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[Gm,Pm,Wcg,Wcp]=margin(G0);[Gm,Pm,Wcg,Wcp];
w=logspace(-1,3);[m,p]=bode(G0,w);
subplot(211),semilogx(w,20*log10(m(:)))
subplot(212),semilogx(w,p(:))
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subplot(211),semilogx(w,20*log10([m(:),m1(:)]))
subplot(212),semilogx(w,[p(:),p1(:)])
G_c1=feedback(G0,1);G_c2=feedback(G_o,1);
[y,t]=step(G_c1);y=[y,step(G_c2,t)];
figure,plot(t,y)
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)(
+
=
s
K
sG
p
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
K=1
K=2
K=4
K=6
K=8
? ? , ≥
H"d|á
3?? b]
HVm>ìY? wL9 V[?
C? ,
Q
9v
H"d×
?
{9v"dY?
????
99
v ,
Q
H"dá
3????X?×? b Vn?B?¨ ,
Q
?
? ?"d￥? ?·S
^??
a￥ b
??? 1 * %n?
1 ès ?? —s￥T¨
])(
1
)([)(
∫
+= dtte
T
teKtu
i
P
YV1?1 è?? ?1 ès?? V[?C1
P F U
1 è?? ? ,
Q
∞?" ?i ?v￥ ??5
{ V U
9?v?????Y"d￥?
?? ?9
P3? ?N)?v?
??? 7 ??¨ 1 *?? ?T1 p F U
5 e?
{ V U
? ¤?B?è′V7
P
{ D U
×?? ù?￥′ b Q
V?
????
? ? A1 è?? ? V??B??
? ,
Q
7 1 *??
5
????
? ,
Q
? 5
J
?"??
29? ?^¤? ?X￥
? aá
?? ?·S b
∫
dtte
T
i
)(
1
y 1 *?? B???
^B?taì? b
)
1
()(
sT
sT
KsG
i
i
pc
+
=
??? 1 * %n?
0 2 4 6 8 10 12 14 16 18 20 22 24
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ti=1.5
Ti=0.7
Function PI
G=tf(1,[1,3,3,1]);
Kp=1;Ti=[0.7:0.1:1.5];
for i=1:length(Ti)
Gc=tf(Kp*[1,1/Ti(i)],
[1,0]);
G_c=feedback(G*Gc,1);
step(G_c),hold on
End
axis([0,20,0,2])
? ?¨ 1 * e?"d￥
×
?μ1
,
??? 1 * %n?
1 %? 1 * %?? —±s￥T¨
? 1 * %?? ￥ 5
J
∞
Hn?
??1B? 1 %?? ?M??
B?? -n?
?"d￥Y?
?￥? ?
^μ ?￥ b?
L=
￥ e?"d??B?¨ 1 % e?￥"d?
eyμ
Z
?B
^B±sì?
L=?íE
LC]
H ??¨ 1 % e? 5"d
òì??￥ ?? ?? (|"d￥
{á
3?v￥o?- K
M?| byN9? ??"d?
?? ?￥??? ? b
)
1
1
1()(
N
T
s
sT
sT
KsG
d
d
i
pc
+
++=
L=￥ 1 * % e? ￥.?f
?
/B?v? bA ?? / ∞
H
T'1 ?X￥ 1 * % e? b
??? 1 * %n?
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
N=1
N>1
Function PID
N=[100,1000,10000,1:10];
G=tf(1,[1,3,3,1]);
Kp=1;Ti=1;Td=1;
Gc=tf(Kp*[Ti*Td,Ti,1]/Ti,[1,0]);
G_c=feedback(G*Gc,1);step(G_c), hold on
for i=1:length(N)
mn=Kp*([Ti*Td,0,0]+conv([Ti,1],[Td/N(i),1
]))/Ti;
cd=[Td/N(i),1,0];
Gc=tf(mn,cd);
G_c=feedback(G*Gc,1);
step(G_c)
end
axis([0,20,0,2])
??? 1 * %n?
= Zieloger-Niclosls??
T
Zieloger-Niclosls??
T
^B??{μ
Hüì?￥B¨"d74
￥
L¨ü
T b
Ts
eK
sG
s
+
?
=
?
1
)(
τ
?
H×Y? wL￥??
k
y(t)
L
T
t
?
H×Y?￥ PID?
???ZEμ
?
?BE
!X$ e` 7ì"d
@
FB?¨?|YV
LZE?
Y??| b
? ?T¤
?
?a L -? 5 B L - 5
5 V? V ?? PID e? ￥μ1?
? b
??? 1 * %n?
?=E
!"d1oμ1 è e?￥>ì"d? K
p
9v? ,
Q
’
H>
ì"d ? ?á
3???? b
?=E]"
a¨?"d ?X?
￥"d b?
^N=E ?¨?9μ?
?y1? "di?D
?"d ?
￥ è ??BEíE?? 7ì.?f
??cs[￥"d?=EüíE
°¤?¨?=¨"d ?
üíE ?¨ Zieloger-NicloslsEé??
? b
)4(
200
)(
0
+
=
ss
sG
y(t)
P’
??? 1 * %n?
'
p
K
'
p
K
'
p
K
'
P
'
P
'
P
¨Y??? ??????
Kp Ti Td Kp Ti Td
P1/% ?00.5 ? 0
PI 0.9/% 3L 0 0.45 0.833 0
PID 1.2/% 2L L/2 0.6 0.5 0.125
?? ??
V 61 PID?
???V
??? 1 * %n?
è 6 10 μB
o?"d￥ 7ì.?f
?1
)4)(3)(2)(1(
1
)(
0
++++
=
ssss
sG
p
!9B? e?
P"d￥×
?ê?μ1
, b
3 ?¨ Zieloger-Niclosls??
T?BE b
? e 7ì"d￥.
?f
? ?¨ MATLAB?
? ¨Y? wL
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
t2=2.72
t1=0.76
g=tf(10,conv([1,1],conv([1,2]
,conv([1,3],[1,4]))));
step(g);k=dcgain(g)
k=0.4167
??? 1 * %n?
?m Ví
?¤? B¨üt"d￥?
?
96.176.072.2,76.0,4167.0 =?=== TLk
??ú¨í
?B¨￥ZEg V¤?
083.224/50,4167.024/10 ==== Tk
¤? PI? PID e? ￥?
?
2.6
1
,1616.0 ====
a
K
T
L
k
p
α
2.6)( =sG
c
AP e? e?
BPI e?
e?
28.23,57.5
9.0
==== LTK
ip
α
)
28.2
1
1(57.5)(
s
sG
c
+=
e?
CPID e?
38.0
2
,52.12,425.7
2.1
======
L
TLTK
dip
α
)38.0
52.1
1
1(425.7)( s
s
sG
c
++=
??? 1 * %n?
"d >ì.?f
? # ¨Y? ?/
24.85) 9.279s (s^2 3.457) 0.7215s (s^2
1.8947
++++
G_c1(s)=
23.59) 9.07s (s^2 2.773) 0.5561s (s^2 0.3735)(s
0.4386)(s 55.7053
+++++
+
G_c2(s)=
10.14) 0.5727s (s^2 1.711) 2.6s (s^2 6.827)(s
1.316)^2(s 68.4
+++++
+
G_c3(s)=
??? 1 * %n?
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 * % e?
1 * e?
1 e?
¨Y?
??? 1 * %n?
è 6 11 μB?êQ ?"d￥ 7ì.?f
?1
)5)(1(
1
)(
0
++
=
sss
sG
1 p
!9B?1 ó e?
P"d￥×
?
?μ1
, b
3 ??"d 7ì?i sì?íE?¨?BE by7?¨
Zieloger-Niclosls??
T ?=E b
n5
7 5
J
∞ 5
E
5>ì"d￥.?f
? 1
p
p
Ksss
K
sR
sC
+++
=
56)(
)(
23
YV ?
? ? ?^¤??
H>ì"dá
3?? ??
?? N
H?? ?
q ? ù b
30
'
==
pp
KK
5=
c
ω 81.2
2
' ==
c
P
ω
π
??? 1 * %n?
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
1
-10 -8 -6 -4 -2 0 2
-8
-6
-4
-2
0
2
4
6
8
Kp'=30
??? 1 * %n?
? ZN?=??E ' VsY¤? PI? PID e? ￥?
?
(A) PI e?
33.2'833.0,5.1345.0
'
==== PTKK
ipp
)
33.2
1
1(5.13)(
s
sG
c
+=
35.0'125.0,4.1'5.0,186.0
'
====== PTPTKK
dipp
)35.0
4.1
1
1(18)( s
s
sG
c
++=
(B) PID e?
?
?
!9￥ e? sY¤? M?￥>ì"d
2.248) 0.02925s (s^2 0.4683)(s 5.502)(s
0.4292)(s 13.5
++++
+
G_c1(s)=
??? 1 * %n?
G_c2(s)=
2.769) 0.739s (s^2 1.122)(s 4.139)(s
1.429)^2(s 6.3
++++
+
n?a￥ ¨Y? wLm
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
2
2.5
? ¨ 1 * % e?
?¨ 1 * e?
??? 1 * %n?
? ?×E ￥??
?T
L
?
^? ?
qY?E¤?￥5 V5? ?￥
NyquistmVm? V[ ?^¤?"d￥g M ?
q D"d￥ K
9m ?
7 ]"VV ó￥ü
T V[¤
? 1 * % e? ?￥?
? b
c
K
cc
T ωπ /2=
c
ω
Là
′à
m 637 1
fm
c
K
1
?
P¨ MATLABé?
!9
H
? 7ì.?f
? |"d￥ K
9m ,
D
?g M ?
q ω
D
'
[Kc,pp,wc,wp]=margin(g) ?a
?
???é?
!9 b
??? 1 * %n?
V 62ZN ?×??E
c
K5.0
c
T8.0
e? ??
P
0
PI
0
PID
c
K4.0
c
K6.0
c
T5.0
c
T12.0
p
K
i
T
d
T
∞
?
B? e? ￥ ? ￥?ZE
?TX?"d￥ ?.?f
?YV? ? ? e?
P¤>
ì"d￥ ? V[M??·?￥ê?V7
P"d￥?
?? ?¤?
? ???ZE?1 ? ￥?E b
è 6 12 μB e?"d ?m ? B C 1 p
!9B? e?
P"d×? b
R(s) +
_
C(s)
bs
as
+
+
3
1n? ->ì"d￥ ?
0=++? bsas 0
2
ab
s
?
= >
y7 e?"d?×? b
?
B? e? ￥ ? ￥?ZE
2 e?` -1 óB?B¨ 8?ì? D>05>ì"
d ?
?
H"d V[×? b?N?
?￥ê4G
?? B a C
V ?í?
a￥′ by7 Vê4 e?
D a E 5μ+?Z?
?
H"d×
? b
cs+
1
0)(
2
=++??+ bsacsacs
01>+?ac
0>?acb
cs
ds
sG
c
+
+
=)(
0)()(2
2
=?+?+++ acbdsacdbs
acdb >++
b
a
c
d
>
9 ' è?? e 7ì"d?×? y7? ?YVe?￥
, ?MhZ
Té? e? ￥
!9 ey ? e? ￥?
? 8
LC?íE
*
1 ?V7 V ??án?a￥"d ˉ?×? b
?
B? e? ￥ ? ￥?ZE
è 6 13 X?B?êQ ? e?"d￥ 7ì.?f
?
)8)(2(
10
)(
0
++
=
sss
sG
1 p
!9B1 ón?
?
Pn?a"d￥á
?
?μ"
d >ì?? ? ) b
)(sG
c
1
5
?
= sK
v
322 js ±?=
3
1
n5YVTn? -"d￥ ?Em
/B: b V[
?C ?? ?1 b1
P?? ?_P êM
R?¨ ? -n?
? b
638.0902.0
2,1
js ±?=
2
7? -n?
? V?¨ ??"
?
E ??M1?
?
bs
as
a
b
K
b
s
a
s
KG
ccc
+
+
??=
+
+
=
1
1
5
16
10
)()(lim
0
0
=?=??=
→
cc
s
v
KsGsGsK 8=
c
K
?
B? e? ￥ ? ￥?ZE
))()(322)(322()(10))(1610(
21
2
δδ ++?+++=+++++ ssjsjsas
a
b
Kbssss
c
))(164(
21
22
cscsss ++++=
? δ
aδ
a D
a D
1??"
? béB? V¤
a
a
b
Ks
a
b
Kbsbsbs
cc
10)1016()1016()10(
234
+++++++
43 2
121212
(4 ) (16 4 ) (4 16 ) 16scs ccsccsc=++ + ++ + + +
'
1
410 cb +=+
12
4161016 ccb ++=+
12
1641016 cc
a
b
Kb
c
+=+
2
1610 cbK
c
=
8=
c
K
3.33=a
24=b
30
1
=c 120
2
=c
¤?
? -n?
?￥.?f
?
24
33.3
57.66
+
+
=
s
s
G
c
l2
1n?￥ "￥ a
a¨S?#ZE b
2n?
?￥2ê4 b
3n?￥'é?B??? b
4PIDn?￥e ? aZE??
??? b
