = kpTp/(2Ti)
Mg Ti
�
10Mg/20 � kpTp = 2Ti Ti
�
Ti = 10Mg/20 � kpTp/2 = 8.957 s
G0(s)
�gc
|G0(j�gc)| = 1
�
|G0(j�)| = |G0(j�)| "�
Ti �gc
2 3
TiTp �gc + Ti�gc - kp = 0
3
�
kp Tp Ti = Ti �gc +44.4444�gc -
148.8525 = 0 �gc
�
�gc = 2.836 rad � s-1 G0(j�)
�
arg G0(j�gc) = -2.375 rad = -136.08ć%
Tm = 0 Mp = 43.92ć%
Tm e" 0 arg G0(j�gc)
�
arg G0(j�gc) = arg G0(j�gc) - �gcTm
 Tmmax
�
Ą + arg G0(j�gc)
Tmmax = = 0.2702 s.
�gc
Tm = Tmmax
Mg = 12 dB
kp
|G0(j�gc)| = = 10-Mg/20.
2 2
�gcTi(1 + �gcTp )
Ti
kp � 10Mg/20
Ti = = 35.66 s.
2 2
�gc(1 + �gcTp )
�
Ti = 8.957 s
Tm = 0 s
Tm = 0 s
 T1 > 0 T2 > 0
Ż Ż
T1 = 0.1 s T2 = 0.0125 s
kc
Mg = 8 dB
T1 T2
kc + s + (T1 + T2)s2 + T1T2s3 = 0.
kcmax
T1 + T2 1 1
kcmax = = + .
T1T2 T1 T2
Ż Ż
T1 T2
kc max = 90 s-1 kc
Mg = 8 dB
kc = kcmax � 10-Mg/20 = 35.8296 s-1.
kc > 0
T1 + T2 > kcT1T2.
kc = 35.8296 s-1
Ż
T1 = T1
T2
Ż
T1
0 < T2 < = 0.0387 s.
Ż
kcT1 - 1
Gp(s) = (1 + Tzs) � G0(s) oraz Tzmin d" Tz d" Tzmax.
G0(s)
Tz
Gp(s)
Gp(s) = p(s) � (1 + "(s) � Wm(s))
p(s) "(s) " RH"
|"(j�)| d" 1 "�
Wm(s)
Ż
p(s) = (1 + Tzs) � G0(s)
Tzmin + Tzmax
Ż
Tz =
2
Tz
Ż
(Tz - Tz)s
= "(s) � Wm(s).
Ż
1 + Tzs
Ż
Tz = 0
Wm(s) = Tzmaxs
"(s) = " " [-1, 1]
Ż
Tz = 0

Ż
2Tz - Tzmin - Tzmax Tzs
� = "(s) � Wm(s).
Ż
Tzmin + Tzmax 1 + Tzs
Tzmax - Tzmin
rTz =
Tzmin + Tzmax
Tz
Ż
2Tz 1 rTzTzs
- � = " � Wm(s).
Ż
Tzmax - Tzmin rTz 1 + Tzs
Tzmin, -1 Tzmax, 1
Ż
rTzTzs
Wm(s) = .
Ż
1 + Tzs
Ż
Tz = 0 "(s) = " " [-1, 1]
Ż
Tz
"Tzs
Wm(s) =
Ż
1 + Tzs
Tzmax - Tzmin
"Tz = .
2
Ż Ż
Tz = 0 "Tz = rTzTz

Ż
P (s)
Ż
P = P (s) = P (s) � e-T0s, 0 d" T0 d" T0max
T0
Ż
Pm = P (s) � (1 + "(s) � Wm(s)), " d" 1
"
Wm(s) " RH"
P �" Pm
P (s) " P
T0 "(s) " H"
1 + "(s) � Wm(s) = e-T0s
Wm(s) " RH"
P (j�)
- 1 d" |Wm(j�)|, "�, T0
Ż
P (j�)
e-jT0� - 1 d" |Wm(j�)|, "�, T0.
D(�) = D(�, T0) = e-jT0� - 1
D(�) = 2(1 - cos(T0�)), min D(�) = 0 oraz max D(�) = 2.
D(�) � (T0) = Ą/T0
T0 > 0 0 d" � d" � (T0)
2
cos(T0�) H" 1 - T0 �2/2 T0� " O(0)
� (T0) � (T0max) = Ą/T0max
ks
Wm(s) =
1 + T0maxs
 k > 0 D(�, T0max)
d" |Wm(j�)| "�
k�
e" 2
2
1 + T0max�2
�=� (T0max )
k
k e" 2 1 + Ą-2 � T0max.
Wm(s) = Wm = 2
e-jT0� - 1
1
Gp(s) = , T1 = 0.2 s, T2 = 0.04 s
(1 + T1s)(1 + T2s)
Gc(s) =
kc/s Gs(s) = 1/(1 + T0s) T0 e" 0
Mg = 10 dB
T0 = 0 s
kc
T0
 T0 = 0 s
T1 + T2
kc = 10-Mg/20 � = 9.4868 s-1.
T1T2
T0 e" 0
2
a0 + a1T0 + a2T0 < 0
a0 = T1T2(kcT1T2 - T1 - T2)
a1 = (T1 + T2)(2kcT1T2 - T1 - T2)
a2 = (T1 + T2)(kc(T1 + T2) - 1).
T0max = 0.10855 s
T0 =
0 s) T0 = T0max/2
T0 = 0.05 s T = 0.2 s
e-Tms Tm e" 0
Tm
kc
� = 0.2
 Mp
kc Tmmax
Tm = 0)
0 < Tm < Tmmax
kc
Tmmax
Mp
Tm = 0
kc = T0/(4ś2T ) ś
ś = 0.4559
kc = 0.3006 Mp = 48.14ć%
k = kc/T0 = 6.0141 s-1
"
Ą 1
2
- arctan ( 1 + 4k2T - 1)
2 2
Tmmax = � T = 0.1876 s.
"
1
2
( 1 + 4k2T - 1)
2
"
1
2
( 1 + 4k2T - 1)
2
�n = = 4.4794 rad � s-1.
T
Tm = Tmmax
2 2
kc = �gcT0 1 + �gcT = 0.10346
�gc = 1.9304 rad�s-1
Ą
- arctan(�gcT ) - Mp
2
�gc = .
Tmmax
Tm = 0 kc = 0.3006
Tm = Tmmax/2 = 0.0938 s kc = 0.3006
Tm = Tmmax = 0.1876 s kc = 0.10346
1
Gp(s) = � G0(s), pmin d" p d" pmax
s - p
G0(s)
p
1/(s - p)
p(s)
Gp(s) =
1 + "(s) � Wi(s)
p(s) "(s) " RH"
|"(j�)| d" 1 "� Wi(s)
1 pmin + pmax
p(s) = � G0(s), gdzie p = .
Ż
s - p 2
Ż
p = 0 Wi(s) = pmax/s "(s) = " " [-1, 1]
Ż
p = 0 "(s)
Ż
rpp pmax - pmin
Ż
Wi(s) = , gdzie rp = .
s - p pmin + pmax
Ż
pmax - pmin 1
Wi(s) = � .
2 s - p
Ż
1
Gp(s) = � G0(s), Tpmin d" Tp d" Tpmax
1 + Tps
G0(s)
Tp
Gp(s)
1 Tpmin + Tpmax
Ż
p(s) = � G0(s), gdzie Tp = .
Ż
1 + Tps 2
Ż
Tp = 0 Wi(s) = Tpmaxs "(s) = " " [-1, 1]
Ż
Tp = 0 "(s)

Ż
rTpTps
Tpmax - Tpmin
Wi(s) = , gdzie rTp = .
Ż
1 + Tps Tpmin + Tpmax
Ż
Tp
Tpmax - Tpmin s
Wi(s) = � .
Ż
2 1 + Tps
 Wi(s)
(1 + Tzs) � G0(s)
(s - z) � G0(s) z
Gp(s) = k � G0(s) kmin d" k d" kmax
G0(s)
kmin + kmax = 0

kmin + kmax � kminkmax kmax - kmin
Ż
k = , k = 2 oraz rk =
2 kmin + kmax kmin + kmax
" " R |"| d" 1
Ż
Gp(s) = kG0(s) � (1 + " � rk)
�
kG0(s)
Gp(s) = .
1 + " � rk
Ż � Ż �
k = k kG0(s) kG0(s)

ą
ą " [ąmin, ąmax] ąmin < ąmax
ąmin + ąmax ąmax - ąmin
ą = oraz rą = , ąmin + ąmax = 0
Ż
2 ąmin + ąmin
ą
 ą = 0
Ż
rą > 1
1
Gp(s) = , amin d" a d" amax.
1 + as + s2
a =

+ "a � "

= (amin + amax)/2 "a = (amax - amin)/2
-1 d" " d" 1
p(s)
Gp(s) =
1 + " � p(s) � Wi(s)
p(s) Wi(s)
1
p(s) = , oraz Wi(s) = "as.
1 +

s + s2