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Chng 3 : iu khin bn vng
Trang 174
Hnh 3.15: Biu Bode Bin h thng SISO Mc khc, hnh 3.14 l cc gi tr tr suy bin ca h a bin. Ch rng, theo th ny khng th d dng bng trc quan tc thi nhn thy cch lin kt h SISO .Nhng ng bao bo m s bn vng c a ra trong h thng h MIMO di dng gi tr tr suy bin cc tiu l ln ti tn s thp ( cho cht lng bn vng) v gi tr tr suy bin cc i l nh ti tn s cao (cho n nh bn vng )..
Hnh 3.16:Cc gi tr tr suy bin ca h thng
Chng 3 : iu khin bn vng
Trang 175
3.3.2 Hm nhy v hm b nhy Kho st c tnh ca h thng hi tip in hnh, t a ra tng thit k tha hip gia mc tiu cht lng v iu khin bn vng nhm tha mn cc yu cu thit k.
Xt h thng hi tip m nh hnh 3.17, trong id l nhiu u vo, d l nhiu u ra, n l nhiu o.
Hnh 3.17: S h thng hi tip m
Lu : lin h vi phn l thuyt iu khin kinh in, trong mc ny ta phn tch s iu khin hi tip m, vi b iu khin l K ( K = -K m hnh hi tip dng) Cc quan h truyn t ca h thng vng kn c th hin qua cc biu thc sau:
y = dKG
dKG
Gn
KGKG
rKG
KGi
11
11
1
++
++
+
+
u = dKG
KdKG
KGn
KGK
rKG
Ki
1
1
1
1
+
+
+
+
Gu = dKGKd
KGn
KGK
rKG
Ki
1
11
1
1
+
++
+
+
e = dKG
dKG
Gn
KGr
KG i 11
111
11
+
+
+
+
n
y Guu e
r
-
+ +
+
K G
di d
Chng 3 : iu khin bn vng
Trang 176
nh ngha cc hm nhy, hm b nhy v li vng nh sau:
- Hm nhy : KG
S1
1+
=
- Hm b nhy : KG
KGT1
+=
- li vng: KGL =
Cc ng thc trn c vit gn li:
SdGSdTnTry i ++= (3.156)
SdKTdSnKSrKu i = (3.157)
G iu KSr KSn Sd KSd= + (3.158)
SdGSdSnSre i = (3.159)
T (3.156) (3.159), ta c th rt ra cc mc tiu cht lng ca h thng vng kn.T phng trnh (3.156) ta thy rng: - gim nh hng ca nhiu u ra d ln u ra y, hm nhy S cn phi nh. - gim nh hng ca nhiu o n ln u ra y, hm b nhy T cn phi nh. Tng t, t phng trnh (3.158), lm gim nh hng ca nhiu u vo di, hm nhy S cn phi nh. Nhng t nh ngha ,hm nhy v hm b nhy c quan h rng buc nh sau:
S + T = 1 (3.160) Do , S v T khng th ng thi nh. gii quyt mu thun ny, ngi ta da vo c tnh tn s ca cc tn hiu nhiu. Nhiu ti d, di tp trung ch yu vng tn s thp, cn nhiu o n tp trung ch yu vng tn s cao.
Chng 3 : iu khin bn vng
Trang 177
Nh vy, h t b nh hng bi d, th S v GS cn phi nh trong vng tn s m d tp trung, c th l vng tn s thp. Tng t, iu kin h t nhy i vi nhiu di l |S| v || SK nh trong vng tn s m di tp trung, c th l vng tn s thp. Ta c:
1|||1|1|| ++ KGKGKG Suy ra:
1||1
11
1||1
+
+ KGKGKG
, nu | KG |>1
hay:
11
11
+ L
SL
,nu L >1
T , ta thy:
S 1
Hn na, nu L >> 1, th:
GS ||1
1 KKGG
+=
|| SKGKG
K 11
+=
Nh vy, i vi u ra y: - gim thiu nh hng ca d, li vng L phi ln (ngha l |L|>> 1) trong vng tn s m d tp trung;
- gim thiu nh hng ca di, bin b iu khin phi ln K 1>> trong vng tn s m di tp trung.
Tng t, i vi u vo (u G )
Chng 3 : iu khin bn vng
Trang 178
- gim thiu nh hng ca di, L phi ln (ngha l |L|>> 1) trong vng tn s m di tp trung. - gim thiu nh hng ca d, bin i tng (khng thay i c trong thit k iu khin) phi ln (|G|>> 1) trong vng tn s m d tp trung. Tm li, mt trong nhng mc tiu thit k l li vng (v c li ca b iu khin, nu c) phi ln trong vng tn s m d v di tp trung, c th l vng tn s thp. Sau y, ta xt nh hng ca sai lch m hnh ln h thng hi tip. Gi s m hnh i tng c sai s nhn l (I + )G, vi n nh, v h thng kn n nh danh nh (n nh khi =0). H thng kn c sai s m hnh s n nh nu:
det ( ) KG 1(1 ++ )=det
+
++
KGKGKG
1
1)1( =det(1+ KG )det(1+ )T
khng c nghim na phi mt phng phc. Ta thy rng, iu ny s c tha nu nh T nh, hay |T| phi nh vng tn s m tp trung, c th l vng tn s cao. rng, nu |L| rt ln th |T| 1 v |S| 0. Do , t (3.156) ta thy nu nh ( )L j ln trong mt di tn s rng, th nhiu o n cng s truyn qua h thng trong vng tn s , ngha l:
y= SdGSdTnTr i ++ (r - n) v rng nhiu o n tp trung ch yu vng tn s cao. Hn na, nu li vng ln ngoi vng bng thng ca G, ngha l ( )L j >>1 trong khi
( )G j
Chng 3 : iu khin bn vng
Trang 179
Phng trnh trn cho thy nhiu ti v nhiu o s c khuych i ln khi m vng tn s m n tp trung vt ra ngoi phm vi bng thng ca G,
v i vi di tn s m ( )G j 1.
Tng t, bin ca b iu khin, | K |, khng c qu ln trong vng tn s m li vng nh nhm trnh lm bo ha c cu chp hnh. V l khi li vng nh ( ( )L j KG , || K >>1 - m bo tnh bn vng v c kh nng trit nhiu o tt trong mt vng tn s no , c th l vng tn s cao ( ,h ),h thng cn phi c :
1||
Chng 3 : iu khin bn vng
Trang 180
tin v c tnh ca nhiu ti, nhiu o, sai lch m hnh.
Hnh 3.18: li vng v cc rng buc tn s thp v tn s cao.
Nhng iu phn tch trn y l c s cho mt k thut thit k iu khin: l nn dng vng (loop shaping). Mc tiu nn dng vng l tm ra mt b iu khin sao cho li vng |L| trnh c cc vng gii hn (xem hnh 3.18) ch nh bi cc iu kin v cht lng v bn vng.
3.3.3 Thit k bn vng H 3.3.3.1 M t khng gian H v RH
Khng gian vector Hardy c chun v cng, k hiu l H, l khng gian cc hm phc G(s) ca bin phc s (s C) m trong na h mt phng phc bn phi (min c phn thc ca bin s ln hn 0) tha mn: - l hm gii tch (phn tch c thnh chui ly tha), v - b chn, tc tn ti gi tr M dng no ( )s MG c phn thc dng.
Tp con c bit ca H m trong iu khin bn vng rt c quan tm l tp hp gm cc hm G(s) thc - hu t (real-rational) thuc H, tc l cc hm hu t phc G(s) H vi cc h s l nhng s thc dng
0 1
1
( )1
m
m
n
n
b b s b ss
a s a s
+ + +=
+ + +G
trong ai,bj R, k hiu l RH. Trong l thuyt hm phc, ngi ta ch ra c rng: mt hm thc hu t G(s) bt k s thuc RH khi v ch khi - lim ( )
ss
< G , hay ( )G b chn (khi mn),c gi l hm hp thc v
- G(s) khng c cc trn na kn mt phng phc bn phi. Ni cch khc G(s) khng c im cc vi Re(s) 0.Mt hm G(s) c tnh cht nh vy gi l hm bn.
Chng 3 : iu khin bn vng
Trang 181
Nu hm truyn hp thc G(s) khng nhng na h bn phi mt phng phc b chn khi s m cn tha mn (khi m
Chng 3 : iu khin bn vng
Trang 182
=
DCBA
G (3.166)
trong : 1( ) ( )s sI = +G C A B D . xc nh phn tch coprime bn tri, trc tin ta cn phi tm nghim ca phng trnh Riccati sau:
1 1 1 1( ) ( ) ( ) 0 + + =A BD R C Z Z A BD R C ZC S CZ B I D R D B (3.167) trong *DDIR + . Phng trnh ny c tn l Phng trnh Riccati lc tng qut (GFARE Generalized Filter Algebraic Riccati Equation). Sau p dng nh l 3.3 tnh N , M . nh l 3.3:
Cho
=
A BG C D . Phn tch coprime bn tri chun ca G c xc nh
nh sau:
1 2 1 2
+ + =
A HC B HDN R C R D ; 1 2 1 2
+ =
A HC HM R C R
(3.168)
trong Z l nghim xc nh dng duy nht ca GFARE, = +R I DD , v 1( ) = +H ZC BD R . Sai s m hnh phn tch coprime bn tri Sau y, ta nh ngha sai s m hnh phn tch coprime bn tri. Gi s G l m hnh i tng, ( N , M ) l mt phn tch coprime bn tri ca G. H c sai s m hnh phn tch coprime bn tri chun c nh ngha nh sau:
1( ) ( )M N = + + G M N (3.169) trong N, M RH l cc hm truyn cha bit th hin phn sai s trong m hnh danh nh. H m hnh c sai s l mt tp G nh ngha nh sau:
[ ]{ }1( ) ( ) : ,M N M N
= + +
Chng 3 : iu khin bn vng
Trang 183
Hnh 3.19: Biu din sai s m hnh phn tch coprime bn tri
Mc tiu ca iu khin bn vng l tm b iu khin K n nh ha khng ch m hnh danh nh G, m c h m hnh G .
u im ca cch biu din sai s m hnh trn y so vi biu din sai s cng v sai s nhn l s cc khng n nh c th thay i do tc ng ca sai s m hnh
3.3.3.3 Bi ton n nh bn vng H: Xt h hi tip hnh 3.20
Hnh 3.20: S phn tch n nh bn vng vi m hnh c sai s LCF
nh l 3.4:
N
N
+
-
1M
+
M
+
N
K
w u
y d
+
+ +
1M
N M
+ +
+
Chng 3 : iu khin bn vng
Trang 184
1=G M N l m hnh danh nh; 1( ) ( )M N = + + G M N l m hnh
c sai s; ( M , N ) l phn tch coprime bn tri ca G; M , N , M , N RH. H n nh bn vng vi mi [ ]M N tha
[ ] 1M N
< nu v ch nu:
a.H (G, K) n nh ni, v
b. 1 1( )
KI GK M
I
(3.171)
nh l 3.4 c th pht biu mt cch tng ng di dng mt bi ton ti u nh sau:
nh l 3.5: i tng 1( ) ( )M N = + + G M N , vi [ ] 1M N
< , n nh
ha bn vng c nu v ch nu:
1 1inf ( )
K
KI GK M
I
(3.172)
trong infimum c thc hin trong tt c cc b iu khin K n nh ha G.
Bi ton n nh bn vng Cho trc gi tr , tm b iu khin K (nu tn ti) n nh ha i tng danh nh G, v tha:
1 1( )
KI GK M
I
(3.173)
trong ( N , M ) l phn tch coprime bn tri ca G. V theo nh l 3.4, nu tm c b iu khin K, th K s n nh ha i tng c sai s G, vi [ ] 1M N
= < .
Nu pht biu di dng mt bi ton ti u H (i vi h thng hnh 3.20) th ta c bi ton ti u H nh sau:
Chng 3 : iu khin bn vng
Trang 185
Bi ton ti u H Tm b iu khin K (nu tn ti) n nh ha i tng danh nh G v cc tiu ha chun H sau y:
1 1( )
KI GK M
I
(3.174)
trong ( N , M ) l phn tch coprime bn tri ca G. Bi ton ti u H phc tp ch phi thc hin cc tiu ha chun (3.174) trong iu kin tn ti b iu khin K n nh ha h thng. gii quyt vn ny, thng thng ngi ta gii bi ton n nh bn vng vi mt gi tr cho trc, ri sau thc hin qu trnh lp tm gi tr min. Glover v McFarlane s dng bi ton m rng Nehari (Nehari extension problem), v dng phn tch coprime chun ca m hnh i tng tm ra li gii khng gian trng thi cho bi ton ti u H m khng cn phi thc hin qu trnh lp tm min. Hn na, t cch tip cn ny, tc gi c th tnh c d tr n nh cc i max ( = min1 ) mt cch chnh xc. Phn sau y ch trnh by mt s kt qu chnh m Glover v McFarlane thc hin. nh l 3.6: B iu khin K n nh ha h thng v tha
1 1( )
KI GK M
I
(3.175)
nu v ch nu K c mt phn tch coprime bn phi: 1=K UV vi U, V
RH tha
( )1 221
+
UNVM
(3.176)
nh l 3.7: a. Li gii ti u ca bi ton n nh bn vng i vi m hnh phn tch coprime bn tri chun cho kt qu:
Chng 3 : iu khin bn vng
Trang 186
{ } 1 221 1inf ( ) 1 H
= K
KI GK M N M
I
(3.177)
trong infimum c thc hin trong tt c cc b iu khin n nh ha h thng. b. d tr n nh cc i l
{ } 1 22max 1 0H = > N M (3.178) c. Cc b iu khin ti u u c dng: 1=K UV , vi U, V RH tha
H
+ =
UN N MVM
(3.179)
Cc nh l trn cho ta nhng nhn xt sau: - d tr n nh cc i c th c tnh trc tip t cng thc (3.178) - Vic xc nh b iu khin ti u H c th c thc hin thng qua bi ton m rng Nehari (Nehari extension). Bi ton ti u con d tr n nh cc i cho ta mt cn di ca , l min = 1/max. Vic gii bi ton ti u H vi > min cho kt qu l mt tp cc b iu khin n nh ha K sao cho
1 1( )
KI GK M
I
(3.180)
y chnh l bi ton ti u con (suboptimal problem). Li gii dng khng gian trng thi ca bi ton ny c xc nh theo cc bc nh sau : Bc 1: Gii hai phng trnh Riccati GCARE v GFARE. Phng trnh GCARE (Generalized Control Algebraic Riccati Equation) c dng:
1 1 1 1( ) ( ) ( ) 0 + + =A BS D C X X A BS D C XBS B X C I DS D C (3.181)
Chng 3 : iu khin bn vng
Trang 187
trong : DDIS += . Phng trnh GFARE l phng trnh trnh by trn.
1 1 1 1( ) ( ) ( ) 0 + + =A BD R C Z Z A BD R C ZC S CZ B I D R D B trong += DDIR .
Bc 2: Tnh gi tr nh nht c th t c. 1 2
min max(1 ( )) = + ZX trong ( )max l tr ring ln nht, X v Z ln lt l nghim ca GCARE v GFARE.
Bc 3: Chn min > . Thng thng, chn ln hn min mt cht; chng hn, min1.05 = .
Bc 4: B iu khin trung tm c biu din trng thi c xc nh nh sau
2 1 2 11 1
0
( )
+ + +=
A BF W ZC C DF W ZCK
B X D (3.182)
trong X v Z l ln lt l nghim ca cc phng trnh GCARE v GFARE,
1( ) = +F S D C B X , v 21 ( )= + W I XZ I . Cng thc tnh min bc 2 c dn ra t cng thc (3.177) trong nh l 3.7. Nu ( N , M ) coprime bn tri chun th
H N M c th c
xc nh t nghim ca hai phng trnh Riccati GCARE v GFARE nh sau:
( )2 1max ( )H = + N M XZ I ZX (3.183) T ta suy ra gi tr min:
1 1 2min max max(1 ( )) = = + ZX
y chnh l cng thc tnh min bc 2.
Chng 3 : iu khin bn vng
Trang 188
Ta thy rng i vi bi ton n nh bn vng cho m hnh phn tch coprime bn tri chun, ta ch cn tm nghim ca cc phng trnh GFARE v GCARE l tnh c gi tr min m khng cn phi thc hin th tc lp . Trong bc 3, ta chn min > nhm bo m s tn ti ca b iu khin c kh nng n nh ha h thng.
Trong trng hp bi ton ti u, min = , th ma trn W1 trong (3.182) suy bin. V do , (3.182) s khng p dng c. Tuy nhin nu ta chn gn min (v d min1.05 = ) th kt qu bi ton ti u con v bi ton ti u s khc nhau khng ng k.
3.3.4 Nn dng vng H 3.3.4.1 Th tc thit k nn dng vng
H :
(LSDP Loop Shaping Design Procedure) Nn dng vng
H (
H loop shaping) l mt k thut thit k do
McFarlane v Glover xut nm 1988. K thut thit k ny kt hp tng nn dng vng (phn hm nhy v hm b nhy) v bi ton n nh bn vng
H . Nn dng vng thc hin s tha hip gia mc tiu cht
lng v mc tiu n nh bn vng, trong khi bi ton ti u
H m bo tnh n nh ni cho h vng kn. K thut thit k gm hai phn chnh: a. Nn dng vng: ch nh dng hm truyn h ca i tng danh nh.
b. n nh bn vng
H : gii bi ton n nh bn vng
H dng phn tch coprime cho i tng c nn dng trn. Th tc thit k nn dng vng (LSDP) Gi s m hnh danh nh ca i tng G, b iu khin cn tm l K Bc 1: Chn cc hm nn dng W1,W2. Tnh Gs: Gs = W2GW1. (Lu l chn W1,W2 sao cho GS khng cha cc ch n (zero cc khng n nh kh nhau))
Chng 3 : iu khin bn vng
Trang 189
Bc 2: Tm nghim Xs,Zs ca GCARE v GFARE ng vi GS.
Tnh ( )( ) 2/1maxmin 1 SS XZ += , trong max (.) l tr ring ln nht Nu min qu ln th tr v bc 1. (Thng thng 1< min min , tng hp b iu khin K sao cho
(Vic xc nh
K c trnh by phn 3.3) Bc 4: B iu khin K cn tm c tnh theo cng thc:
K = W1 K W2
Th tc thit k c minh ha trong hnh 3.21
~
11)( ss MKGIIK
Chng 3 : iu khin bn vng
Trang 190
Hnh 3.21: Th tc thit k
H loop shaping
Nhn xt: -Khc vi phng php thit k nn dng vng c in (nn dng hm S v T), y ta khng cn quan tm n tnh n nh vng kn, cng nh thng tin v pha ca i tng danh nh, v iu kin n nh ni c m bo trong bi ton n nh bn vng
H bc 3.
- Th tc thit k s dng thch hp cho cc i tng n nh, khng n nh, cc tiu pha, khng cc tiu pha; i tng ch cn tha mn yu cu ti thiu cho mi thit k l khng c cc ch n. C th l nu i tng khng cc tiu pha th cc hn ch v cht lng iu khin vn th hin trong th tc thit k qu gi tr ca min .
1W G 2W
1W 2W G
K
sG
G
1W 2W K
K
Chng 3 : iu khin bn vng
Trang 191
3.3.4.2 S iu khin: Trn y ta ch quan tm n vng iu khin, khng quan tm n v tr tn hiu t c a vo vng iu khin nh th no. Thng thng, tn hiu t a vo vng iu khin nh hnh 3.22 vi hi tip n v.
Hnh 3.22: S iu khin hi tip n v Nu b iu khin K t c t th tc nn dng vng
H , th
K v cc
hm nn dng W1, W2 c th c tch ra ring r, v nh ta c th c cc s iu khin khc nhau. Hnh 3.23 l s iu khin vi b iu khin thit k theo th tc LSDP. Ta c th thay i s ny mt cht nh hnh 3.24, m khng lm thay i dng vng L.
Hnh 3.23: S iu khin hi tip n v vi b iu khin t c
t LDSP
G K y
r
-
+
y W2 r
-
+
K
W1 G
Chng 3 : iu khin bn vng
Trang 192
Hnh 3.24: S iu khin ci tin vi b iu khin t c t LDSP
Khi tn hiu t c a vo h thng ti v tr gia hai khi
K v W1, ta cn b sung mt b tin b chnh m bo li tnh bng 1 (hnh 3.24). Hm truyn vng kn t tn hiu t r n u ra y tr thnh:
y(s)= )()0()0()()(1)()(
21 srWK
sKsGsWsG
(3.184)
trong :
)()(lim)0()0(02
sWsKWK ss
= (3.185)
Theo kinh nghim, iu khin theo s hnh 3.24 s cho p ng qu tt hn; iu khin theo s hi tip n v nh hnh 3.23 thng cho p ng qu , c vt l ln. Nguyn nhn l trong s 3.24 tn hiu t khng trc tip kch thch c tnh ng ca
K . Theo th tc thit k
LSDP,
K li c xc nh qua li bi ton n nh bn vng, trong ta khng th trc tip can thip vo v tr im cc zero c, m mi c tnh mong mun ta ch c th a vo h thng thng qua cc hm nn dng W1 v W2.
3.3.4.3 La chn cc hm nn dng W1,W2: Vic la chn cc hm nn dng trong th tc thit k LSDP ni chung l da vo kinh nghim ca ngi thit k. Tuy nhin, i vi tng i tng c th, ngi ta thng a ra cc hng chn hm nn dng thch hp. Thng thng, W2 c chn c dng ma trn ng cho vi cc phn t trn ng cho l cc hng s nhm t trng s ln cc tn hiu ra ca i
y )0()0( 2WK
K
r
2W
1W G +
Chng 3 : iu khin bn vng
Trang 193
tng. W1 thng l tch ca hai thnh phn: WP v WA; trong , WA l b tch knh (decoupler), WP c dng ng cho c chn sao cho tha hip cc mc tiu cht lng v n nh bn vng ca h thng, v thng c cha khu tch phn m bo sai s xc lp bng 0. i vi h SISO, vic la chn cc hm nn dng n gin hn: W2 thng c chn bng 1, v W1 c chn sao cho tha hip c cc mc tiu cht lng v n nh bn vng ca h thng.
3.4 Thit k ti u H2
3.4.1 t vn Xt h thng n nh
RttCxty
tBwtAxtx
=
+=
)()()()()( (3.186)
H thng c ma trn hm truyn H(s) = C(sI-A)-1B. Gi s rng tn hiu w l nhiu trng vi hip phng sai { } )()()( WtwtwE T =+ .Ng ra y ca h thng l mt qu trnh nhiu tnh vi ma trn mt ph. )()()( jWHjHS T = (3.187) Do tr trung bnh ng ra ton phng :
{ } djHWjHtracedStracetytyE T )(~)(21)(
21)()(
+
+
=
= (3.188)
y ta k hiu )(~ jH =HT(- j ) Ta c :
+
= djHjHtraceH )(~)(21
2 (3.189)
Gi l chun H2 ca h thng .Nu nhiu trng w c mt W = I th tr trung bnh ca ng ra ton phng )}()({ tytyE T tng ng vi bnh phng ca chun H2 ca h thng
3.4.2 Ti u H2
Chng 3 : iu khin bn vng
Trang 194
Vn ti u H2 c th hin di dng ma trn chuyn i. Chng ta gi s rng Q = I, v R = I, phim hm cht lng LQG l )]()()()([lim tututztzE TT
t+
(3.190)
S gi s ny khng lm mt i tnh tng qut bi v bng cch bin i thang t l cc thng s z v u ch tiu cht lng lun c th chuyn thnh hnh thc ny . Cho h thng vng h : wBuAxx ++= (3.191) Dxz = (3.192) vCxy += (3.193) C ma trn chuyn i uBAsIDwAsIDz
sGsG
)(
1
)(
1
1211
)()( += (3.194)
vuBAsICwAsICysGsG
++= )()(
1
2221
1)()( (3.195)
Kt ni h thng nh hnh (3.25) vi mt b iu khin Ce chng ta c cn bng ca tn hiu :
Hnh 3.25 : H thng hi tip vi ng vo v ng ra nhiu lon
vCGCIwGCGCIuvuGwGCyCu
sH
ee
sH
ee
ee
)(
122
)(21
122
2221
2221
)()()(
++=
++==
(3.196)
T uGwGz 1211 += ta c :
vCGCIGwGCGCIGGzsH
ee
sH
ee )(
12212
)(21
1221211
1211
)()( ++= (3.197)
eC G
y v
+ +
u w
-
z
Chng 3 : iu khin bn vng
Trang 195
Hay
=
v
w
sHsHsHsH
u
z
sH
)(2221
1211
)()()()(
(3.198)
T (3.198) theo ta c :
2
2
)(~)(21
))()(
)()((lim))()()()((lim
H
djHjHtrace
tu
tz
tu
tzEtututztzE
T
t
TT
t
=
=
=+
+
pi
(3.199)
V vy gii quyt vn LQG l cc tiu ho chun H2 ca h thng vng kn hnh (3.25) vi (w,v) nh ng vo v (z,u) nh ng ra. Cu hnh ca hnh (3.25) l trng hp c bit ca cu hnh hnh (3.26). hnh (3.26)v l ng vo m rng (w v v trong hnh (3.25)).Tn hiu z l tn hiu sai s (l tng bng 0)(z v u trong hnh (3.25)).Thm vo u l ng vo iu khin v y l ng ra quan st .G l i tng tng qut v Ce l b iu khin .
Hnh 3.26: Vn chun H2
3.4.3 Vn chun H2 v li gii ca n
G
eC
z v
y u
Chng 3 : iu khin bn vng
Trang 196
Vn ti u chun H2 l la chn b iu khin K hnh (3.26) : a. n nh vi h thng vng kn v
b. Cc tiu ho chun H2 ca h thng vng kn (vi v l vo, z l ng ra) S hnh 3.26 c m t bi h phng trnh trang thi sau:
)()()()( 21 tuBtvBtAxtx ++= (3.200) )()()()( 12111 tuDtvDtxCtz ++= (3.201) )()()()( 22212 tuDtvDtxCty ++= (3.202) Vn ti u H2 c th c gii quyt bi vic dn ti vn LQG. Gii quyt vn ti u H2 nh th l vn LQG. l , cc tiu ho : { })()( tztzE T (3.203) Gi s rng v l nhiu trng ng vo vi ma trn mt V=I.
Hi tip trng thi:
u tin, xem xt li gii vi hi tip trng thi .Kho st hai phng trnh :
)()()()( 21 tuBtvBtAxtx ++= (3.204) )()()()( 12111 tuDtvDtxCtz ++= (3.205) Nu D11 0 th ng ra z c thnh phn nhiu trng . iu ny c th lm cho trung bnh ng ra ton phng (3.203) khng xc nh. V vy chng ta gi s rng D11 = 0 . Di s gi s ny chng ta c :
[ ] [ ]
=
=+= )(
)()()()()()( 012112121 tu
tzDI
tu
txCDItuDtxCtz (3.206)
vi z0(t) = C1x(t). Do
[ ] 00 1212
12 00
12 12 12
( ){ ( ) ( )} ( ) ( ) ( )( )( ) ( ) ( )
T T TT
T TT T
I z tE z t z t E z t u t I D
D u t
I D z tE z t u t
D D D u t
=
=
(3.207)
Chng 3 : iu khin bn vng
Trang 197
y l vn b iu chnh tuyn tnh vi thnh phn cho ng ra v ng vo .N c li gii nu h thng )()(,)()()( 102 txCtztuBtAxtx =+= l n nh v tm c, ma trn trng lng
121212
12
DDDDITT (3.208)
L xc nh dng. iu kin cn v cho (3.208) l 1212 DDT khng suy bin .Li gii ca vn iu chnh l lut hi tip trng thi: )()( tKxtu = (3.209) Hi tip ng ra :
Nu trng thi l khng c gi tr cho hi tip th cn c c lng vi mt b lc Kalman. Xem xt hai phng trnh : )()()()( 21 tuBtvBtAxtx ++= (3.210) )()()()( 22212 tuDtvDtxCty ++= (3.211) Phng trnh th hai c th tr thnh dng chun cho b lc Kalman nu coi y(t) D22u(t) nh l bin quan st hn l y(t).Nu biu th nhiu s quan st l )()( 21 tvDtv = th :
)()()()( 21 tuBtvBtAxtx ++= (3.212) )()()()( 222 tvtxCtuDty += (3.213)
Xc nh mt h thng nhiu vi nhng thnh phn nhiu tng quan cho chng ta c:
[ ] [ ])(
)()()()()()(
212121
21
2121
=
+
=
+
+
T
T
TTTT
DDDDI
DItvtvD
IEtvtv
tv
tvE
(3.214)
Gi s rng h thng )()(,)()()( 21 txCtytvBtAxtx =+= l n nh v tm c , v ma trn mt
T
T
DDDDI
212121
21 (3.215)
Chng 3 : iu khin bn vng
Trang 198
xc nh dng. iu kin cn v cho(3.215) l TDD 2121 khng suy bin Khi s tn ti mt b lc Kalman : )]()()([()()()( 2222 tuDtxCtyLtuBtxAtx ++= (3.216) Ma trn li L c tm t th tc thit k b lc Kalman. Vn hi tip ng ra c ly : )()( txKtu = (3.217) K ging nh li hi tip trng thi (3.209) Xem xt vn ti u H2 cho i tng tng qut : )()()()( 21 tuBtvBtAxtx ++= (3.218) )()()( 121 tuDtxCtz += (3.219) )()()()( 22212 tuDtvDtxCty ++= (3.220) Gi s : H thng )()(,)()()( 22 txCtytuBtAxtx =+= l n nh v tm c .
Ma trn
212
1
DCBsIA
c hng y cc hng ngang cho mi
js = v D21 c hng y cc hng ngang
Ma trn
121
2
DCBsIA
c hng y cc ct cho mi js = v
D12 c hng y cc ct
Di nhng gi s ny b iu khin hi tip ng ra ti u l )]()()([)()()( 2222 tuDtxCtyLtuBtxAtx ++= (3.221) )()( txKtu = (3.222) Ma trn li hi tip trng thi v b quan st l :
1 112 12 2 12 1 2 1 21 21 21( ) ( ) , ( )( )T T T T T TK D D B X D C L YC B D D D = + = + (3.223)
Ma trn i xng X,Y l nghim xc nh dng duy nht ca phng trnh i s Riccati:
0)())((0)())((
12121
2121211211
11221
1212121211
=++++
=++++
TTTTTT
TTTTTT
BDYCDDDBYCBBAYAYCDXBDDDCXBCCXAXA
(3.224)
Chng 3 : iu khin bn vng
Trang 199
3.5 ng dng trong MABLAB
3.5.1 LQG h l xo m Xt h thng l xo m nh hnh v sau:
Vi cc thng s ca h thng nh sau:
M=1 m=0.1
b=0.0036 k=0.091
Bin trng thi ca h thng: [ ]Tx d d y y=
Phng trnh Bin Trng Thi ca h lin tc:
x Ax Buy Cx Du
= +
= +
vi ma trn Bin Trng Thi c cho nh sau:
0 1 0 0
0 0 0 1
k b k bm m m mA
k b k bM M M M
=
;
0001
B
M
=
Chng 3 : iu khin bn vng
Trang 200
1 0 0 00 0 1 00 0 0 0
C
=
; 001
D
=
Thi gian ly mu: T=0.4(s) Kho st h thng trn dng phng php LQG. S khi ca mt b iu khin LQG nh sau:
T s khi trn, ta thy rng cu trc ca b iu khin LQG chnh l b iu khin LQR kt hp vi b c lng Kalman v c xt n nhiu qu trnh w(k) v nhiu o lng v(k). Phng trnh Bin Trng Thi ca h ri rc khi c xt n nhiu nh sau:
( 1) ( ) ( ) ( )( ) ( ) ( ) ( )
x k x k u k w ky k Cx k Du k v k
+ = + +
= + +
vi lut iu khin: ( ) ( )u k Kx k=
S m phng h thng:
Chng 3 : iu khin bn vng
Trang 201
KT QU: p ng ca h thng
Chng 3 : iu khin bn vng
Trang 202
x0
x
y y0
1
2
3
0
3.5.3 Thit k H cnh tay mm do Xt thanh ng cht, khi lng phn b u, chiu di l L. Thanh c
chia thnh 3 phn t c di bng nhau 3Lh = .
Hnh 3.27 : Thanh mm do c chia thnh 3 phn t
Chiu di thanh: l = 0.98 m
Chng 3 : iu khin bn vng
Trang 203
Khi lng thanh: m = 0.35 kg
cng bin dng: EI = 72.2 N.m2
Qun tnh trc ng c: IH = 0.025 kg.m2 Dng phng php phn t hu hn v phng trnh Euler-Lagrange m
hnh ha cnh tay mm do.
nh ngha vect trng thi nh sau:
1 2 3 1 2 3
Tq q q q q q = x
trong :
: gc quay ca trc motor
dtd =
qi : chuyn v ( bin dng) ca nt i
dtdq
q ii =
M hnh biu din trng thi ca i tng(n=3) c dng nh sau:
12 13 14 11
22 23 24 21
32 33 34 31
42 43 44 41
0 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0
Tu u u u
u u u u
u u u
u u u u
= +
x x
[ ]3 0 0 1 0 0 0 0y l= x Hm truyn t ca i tng:
Chng 3 : iu khin bn vng
Trang 204
DBAsICsG += 10 )()(
0 2 2 2 2990679.4792 (s+598.2) (s-598.2) (s+167.2) (s-167.2)( )
s (s + 1.309e004) (s + 1.215e005) (s + 93.16s + 8.678e005)s =G
Thng thng ,trc khi a m hnh vo s dng ,cn phi sa i m hnh da trn biu Bode.
2 2 2 2
990679.4792 (s+598.2) (s-598.2) (s+167.2) (s-167.2)(s+1e-006) (s + 11.44s + 1.309e004) (s + 34.86s + 1.215e005) (s + 93.16s + 8.678e005)
( )s =G
Ta s dng G(s) lm m hnh danh nh v s dng th tc LSDP thit k b iu khin S m phng:
Ch thch cc khi trong s :
Chng 3 : iu khin bn vng
Trang 205
Step Khi to tn hiu t l hm nc thang n v.
Gain B tin x l, h s khuch i = - 2(0) (0)K W . W1, W2 Cc hm nn dng ch nh trong th tc LSDP
Kinf B iu khin
K t c sau bc 3 ca th tc
LSDP.
Flexible Link
Khi gi lp i tng iu khin. L ra khi ny
ch c mt u vo mt u ra, nhng phn hot
hnh (animation) cn ly trng thi ca i tng v, nn khi ny cn c cc u ra ph q
(chuyn v nt) v Theta (gc quay ca trc). Disturbance Khi to nhiu ti, pht tn hiu c dng hm nc
m.
Noise Khi to nhiu o.
Load Np d liu t file loaddata.m m phng.
Design W1 (Raw) Kch hot cng c h tr thit k s b hm nn dng W1.
Design W1 (Fine) Kch hot cng c h tr thit k cho php tinh chnh hm nn dng W1.
Plot G/W1/Gs/L/ST Khi nhp kp chut vo nhng khi ny, Matlab s v biu Bode cc hm tng ng.
Info Hin th thng tin h thng ln Workspace.
Cng c h tr thit k Design W1 (Raw)
Chng 3 : iu khin bn vng
Trang 206
Cng c ny c sa li t cng c shapemag.m ca MATLAB cho tin s dng vi phn m phng iu khin trong lun n ny. Design W1 (Raw) c giao din nh sau:
S dng: Ngi thit k ch nh cc im gy (im ch nh), cc im ny s t ng c ni vi nhau bng cc on thng to nn dng ch nh, sau in bc mong mun ca W1 vo Bc ca W1, v nhn nt Xp x MATLAB pht sinh W1. Sau khi c c W1, ngi thit k cn phi tinh chnh li hm ny bng cng c Design W1 (Fine).
Design W1 (Fine)
Chng 3 : iu khin bn vng
Trang 207
Cng c ny c xy dng da trn giao din ha ca JF Whidborne v SJ King. Design W1 (Fine) c th c s dng c lp, hoc s dng tinh chnh dng ca W1 t c sau khi s dng Design W1 (Raw). Design W1 (Fine) c giao din nh sau:
S dng: Nu Design W1 (Fine) c s dng c lp, th lc khi ng W1 = 1; nu c s dng sau Design W1 (Raw), th W1 s tha k kt qu t c t Design W1 (Raw). Ngi thit k c th thm/bt cc-zero, dch chuyn cc-zero thm/bt khu tch phn, hay thay i li ca W1 bng cc cng c bn phi giao din. Sau cng, ngi thit k nhn nt Tnh Kinf tng hp b iu khin. Hp thoi xut hin sau khi nhn nt Tnh
Chng 3 : iu khin bn vng
Trang 208
Kinf cho bit thng tin v gi tr min t c, v a ra 3 la chn cho ngi dng chn.
Nhn nt Tr v quay li giao din Design W1 (Fine) hiu chnh W1. Nhn nt p ng nc nu mun xem p ng vi u vo hm nc thang
n v ca i tng danh nh. Nhn nt M phng, kt qu thit k (Kinf, W1) s c chuyn vo Workspace chy m phng. Load Khi nhp kp chut ln khi Load, Simulink s gi loaddata.m. Tp ny cha ton b thng s thit k ca h thng. Ngi thit k c th t hm nn dng W1 vo tp ny nu mun thit k bng s
Sau bc thit k, Kinf v W1 c np vo Workspace di dng m hnh
trng thi. chy m phng, nhn nt .
Kt qu m phng:
Chn hm nn dng: 2
1 2
150.51 (s + 0.9)( )s (s + 10) ( 2)s s= +W
Kt qu thit k:
Chng 3 : iu khin bn vng
Trang 209
Gi tr nh nht: min = 3.68.
Chn min1.05 = = 3.86
B iu khin t c: 2
2 2
-131.7524 (s+10.17) (s+9.817) (s+2.146) (s + 1.103s + 0.4046)(s+35.13) (s+16.61) (s + 1.799s + 0.8091) (s + 11.61s + 96.32)
=K
p ng ca h thng:
r(t) l tn hiu t, y(t) v tr u mt, u(t) l in p iu khin, q(t) l dch chuyn ngang ca u mt.
Hnh 3.28: p ng qu ca h thng
Chng 3 : iu khin bn vng
Trang 210
CU HI N TP V BI TP
1. Khi nim iu khin bn vng 2. Chun tn hiu 3. Chun ma trn 4. nh ngha vt ma trn ,tnh cht, tr suy bin ca ma trn- li
chnh. 5. Khi nim n nh ni , n nh bn vng v nh l li nh 6. iu khin bn vng LQG (S nguyn l , b quan st,b lc
Kalman , gii thut thit k) 7. Biu Bode cho h a bin 8. Hm nhy v b nhy 9. Sai s m hnh phn tch coprime 10. Thit k bn vng
H
11. Nn dng vng
H 12. Thit k ti u H2 13. Cho h thng:
i tng G(s) c m t:
xzuxx
=
+
=
10
00
01
,
100100
000030010
V b iu khin K(s)=2I2 a. Tm li vng a bin GK(j ) b. Tm hm nhy v hm b nhy c. Tm hm truyn vng kn t r(t) n z(t) v cc cc ca vng
kn 14. Thit k LQG dng Matlab m phng m hnh con lc ngc
K r(t)
z(t)
-
G + +
+
+
+
n(t)
d(t)
u(t) s(t)
Chng 3 : iu khin bn vng
Trang 211