Nqtilen cuu & Trao d 6 i \
CHINH DjNH BO DIEU KHIEN P H A N LY
B A N G PHUONG PHAP M O M I
DAMIRVRANCIC Bo mon Dieu khien va Tu dong hoa - Vien Jozet Stefan, Slovenia
Phan Ion cac phiromg phap chinh djnh bo dieu chinh PI da bien hien nay deu yeu cau phai co duoc mo hinh qua trinh t o t thi moi c6 kha nang cho ket qua nhu mong muon. Viec mo hinh hoa va nhan dang cac mo hinh qua t r i n h mot each chinh xac thudng chiem rat nhieu thdi gian va can phai thuc hien cac t h u nghiem tren pham vi rong. Trong bai bao nay, chiing toi se gidi thieu mot phuong phap mdi ve nhan dang tham so cua mo hinh qua trinh dua tren viec tich hgp nhieu dap ung budc nhay. Phuong phap nay da chimg t d su thanh cong trong viec nhan dang tham so cho bo dieu chinh phan ly. Cac thi nghiem thuc hien tren hai md hinh qua trinh cho thay phuong phap nay cho ket qua chinh djnh tot hon cac phuong phap chinh djnh thdng thudng khac.
TOng quan
Rat nhieu he thong trong hoa chat va trong cac qua trinh cdng nghiep la cac he da bien. Trong phan Ion cac he thong d6, sii tac dpng xen kenh giiia dau vao va dau ra thutjng la thap, do vay c6 the su dung cac bp dieu khien don bien (SISO) thong thudng.
Tuy nhien neu mpt he thong 6a bien ton tai su xen kenh manh giua dau vao va (3au ra.ta se phai sir dung bp dieu khien da bien de co the dat dupe dap ung mong muon.
Co hai bp dieu khien thong ditng la bp dieu khien da bien (Hinh 1) va bp dieu khien phan ly (Hinh 2).
Cac bp dieu khien nay thuang bao gom cac thanh phan PI hoac PID nen cau true CLJa n6 kha 6an gian va cung cap kha nang thoa hiep giua dap ung va tinh ben vung.
Mpt vai huang tiep can gan day c6 xu huang lam cho cac buac chinh dinh trd nen dan gian va than thien han. Wang va dong su [10] da de xuat mpt phuang phap \u chinh cho bp dieu khien PID da bien dua tren kich thich ra le. Mo hinh qua trinh cti dupe tu dac tinh tan so tai hai diem b4ng each su dung phan hoi ra le eo bias. Cac tham so bp dieu chinh PID da bien duac tinh toan dua tren viec lua chpn he so khueeh dai va pha. Wang ya dong su [11] da giai thieu mpt huang tiep can de ehinh dinh cac vong dieu chi'nh cua he thong co hai vao - hai ra (TITO) b§ng each su dijng bp dieu khien phan ly.
Viee chinh dinh bp dieu chinh va bp phan ly dupe
^
w.
^
-rO
'<>
controller process
4 •
Hinfi 1. Qua trinti da bien (TITO) va bq dieu chinh da bien
^
- ^ ^
V, *
>6
controller
^ • SSL—^
^
process
J^
Hinh 2. Qua trinh da bien (TITO) va bq dieu chinh phan ly (Bq dieu chinh la c, va C2, bq phan ly la d,
va d2)
so 6 (82) 2007 T j dong hoa ngay nay 45
Nghien ciju & Trao doi
dt/a tren cac mo hinh bac 1 va bac 2 co tre co duac tii cac dap iing buoc nhay cua qua trinh. Bang phuang phap nay chung ta se phai xac dinh he so khueeh dai, pha va tham so chinh dinh ket hap.
Gan day, mpt phuang phap mai de chinh dinh cho cac qua trinh da bien dua tren viec tich hop nhieu dap ij'ng cua qua trinh da duac phat trien va nghien cuu rpng rai [7]. Y tuang chinh cCia phuang phap do Lieslefito [3] de xua't la dan gian hoa cac thij tuc tinh toan tham so cho bp dieu chinh PI da bien. Cu the la viec mo hinh hoa va nhan dang qua trinli duac thay the bang each thue hien tich hap nhieu lan thay ddi trang thai xac lap ciJa qua trinh. Do vay viec tinh toan tham sd cua cac qua trinh da bien duac dua tren nhieu thanh phan tich phan thay vi theo cac tham sd cua ham truyen. Viec tinh toan tham sd nay la dan gian han rat nhieu vi nd cd the su dtjng ket qua CLia nhieu kieu thay ddi vdng hd trang thai xac lap eija qua trinh.
Trong bai bao nay, phuang phap dupe md rdng cho bo dieu khien phan ly. L/u diem CLia viec su dung bd dieu khien phan ly so vdi bd dieu khien da bien PI la viee phan ly va viec chinh dinh duac tach ra thanh hai nhiem vu rieng biet. Do vay viec tinh toan cac tham sd cho bd dieu khien phan ly trd nen minh bach han va trong mdt sd trudng hop viec phan ly cho hieu qua tdt han.
Bo di6u khj6n phan ly
Hinh 2 md ta mdt he thdng da bien dan gian cd hai vao - hai ra, mdt bd dieu khien phan ly bao gdm hai bd dieu chinh SISO (c^ va C2) va hai bd phan ly (d., va d^).
Muc tieu dieu khien dau tien la phan ly he da bien bang cac bd phan ly di va 62- Mue tieu nay cd the dupe thuc hien bang each chpn eac ham truyen cho bd phan ly nhu sau:
rf ls) = -l-ii =
( • )
f;.,(5 I (1)
Neu bieu thuc (1) la dung, tac ddng cheo giua dau vao Vi va V2 va dau ra y^ va y2 se khdng tdn tai. Trong trudng hpp nay dau ra cua qua trinh la:
V ( : i = r , l s l g . f v l r , ( s l y , f j i = C-.I.S )g,i': k',(>)
Trong dd:
.ir.(sl = , t r . , l ; ! - r / , ( s ) ? j , ( s l
e J ^ l = ?.,.(sj-c/-,(si.er,.(s!
(2)
(3)
Mijc tieu dieu khien thu hai la ehinh dinh eac bd dieu chinh Cl va C2 eho cac qua trinh g^ va g2 tuang ung.
Mdt dieu bat Ipi la trong thuc te rat it khi ta cd the cd duac ham truyen cua cac qua trinh mpt each chinh xac.
Va khi dd trong phan Idn cac trudng hop, chijng ta chi co the su dung cac phep do dan gian nhu dap iing budc nhay eua qua trinh. Trong trudng hap dd viec chinh dinh bd dieu khien phan ly cd the thay doi cho phu hpp. Viec nay ed the duac thue hien bang each su dung phuang phap tich phan nhieu lan [5,6] de tinh toan cac tham so md hinh bd phan ly (d^ va d2) va phuang phap chinh dinh MOMI (Magnitude - Optimum - Multiple - Integration) de chinh dinh tham sd bd dieu ehinh c^ va e2.
Chinh dmh tham sd bp didu Khi6n phan ly
Cac tham sd sau (Ao..A|^) cd the dupe tinh toan bang each tich phan dap ung budc nhay vdng hd cua qua trinh (y(t)), sau khi thay luang thay ddi cua budc nhay AU vao dau vao cua qua trinh tai t = 0 [5,8] ta cd:
!.(:/:)
(4)
Trong dd:
i . i f i = l | r ) - , i n i
(5)
1 J r i = I [.i;_. -y;_,irli<-(r
AQ la he sd khueeh dai tinh cua qua trinh.
Neu cd the nhan dang duac qua trinh va nd cd ham truyen nhu sau:
G^\s}=K,, '• - ^ '^ c-'- (6) 1 -1- (7• y - r?. s ' + — - a „ . ? ' ' '
Thi ta ed the tinh toan cac tham sd dua tren cac tham sd cua dae tinh [5,6] nhu sau:
-J-, =A.- , -ii=^P?.^'h-h-T.,:)
I ?i I (7)
- , T,..'b..
Cac tham so (7) cung cd the duac do trong mien thai gian tu thay doi trang thai xac lap cua qua trinh [8],
Dua vao cac tham sd do duac, ta cd the uac lupng cac tham so ham truyen cua qua trinh tu (7). So lupng cac tham sd ham truyen duac udc luang bang sd lupng cac tham sd do duac. Ta cd the tinh duac md hinh ham truyen bac hai vdi hai diem cue va mpt diem "khdng" tijf cac tham sd do duac nhu sau:
46 Automation Today
so 6 (82) 2007
•Nghien
CLJUf^ T r a o d o i
A. A:
A., (8)
(7, =
t'. = CL A.
A._
A.
Ta cung cd the tinh toan cac bd phan ly d^ va d2 tu cac tham sd do dupe cua qua trinh g^^ den g22. Cac tham sd cua bd phan ly d^ ed the dupe tinh toan b§ng each su dung bieu thuc sau:
= V I
Trong dd:
(9)
. 1 ,
- • * ' : : '
, 1 - |
A.
A.,
-^:,.
.J "
2.-J. . 4 , . .
' A . ^•
.J,,.
-J,- "
(10)
A:
Cac he sd 21 va 22 bieu hien cho tham sd cua cac qua trinh g2i va 022 tuang ung. Doi vdi bp phan ly d2, chi sd di trong bieu thue (9) cd the thay the bang chi sd d2 trong khi dd he sd 21 va 22 trong bieu thuc (10) se duac thay bang 12 va 11 tuang ung. Tu cac tham sd cd duac (9), bd phan ly bac hai (d^ va d2) cd the dupe tinh toan tu (8).
Trong mdt sd trudng hpp, khdng the dat dupe bd phan ly vat ly (he sd khueeh dai cao tan qua cao hoac bang vd cung). Trong trudng hpp nay, chung ta se them bp lpc bac 1 vao bd phan ly nhu tren Hinh 3. Thdng thudng hang sd thdi gian cua b<? lpc se duac chpn nhu sau:
T, = 0; if il. £ 4 , T, = 0 . 2 5 i'l.
(11)
4,'),
Neu su suy hao eua eac diem cue qua thaip, ta se phai gidi han tham sd a2. Thdng thudng, ta se phai chpn giai gidi han tham sd nhu sau:
«, ^0.5«,- ^ (12) Neu 82 da bi thay dd'i theo bieu thuc (12), ta se phai
e.
1
l+sTf
-rO
i-^r.. • ^ controller
Hinh 3.Ket hop su dung bq Iqc bac mot voi bq phan ly d^ va d2
tinh toan lai tham sd al theo bieu thuc (7):
A +,A 2A.
(13) Sau khi cd dupe cac bd phan ly, ta se phai thuc hien ehinh dinh lai cac bd dieu chinh c^ va C2. Chpn caiu true bd dieu chinh PI:
v ( H = A" ei i I (14)
Trong dd e(s) la sai lech dieu chinh va v(s) la dau ra eua bd dieu chinh (xem Hinh 2). Viec tinh toan tham sd cua cac bd dieu chinh cd the dupe thuc hien bang each SLl dung phuong phap chinh djnh MOMI [6-9]. Phuang phap chi yeu cau cd cac tham sd do dupe cua cac qua trinh g^ va g2 (3) de tinh toan tham sd bp dieu chinh.
K.
2[A.A.. -A.A: I - 0 "^
(15)
J^a.
A;
Cac tham sd cua qua trinh g^ va g2 (3) cd the dupe tinh toan tu cac tham sd cua qua trinh con g^i den g22 va tham sd eua bd phan ly d^ va d2 (9) nhu sau:
..-=C
n
-Ik
(16)
Cae tham sd cua bd dieu chinh c^ va e2 se dupe tinh toan bang each thay the A^, trong (15) bang A^gi va Ang2 trong (16).
Neu da them bd lpc vao trong bd phan ly, cac tham sd Apg! va Apg2 se dupe thay ddi nhu sau:
so 6 (82) 200 Jo dong hoa ngay nay 4 7
Nghien cau & T r a o d b i
(17)
Nhu vay ta cd quy trinh chmh dinh bd dieu chinh phan ly se thuc hien nhu sau:
• Tao mdt lupng thay dd'i bude nhay AU tai dau vao thu nhat cua qua trinh va do dap ung cua tat ca eac dau ra. Tu day ta cung se cd duac dap ung ctja cac qua trinh con g^^ va g2i. Lap lai quy trinh cho dau vao thu hai ta se thu dupe dap ung cua cac qua trinh con gi2 va g22-
• Tinh toan he sd khueeh dai qua trinh KPRij (Agy) va cac tham sd (A^jj, A21J va A3JJ) bang each su dung tich phan rdi rac theo bieu thuc (4).
• Tinh toan cac tham sd cua bd phan ly tu (9).
• Tinh toan ham truyen cua bd phan ly (d-| va d2 trong Hinh 2) tu cac tham sd ed duac trong budc 3 va bieu thuc (8).
• Neu can thiet, them eac bd lpc va/hoac thay doi ham truyen eua bd phan ly theo cac bieu thuc (11) den (13).
• Tinh toan cac he sd eua dac tinh cua cac qua trinh gi vag2tu(16)va(17).
• Tinh toan tham sd cua bd dieu chinh PI (c^ va C2 trong Hinh 2) theo bieu thue (15) dua tren cac tham sd cd dupe tu cac bude tren.
Chung ta cd the tai bd cdng cu Matlab de thuc hien viec tinh toan tham sd bd dieu chinh phan ly tu [9].
Luu y: neu he sd khueeh dai K tinh ra (15) la qua cao (dac biet ddi vdi cac qua trinh ed bac tha'p), ta cd the gidi han den mdt gia tri nhat dinh. Sau dd ta se_ phai dua tren he sd khueeh dai mdi eua bd dieu khien de tinh lai hang sd thdi gian tich phan theo (15).
Mpt s6 Ui du utig dung PhUdng phap MOMI
Chung tdi gidi thieu hai vi du sau day de minh hpa cho phuang phap chinh dinh bd dieu chinh phan ly.
Truang hap 1:
Vi du dau tien dupe thuc hien vdi md hinh da bien cua Menani va Koivo [4]:
G ( s ) = -
1 (l + 0 , l s ) ( l + 0 . 2 s | "
0.5 - 0 , 1 s
1
- 1 2.4
(18)
Theo quy trinh chinh dinh d tren, ta se thue hien hai dap ung budc nhay tai hai dau vao cua qua trinh.
Dau tien ta tinh duac he sd khueeh dai tinh cua qua trinh nhu sau: KPR11 = 0.5, KPR12 = -1, KPR21 = 1, KPR22 - 2.4. Sau dd ta se tinh toan dupe cac tham sd Al den A3 dua tren cac dap ung cua cac qua trinh con b§ng tich phan rdi rac nhu md ta trong phan 3. Cac gia tri tinh toan cho g i i : A1=0.3, A2=0.115, A3=0.036, cho
g12: A1=-0.5, A2=-0.17, A3-0.049, cho g21: A1=0.5, A2^0.17, A3=0.049, va cho g22: A1=2.4, A2=1.608, A3=0.9216.
Cac bp phan ly sau dugc tinh toan dua tien (8):
(fi(^)= 0.417(1-0.55] (19)
Vi cac bd phan ly tinh dupfc khong phai ti> thi/c nghiem (Do thanh phan thuan vi phan), ta se phai su dung them bp lpc theo bieu thuc (11).
r , = 0 . 1 2 5 : T, = 0 . 0 2 5 ^^^^
Tham sd cija bp dieu chinh PI se duac tinh theo eac cdng thuc tu (15) den (17):
A'l =0.906: K._^ =3.209
A% =0.2S1: K, =0.524 (21) Dap ung vdng kin dua tren cac thay doi diem dat cua ca hai dau vao cua bd dieu chinh dupe cho tren Hinh 4. Dap ung nay dupe dem so sanh vdi dap ung d^t dupe eua Menani va Koivo [4]. Bd dieu chinh PI da bien eua Menani va Koivo dupe tinh toan dua tren phuang phap tinh toan thiet ke bd dieu ehinh MOMO cua Maciejowski cho md hinh qua trinh ed duac tu phuang phap kich thich ra le.
Ket qua eho thay dap ung vdng kin cua phuang phap MOMI eho ket qua rat tdt, dap ung kha nhanh va thuc hien phan ly ly tudng.
The proposed mefhod
Menan and Koivo. 1096
Tnoels]
Hinh 4. Dap ung vong kin cua qua trinh khi su dung bq dieu khien phan ly cua tac gia de xuat (hinh tren) va khi su dung cac tham so do tVlenani va Kqivo de xuat (hinh duoi).
Truang hap 2:
Vi du thu hai dupe thuc hien tren md hinh qua trinh cdt Chung cat methanol-nudc do Ho va ddng su [2] va Menani va Koivo [4] de xuat.
48 Automation Today
SO 6 (82) 2007
Nghien
CLTU& T r a o d o i
1 2 . S f ' lS.9e~
l - K r . J s 6.6c' '' I 1 - 1 0 . P i
I + 21.S - 1 9 . 4 « ' ' ' - '
l + ]4,4,s
(22)
Thu tuc chinh dinh duoc thuc hien nhu vi vu tren, Cac he sd khueeh dai tinh eua qua trinh tinh duac nhu sau: /<pR,p12,8, /<pR,^-18.9, Kp^2r6.6, /<pR2^-19.4, Cac tham sd cua g ^ : A-,=226.56, A^3.79e3, /I3=6,329e4, cho gi2. /^r-453.6, /1^-9.6106e3, A:^-2.0^9^e5, cho g2{.
/^,:=118.14, /^^1.4494e3, A2^^.618e4, va cho g22. A.,=- 337.56, A:^-4.9482e3, /l3=-7.134e4.
Cac bd phan ly tinh dupe theo (8):
rf,ls.l =
rf-(i) (l
-0.34(1- - 1 2 , 7 . s - -1.4SI1
- i ; -50 + 1"
2: 1 5 J ' ! 4s 1
(23)
( l - 2 3 , 7 . s + 4 S . 5 S -
Cac tham sd cua bd dieu chinh PI tinh theo (15) va (16):
K. =0.064: K.^ =0,013 (24)
A', -0.052; A.', = - 0 . 0 0 9 8
Ket qua bd dieu ehinh PI phan ly mdi se dupe dem so sanh vdi bd dieu ehinh PI cua Ho va ddng su [2]. Bd dieu chinh cua Ho va ddng su [2] dupe tinh toan dua tren cac tham sd ve pha va he sd khuyech dai. Md hinh qua trinh cLia Ho [2] dupe tinh toan theo phuang phap binh phuang nhd nha't.
The propos«d rrtelhtod
Hinh 5. Dap iing vong kin khi su dung bq dieu chinh phan ly theq phuong phap moi (hinh tren) va khi su dung tham so cua Ho va dong su [2] (hmh dudi)
Dap U'ng vdng kin trong ca hai truang hap tren Hinh 5 cho thay phuang phap de xua't mdt lan nCra cho ket qua dap ung vdng kin kha td't. Ndi each khac, dap ung cua Ho va ddng su [2] nhanh han nhung dp phan ly lai ketn hon phuang phap mdi.
Ketluan
Trong bai bao nay, chung tdi da de xuat mdt phuang phap chinh dinh mdi dan gian han cac phuang phap thdng thudng. Net mdi trong phuang phap de xuat sir dung de thiet lap tham sd cua qua trinh (tinh toan tham sd
\), la khdng yeu cau phai md hinh hda qua tnnh mdt each chinh xac va ve can ban don gian hda dupe giai doan nhan dang (dap u'ng budc nhay vdng hd). Viec tinh toan cac tham sd (A^) va cac budc cdn lai cua qua trinh chinh dinh cd the thuc hien m(3t each tu ddng, do vay tat ca cac phep tinh se khdng can phai rdi rac hda,
Ket qua md phdng chi ra rang dap ung cua phuong phap de xua't la kha td't neu so sanh vdi mdt vai phuang phap ehinh dinh thdng thudng. Cac phuang phap chinh dinh thdng thudng yeu cau cao han trong giai doan md hinh hda va nhan dang qua trinh. Tuy nhien, nhieu thdng thaip va su phi tuyen cua qua trinh cd the anh hudng den dp chinh xac cua cac tham sd tinh ra. Qua trinh da bien do vay se phai cd cac qua trinh eon dn dinh (tait ca cac diem cue phai nam ben tay trai cua true ao). Cd the xuat hien su suy giam dap ung vdng kin neu ham truyen qua trinh con cd bac Idn han.
Diem nhain manh chinh cua bai bao nay la su' don gian trong thu tuc chinh dmh cho bo dieu chinh phan ly, do vay cau true bd dieu chinh phan ly duac lua chpn sao cho dan gian nhat cd the. Tuy nhien, chung ta hoan toan cd the cai thien dap ung vdng kin bang each thay bd dieu chinh PI bang bd PID [10, 11] hoac bang each su dung ham truyenkhac cho bd phan ly d^ va d2 (vi du khau bac nha't ed tre nhu trudng hop 2).Q
P H A M H O N G S O N - (luac dich)
Tai lieu tham khao:
[I] Astrom, K. J., and T. Hagglund (1995). PID Controllers: Theory, Design, and Tuning, Instrument Society of America, 2nd edition.
[2] W.K. Ho, T.H. Lee and O.P. Gan, Tuning of multiloop PID controllers based on gain and ptiase margin specilicalions, in: Proc. IFAC 13tti Triennal Worid Congress, San Francisco, Vol. M. pp. 211-216 (1996).
[3] Liesletito, J., "MlfVIG controller design using SISO controller design metti- ods", Proc. IFAC 13tti Triennal World Congress, San Francisco, Vol. C, pp. 169-173, 1996.
[4] S. Menani and H.N. Koivo, Relay tuning of multivariable PI controllers, in:
Proc. IFAC IStti Triennal World Congress, San Francisco, Vol. K, pp.
139-144(1996).
[5] V. Strejc, Auswertung der dynamisctien Eigensctiaften von Regelstrecken bei gemessenen Bin- und Ausgangssignalen allgemeiner Art, Z. Messen, Steuem, Regein 3 (1), 7-10 (1960).
[6] VranCic, D., S. Strminik, andO. Juricic, "A magnitude optimum multi- ple integration mettiod for filtered PID controller", Automatica, 37, pp.
1473-1479, 2001a.
[7] Vrancic, D., J. Lieslehto, S, Strmtnik, "Designing a IVIIIVIO PI controller using ttie multiple integration approacti". Process Control and Quality, Vol. 11, No. 6, pp. 441-550, 2001b.
[8] Vrancic, D., J. Kocijan, S. Strmenik, "Improving PID Controller Disturbance Rejection by Means ot Magnitude Optimum", Submitted to ttie 4th Asian Control Conference, Singapore, 2002.
[9]Vran6ic, D "Matlab Toolset for Decoupling Controller", Available on tittp://www-e2.ijs.si/damir.vrancic/tools,html.
[10] Wang, Q.G., B. Zou, T.H. Lee, Q. Bi, "Auto-tuning of Multivanable PID Controllers trom Decentralized Relay Feedback", Automatica, Vol, 33, No. 3, pp. 319-330, 1997.
[ I I ] Wang, O.G., B. Huang, X. Guo, "Auto-tuning ol TITO decoupling con- trollers Irom step tests", ISA Transactions, Vol, 39, pp. 407^18, 2000
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