, - 1- , n£(m-n).
W() ; i:
:
1. w wi=1/Ti. .
2. 20lgk -20/ , -40 / , +20 / , , +40 / , , ..
3. -20*u / (u Ti), , +20*u / .
4. () -20*u / (+20*u /) L(w).
( ) .
:
1. wi=1/Ti, : 0 /, -40 / +40 / .
2. - .
3. , .
.
.
1. .
, 1:
.
-, :
,
,
.
- , - . - , - , .
2. .
. . ,
/, /,
/, /.
lgω, :
, ,
, .
3. .
:
|
|
, ( ) , k=10. , ( )
20lgk=20lg10=20
:
, -20 /,
, :
+20 /, , :
+20 /, , +20 /. , 9,6+10,4 20 :
, -40 /, , -20 /:
3
(. 5) .
_ |
r |
. 5.
:
W(p) ; W(p) ;
:
(1+1)(2+1)(3+1)+k1k2k3=0;
:
1233+(12+13+23)2+(1+2+3)+(1+k1k2k3)=0;
:
0=T1T2T3;
a1=12+13+23;
2=1+2+3;
3=1+k1k2k3;
:
03+12+2+3=0;
3- :
1) 0>0, a1>0, a2>0, a3>0 ( );
2) a1a2>a0a3 ( );
, ;
, , :
;
.
4
1910 . , .
1942 J.G. Ziegler N.B. Nichols (, . , -) :
, ;
Ti Td .
D = 0,2-0,3. D , ,
|
|
.
: , . .
1. .
.
- .
:
1. , Simulink:
2. ki/p a Simulink (. 2), ki = max[dh(t)/dt].
3. a L .
4. 1 [ ].
1. -
k | T | T | |
1/ a | |||
0,9/ a | 3L/k | ||
1,2 a | 2L/k | 0,5Lk |
:
.
5. Simulink .
1
- .
1. Simulink (. 1).
, , , , .
. 1. Simulink
2. : a = 0,145 - ( hmax = 3,704 t. = 0,1205), L = 0,04- , . .2.
3. Matlab :
a=0.145;
L=0.04;
k=1.2/a
Ti=2*L/k
Td=0.5*L*k
ki=1/Ti
kd=Td
:
k = 8.2759, Ti = 0.0097c, Td =0.1655c, ki = 103.4483c-1, kd =0.1655c.
. 2. : 1 - , 2 , 3 - , (k = 3,704), 4 -
4. Simulink (. 3).
. 3 ( ) , , .
|
|
. 3. Simulink
5. , . 3, : 1 , 2 , 3 , 4 - (k = 15, Ti = 0,013c, Td = 0,525c)
. 4. , . 3
2. .
.
-
.
:
1. ,
-180.
2. .
3. .
4. . 2 ,
.
2. -
k | T | T | |
0,5/ | |||
0,4/ | 0,8 /k | ||
1,6/ | 0,5 /k | 0,125 k |
:
.
5. Matlab
,
.
2
- .
1. Matlab .
numo=[1];
deno=[10 11 1];
Wo=tf(numo,deno)
[nums,dens]=pade(0.3,2)
Ws=tf(nums,dens)
Wir=Wo*Ws
nyquist(Wir)
%margin(Wir)
2. =1,87-1 =|Real|=0,0265.
3. =1,84-1 =10-31,7/20= 0,026.
, , , .
4. : =1,84-1 =10-31,7/20= 0,026. 2 .
k180=0.026;
w180=1.84;
T180=2*pi/w180
kn=0.6/k180
Tu=0.5*T180/kn
ku=1/Tu
Td=0.125*T180*kn
kd=Td
: kn = 23.0769, ku = 13.5159, kd = 9.8503.
5. .
kp=23.0769;
ki=13.5159;
kd=9.8503;
numo=[1];
deno=[10 11 1];
Wo=tf(numo,deno)
[nums,dens]=pade(0.3,2)
Ws=tf(nums,dens)
numi=[ki];
deni=[1 0];
Wi=tf(numi,deni)
numd=[kd 0];
dend=[1];
Wd=tf(numd,dend)
Wsr=minreal(Wo*Ws/(1+Wo*Ws))
step(Wsr)
%Wsrpid=minreal((kp+Wi+Wd)*Wo*Ws/(1+(kp+Wi+Wd)*Wo*Ws))
|
|
%step(Wsrpid)
6. () , .5 Matlab.
7. , % % .
.
3. CHR (Chien Hrones Reswick).
20% - . , .
CHR . , , , .
, .
:
.
: .
:
, ,
.
3 4.
3
CHR
20% - | ||||||
k | k | T | ||||
0,3/ | 0,7/ a | |||||
0.35/ a | 1,2 /k | 0,6/ a | 1,0 /k | |||
0,6/ a | 10 /k | 0,5 k | 0,95/ a | 1,4 /k | 0,47 k |
4
CHR
20% - | ||||||
k | k | T | ||||
0,3/ | 0,7/ a | |||||
0.6/ a | 4 /k | 0,7/ a | 2,3 /k | |||
0,95/ a | 2,4 /k | 0,42 k | 1,2/ a | 2,0 /k | 0,42 k |
3
- CHR.
1. :
= =0,143.
2. 3
( ):
kp = 0,6/ a = 4,196;
= 10 /kp =10*0,4/4,196 = 0,95c; ku = 1/ = 1,0526c-1;
= 0,5 kp = 0,5*0,4*4,196 = 0,839c; k = = 0,839c.
3.
.
kp=4.196;
ki=1.0526;
kd=0.839;
numo=[1];
deno=[2.8 1];
Wo=tf(numo,deno)
[nums,dens]=pade(0.4,2)
Ws=tf(nums,dens)
numi=[ki];
deni=[1 0];
Wi=tf(numi,deni)
numd=[kd 0];
dend=[1];
Wd=tf(numd,dend)
Wsr=minreal(Wo*Ws/(1+Wo*Ws))
step(Wsr)
%Wsrpid=minreal((kp+Wi+Wd)*Wo*Ws/(1+(kp+Wi+Wd)*Wo*Ws))
%step(Wsrpid)
4. () , .3 Matlab.
5. , % % .
6. Simulink
.
4. () .
, , () () (. 1).
.1. , () ()
, , , , (.2): , , . . , , .
|
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. 2.
, , (. 3) . , .
. 3.
:
. (1)
, .
, , ,
. (2)
n = 28.
(3)
, (4)
, (5)
(2) n =2.
, (3)
(6)
, (7)
-
( ), ; -
.
(3)
(8)
(9)
.
(n > 2) (3).
(10)
, (11)
= - , () = ; -
; (12)
. (13)
, , (11) (. 1).
1
(2), , , 0,7 ( ).
. , , :
, (14)
n .
(12) k, , , : - , - .
(3)
. (15)
, ,
(16)
, (10) p. . . , , , , , p.
4
, ,
, (17)
1 , 2 .
(18)
k, .
. (19)
, , .
, (14):
. (20)
(12) ( ) :
-1. (21)
, . 1, 2:
, (22)
. (23)
, .
:
; (24)
; (25)
; (26)
; (27)
(28)
, (29)
:
; (30)
; (31)
; (32)
. (33)
, , ,..
. (34)
. 2 , ( , ).
2
, , ,
, (35)
(30) (32) (31) (33) . . , .
- , :
; (36)
; (37)
, (38)
,
(. 4, . 3).
. 4. (1) (2) (3)
3
s - (. 4)
, , , .
. 4 , . R(p) F(p) .
(0 20%) .
, .
, ( ), , .
. (39)
4
, , :
1. .
2. .
3. , , .. .
4. , , .. .
5. y :
, (40)
0,85 , 1,45 .
6. :
, (41)
:
, (42)
( ).
7. . , - .
5. .
, . . , 2% . , , . 1. :
1. = 0.
2. .
3. 0,1% < 2%.
4. < 2%.
(3) (4) , 2% , .
. 1. (
)
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. (1)
:
. (2)
, :
. (3)
(3) . . .. , . , . 1, , 100% . (3) . . , , 1,2 , . 1
.
:
.
1
(1).
. (1), .
5
, .
,
.
.
.
.1 : .
( 2%) 2 , , , . :
.
, s = 2,84, z = 1,34 k = 6,14. ,