chuong 3_dieu khien ben vung_2

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


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


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.


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 )


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


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||


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.


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


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

= + +


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

+ +

+


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:


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:


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)


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.


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))


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


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


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


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 +


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


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


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


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)


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)


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)


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

=


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:


Trang 201

KT QU: p ng ca h thng


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


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:


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 :


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)


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)


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


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:


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


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)


Trang 211

chuong 3_dieu khien ben vung_2

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