sổ tay cdt chuong 31-dk thich nghi va phi tuyen

8/6/2019 s tay cdt Chuong 31-Dk Thich Nghi Va Phi Tuyen

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31Thit k iu khinthch nghi v phi tuyn

Maruthi R.AkellaT he University of Texas at Austin

31.1 Gii thiu ............................................................1

31.2 L thuyt Lyapunov v h bt bin vi thi gian ..........................................................................................231.3 nh l Lyapunov i vi h bin i theo thi

gian ...................................................................................331.4 L thuyt iu khin thch nghi ..........................431.5 Cc h thng iu khin thch nghi phi tuyn .. ..731.6 V d v iu chnh thch nghi t th bay cho tu

v tr ................................................................................931.7 iu khin thch nghi phn hi u ra ..............1131.8 Cc b quan st thch nghi v iu khin phn

hi u ra ........................................................................12

31.9 Nhn xt ............................................................13

31.1 Gii thiuThch thc ln nht cho l thuyt iu khin hin i l em li s thc hin c th chp nhn

c trong khi gii quyt cc m hnh khng r, phi tuyn cao, v cc cm bin gi r di mts iu kin hot ng khc nhau. Cc kh khn khng ch gp cc lp n l ca h thng vchng xut hin khng r trong cc ng dng cng nghip. Lc no cng vy, cc h thng nylun cha mt s lng ln m hnh v tham s khng xc nh (lun thay i) v vy b iukhin c nh khng th tip tc t c s n nh v cc ch tiu cht lng. Khng c ligii hp l no cho bi ton nh vy m em li s ha hp gia l thuyt iu khin phi tuyn,cc thnh phn thch nghi, v x l thng tin. Cc tha s ng sau s xut hin v pht trin calnh vc iu khin thch nghi, thc y mnh m cc ng dng thc t nh l iu khin qu trnhha hc v thit k my li t ng cho my bay, chng hot ng n nh, c chng t qua sa dng ca tc v cao.

Mt nh ngha c chp nhn rng ri i vi h thng thch nghi l: n l mt h vt lc thit k trn quan im thch nghi [1]. Tt c tnh n nh hin nay v cc kt qu hi t,

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S tay C in t

trong lnh vc l thuyt iu khin thch nghi, xoay quanh gi thit chnh l cc tham s cha bit phi xut hin tuyn tnh trong m hnh cha cc c tnh phi tuyn bit. Da trn cc khinim, trong sut qu trnh to nn tham s t c lng nh l cc bin trng thi, v vy s m rng chiu khng gian trng thi ca h thng ban u. Mt cch t nhin, li gii cho bi toniu khin thch nghi i vi c h ng hc tuyn tnh v phi tuyn u dn n cng thc bini theo thi gian mt cch phi tuyn trong s c lng ca cc tham s cha bit c cpnht s dng d liu vo ra. Mt phng php thch nghi tham s (ch yu l phi tuyn) c sdng cp nht cc tham s trong lut iu khin. Thm ch tnh phi tuyn da vo phn hithch nghi, cn chc chn rng tnh n nh ca h kn c duy tr. V vy mt thc t hin nhinl lnh vc iu khin thch nghi v tnh n nh ca h phi tuyn thc cht u lin quan n mtvn v mi kt qu nghin cu trong lnh vc ny u thun li cho vic nghin cu lnh vckia. Nhiu cng thc trong l thuyt v tnh n nh phi tuyn c th c tn dng nh l phng php trc tip Lyapunov v cc phng php da trn th ng (passivity-based methods). Chngta s trnh by cc kin thc ton hc quan trng v cc cng c phn tch trc nghin cu tnhn nh ca h ng hc phi tuyn.

31.2 L thuyt Lyapunov v h bt bin vi thi gianPhng php trc tip Lyapunov l mt trong nhng phng php ph bin nht c chp

nhn rng ri chng minh tnh n nh ca h kn trong phm vi iu khin thch nghi. N

khng b hn ch v phm vi h thng( ), (0) 0 x f x f = =& (31.1)

R rng ( ) 0 x t = l mt nghim. iu kin tn ti duy nht nghim phng trnh (31.1)l ( ) f x b chn (locally Lipschitz), tc l,

( ) ( ) f x f y L x y (31.2)

vi mi x v y trong ln cn im ban u. Chng ta cn nghim phng trnh (31.1) n nh vis c mt ca nhiu. Trc khi trnh by v nh l nh ca Lyapunov, ta trnh by mt s nhngha quan trng.

nh ngha:n nh Lyapunov

Nghim x(t) = 0 ca phng trnh (31.1) c gi ln nhtheo tin chun Lyapunov nu vimi 0 > , lun tn ti ( ) 0 > sao cho tt c cc iu kin tha mn(0) x < ta lun c(0) x < vi [0, )t . Nghim ln nh tim cnnu n l n nh ti v lun tn ti

0 > sao cho tt c cc iu kin u tha mn(0) x < ta lun c tnh cht

lim ( ) 0t

x t

=

Nghim ln nh tim cn ton ccnu n n nh tim cn ti tt c cc iu kin u. Ccnh ngha ny lin quan n s n nh ca cc nghim ring ca phng trnh (31.1) vi iukin u v khng n nh vi cc phng trnh vi phn.

nh ngha: Hm xc nh dng v hm bn xc nh dng

Mt hm lin tc V: n R R > c gi lhm xc nh dng nu (i) (0) 0V = v (ii)( ) 0V x > vi mi 0 x . Mt hm lbn xc nh dng nu iu kin (ii) c thay bng( ) 0V x vi mi 0 x . nh l:nh l v tnh n nh ca Lyapunov cho h bt bin vi thi gian

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Thit k iu khin thch nghi v phi tuyn

Nu lun tn ti mt hm xc nh dng V:n R R > sao cho o hm theo thi gian ca V

vi nghim ( ) x f x=g

c cho bi

( )T T d V V V x f x

dt x x = =

&

l bn xc nh m, khi nghim( ) 0 x t = ca phng trnh (31.1) l n nh. Trong trng hp

ny, nghim hi t trong min n{x R : ( ) 0}V x =g

. NuV l xc nh m, khi nghim s

l n nh tin cn. Hn na, nuV l xc nh m v ( )V x > khi x > , khi nghim l

n nh tim cn ton cc. Hm( )V x c gi l hm Lyapunov cho h c miu t trong phng trnh (31.1).

Nhn xt:nh l Lyapunov, mc d i t trng thi n gin, nhng c ng dng rt lntrong vic phn tch tnh n nh ca h phi tuyn. Tuy nhin, v nh l ny ch cung cp iukin di dng hm Lyapunov, nn chng ta thng gp phi vn kh trong vic i tm hmLyapunov ph hp. Trong trng hp c bit khi phng trnh (31.1) l mt h tuyn tnh nnh,

m x A x=&

mt hm Lyapunov ton phng T V x Px= tn ti khi P l mt ma trn xc nh dng i xngtha mn phng trnh c gi l phng trnh Lyapunov

T m m A P PA Q+ = (31.3)

vi Q l mt ma trn xc nh dng i xng bt k. Ngoi ra, khng cn cch tng qut no xy dng hm Lyapunov cho h phi tuyn. Ging nh quy tc ngn tay ci, trong trng hp lcc h c kh, u tin cn xc nh cc i lng kiu nng lng.

31.3 nh l Lyapunov i vi h bin i theo thi gianBy gi chng ta xt tnh n nh ca nghim phng trnh vi phn bin i theo thi gian

(nonautonomous)( , ), (0, ) 0 0 x g x t g t t = = & (31.14)

Hm g c gi thit l lin tc trn tng on i vi t v b chn Lipschitz (locally Lipschitz)vi ln cn nghim( ) 0 x t = . iu ny m bo s khi to trng thi cn bng ca phng trnh(31.4). nghin cu v tnh n nh cn bng ca h thng khng dng (nonautonomous), phinhn ra rng mi nghim ca phng trnh (31.4) khng nhng ph thuc vo thi giant m cn ph thuc vo thi gian khi to0t . V vy, chng ta cn xem li nh ngha v tnh n nh trcy.

nh ngha: Tnh n nh bt bin Lyapunov Nghim x(t)=0 ca phng trnh (31.4) ln nh bt binnu vi mi 0 > , lun tn ti

( ) 0 > khng ph thuc vo thi gian khi to t0 tc l:0 0( ) ( ) 0 x t x t vi t t < <

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S tay C in t

Nghim ln nh tim cn bt binnu n l n nh bt bin v mt hng s dng khng ph thuc vo t0 tc l ( ) 0 x t > khi t > vi mi 0( ) x t < . Nghim ln nh timcn bt bin ton ccnu n l n nh tim cn bt bin vi mi iu kin u.

nh l v tnh n nh cho h phi tuyn khng dng yu cu xc nhlp hm K xc nh. nh ngha: Lp hm K Mt hm lin tc :[0,a) [0, ) c gi l thuclp hm K nu n n iu tng v

(0) 0 = . N c gi l thuc lp hm K , hay tin ti v cng, nua = lun tn ti( )r khi r . nh l: nh l v s n nh ca Lyapunov cho h bin i vi thi gianXt min n{x R : } D x R= cn bng vi (0) 0 x = ca phng trnh (31.4). Nu lun tn

ti mt hm v hng: nV R xR R+ vi cc o hm ring lin tc tc l

i. 1 2( ) ( , ) ( ) x V x t x xc nh dng v gim

ii. 3( , ) ( )T V V V g x t x

t x

= + &

vi mi 0t , trong 1 , 2 v 3 l lp hm K, khi im cn bng0 x= l n nh timcn bt bin.

Nhn xt : Ch rng trnh by tnh n nh ca h ph tuyn khng dng, cn gii hn hm( , )V x t bi cc lp hm K khng ph thuc vo thi gian t. Mt cch gii quyt chi tit ca tt c

cc nh ngha v chng minh ca nh l ny bn c c th xem trong Slotine v Li [2] v Khalil[3].

Nhn xt:Trong nhng nm gn y, nh l o Lyapunov t c mt s kt qu tt. Cth l tt c cc h thng n nh bt bin (hoc n nh tim cn bt bin), lun tn ti mt hmLyapunov xc nh dng c o hm theo thi gian l bn xc nh m (xem Sastry v Bodson[4]). Cc kt qu ny rt hu ch cho cht lng ca h kn bi v n cho php c lng mt cchr rng cc tc hi t cho nhiu trng hp ca h iu khin thch nghi phi tuyn.ng dng ca nh l n nh ca Lyapunov cho h khng dng vt ra khi iu khin thch

nghi, n thng dn n o hm bn xc nh m ca hm Lyapunov. V vy, vic phn tch tnhn nh tim cn l mt bi ton kh hn nhiu v kt qu sau y, c coi nh l b Barbalat,rt hu ch cho trng hp ny.

B :BarbalatXt mt hm lin tc bt bin: R R tp xc nh l tt c cc gi tr thc0t . Nu

0lim ( )

t

t s ds

tn ti v c gii hn, th( ) 0t khi t .

Nhn xt:Mt h qu ca b ny l nu2 L v L , th ( ) 0t khi t (XemSlotine v Li [2] v Tao [3] xem trnh by v chng minh).

31.4 L thuyt iu khin thch nghiKhc vi mt b iu khin c nh (fixed) hay b iu khin truyn thng, mt b iu khin

thch nghi l mt b iu khin c cc tham s c th chnh c v mt c ch iu chnh. Sauy l cc khi nim c bn cn cho mi trnh by v l thuyt iu khin thch nghi.

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Bi ton iu chnh v bi ton iu khin bmMc ch ca mi bi ton iu khin l duy tr gi tr u ra lun gi mt gi tr mong mun

hoc nm trong gii xc nh (c th chp nhn c) ca gi tr mong mun. Nu cc gi trmong mun ny l hng s vi thi gian, th chng ta c bi ton iu khin iu chnh, cn li l bi ton iu khin bm.

Nguyn l tng ng chc chn (Certainty Equivalence Principle) Nguyn tc ny l nn tng ca hu ht cc phng php thit k iu khin thch nghi v

nhn c v tr to ln trong hai thp k qua [4, 6, 7]. B iu khin thch nghi da trn phng php ny c c bng cch thit k mt lut iu khin c lp t c mc ch iu khingi thit l bit tt c cc tham s ca m hnh cha bit (trng hp tin nh), cng vi mtlut cp nht tham s, thng l mt phng trnh vi phn to ra cc c lng tham s trctuyn s dng thay th cc tham s cha bit trong lut iu khin. Nh l mt b iu khin hon thin vi kh nng bm u ra trong trng hp cc tham s m hnh bit chnh xc. Hin ticha xc nh c cc tham s, c cu chnh nh s chnh cc tham s b iu khin sao chomc nh bm c thit k mt cch tim cn. Vn ch yu trong thit k b iu khin thchnghi l to ra c cu chnh nh (lut cp nht tham s) n s m bo rng h thng iu khinvn n nh v sai lch bm u ra hi t v 0 khi cc gi tr tham s c cp nht.

iu khin thch nghi trc tip v thch nghi gin tipC hai phng php khc nhau trong iu khin thch nghi cho cc m hnh c cha cc tham

s cha bit hay cha xc nh. Phng php th nht c gi l phng php trc tip, cctham s ca b iu khin c chnh trc tip bi c cu thch nghi bng cch ti u cc ch tiucht lng xc nh c trc da trn u ra. Phng php th 2 l phng php gin tip,cc tham s m hnh c c lng trc tip v c cp nht bng lut thch nghi v cc gi trc c lng ny th c s dng tnh ton cc tham s b iu khin. iu khin thchnghi trc tip khng cn thm s tnh ton ny. Do , iu khin thch nghi gin tip l thch nghitham s m hnh, ngc li iu khin thch nghi trc tip l thch nghi kt qu u ra. Qu trnhnhn dng tham s m hnh trong phng php gin tip l hin (explicit) trong khi trong phng php trc tip l ngm (khng r rng). V vy, chng cn c gi l phng php ngmv phng php hin. Trong c hai trng hp ny th cu trc b iu khin vn ging nhau vc xc nh t nguyn tc cn bng iu ho.

iu khin thch nghi c m hnh tham chiuCu trc bi ton iu khin MRAC (Model Reference Adaptive Control) gm 4 phn: (i) m

hnh cha cc tham s cha bit, (ii) m hnh theo di ph hp xc nh cc c tnh u ramong mun, (iii) mt lut iu khin phn hi cha cc tham s c th iu chnh, v (iv) mt c cu thch nghi cp nht cc tham s c th chnh vi lut iu khin trn. Nh hnh 31.1.

M hnh i tng c gi thit l c cu trc bit vi cc tham s cha bin. Trongtrng hp cc h tuyn tnh, ngha l s im cc v im khng gi thit l bit, nhng v trchnh xc ca cc im cc v im khng th cha bit. i vi h phi tuyn, cu trc ca cc phng trnh nh hng n chuyn ng c gi thit l bit, nhng mt vi tham s xuthin tuyn tnh trong cc phng trnh ny th cha bit. M hnh theo di xc nh u ra ltng nhn c t m hnh l mt kt qu ca u vo tham chiu ngoi. N cung cp p ngca m hnh l tng cho c cu thch nghi tm bm theo trong khi cp nht tham s c lng.

Chn m hnh theo di l trng tm ca thit k MRAC v mi s la chn c th chp nhnv c bn phi tho mn hai yu cu. Yu cu th nht l m hnh theo di phi phn nh chnhxc cc ch tiu cht lng ca h kn. nh l rise time, setting time, qu iu chnh (overshoot)

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S tay C in t

v cc ch tiu cht lng khc. Yu cu th hai c gi thit l cu trc ng hc m hnh, hotng u ra m hnh tham chiu nn c thit k tim cn bi h thng iu khin thch nghi dnn phi cn thm cc iu kin v quan h ca m hnh tham chiu v cc iu kin kch thchlin tc trn u vo tham chiu. Cu trc b iu khin c ra lnh bng nguyn l tng ngchc chn v c 2 th tc cp nht tham s trc tip v gin tip u c th c chn vi m hnhiu khin MRAC. Phng php ny gii quyt c rt nhiu bi ton vi h lin tc.

Hnh 31.1Cu trc iu khin thich nghi c m hnh tham chiu

Hnh 31.2Cu trc ca b iu khin t chnh

B iu khin t chnh (STC)Tri vi iu khin MRAC, khng c m hnh trong thit k b iu khin t chnh. Biu

ca thit k STC nh hnh 31.2. Trong cng thc ny, cc thng s ca tham s h thng c clng trong thi gian thc, ph thuc vo n l phng php trc tip hay gin tip. Cc clng ny v th m c s dng nh l chng bng cc tham s thc (thit k tng ngchc chn). Vic c lng tham s bao hm vic tm ra b tham s ph hp nht da trn d liuvo/ra ca h thng. iu ny th khc vi kiu thch nghi tham s MRAC c lng tham sc cp nht bng cch thit k bm tim cn sai lnh gia h thng tht v m hnh theo di.

Trong nhiu trng hp phng php c lng STC, cng c th nh gi tiu chun chtlng ca vic c lng tham s, n c s dng trong thit k b iu khin. Nhiu s kt hp

khc nhau ca cc phng php c lng c chp nhn v cung cp cho c h lin tc v hri rc. Do c s tch bit gia c lng tham s v iu khin trong STC, nn linh hot hntrong thit k. Tuy nhin, tnh n nh v s hi t th kh chng minh v cc iu kin kht khehn v yu cu ca cc tn hiu u vo (kch thch lin tc) m bo hi t tham s. Nh ni trn, thit k STC xut hin trong vic nghin cu bi ton iu chnh ngu nhin vnhiu l thuyt dnh cho h ri rc s dng phng php gin tip. Mc d dng nh c s khc

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nhau gia MRAC v STC, nhng mt s tng ng trc tip tn ti gia cc bi ton t 2 mintrn.

31.5 Cc h thng iu khin thch nghi phi tuyni vi hu ht cc trng hp tng qut ca h phi tuyn, c rt nhiu l thuyt b gii hn

bi lnh vc iu khin thch nghi. Thm ch lnh vc ny rt c quan tm nh vo cc ng

dng tim nng trong s a dng ca cc h c phc tp. Nhiu kh khn trong nghin cu lthuyt tn ti bi v thiu cc cng c phn tch tng qut. Tuy nhin, cc trng hp c bitquan trng s c trnh by y, v chng ta tng kt cc iu kin m cc lp h thng thamn:

1. Cc tham s cha bit trong h phi tuyn c tham s ho mt cch tuyn tnh2. Vec t trng thi hon ton o c3. Khi cc tham s cha bit c gi thit l bit, u vo iu khin c th b qua tt

c cc tnh cht phi tuyn theo hng tuyn tnh ha phn hi v cc thnh phn nghc bn trong cn li phi n nh. Sau thit k thch nghi thc hin c bngnguyn l tng ng tuyt i.

By gi chng ta gii thiu mt phng php MRAC in hnh gii quyt trng hp mhnh h phi tuyn khng bit cc tham s bn trong n. Xt h phi tuyn

( ) x f x u = +& (31.5)

trong l mt tham s ma trn hng cha bit, v f l mt hm vec t phi tuyn c o hm v bit. Tng t vi phng php MRAC, chng ta gi s rng n l l tng c trng thi x bm tim cn vi trng thi xm ca h tham chiu tha mn

m m m x A x r = +& (31.6)

trong r(t) l u vo tham chiu c gii hn v lin tc trn tng on v Am l ma trnHurwitz. a vo mt vec t sai s me x x= v vy ng hc ca sai lch c th c thit lp bng cch tr hai phng trnh (31.5) v (31.6) nh sau:

( ) m me f x A x r u = +& (31.7)

Nu tham s c gi thit l bit, chn u vo iu khin ( )mu A x r f x = + lm chong hc ca sai lch c cu trc sau:

me A e=&

l ra thit k c mc tiu iu khin. Tuy nhin, khng th chn c lut iu khin v cha bit. Do chng ta vn gi li cu trc ny tm lut iu khin, thay bng c lng bin i vi thi gian do vy lut iu khin thch nghi da trn nguyn l tng ng tuyt ith cho bi cng thc

( )mu A x r f x = + (31.8)

Ghp vo lut iu khin trong phng trnh (31.7) cho ta ng hc ca h kn nh sau:

( )me A e f x = %& (31.9)

trong chng ta a ra bin~ ( )t miu t sai lch c lng tham s~

( )t . C 2 vic

cn li phi lm l: (i) ch r tnh n nh v hi t tim cn ca( )e t tin ti 0 khit , (ii)

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S tay C in t

a ra mt c cu chnh nh tham s cho~ ( )t . Chng ta hon thnh c hai vic trn bng cchchn phng php Lyapunov. Gi thitm A l Hurwitz, i vi mi cch chn ma trn xc nhdng v i xng Q, lun tn ti mt ma trn xc nh dng, i xng P tha mn phng trnhLyapunov cho bi (31.3). Chn hm Lyapunov di dng mt ma trn P,

1[ ]T T V e Pe tr = + % % (31.10)

trong l ma trn t l hc xc nh dng i xng. o hm V theo thi gian v kt hpvi nghim ca phng trnh (31.9) ta c1( ) 2 ( ) 2 [ ]T T T T m mV e PA A P e e P f x tr

= + + &% % %& (31.11)

S dng cc php ng nht vt ma trn, [9] ta c biu thc sau

( ) [ ( ) ] [ ( ) ] [ ( )]T T T T T e P f x tr P f x e tr f x e P tr Pef x = = =% % % %

v vy chng ta c th kt hp 2 biu thc bn v phi ca phng trnh (31.11) nh sau:1( 2 [ { ( )}]T T T m m

Q

V e PA A P e tr Pef x

= + + &% %&1 4 2 4 3 (31.12)

Vi l hng s, 0

= v ^ ~

= . V vy, nu lut thch nghi cp nht~

c chn l:

( )T Pef x = & (31.13)

th o hm ca hm Lyapunov trong phng trnh (31.12) tr thnhT V e Qe= & (31.14)

n l hm bn xc nh m, nhng n khng xc nh m. iu ny dn n( ) (0)V t V vi mi0t v v the v ^ phi c gii hn. iu ny dn n m x e x= + cng c gii hn.

Ngoi ra ( ) 0V t v (0) 0V , ngha llim ( )t V t V = tn ti v c gii hn.

0 0( ) ( ) ( ) (0)T V d e Qe d V V

= = &

dn n 2 1e L L . T phng trnh (31.9), hin nhin rnge L

. V vy, chng ta c th dng b Barbalat chng minh( ) 0e t khi t . Lu rng, mc d, sai lch tham s

~ ^

= khng cn thit phi hi t v 0. Nhng vic hi t ca tham s tht c th ch xut hinkhi u vo tham chiu r(t) tha mn tnh quan st bt bin v cc iu kin kch thch lin tc [4].

Nhn xt:Ch trong thit k MRAC trn trong khi s n nh v s hi t ca sai lch bmc m bo cho mi gi tr cam A , Q v , cht lng ca b iu khin s ph thuc rtnhiu vo t l hc (learining rate) . T l hc nh hn ngha l s thch nghi s chm hn dnn sai lnh bm ln v s tm thi ln (transients). Ngc li, gii hn trn ca t l hc cgii hn bi s c mt ca ng hc phi m hnh. Bi v mt gi tr qu ln cho t l hc s dnti c lng tham s dao ng cao, n c th kch thch bt li cho ng hc i tng phi mhnh tn s cao.

Nhn xt:Phng php thit k b iu khin c da trn 3 bc ch yu trn: (i) tm cutrc b iu khin thch hp vi t tng (spirit) tuyn tnh ha phn hi, (ii) o hm ng hc

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sai lch bm thnh phn ph thuc vo phm vi sai lch tham s, v (iii) tm mt hm Lyapunov ph hp, s dng nhn c lut cp nhp tham s sao cho sai lch bm s v 0. Trong trnghp xc nh cu trc b iu khin tham s bit th bc ch yu nht nm trong thit k thchnghi no bi v phn ln quan im cho rng tuyn tnh ha phn hi thich nghi khng th lunlun ng dng c vo cc h thng tuyn tnh ha phn hi trong trng hp tham s bit.iu ny xy ra bi v cc o hm bc cao ca c lng tham s xut hin trong lut iu khinvi h thng bc cao hn, lm kh ng dng nguyn tc tng ng tuyt i. Cc phng phpmi khc khng ging vi nguyn tc tng ng tuyt i nh cun chiu (backstepping) tch phn, suy gim phi tuyn, v cc hm chnh [10]. Trong cc phng php ny, cc lut thch nghic lng cc tham s i tng cha bit mt cch trc tip, do cho php tn dng ton bkin thc trc v v vy loi tr kh nng trn tham s ha gii thiu trong phng phpMRAC trc tip truyn thng. Phng php thit k v kim chng n inh t c thng quaqu trnh quy [10, 11], Tng quan v vn ny c th thy t Kokotovic [12] v cc kt quca nhm nghin cu ca ng sau ny.

Nhn xt:Vi thc t l lun tn ti cc sai lch m hnh v cc hiu ng ca nhiu cha bit,cng vi cc tham s cha bit, cc gii php iu khin thch nghi phi hng ti bi ton bnvng. Sai lch tham s lun khng bit, o hm theo thi gian hm Lyapunov lun bn xc nhm. iu ny dn n phng trnh h kn khng n inh theo hm m, thm ch cng khng nnh tim cn bt bin. Nhiu phi m hnh hoc nhiu ngoi lmV

bt nh ngay lp tc, v hu

ht cc phng php bin i chng minh bn vng n nh c c nh m boV

l mngoi khong ln cn ca trng thi cn bng. Bng vic a vo thm mt s hng trong lut iukhin thch nghi trong phng trnh (31.13), (xem nh l b bin i ), Ioannou [7] thc hintnh n nh bn vng. Phng php ny, mc d rt ph bin, nhng c nhiu mt hn ch khi ccnhiu b loi b, sai lnh bm khng th hi t v 0. vt qua vn ny, cc hng khcnh thay i [6] c xut bo m bn vng trong thit k thch nghi.

31.6 V d v iu chnh thch nghi t th bay cho tu v trXt bi ton ca tu v tr vi mt t th ban u khc khng v vec t vn tc gc ca vt

c a v vec t v tr 0. Bi ton iu chnh t th thch nghi i tng cng ny da trn phng php tuyn tnh ha phn hi c gii quyt bi Schaub, Akella, v Junkins [13]. Cc phng trnh ch yu c miu t bi phng trnh quay Euler ca s chuyn ng v ng hch kn tuyn tnh mong mun (LCLD linear closedloop dynamics) c th hoc nh l dng PDhoc dng PID[13, 14]. Gi thit ch c c lng th m men ca ma trn qun tnh c l bit. Lut iu khin thch nghi c trnh by bao gm mt s hng phn hi nguyn trong nghc ca h kn mong mun v thit k n nh tim cn ngay khi c nhiu ngoi phi m hnh.

Kt qu m phng c minh ha nh hnh 31.3. vec t t th (attitude vector) c xc nhdi dng tham s Rodrigues bin i (MRP modified Rodrigues parameter) m thnh phn can i nh hnh 31.3(a). Cha c b chnh nh, h h vn n nh tim cn. Tuy nhin, sai lch tth tc thi khng tt bng LCLD mong mun. Khi c c cu chnh nh, cht lng rt gn viLCLD l tng.

9


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S tay C in t

(a) vec t t th MRP (b) Bin vec t t th

(c) vec t iu khin u (Nm) (d) c lng m men ngoi thch nghi (Nm)

(e) Sai lch r (f) Sai lch t g g

Hnh 31.3S n nh ca vt cng khi thc hin LCLD trong iu kin qun tnh ln v khng bit nhiu bn ngoi

Hnh 31.3 (b) minh ha bin ca vec t sai lch t th MRP trn th logarit. Sai s tcthi ln ca h h, lut iu khin thch nghi t do c hin th trong vng 20s u ca qu trnh

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cng vi cc c tnh hi t cui cng tt. S biu din ng hc h kn tuyn tnh li c ch ra bng ng chm chm. Hai phin bn ca lut iu khin thch nghi c so snh y, chngch khc l c v khng c nhiu ngoi cng c c lng thch nghi. Trong hnh ny c 2 lutthch nghi xut hin lm cho ng hc h kn tuyn tnh rt tt trong 40s giy u ca qu trnh.Sau lut thch nghi khng hc nhiu bt u suy yu mt t l chm hn, thm ch hn c gii php h h (khng thch nghi). Mc d c nhiu ngoi, nhng s thch nghi ci thin ng k t lhi t cui cng. Lu , mc d, cng khng phi trng hp thch nghi bt u nhn c ttrng hp ng hc h kn tuyn tnh l tng nhng bin sai lch t th MRP b suy gimxung xp x 10-3. iu ny tng ng vi sai lch gc quay ch yu khong xp x 0.230. Vi sthch nghi nhiu ngoi, sai lch bm so vi LCLD xut hin gn vi bc 2 ca bin nh hn.

Hiu sut ca lut iu khin thch nghi c th b thay i ln bng vic chn t l hc khcnhau. Tuy nhin, t khi ma trn qun tnh khi to ln v sai lch m hnh nhiu ngoi xut hin,t l hc b gim trnh cc m men chuyn tip ln. Cc thnh phn vec t m men iukhin iu cho vi trng hp th nh hnh 31.3(c). Cc m men h h khng n gn m menLCLD l tng trong khong thi gian tc thi. Cc m men c yu cu bi trng hp thchnghi khc th rt ging nhau. S khc nhau ch l trng hp hc nhiu ngoi gy ra vi daong ca iu khin trng hp ng hc h kn tuyn tnh. Tuy nhin, ch rng vi t l hcthch nghi c chn th lut iu khin hoc chng t cc m mn t tr tc thi gn vi m menLCLD l tng. Hnh 31.3(d) minh ha c lng nhiu ngoi thch nghi e F tht vy tim cnnhiu ngoi sinh thc bng cch gim t l hc thch nghi nhiu ngoi v gi cc sai lch clng thch nghi tc thi trong di hp l.

Hnh 31.3(f) minh ha sai lch thc hin tuyt i (absolute performance error). C 2 trnghp c c cu chnh nh v khng c c cu chnh nh. Ta thy trong trng hp c c cu thchnghi sai lch gim ng k so vi trng hp khng c c cu thch nghi.

Mc ch ca iu khin thch nghi trnh by trong v d ny l tin ti LCLD mongmun. Cc hnh trn minh ha rng kt qu ton b phn cn li ca h thng n nh tim cn.Hnh 31.3 (e) minh ha sai lch hiu sut tuyt i gia chuyn ng tht( )t v chuyn ngtham chiu tuyn tnh l tng( )r t . Hnh ny minh ha li sai lch hiu sut ln kt qu ny tvic s dng lut iu khin h h vi m hnh h thng cha chnh xc. Thm c cu thch nghici thin vic thc thi tc thi bm m bc 2 ca bin . Khng c vic hc nhiu ngoi, t lsuy gim sai lch thc hin cui cng tri phng. Sai lch ny s suy gim v 0. Tuy nhin, vi cc

h s hc cho n lm t l ny chm hn c khi khng dng c cu thch nghi. Thm vic hcnhiu ngoi ci thin ng k s suy gim sai lch thc hin cui cng t khi h thng t cmt m hnh chnh xc ca nhiu hng thc. Nu c lng m hnh ban u cng chnh xc, t lhc thch nghi cng cao, kt qu thm ch cn tt hn hiu sut theo ri ca LCLD. Vi s m phng ny n c th bm LCLD mong mun rt tt ngay c vi h thng ln c m hnh khng r.

31.7 iu khin thch nghi phn hi u raTri vi cc phng php khng gian trng thi, phng php vo-ra coi i tng nh l mt

hp en bin i cc u vo c cung cp sang khng gian u ra tng ng. L thuyt n nhcho h phi tuyn nhn t gc vo-ra th rt quan trng trong trng hp thit k iu khin phnhi u ra thch nghi. Gii php cho bi ton thit k b quan st thch nghi cn phi c lngtrng thi ca h thng vi cc tham s cha bit ang t c tin b theo cch gii quyt biton iu khin phn hi u ra. Gn y c nhng s tin b trong gii php ny th tcthit k b quan st thch nghi phi tuyn c m rng hn vi trng hp tng qut cc h skhuych i ca cc tham s cha bit ph thuc vo ton b trng thi, v khng hiu chnh ctrn phn o [15].

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S tay C in t

Trong phm vi ln, cc kt qu hu hiu c lm bng cch kt hp cc iu kin nh l b ng (passivitylike) vi cc iu kin kch thch lin tc thng thng. Ni dng ch yu trongmc ny l khi nim ca tnh b ng, n gn nh l mt trnh by vn tt v tng tiu thnng lng trong c h thng tuyn tnh v phi tuyn. H b ng c hu ht trong cc ng dngc kh v k thut in. Mt h thng c kh gm khi lng, tnh n hi v gim chn nht lmt v d in hnh cho h b ng. Sau y l cc nh ngha.

nh ngha: Cht tn hiu

Cho Y l min xc nh ca hm gi tr thc c nh ngha trong khong[0, ) . x l mt phn t ca Y. Khi hm cht ca x ti T>0 c nh ngha

( ) 0( )

0T x t t T

x t t T

=

>

nh ngha: Khng gian m rng Nu X l mt tp con tuyn tnh ca Y, thkhng gian m rng e X c nh ngha bi tp

{ : 0}T x Y x X T vi vi gi tr T

Tp m rng2 L c bao hm bi2e L

nh ngha: Tch v hng ca 2 tn hiuTnh v hng ca 2 tn hiu bin i theo thi gian2, e x y L c nh ngha l

0 0( ) ( ) ( ) ( )

T T T x y x y d x y d

= = nh ngha : H thng th ngMt h thng vi u vo( )u t v u ra ( ) y t l th ng nu

0 y u

Hl th ng tuyt i u vonu 0 > tha mn2 y u u

H c gi lth ng tuyt i u ranu 0 > tha mn2 y u y

31.8 Cc b quan st thch nghi v iu khin phn hi u raBy gi chng ta trnh by bi ton quan st thch nghi phi tuyn theo cng thc ca Besancon

[15]:( , , ) ( , , )( )

x f x u t g x u t y h x

= +=

&(31.15)

trong hm f v g lC i vi tt cc tham s ca n v l mt tham s hng cha bit. Cc bin x, u v y theo th t biu th vec t trng thi, vec t u vo, vec t u ra. Cc tn hiu uvo c th c gi s ph thuc vo mt s hm o c v b gii hn. Bng mt b quan stthch nghi, chng ta a n bi ton khi phc c lng trng thi^ ( ) x t s dng u vo u v

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u ra y vi s c mt ca tham s cha bit v vy^

lim ( ) ( ) 0t x t x t = . iu kin tn ti

mt b quan st th sn c, [15] n c th c pht biu nh sau. Nu tn ti mt b quan st phhp trong trng hp m bit, v nu trong trng hp xc nh b quan st nh l sai lchca tham s c to ra v h thng sai lch c lng trng thi l b ng gia u vo v sailch u ra ^( )h x y , th c th thit k b quan st trng thi tim cn thn ch cha bit . Thm

na, s i hi th ng, hi t sai lch tham s ny, nh bnh thng, cn kch hot lin tc hni vi u.Kt qu tuyt vi ny c cc ng dng ngay lp tc trong bi ton theo di nh v tu khng

gian m khng c gi tr o vn tc gc [16]. Hin nay ni ting v phng trnh ca bi ton iukhin nh v vt rn di dng iu kin th ng tha mn vec t MRP [17, 18] gia vec t vntc gc v vec t MRP. Mt kt qu rt th ng quan trng trong trng hp ny l trong thc tlut iu khin phn hi cho iu khin nh v c thc hin bng cch xy dng da trnLyapunov khng cn o vn tc gc. Kt qu lut iu khin cung cp hu ht s n nh tim cnton cc trong hng i ca Tsiontras [18].

31.9 Nhn xt Nh ni, vic pht trin v ng dng ca l thuyt iu khin thch nghi hin i cho h phi

tuyn tng qut thng qua hng tip cn trit hc ca cc phng php lun h tuyn tnhang tn ti rng ri. Trong nhiu trng hp b gii hn nh l l thuyt t chnh, cch tip cnca cc phng php tuyn tnh song song ny thnh cng ln. Tuy nhin, t c thnhcng nh trong cc lnh vc nghin cu khc nh l bm qu o, tng hp b iu khin, v khi phc trng thi th rt kh.

Cc h thng phi tuyn xut hin trong s bin i theo cc cch a dng, v khng phi tt cchng u c th iu khin c bng cch m rng n gin ti phng php iu khin thchnghi tuyn tnh. Mt phng php ha hn cho mc ch nghin cu xa hn l chun ha vicnghin cu cc h thng c kh, do gii hn lp cc h phi tuyn nghin cu, v v vy cth gii thiu cu trc v cc iu kin rng buc thm. Ngc li trong trng hp iu khin phn hi u ra cho h phi tuyn tng qut, cc thit k ring ca b quan st trng thi v b iukhin khng cn m bo tnh n nh nh trong thit k tch hp. (khng theo nguyn l tch),

cc phng php c cu trc s dng bin i trng thi c a ra gip ly li cc tnh chtring bit trong nhiu trng hp [15]. Vi kt qu ny, cc phng php c cu trc ni trncng c th tnh b iu khin bm ton cc v bn ton cc da trn phn hi u ra (trngthi tng phn). Theo cch tip cn ny rt c th c tim nng cung cp cha kha giquyt vi bi ton khc xut hin khng thuc cc h c in cc m chng ta cn cha bit.

Ti liu tham kho[1] Narendra, K. S., Parameter adaptive controlThe End or The Beginning?,

Proceedings of the 33rd Conference on Decision and Control . Lake BuenaVista, Florida, December 1994.

[2] Slotine, J. E. and Li, W., Applied Nonlinear Control . Prentice-Hall, EnglewoodCliffs, NJ, 1991.

[3] Khalil, H. K.,Nonlinear Systems . Macmillan, New York, NY, 1992.[4] Sastry, S. and Bodson, M., Adaptive Control: Stability, Convergence and

Robustness . Prentice-Hall, 1989.[5] Tao, G., A simple alternative proof to the Barbalat Lemma,IEEE Transactions on

Automatic Control, Vol. 42, No. 5, May 1997, p. 698.13


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S tay C in t

[6] Narendra, K. S. and Annaswamy, A. M.,Stable Adaptive Systems . Prentice-Hall,1989.

[7] Ioannou, P. A. and Sun, J.,Stable and Robust Adaptive Control . Prentice-Hall,Upper Saddle River, NJ, 1995, pp. 85134.

[8] Astrom, K. J. and Wittenmark, B., Adaptive Control . Addison-Wesley, Reading, MA,1995.

[9] Gantmacher.The Theory of Matrices, Vol I. Chelsea Publishing Company, NY, 1977, pp. 353354.

[10] Krsti , M., Kanellakopoulos, I., and Kokotovi , P. V., Transient performance improvementwith a new class of adaptive controllers,Systems & Control Letters, Vol. 21, 1993, pp. 451461.

[11] Krti , M., Kanellakopoulos, I., and Kokotovi , P. V., Nonlinear design of adaptivecontrollers for linear systems,IEEE Transactions on Automatic Control, Vol. 39,1994, pp. 738752.

[12] Kokotovic, P. V., The joy of feedback: nonlinear and adaptive control,IEEE ControlSystems Magazine, Vol. 12, No. 3, 1992, pp. 717.

[13] Schaub, H., Akella, M. R., and Junkins, J. L., Adaptive control of nonlinear attitudemotions realizing linear closed loop dynamics, Journal of Guidance, Control and

Dynamics, Vol. 24, No. 1, Jan.Feb. 2001.[14] Akella, M. R., Schaub, H., and Junkins, J. L., Adaptive realization of linear closed looptracking dynamics in the presence of large system model errors, Journal of

Astronautical Sciences, Vol. 48, No. 4, 2000.[15] Besanon, G., Global output feedback tracking control for a class of Lagrangian systems,

Automatica, Vol. 36, 2000, pp. 19151921.[16] Akella, M. R., Rigid body attitude tracking without angular velocity feedback,Systems

& Control Letters, Vol. 42, No. 4, 2001.[17] Lizarralde, F. and Wen, J. T., Attitude control without angular velocity measurement: a

passivity approach,IEEE Transactions on Automatic Control, Vol. 41, No. 3,1996, pp. 468472.

[18] Tsiotras, P., Further passivity results for the attitude control problem,IEEETransactions on Automatic Control, Vol. 43, No. 11, 1998, pp. 15971600.

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