o o Q o T . C R T . Problem 02- 01 ) 4 ( K 30000 1 2 K 80 4 1 V 1 K T 2 K 2 V o T Q ) s ( G ) T T ( o e V p K r V ) 1000 5 . 0 ( R 1 2000 C 1 1 1 RCs R ) s ( G Reconsider Problem 01-01. a) Find the transfer function H(s) of the closed loop control system (K p is a symbol) b) Determine the value of Kp to obtain steady state error of %2 to a unit step input. c) Find the step response of the control system (K p =65.3333) d) Find the settling time of the closed loop control system? Answer: e) Find the sensitivity of steady state error to a disturbance. ) 1 K 75 . 0 ( s 4 K 75 . 0 ) s ( H ) a p p 3333 . 65 K ) b p s ) 5 . 12 s ( B 5 . 12 s B A s B ) 5 . 12 s ( A ) 5 . 12 s ( s 25 . 12 ) 50 s 4 ( 49 s 1 ) s ( V ) c 2 98 . 0 e 98 . 0 ) t ( v t 5 . 12 2 5027 . 0 t , 0255 . 0 t , 08 . 0 ) d ss 50 05 . 0 e ), s ( H s 1 ) s ( V , 50 s 4 ) 1 s 4 ( 05 . 0 ) s ( H ) e ss 2 d d Sensitivity= % 0.1
Transcript
oo
Q
oT.
C
R
T.Problem 02-01
)4(K30000 1
2K804
1V1K T
2K 2VoTQ
)s(G)TT( oeV
pKrV
)10005.0(R1 2000C
11
1RCs
R)s(G
Reconsider Problem 01-01.
a) Find the transfer function H(s) of the closed loop control system (Kp is a symbol)
b) Determine the value of Kp to obtain steady state error of %2 to a unit step input.
c) Find the step response of the control system (Kp=65.3333)
d) Find the settling time of the closed loop control system?
Answer:
e) Find the sensitivity of steady state error to a disturbance.
)1K75.0(s4
K75.0)s(H)a
p
p
3333.65K)b p
s)5.12s(
B5.12sBA
s
B
)5.12s(
A
)5.12s(s
25.12
)50s4(
49
s
1)s(V)c 2
98.0e98.0)t(v t5.122 5027.0t,0255.0t,08.0)d ss
50
05.0e),s(H
s
1)s(V,
50s4
)1s4(05.0)s(H)e ss2d d
Sensitivity= % 0.1
Solutions of Week 2:
002.01000*5.01
R
12000
C
05.0804
K2 75004
30000K1
1s4002.0
1RCsR
)s(G
)s(T)K75.01(s4
05.0s2.0)s(V
)K75.01(s4K75.0
)s(T
1s4K75.0
1
05.0)s(V
1s4K75.0
1
1s4K75.0
)s(V
)s(T05.0
1s4002.0
7500K1
05.0)s(V
05.01s4
002.07500K1
05.01s4
002.07500K
)s(V
)s(TK)s(GKK1
K)s(V
K)s(GKK1K)s(GKK
)s(V)a
0p
rp
p0
pr
p
p
2
0
p
r
p
p
2
021p
2r
21p
21p2
Transfer function between Reference and OutputTransfer function between disturbance and output.
Problem 02-01:
Open-loop system
Closed-loop system
Solutions of Week 2:
)K75.01(s4K75.0
)s(V)s(V
)s(H
)b
p
p
r
2
s1
)s(Vr s1
)K75.01(s4K75.0
)s(Vp
p2
02.0v1e ss2ss
98.0e1v ssss2 98.0K75.01
K75.0s1
)K75.01(s4K75.0
slimvp
p
p
p
0sss2
333.65015.098.0
K
K75.0K735.098.0
p
pp
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
v2ss
c) Step response of closed loop system v2(t)
)5.12s(sB5.12s)BA(
sB
5.12sA
)5.12s(s25.12
s1
50s449
)s(V2
98.0
5.1225.12
B25.12B5.12
98.0BA0BA
s98.0
5.12s98.0
)s(V2
98.0e98.0)t(v t5.122
Solutions of Week 2:
The coefficients A and B can be determined by applying Residue Theorem.
98.05.12
25.12s)5.12s(
25.12)0s(limB
98.05.12
25.12s)5.12s(
25.12)5.12s(limA
0s
5.12s
d) The settling time of the closed loop control system ts
The eigenvalue of the closed loop control system is s=-12.5. Then the time constant is calculated as
s5027.008.0*1415.3*22ts08.05.12
1s1
s
e) Sensitivity of the steady state error to disturbance (The fraction of the steady state error due to disturbance in the form of a unit step function)
50s405.0s2.0
)1K75.0(s405.0s2.0
)s(T)s(V
)s(Hp0
2d
disturbance
For unit step disturbance
s1
50s405.0s2.0
)s(V2
001.05005.0
V ss2 ess=0.001
Sensitivity %0.1
Problem 02-02 (Kuo, s.195)
054.0u wM65 wT
1s10
1
AoT
1s2
1
pKeT
rT
sT
Reconsider Problem 01-02.
a) Find the transfer function of the open loop system [H1(s)].
b) Find the eigenvalues of the open loop system.
c) Find the transfer function of the closed loop control system H(s) (Kp is a symbol).
d) Find the eigenvalues of the closed loop control system (Kp=9).
e) Write the form of the step response of the closed loop control system.
f) Determine the steady state error.
g) Find the damping ratio, undamped frequency, time increment Δt and settling time.
)1K51.3(s12s10
K51.3)s(H)c
p2
p
)1s2)(1s10(
51.3)s(H)a 1 5.0p,1.0p)b 21 i7026.16.0p)d 2,1
9693.0)t7026.1cos(eA)t(T)e 1t6.0
1s
1.3)%f s472.10t,s174.0t,Hz2873.00f,3321.0)g sso
Answer:
Solutions of Week 2:
a) Transfer function of open loop system
1s12s1051.3
1s21s1065*054.0
)s(H 21
Open systemClosed system
b) Eigenvalues of open loop system
5.0s,1.0s 21
c) Transfer function of closed loop system
)1K51.3(s12s10
K51.3
1s21s1065*054.0*K
1
1s21s1065*054.0*K
)s(Hp
2p
p
p
c) Eigenvalues of closed loop system for Kp=9.
59.32s12s1059.31
)s(H 2
>>roots([10 12 32.59])
s1,2=-0.6±1.7026i
Problem 02-02:
Solutions of Week 2:
e) The form of the response of the closed loop system to a unit step input.
59.32s12s1059.31
)s(T)s(T
)s(H 2R
s
s1
)s(TR s1
59.32s12s1059.31
)s(T 2s
sC
)i7026.16.0(sbia
)i7026.16.0(sbia
)s(Ts
9693.059.3259.31
s1
59.32s12s1059.31
slimC 20s
9693.0)t7026.1cos(Ae)t(T t6.0s
e) Steady state error :
07.3%e0307.09693.01T1e sssssss f) Damping ratio, undamped natural frequency, time step (increment) and settling time.
Re
Im
-0.6
1.7026ω0
α
3324.08052.1
6.0
7026.16.0
6.0cos
22
Hz2873.02
f
s/rad8052.17026.16.0
00
220
s47.103324.0
48.3Tt,s174.0
2048.3
20T
t
s48.38052.122
T
0s
0
00
Reconsider Problem 01-03.
Problem 02-03
34.5K,0034.0K 21
1Q
2QRC
dQ
1s1758
140)s(G
1V1K
1Q)s(G 2K 2V
1h
dQ
)s(GceV
rV
a) Find the transfer function of the closed loop control system H(s). (Gc=Kp is a sembol)
b) Determine the value of Kp in order to obtain the sensitivity to disturbance as % 4.
)1K54.2(s1758
K54.2)s(H)a
p
p
9.7357K)b p Answer:
Solutions of Week 2:
Open systemClosed System
a) Transfer function of closed loop system for Gc(s)=Kp
K1=0.0034
K2=5.34
1s1758140
)s(G
)s(QGKKK1
GK)s(V
GKKK1GKKK
)s(V d21p
2r
21p
21p2
34.5
1s1758140
0034.0K1
34.51s1758
1400034.0K
)s(V)s(V
)s(Hp
p
r
2
)1K54.2(s1758
K54.2)s(H
p
p
Problem 02-03:
)s(QGKKK1
GK)s(V d
21p
22
b) Kp value for the sensitivity of 4%.
s1
34.51s1758
1400034.0K1
34.51s1758
140
)s(Vp
2
s1
)1K54.2(s17586.747
)s(Vp
2 9.7357K04.0
1K54.26.747
V pp
ss2
Problem 02-06
)s(t
)t(c
1
25.1
0
The step response of a system is given in the figure. Determine a 2nd order tranfer function for this system.
Answer:
Kuo-91 (Sh.389)
Problem 02-07 Kuo-91 (Sh.389)
)s(R K
tK
)s2.01(
100)s(G
)s(C
s20
1
Determine the K and Kt values for the system whose block diagram is given in the figure
a)To obtain the maximum overshoot as 6% and the steady state error as 4.8%.
b)Find the steady state error to a unit step input.
c)Write the Matlab code in order to plot the step response of the system.
Answer:
09.0
533.0K 027.0K t
29905s62.139s
29905)s(H
2
4037.0 s/rad933.1724037.0*09.0
2209.0 n
n
Solutions of Week 2:
Problem 02-07:
K and Kt values for maximum overshoot 6% and steady state arror 4.8%.
