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3. Reglarea PID
3.1. Structura regulatoarelor PID
3.1.1 Regulatorul PID continuu
P I D P ro c e s ε
+ r u y
Forma standard
(algoritmul ISA) )1
1()( d
i
RR sTsT
KsG
KR : factorul de proporţionalitate
(BP=100/Kr : banda de proport.)
Ti : timpul de integrare
Td : timpul de derivare
Forma serie
(cu interinfluenţă) )1)(1
1()('
'
'd
i
RR sT
sT
KsG
P
D I
+ + + + u ε
''
''
''
'
''
'
;
di
di
ddii
i
di
RR
TT
TTTTTT
T
TTKK
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Forma paralelă d
i
R sks
kksG )(
P
I
D
+
+ + ε u
PID cu ponderarea
referinţei (regulatoare cu
două grade de libertate) ])(
1[)(
0dt
dTd
Tktu
dt
d
i
pR
ybrp ycrd yr
- b = c = 0 regulator I – PD
- b = 1 c = 0 regulator PI – D
-
G R s p P ro c e s + r
n
y
-
G R y
-
)1
1()(
)1
()(
d
i
RRy
d
i
RRsp
sTsT
ksG
csTsT
bksG
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limitarea amplificării componentei derivative
tTaKdt
dnTkutan dRdRd cossin
)10(208;/1
NNNsTd
sTdD
dt
dyTk
dt
dD
N
TdD dR
PID incremental
(de viteză)
d t
d u
R e g u la to r
Ns T d
T dks R
/1
2
Rs K
i
R
T
k
+
+ +
s
1
ε d
u ε p
ε
I e x t e r io r
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3.1.2 Regulatorul PID numeric
Discretizare
componenta proporţională (P): )]()([)()( kykbrkkPybrkPRR
componenta integrală (I)
t
i
R
i
R
T
k
dt
dId
T
ktI
0)()(
metoda dreptunghiului în avans
)()()1()()()1(
kT
TkkIkIk
T
k
T
kIkI
i
R
i
R
metoda dreptunghiului
)()1()()()1()(
kT
TKkIkIk
T
k
T
kIkI
i
R
i
R
metoda trapezelor (Tustin)
)]1()()1()(21
kbkbkIkIii
)]1()([2
)1()( kkT
TKkIkI
i
R
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componenta derivativă
dt
dyTkD
dt
dD
N
T
dR
d
metoda dreptunghiului în avans
)]()1([)()1()1()()1(
)()()1(
kykyNkkDT
NTkD
T
kykyTkkD
T
kDkD
N
T
R
d
dR
d
metoda dreptunghiuului
metoda trapezelor (Tustin)
)]1()([2
2)1(
2
2)( kyky
NTT
NTkkD
NTT
NTTkD
d
dR
d
d
)]1()([)1()( kykybkDkDdd
1d stabilitate ; 0
dpentru Td mic cu drept în avans şi Tustin de evitat
)]1()([)1()()()1()(
)()1()(
kykyNTT
NTkkD
NTT
TkD
T
kykyTkkD
T
kDkD
N
T
d
dR
d
d
dR
d
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Notă: metoda dreptunghiului în avans instabilitate pentru Td mic
metoda dreptunghiului pentru: Td mic
metoda trapezelor (Tustin) : rezultate bune
PID )()()()()()(111
krzTkyzSkuzR regulator 2-GDL
2
2
1
10
1-
2
2
1-
10
1-
11-1
)(
)(
)1)(1()(
ztzttzT
zszsszS
zzzRd
)2()1()(
)2()1()()2()1()1()(
210
210
krtkrtkrt
kyskyskyskukukudd
didR
ididR
iR
ddidR
dididR
diR
bbkt
bbbkt
bbkt
bbks
bbbks
bbks
22
211
10
22
211
10
)1(
2)1(
Drept în avans Dreptunghi Tustin
bi1 0
i
R
T
Tk
i
R
T
Tk
2
bi2
i
R
T
Tk
0
i
R
T
Tk
2
d d
T
NT1
NTTd
Td
NTTd
NTTd
2
2
bd kRN
NTTd
TdNkR
NTTd
TdNkR
2
2
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3.1.3 Regulatoare PID comerciale
U(s), E(s), R(s) şi y(s) Transformatele Laplace pentru u, , r. şi y
I. Forma standard (ISA) : )](/1
1[ ycR
NsTd
sTdE
sTybRkU
i
R
II. Forma serie :
]/'1
'1)
11(
/'1
'1)
1[(
'
'y
NdsT
dsT
sTR
NsTd
scTd
sTbkU
ii
R
III. Forma paralelă: )(
)/(1)( ycR
Nskd
skE
s
kybRkU
di
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- modelul dinamic al procesului
- performanţele impuse
3.2 Proiectarea regulatorului PID
P I D P r o c e s
+r u
y
+
+
_
P
Proiectare KR , Ti , Td (N)
![Page 9: FXLQWHULQIOXHQ r ' ' T T G s K T T T T R R sT · 3. Reglarea PID 3.1. Structura regulatoarelor PID 3.1.1 Regulatorul PID continuu P ID P ro ces r + 0 u y Forma standard](https://reader030.vdocuments.pub/reader030/viewer/2022040313/5e098f77df33047b2c1ff56b/html5/thumbnails/9.jpg)
R e g u la t o r K R T i T d
P 1 /a - -
P I 0 .9 /a 3 L -
P I D 1 .2 /a 2 L L /2
u =
sL
f esL
asG )(
y
tL
a
4
1d
3.2.1 Acordarea experimentala
• Metoda Ziegler – Nichols bazată pe răspunsul la semnal treaptă
![Page 10: FXLQWHULQIOXHQ r ' ' T T G s K T T T T R R sT · 3. Reglarea PID 3.1. Structura regulatoarelor PID 3.1.1 Regulatorul PID continuu P ID P ro ces r + 0 u y Forma standard](https://reader030.vdocuments.pub/reader030/viewer/2022040313/5e098f77df33047b2c1ff56b/html5/thumbnails/10.jpg)
PID P : Ti = , Td = 0 KR (Kc , T0): limita de stabilitate
K R )( sG f
r +
_
ε u y
)j(Gf
N y q u is t D ia g ra m
R e a l A x is
Ima
gin
ary
Ax
is
- 1 -0 .5 0 0 .5 1 1 .5 2
-1 .5
-1
-0 .5
0
0 .5
1
1 .5
)j(Gf
)j(Gf 0
N y q u is t P lo t
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
1 .4
1 .6
1 .8
2
T im e ( s e c )
y
t
r = 1
T 0
Regulator KR Ti Td
P 0.5Kc - -
PI 0.4Kc 0.8T0 -
PID 0.6Kc 0.5T0 0.125T0
Notă: )(
1
0jG
K
f
c
• Metoda Ziegler-Nichols bazată pe răspunsul la frecvenţă
![Page 11: FXLQWHULQIOXHQ r ' ' T T G s K T T T T R R sT · 3. Reglarea PID 3.1. Structura regulatoarelor PID 3.1.1 Regulatorul PID continuu P ID P ro ces r + 0 u y Forma standard](https://reader030.vdocuments.pub/reader030/viewer/2022040313/5e098f77df33047b2c1ff56b/html5/thumbnails/11.jpg)
• Metoda “reaction curve” (raspuns la semnal treapta)
Regulator KR Ti Td Regulator KR Ti Td
P Lk
T
f
f
- - P ff
f
T
L
Lk
T
31 - -
PI Lk
T
f
f9.0
3L - PI ff
f
T
L
Lk
T
129.0
LT
LTL
f
f
209
)330(
-
PID Lk
T
f
f2.1
2L 0.5L PID ff
f
T
L
Lk
T
43
4
LT
LTL
f
f
813
)632(
LT
LT
f
f
211
4
model sL
f
f
fe
sT
ksG
1)(
0
0
12
01
uu
yyk
ttT
ttL
f
f
Ziegler – Nichols Cohen - Coon
y∞
t
y0
t1 t2 t0
y(t)
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3.2.3 Acordarea prin tehnici de optimizare
Metoda Zhuang – Atherton
diRTTk ,,
0
2,),()( dttetI
n
n = 0, 1, 2 rezultă criteriile ISE, ISTE şi IST2E n = 0, 1, 2 rezultă criteriile ISE, ISTE şi IST2E
1
1
2 2
, ,
b
f
R i
f f
f
Ta Lk T
Lk Ta b
T
31
3
22
1,
)/(,
b
f
fd
f
f
i
b
ff
RT
LTaT
TLba
T
TT
L
k
ak
Regulator PI:
Regulator PI:PI:
![Page 13: FXLQWHULQIOXHQ r ' ' T T G s K T T T T R R sT · 3. Reglarea PID 3.1. Structura regulatoarelor PID 3.1.1 Regulatorul PID continuu P ID P ro ces r + 0 u y Forma standard](https://reader030.vdocuments.pub/reader030/viewer/2022040313/5e098f77df33047b2c1ff56b/html5/thumbnails/13.jpg)
3.2.3 Acordarea prin tehnici de optimizare
Criteriile modulului si simetriei
Răspunsul sistemului de reglare răspunsul pentru frecvenţe joase
0/;1)0( 00
nndjGdG (pentru 0)
Criteriul modului: performanţe optime pentru r = în prezenţa perturbaţiilor
Criteriul simetriei: performanţe optime pentru r = în prezenţa pe r t u r b a ţ i i l o r
Proces: f
f
sT
k
sG1
)(1
321
21
21
21
;
111
)(
;
11
)(
3
3
2
fff
fff
f
ff
ff
f
TTT
sTsTsT
k
sG
TT
sTsT
k
sG
![Page 14: FXLQWHULQIOXHQ r ' ' T T G s K T T T T R R sT · 3. Reglarea PID 3.1. Structura regulatoarelor PID 3.1.1 Regulatorul PID continuu P ID P ro ces r + 0 u y Forma standard](https://reader030.vdocuments.pub/reader030/viewer/2022040313/5e098f77df33047b2c1ff56b/html5/thumbnails/14.jpg)
21
21
;
11
)(
1
)(
5
4
ff
ff
f
f
f
TT
sTsTs
K
sG
sTs
K
sG
Proiectare : polii procesului compensaţi de zerourile regulatorului astfel încât:
C r i t e r i u l m o d u l u i ( M o d u l u s O p t i m u m ) = B O ( B e t r a g s o p t i m u m )
if
if
ff
opdTmimT
sTsTsG ;
12
1)(
C r i t e r i u l s i m e t r i e i ( S y m m e t r i c a l O p t i m u m ) = S O ( S y m m e t r i s c h e O p t i m u m )
if
if
ff
f
opdTmimT
sTsT
sTsG ;
18
41)(
22
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F.d.t. în circuit închis
BO fs
f
n
ffTt
TsTsT
sG8
%3.4%
707.0
2
1
122
1)(
220
f
f
f
n
fff
f
fff
f
Tz
Tp
T
sTsTsT
sT
sTsTsT
sTsG
4
1
2
1
5.0
2
1
12412
41
1488
41)(
32222330
SO
r = fs Tt 11
%43%
bloc integrator pe canalul referinţei
![Page 16: FXLQWHULQIOXHQ r ' ' T T G s K T T T T R R sT · 3. Reglarea PID 3.1. Structura regulatoarelor PID 3.1.1 Regulatorul PID continuu P ID P ro ces r + 0 u y Forma standard](https://reader030.vdocuments.pub/reader030/viewer/2022040313/5e098f77df33047b2c1ff56b/html5/thumbnails/16.jpg)
)(
1)()(
sGsGsG
f
dRop
GF R M Obs. KRKf Ti Td
G1 1. I BO 0.5 fT
G2 PI BO
2
1
2 f
f
T
T
1
fT
G3 PID BO
32
21
f
ff
T
TT
21ff TT
21
21
ff
ff
TT
TT
G4 P BO
fT2
1
G5 PD BO
22
1
fT
1fT
G2 PI SO A1
2
1
2 f
f
T
T
24 fT
G3 PID SO A1
2
3
321
8
4
f
fff
T
TTT
324 ff TT
32
32
4
4
ff
ff
TT
TT
G4 PI SO
fT2
1
fT4
G5 PID SO
2
2
21
8
4
f
ff
T
TT
21 4 ff TT
2
2
4
4
1
1
ff
ff
TT
TT
Nota: 11
1 1 ff sTsTA
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3.3.2 Metode bazate pe reguli
A c o r d a r e = c o m p r o m is în t r e r e g la r e r a p id ă ş i r e g la r e s t a b i l ă
R e g u l i d e a c o r d a r e
R e g la r e r a p id ă S t a b i l i t a t e
K R
T i
T d
P r o c e d u r a d e a u to a c o r d a r e b a z a te p e r e g u l i “ E x p e r t tu n e r ”
- s e a s te a p tă o v a r ia ţ i e a r e f e r in ţe i s a u a p e r tu r b a ţ i e i
- s e m o n i to r i z e a z ă i e ş i r e a r e g la tă ş i s e e v a lu e a z ă p e r f o r m a n ţe le
- s e m o d i f i c ă p a r a m e t r i i r e g u la to r u lu i a p l i c â n d r e g u l i