sổ tay cdt chuong 23- signal v system

8/6/2019 s tay cdt Chuong 23- Signal v System

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23Tn hiu v h thng

Momoh-Jimoh Eyiomika SalamiInternational Islamic University of Malaysia

Rolf JohanssonLund Institute of Technology

Kam LeangUniversity of Washington

Qingze ZouUniversity of Washington

Santosh DevasiaUniversity of Washington

C. Nelson DornyUniversity of Pennsylvania

23.1 Tn hiu thi gian lin tc v ri rc ..................... .......1

23.2 Bin i z v cc h thng s .................................... .23

23.3 Cc m hnh khng gian - trng thi thi gian ri rc v lin tc .......31

23.4 Hm truyn v bin i Laplace ................ ........ ........41

23.1 Tn hiu thi gian lin tc v ri rc

Cc tn hiu l cc bin vt l hoc cc i lng, o c cc vng khc nhau ca mt h, biu din cc thng tin mong

mun. Trong thc t, c rt nhiu loi tn hiu. Tn hiu in dng dng hoc p l i lng o c d dng nht, v vy cndng cc u o v cc b chuyn i bin i cc i lng khng phi l tn hiu in thnh cc tn hiu in. Cc tn hiuny cn phi c x l bng cc phng php xp x thu c cc kt qu mong mun. Mt vi phng php biu din tnhiu ph hp vi vic x l tn hiu c min thi gian v tn s s c m t trong mc di.

Phn loi tn hiu [1-4]

Cc tn hiu c phn loi thnh cc tn hiu thi gian lin tc (CT) hoc cc tn hiu thi gian ri rc (DT), mi loi ny lic th phn thnh cc tn hiu xc nh hoc ngu nhin. Mt tn hiu xc nh lun c th biu din c dng ton hc, trongkhi , thi im xy ra hoc gi tr ca mt tn hiu ngu nhin khng th on trc c. Mt tn hiu thi gian lin tc, x(t),c mt gi tr ng vi mi gi tr thi gian t, trong khi mt tn hiu thi gian ri rc x(n), s ch c gi tr cc im ri rc, lcc gi tr nguyn n.

1


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S tay C in t

HNH 23.1 Mt vi lp tn hiu: (a) Tn hiu tng t thi gian lin tc, (b) D liu c trch mu, (c) Tn hiu s, (d) Tnhiu ngu nhin

HNH 23.2 Cc tn hiu c chu k: (a) Thi gian lin tc (CT), (b) Thi gian ri rc (DT)

Gn vi cc tn hiu CT v DT l cc tn hiu tng t v s tng ng. Nu bin ca mt tn hiu c th nhn c gitr bt k trong mt di lin tc th n l mt tn hiu tng t. Mt khc, bin ca mt tn hiu s ch c th c mt s gii

hn cc gi tr cc im ri rc. Cc v d ca cc tn hiu thi gian lin tc, thi gian ri rc s v ngu nhin c ch ra trnhnh 23.1. Cc tn hiu xc nh c phn thnh hai loi chnh, l cc tn hiu c chu k v cc tn hiu khng c chu k. Mttn hiu c chu k s c cng gi tr cc thi im cch nhau mt chu k T, tc l x(t) tha mn mi quan h x(t)=x(t+T),-


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Tn hiu v h thngvi iu kin: (t) lin tc t=0. Cng thc ny ch ra tnh cht thay i ca hm delta. Biu ca hm delta c ch ratrong hnh 23.3(a). Bng 23.1ch ra cc thuc tnh iu khin ca hm delta.

BNG 23.1 Cc tnh cht ca hm delta

Tnh cht Biu din ton

Trch mu ( ) ( ) ( )x t t a dt x a+

=

Dch ( ) ( ) ( ) ( )x t t a x a t a =

Cn chnh1

( )b

at b t a a

=

Tch chp ( ) * ( ) ( ) ( ) ( )x t t a x t a d x t a +

= =

HNH 23.3 Cc hm c bit: (a) xung, (b) nhy, (c) dc, (d) parabol

Hm nhy n v (Unit Step)

Hm nhy n v thng dng phn tch ton hc cc tn hiu CT. iu ny c m t trong hnh 23.3(b), v cnh ngha nh:

1 0( )

0 0

tu t

t

>=


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S tay C in t

iu ny c m t trong hnh 23.3(d).

Cc tn hiu thi gian lin tc c bn

Hnh 23.4 ch ra mt vi tn hiu c bn thng gp trong phn tch tn hiu. Mt vi trong s tn hiu c th nhn ctrc tip t cc hm c bit c m t trn. V d, tn hiu xung ch nht n v t -/2 n /2 c th c biu dinnh sau:

( )2 2

t u t u t = +

(23.7)

v iu ny c m t trong hnh 23.4(a).

HNH 23.4Biu din cc hm CT c bn: (a) xung ch nht, (b) xung tam gic, (c) hm sin cHm tam gic k hiu l (t), c nh ngha nh sau:

1 , 1( )

0, cn li

t tt

t


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Tn hiu v h thng( ) ( ) ( ) ( )x n n m x m n m =

( )b

an b na

=

( )* ( ) ( ) ( ) ( )n

x n n m x r n r m x n m

=

= =

Ch rng tnh cht scaling ch c p dng khi c a v b/a l cc s nguyn. Hai tn hiu c bn khc cng thng c

dng phn tch l tn hiu nhy n v v dc n v. Chui nhy n v u(n), c nh ngha l:

1, 0( )

0, 0

nu n

n

=


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S tay C in t

Tch chp v tng quan [2]

Mc d tch chp thng lin quan ti vic nghin cu cc h thng, nhng i khi php tnh ny cng c th cn trong phntch cc tn hiu nhn c t mt h thng vt l. Tch chp ca hai tn hiu thi gian lin tc x(t) v y(t) s nhn c z(t),trong :

( ) ( )* ( ) ( ) ( )z t x t y t x y t d

= = (23.11)

Tch chp khng b gii hn trong min thi gian v n cng c dng xc nh ph tn s lin quan ti tch ca hai tnhiu thi gian. Hm tng quan cho giax(t) vy(t) k hiu lRxy(t), c nh ngha nh sau:

( ) ( ) ( ) ( ) * ( )xyR t x t y t x y t d

= = (23.12)

Khc vi tch chp, y khng c php nh x. Ngoi ra, tr tl tham s o ging nhau gia hai tn hiu . Nu x(t)=y(t), th (23.12) s m t hm t tng quan. Mt s tnh cht ca cc hm tng quan s c a ra trong bng 23.2.

C hai tch phn ca tch chp v tng quan u c th p dng cho cc tn hiu nng lng cng nh cng sut. Vi cc tnhiu cng sut, tch phn nhn c trn chu k Tv kt qu c cn chnh bi 1/T. Phn tch tng quan l quan trng tnhmt ph nng lng cho cc tn hiu ngn, v mt ph cng sut cho c cc tn hiu c chu k v cc tn hiu ngu nhin.

BNG 23.2 Cc tnh cht ca hm tng quan

Tnh cht T tng quan Tng quan choChn/sp li (Even / Reorder) ( ) ( ) xx xxR t R t = ( ) ( ) xy yxR t R t =

Gii hn trn (0) ( ) xx xxR R t , cho mi t ( ) (0) (0) xy xx yyR t R R=

HNH 23.6Cc php tnh c bn ca tn hiu: (a) Tn hiu gc, (b) Php cn chnh, (c) Php dch, (d) Php nh x

Phn tch Fourier ca cc tn hiu CT

y chng ti mi ch bn ti cc phng php phn tch min thi gian tn hiu CT. Php tch phn tch chp c quantm c bit v n c th c dng nghin cu cch mt tn hiu c sa i ging nh n s i qua mt h thng. Cngcn phi xem xt cc phng php phn tch min tn s v phn tch tch chp c th l kh. Hn na, cng thc ca tch chpl da trn s biu din cc tn hiu bng cc hm c dch i. Trong nhiu ng dng, phng php phn tch trong mintn s s thch hp hn chn mt tp cc hm trc giao nh cc tn hiu c bn v phng php ny s lm gim phc tpca tnh ton s cng nh s a ra mt cch biu din bng ha cho cc thnh phn tn s ca tn hiu cho.

Cc hm trc giao c bn [2, 3]

S thun li v mt ton hc nu biu din cc tn hiu bt k nh mt tng c hiu chnh ca cc sng trc giao, dnti php phn tch tn hiu c n gin ha rt nhiu, cng nh ch ra s ging nhau c bn gia cc tn hiu v vect. Xtmt tp hm c bn (t), i = 0, 1, 2,. c gi l trc giao trn khong ( t1, t2) nu:

2

1

*( ) ( ) ( )t

m k kt

t t dt E m k = (23.13)6


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S tay C in t

Xt cc tn hiu c chu k c ch ra trong hnh 23.7, chng c th c biu din nh sau:

1( )

n

t nTx t

T

=

=

Thay vo cng thc (23.17) chng ta c:

0

0

/ 22

/ 2

/ 22

/ 2

0

0 0

1( )

1

1sin( )

sin ( )

T j kf t

k

T

T j kf t

T

c x t e dt

T

e dtT

k fk

f c kf

=

=

=

=

V vy:

02

0 0( ) sin ( ) j kf t

n

x t f c kf e

=

=

Biu ph bin v pha ca cc h s ca chui Fourier phc c ch ra trong hnh 23.7(b) vi /Tthay i.

Tnh cc h s ca chui Fourier phc ca tn hiu c ch ra trn hnh 23.8(a)bng cch dngphng php vi phn. Tnhiu ny c trin khai theo cng thc (23.16). Ly vi phn cng thc ny 2 ln theo ts nhn c:

022

0( ) ( 2 ) j kf t k

k

x t j kf c e

=

=

HNH 23.8 Minh ha phng php vi phn tnh h s ca chui Fourier

N c th c vit nh sau:

02( ) j kf t kk

x t e

=

=

Hnh 23.8(c)ch ra kt qu ca php vi phn x(t). Ch rng nu tn hiu c chu k th o hm ca n cng c chu k. iuny c ngha rngk l h s ca chui Fourier phc cax(t) v c th c tnh t:

0/ 2

2

/ 2

1( )

T j kf t

kT

x t e dt T

= trong :

( ) 2 2 ( )2 2

A T T x t t t t

T

= + +

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S tay C in t

bk = 0 ck c gi tr thc

i xng l

x(t)=-x(-t)

ak = 0/ 2

0

0

4( )sin(2

T

kb x t kf tdt T=

ck c cc gi tr o Pha ca ck l/2 hoc

-/2

i xng chn na sng

( )2Tx t x t = +

a2k v b2kc th c gi tr khc 0

nhng a2k+1=0, b2k+1=0

2

2 1

0

0

k

k

c

c +

=

i xng l na sng a2k+1 v b2k+1 c th c gi trkhc 0 nhng a2k=0, b2k=0

2

2 1

0

0k

k

c

c +

=

BNG 23.4Cc tnh cht ca chui Furier

Tnh cht M t tn hiu Cc h s ca chui Furier

Tuyn tnh ( ) ( )ax t by t + ; a, b l hng s k ka b +

Nhn ( ) ( )x t y t *k k

Tch chp ( )* ( )x t y t k k

Quan h Parseval 1

1

21 ( )t T

t

x t dt T

+

2

k

k

Dch theo thi gian ( )x t = 02j fke

Vi phn( )

n

n

dx t

dt0

( 2 )n kj kf

Tch phn ( )Tx d 10 0( 2 ) , 0kj kf

=

Trong c0 l thnh phn mt chiu (DC), ckv kbiu din bin v gc pha ca thnh phn iu ha th ktng ng.Cng thc (23.22) c gi l dng iu ha ca khai trin Fourier ca x(t). Cc tham s ckv klin quan vi akv bknh sau:

2 2 10

0, , tan

2k

k k k k

k

a bc c a b

a = = + =

Cc tnh cht ca chui Fourier [1, 4]

Hiu tnh i xng ca tn hiu c th n gin ha vic tnh ton cc h s Fourier phc ca n. C nhiu dng i xng c chng minh, m mt s kiu i xng quan trng thng gp trong phn tch tn hiu l:

i xng chnx(t)=x(t)

i xng lx(t)= x(t)

i xng l na sngx(t)= x(t+T/2)

nh hng ca tnh i xng ti vic tnh ton chui fourier c ch ra trong bng 23.3. Cc tnh cht khc ca chuiFourier c tm tt trong bng 23.4, trong kandkl cc h s phc ca chui Fourier cax(t) vy(t) tng ng.

Bin i Fourier

Phng php tn s phn tch cc tn hiu thi gian CT c biu din trong mc trc. Mt phng php khc c gil Bin i Fourier, c dng phn tch cc tn hiu khng c chu k. S khai trin ca bin i Fourier c da trn cccng thc (23.16) v (23.17). Thay (23.17) vo (23.16) s c:

0 0/ 2

2 2

/ 2

1( ) ( )

T j kf j kf t

Tk

x t x e d eT

=

=

(23.33)

Cho T , th 1/T df, kf0 f, v cng thc (23.23) tr thnh:

2

( ) ( )j kft

X f x t e dt

= (23.24a)

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Tn hiu v h thngv

2( ) ( ) j ft x t X f e df

= (23.24b)

Cc tnh cht ca bin i Fourier [5, 6]

Mt s tnh cht c bn ca bin i Fourier thng c dng trong phn tch tn hiu c a ra trong bng 23.5.

Cp bin i Fourier ca cc tn hiu c bn sau cng thng c dng phn tch tn hiu:

( )K K f

1sgn( )t

j f

1 1( ) ( )

2 2u t f

j f

+

0 0 0

1 1cos(2 ) ( ) ( )

2 2f t f f f f + +

xc nh bin i Fourier ca mt tn hiu c chu k cn thc hin mt th tc gm hai bc. Nu x(t) l mt tn hiu c

chu k vi chu k T, thx(t) c th c khai trin Fourier nh sau:02

0

1( ) , j f t k

k

x t c e f T

=

= = (23.25)

p dng tnh cht tuyn tnh vi cng thc ny s c:

0( ) ( )k

k

X f c f kf

=

= (23.26)

l bin i Fourier ca tn hiux(t) c chu k bt k. Cng thc (23.26) gii thch s khc nhau gia ph tn s c a ra biphp phn tch Fourier v mt ph bin c a ra bi php bin i Fourier. Do , ph tn s l mt cch biu din (rirc) ca cki vi kf0, trong khi mt ph bin l cch biu din lin tc ca ph mt bin , m trong trng hp nyn c dng xung, ng hn l s.

BNG 23.5Cc tnh cht ca bin i Fourier

Tnh cht M t tn hiu Bin i Fourier

Tuyn tnh ( ) ( )ax t by t + ; a, b l hng s ( ) ( )aX f bY f +

Chn v l ( ) ( )x t x t =

( ) ( )x t x t = 0

( ) 2 ( ) cos(2 )X f x t ft dt

=

Dch thi gian ( )x t 0

( ) 2 ( ) sin(2 )X f x t ft dt

= Cn chnh thi gian ( )x at 2 ( )j fe X f

o thi gian ( )x t 1 fX

a a

i ngu ( )X t * ( )X f

Tch chp thi gian ( ) ( )x t y t ( )x f

Tch chp tn s ( ) ( )x t y t ( ) ( )X f Y f

iu bin 02( ) j f tx t e ( )* ( )X f Y f

Vi phn thi gian( )

n

n

dx t

dt

0( )X f f

Vi phn tn s ( )nt x t ( 2 ) ( )nj f X f

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S tay C in t

Tch phn ( )x d

1 1( ) (0) ( )2 2X f X f j f +

Tng quan ( ) ( ) ( )xyR y t x t dt

= ( ) ( )Y f X f

Quan h Parseval 2( )x t dt

2

( )X f df

Tng t bin i Fourier ca mt dy xung c dng:

( ) ( )n

p t t nT

=

= (23.27)

l:

1 1( ) ( ),s s

k

P f f kF F T T

=

= = (23.28)

Mt ph nng lng v cng sut [6]

Gi thitx(t) l mt tn hiu khng c chu k c bin i Fourier l X(f), th nng lng ca n s l:

2 2(0) ( ) ( )xxE R x t dt X f df

= = = (23.29)

y l thuyt Parseval v n ch ra rng nguyn l bo tn nng lng theo thi gian v tn s l ng. Ph bin X(f) cth c biu din l:

( ) ( ) ( ) X f X f X f = =

v k hiu

2( ) ( )xxS f X f =

th nng lng tng ca tn hiu s l:

( )xxE S f df

= (23.30)

Trong Sxx(f) biu din s phn b ca nng lng tn hiu nh mt hm ca tn s. Sxx(f) c gi l mt ph nnglng ca tn hiu nng lng hu hnx(t).

Xt mt tn hiu c chu kx(t) c hm t tng quan l:

/ 24

/ 2

1( ) lim ( ) ( )

T

xxTx

R x t x t dt T

= Th,

/ 2 2

/ 2

1(0) lim ( )

T

xxTT

R x t dt T

= (23.31)

l cng sut ca tn hiu. Ging nh vi cc tn hiu nng lng, nh ngha Sxx(f) l bin i Fourier caRxx() th

(0) ( ) xx xxP R S f df

= = (23.32)

v Sxx(f) c gi l mt ph cng sut ca tn hiu c chu k x(t).

Trong nhiu trng hp thc t, cng cn phi phn tch cc tn hiu ngu nhin tnh. Tnh cht ca cc tn hiu nh vy cth c suy ra t cc hm tng quan ca chng. V d: hm t tng quan xx() ca mt tn hiu ngu nhin tnh s gim ti0 khi tng v cc trng hp tr nn khng tng quan vi khong thi gian ln. V vy, xx() = xx() v tn ti bin iFourier.

Do vy chng ta c th vit:

(0) ( ) xx xx

f df

= (23.33)Trong xx(f) v xx(0) tng ng biu din mt ph cng sut v cng sut trung bnh ca mt qu trnh ngu nhin.

Cc tn hiu thi gian lin tc c trch mu

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Tn hiu v h thngCc tn hiu thi gian ri rc (DT) xut hin mt cch t nhin hoc bng cch trch mu cc tn hiu thi gian lin tc (CT);

tuy nhin, dng sau thng gp trong thc t hn. Trong trng hp ny, tn hiu s c nh dng t mt tn hiu CT quaqu trnh bin i tng t-s. Phn u tin ca qu trnh ny l trch mu tn hiu tng t, iu c ngha l bin i x(t)thnhx(nTs), trong Tsl chu k trch mu v nghch o ca n lFs= 1/Ts, l tn s trch mu tnh theo s mu trch/giy.Tn s trch mu phi c chn thch hp trnh mo ph (aliasing), theo cch chc chn rng x(t) c th c khi phct cc mu trch ca n. hiu bit thm v th tc ny, chng ta s kho st qu trnh trch mu min tn s.

HNH 23.9 Qu trnh trch mu xung l tng: (a) tn hiu thi gian lin tc gii hn (bandlimited) v bin i Fourier ca n,(b) dy xung v ph ca n, (c) tn hiu c trch mu v ph ca n

Trch mu xung [6-9]

Xt qu trnh trch mu xung l tng c ch ra trn hnh 23.9, trong x(t) c trch mu bng cch dng mt chuixungp(t), v vy

( ) ( ) ( ) ( ) ( ) ( ) ( ) s s s sn n

x t x t p t x t t nT x nT t nT

= == = = (23.34)

S khc nhau chnh giaxs(t) vx(nTs) lxs(t) bn cht l tn hiu CT vi cc gi tr 0 tr ti cc gi tr nguyn ca Ts, trongkhix(nTs) l mu trch cax(t), l kt qu ca vic trch mu xung.

Bin i Fourier ca cng thc (23.34) s cho

( ) ( ) * ( ) ( ) s s sk

X f X f P f F X f F

=

= = (23.35)

trong X(f) l ph cax(t).

Nhn thy rngXs(f) bao gm mt phin bn ca X(f) c hiu chnh v c lp li mt cch tun hon vi chu k Fs.Nh ch ra trong hnh 23.9(c), hin nhin rng khi s M M F F F th s khng c s chng nhau ca ph, v tn hiu x(t) c th

c khi phc hon ton t xs(t). Tuy nhin, nu s M M F F F < th cc bn ti to ca X(f) s chng ln nhau, dn ti ph bmo v v vy c th khng khi phc c x(t) t phin bn mu trch ca n. V vy, khi phc cx(t) t cc mu trchca n, th tn s trch mu phi l:

s M M F F F

c ngha l:

2s MF F

y c gi l nh l trch mu Nyquist. Tn s trch mu ti thiu 2s MF F= c gi l tn s Nyquist. Vic trch mumt tn hiu tn s nh hn tn s Nyquist s dn n mo ph c gi l mo aliasing. Ngoi ra, vic trch mu mt tn hiu tn s Nyquist nh nht c ngha rng mt b lc thng thp l tng (LPF) vi mt h s 1/Fs v tn s ctFc c th cdng khi phc ph ban u ca n, trong FM Fc Fs -FM.

Gi s chng ta mun khi phc x(t) t cc mu trch ca n. Gi thit rng X(f) l ph ca x(nTs), khng c mo aliasing,nh ch ra trong hnh 23.9(c). Nh vy:

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S tay C in t

1( ),

2( )

0,2

s

sa

s

FX f f

FX f

Ff

=

>

(23.36)

ch rng:

2( ) ( ) s j fT a sX f x nT e

= (23.37)

v vy bin i ngc Fourier s l:

/ 22 ( )

/ 2

1( ) ( )

sin[ ( ) / ]( )

( ) /

ss

s

F j f t nT

a sF

ns

s s

s

n s s

x t x nT e df F

t nT T x nT

t nT T

=

=

=

=

(23.38)

y l cng thc khi phcx(t) t cc mu trch ca n. iu c ngha lx(t) c to ra bng cch nhn hm cdch mt cch thch hpg(t) = sinc(tFs) vi cc mu trch tng ng cax(nTs).

Vic trch mu thc t [8-10]Bn lun trn v vic trch mu da trn cc m hnh trch mu xung c chu k l tng v ni suy di gii hn. Trong thc

t, Cc tn hiu CT khng c gii hn di mt cch chnh xc nh cc tn hiu xung v cc b lc thng thp l tng khngtn ti. Hnh 23.10biu din s khi bin cc tn hiu thi gian lin tc thnh dng ri rc. Tn hiu thi gian lin tc utin c lc, trch mu, lng t ha, v cui cng c m ha thnh cc t c di hu hn c gi l cc bit b. B lcc gi l lc chng mo aliasing (AAF), l mt b lc thng thp gii hn rng di ca tn hiu vo ti Fs/2 trc khitrch mu loi mo aliasing. Trong thc t b lc ny c cc c tnh khng l tng, v vy n phi c thit k a ra suy gim cc tn s trn tn s Nyquist, thng ti mc khng th pht hin c bi b bin i tng t s (ADC).

HNH 23.10 Trch mu thc cc tn hiu thi gian lin tc

Cc tn hiu lc c a n ADC bin i thnh tn hiu DT. ADC c sn mt mch trch mu hot ng vi tc trch muFs; tuy nhin, hm trch mu c rng hu hn tri vi vic trch mu xung bn trn. Php trch mu c th cm hnh ha bng b trch mu xung c rng hu hn c ch ra trong hnh 23.11(b), trong cng trch mu s m vi trong s Tsgiy v ngn mch trong phn cn li ca khong trch mu. yp(t) c biu din nh sau:

( ) sn

t nTp t

=

=

n c th c khai trin chui Fourier nh sau:

2( ) s j kf t

k

n

p t c e

=

=

trong :

sinks s

c c kT T

=

Bin i Fourier ca cc tn hiu trch mu c th c vit nh sau:

( ) ( ) s k snX f c X f kF

== Ch rng ck khng phi l hng s trong biu thc ny (tri vi vic trch mu xung) v gi tr ca n ph thuc vo s cc

iu ha (k) cng nh chu k /Ts. Tn hiu ri rc theo thi gian, x(nTs), c cp ti b lng t ha m mi mu trchc bin i thnh mt trong cc tp hu hn gn nht ca cc gi tr quy nh, l ( ) ( ( ))sx n Q x nT = , trong ( )x n l mu

14


15/46

Tn hiu v h thng c lng t. Qu trnh lng t c ch ra trong hnh 23.11(d) cho b bin i tng t-s trch v gi mu bc 0 (ADC),trong Li l k hiu mc lng t v l bc lng t. Sai s lng t (hoc l n) ca qu trnh ny l:

( )2 2i s i

L x nT L

< < +

HNH 23.11Cc tn hiu trch mu xung vi rng hu hn v ph ca n: (a) Tn hiu c di gii hn v ph ca n, (b)Chui xung c rng hu hn v bin i Furier ca n, (c) Tn hiu trch mu v ph ca n, (d) Qu trnh lng t ha.

HNH 23.12Qu trnh bin i s-tng t

15

HNH 23.13 p ng tn s ca blc khi phc l tng


16/46


17/46

Tn hiu v h thng

HNH 23.15(a) Khi phc ca tn hiu tng t, (b) hiu qu ca lc khng nh sau.

Phn tch tn s ca cc tn hiu thi gian ri rc

Phn tch cc tn hiu thi gian ri rc (DT) trong min tn s tng t nh phn tch cc tn hiu thi gian lin tc (CT).Nh trong phn tch tn hiu CT, cc phng php phn tch ph thuc vo kiu tn hiu. Phn tch tn hiu DT khng c chu ks c xem xt trc.

Bin i Fourier ca cc tn hiu thi gian ri rc [6-8]

Vic phn tch mt tn hiu DT khng c chu k thnh cc thnh phn tn s c tin hnh bng cch dng bin i Fourierri rc (DTFT). Do , DTFT cax(n) s c a ra bi:

2( ) ( ) j fn

n

X f x n e

=

= (23.39)

Khng ging bin i Fourier ca cc tn hiu tng t,X(f) l tn hiu tun hon vi chu kFs; v vy, di tn s ca mt tnhiu DT l duy nht trn khong tn s (Fs/2,Fs/2) hoc, tng ng (0,Fs). Ch rng cng thc (23.39) phi l tng tuyti X(f) tn ti, c ngha rng:

( )n

x n

=

<

Nu ph ca mt tn hiu tn ti th chng ta c th tm thy tn hiu t ph ca n thng qua DTFT ngc. DTFT ngc caX(f) c a ra nh sau:

/ 22

/ 2

1( ) ( )

s

s

F j fn

Fx n X f e df

F

=

Cc tnh cht ca DTFT ca mt tn hiu DT c ch ra trong bng 23.6.

Vi cc tn hiu DT, tch chp ca hai chuix(n) andy(n) c biu din nh sau:

( )* ( ) ( ) ( )mx n y n x m y n m

== (23.40)trong khi hm tng quan cho cax(n) vy(n) c a ra bi:

17


18/46

S tay C in t

*( ) ( ) ( ) ( ) ( )xym

R n x n y n x m y m n

=

= =

Khix(n)=y(n), th t tng quan cax(n) l:

*( ) ( ) ( )xxm

R n x m x m n

=

=

BNG 23.6 Cc tnh cht ca DTFT

Tnh cht M t tn hiu Min tn s

i xng chn (tn hiu thc) 1( ) { ( ) ( )}

2ex n x n x n= + ( ) ( ) cos(2 )e e

n

X f x n nf

=

=

i xng l (tn hiu thc) 1( ) { ( ) ( )}

2ox n x n x n= ( ) ( ) sin(2 )o o

n

X f x n nf

=

=

Tuyn tnh ( ) ( )ax n by n+ ( ) ( )aX f bY f +

Dch thi gian ( )x n m 2 ( ) j fme X f

o thi gian ( )x n ( )X f

Tch chp ( ) ( )x n y n ( ) ( )X f Y f

Tng quan ( ) ( ) ( )xyR n x n y n= ( ) ( ) ( )xyS f X f Y f =

Quan h Wiener-Khinchine ( )xxR n ( )xxS f

Dch tn s 02 ( ) j f ne x n 0( )X f f

iu bin0

( ) cos(2 )x n nf 0 0

1{ ( ) ( )}

2X f f X f f + +

Nhn ( ) ( )x n y n 1 ( ) ( )sF

s

X Y f d F

Vi phn trong min tn s ( )nx n ( )

2

j dX f

df

Sai khc trong min thi gian ( ) ( 1)x n x n 2(1 ) ( )j fe X f

Tng( )

n

m

x m= 2

(0)( )( )

2(1 )s

sj fm

F XX ff mF

e

=

+

Lin hp phc ( )x n * ( )X f

Quan h Parseval( ) ( )

n

x n y n

= 1 ( ) ( )

sFs

X f Y f df F

Bng cch loi php nh x (cho php tch chp), th cc th tc tnh tch chp v tng quan l nh nhau. V vy hiu qutnh ton s cao hn khi dng cng mt thut ton nh gi c hai hm. t c iu ny, th mt trong cc chui cnh x (ch cho phn tch tng quan) c dn ra bi php tch chp, l:

( ) ( )* *( )xyR n x n y n=

v*( ) ( )* ( )xxR n x n x n=

Nng lng ca mt tn hiu khng tun hon c tnh t:

2 *( ) ( ) ( )n n

E x n x n x n

= =

= =

18


19/46

Tn hiu v h thngThay th (23.40) vo cng thc trn s nhn c:

21 1( ) ( )

s sxx

F Fs s

E X f df S f df F F

= = (23.41)

Biu thc ny lin quan n s phn b nng lng ca mt tn hiu khng tun hon theo tn s. i lng /X(f)/2 c gil mt ph nng lng cax(n). DTFT ca mt vi tn hiu thng gp c ch ra trong bng 23.7.

BNG 23.7 DTFT ca mt s tn hiu ph bin

( )x n Biu din trong min tn s, ( )X f

( )n 1

,A n < < 1

( ), s s sk

AF f mF F T

=

=

( )u n2

1( )

21s

sj fk

Ff kF

e

=

+

2 1

n

q

+

sin[(2 1) ]

sin( )

q f

f

+

n

q

2

2

sin ( )

sin ( )

fq

q f

sgn( )n2

2

1 j fe

( )n u n2

1, 1

1 j fe

. Khi , bin i ngc c th nhn c nh sau:

11 ( )2

= kkx X z z dzi (23.51)

Trong ng cong ca tch phn s bao tt c cc im k d ca X(z). Trong thc t, l mt qu trnh chun s dngcc kt qu dng bng; Mt vi cp bin iz chun c th thy trong bng 23.10.

Cc h thng s v d liu ri rc

Vic trch mu tun hon cc tn hiu v vic tnh ton tip theo hoc lu gi cc kt qu yu cu my tnh nh gi cho victrch mu v x l cc chui s kt qu. Mt bin o cx(t) c th s dng ch nh l cc quan st theo chu k cax(t) ctrch mu theo khong thi gian T(chu k trch mu). Chui mu c th c biu din nh sau:

{ } , ( ) ...., 1, 0,1, 2, ...k kx x x kT k = = (23.52)

v iu quan trng l phi bit chc rng chui mu biu din ph hp bin gc x(t); xem hnh 23.16. vic trch mu l ltng, yu cu khong cch trch mu phi rt ngn v hm trch mu phi c biu din bng mt chui cc xung ngn v hn (t) (xung Dirac). V vy, hm thi gian trch mu c biu din nh sau:

( ) ( ) ( ) ( ) ( )

=

= = Tk

x t x t T t kT x t t (23.53)

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24/46

S tay C in t

( ) ( )

=

Tk

t T t kT (23.54)

BNG 23.10 Cc tnh cht ca bin iz

Bin iz ({ }) ( )kf F z=Z

Tch chp ({ }) ({ }) ({ })k k k k f g f g = Z Z Z

({ }) ({ }) ({ })k k k k f g f g = Z Z Z

Dch trc1

({ }) ({ }) ( )k kf z f zF z+ = =Z Z

Dch sau 11

({ }) ({ }) ( )k kf f z F z

= =Z Z

Tuyn tnh ({ }) ({ }) ({ })k k k k af bg a f b g + = +Z Z Z

Nhn 1({ }) ( )k ka f F a z=Z

Gi tr cui 11

lim lim(1 ) ( )kk k

f z F z

=

Gi tr u 0 lim ( )zf F z=

Min thi gian Bin i z

Xung 1, 0{ } 1,

0, 0k kk

z Ck

=

= = Z

Hm bc 0, 0( ) , 1

1, 0 1k kk z

zk z

Z

Hm dc2( ) , 1( 1)k k

z

x k X z zz= = >

Hm m( ) ,kk k

zx a X z z a

z a= = >

Hm sin2

sinsin ( ) , 1

2 cos 1k k

zx k X z z

z z

= = >

+

HNH 23.16 Tn hiu thi gian lin tcx(t) v thit b trch mu a ra chui mu {xk}.

v trong chu k trch mu Tc nhn ln chc chn rng trung bnh trn mt chu k trch mu ca bin gc x v tnhiu trch mux tng ng l cng ln. Mt ng dng trc tip ca bin ri rc x (t) theo cng thc (23.53) s chng trng ph cax lin quan ti bin iz, X(z) nh sau:

( ) { ( ) ( ) exp( ) ( )

=

= = = = i TT kk

X i F x t t Tk x i kT TX e (23.55)

R rng rng, bin gc x(t) v d liu trch mu l khng ging ht nhau, v v vy cn phi xem xt cc hiu ng mo cavic ri rc ha. Xt ph ca tn hiu trch mux (t) nhn c bi bin i Fourier:

( ) { ( )} { ( )}* { ( )}TX i F x t F x t F t = = (23.56)

trong :

2 /

2{ ( )} ( )

2

=

= =

T Tk

TF t k

T (23.57)

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25/46

Tn hiu v h thngsao cho:

2( ) { ( )}* ( )}

=

= = { T

k

X i x t t X i k T

F F (23.58)

Do , bin i FourierX ca bin c trch mu l m rng tun hon ca ph gc X(i) dc theo trc tn s vi mt chuk bng tn s trch mu s =2/T.Mt kt qu quan trng da trn nhn xt ny l thuyt trch mu Shannon pht biu rngbin thi gian lin tcx(t) c th c khi phc li t cc mu trch nu v ch nu tn s trch mu t nht bng hai ln tn scao nht m vi nX(i) khc khng. V vy bin gcx(t) c th c khi phc l:

sin ( )( )

( )

=

=

k

k

t kTTx t x

t kTT

(23.59)

Cng thc (23.59) c gi lphp ni suyShannon, thng c trch dn mc d n ch ng vi cc chui d liu di vhn v yu cu mt b lc khng nhn qu khi phc tn hiu thi gian lin tc x(t) theo php tnh thi gian thc. Tn sn=s/2= /Tc gi l tn s Nyquistv ch ra gii hn trn ca vic trch mu nhiu t do. Mt ph khc khng khngthuc gii hn ny s dn n s giao thoa gia tn s trch mu v tn hiu c trch mu ( aliasing); xem hnh 23.17.

HNH 23.17 Biu din aliasing xut hin trong khi trch mu mt tn hiu sin x(t)=sin2 .0.9t tn s trch mu thiu, 1 Hz(chu k trch mu T = 1) ( th trn). Tn hiu c trch mu s b mo aliasing cc thnh phn chnh ca n ging nh tn

hiux(t)=sin2 .0.1tc trch mu vi cng tc ( th di).

Bin i Fourier ri rc

Xt mt chui di hu hn { }1

0

N

k kx

=bng 0 bn ngoi khong 0kN - 1. c lng bin i z, X(z) ti Nim cch u

nhau trn ng trn n v [ ]exp( ) exp (2 / )kz i T i NT kT = = vi k = 0, 1,, N - 1 s xc nh Bin i Fourier ri rc(DFT) ca tn hiux vi chu k trch mu h vNphp o:

1

0

{ ( )} exp( ) ( )

=

= = = kN

i T

k l k

l

X DFT x kT x i lT X e (23.60)

Ch rng bin i fourier ri rc ch c xc nh cc im tn s ri rc

2, 0,1,...., 1

= = k k k N

NT(23.61)

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26/46

S tay C in t

trong thc t, bin i fourier ri rc phng theo bin i fourier v bin i z cho cc yu cu thc t ca cc php o l huhn. Cc tnh cht tng t u c gi nguyn i vi bin i Laplace ri rc vi z=exp(sT), trong s l bin ca bin iLaplace.

Hm truyn

Xt h thi gian tuyn tnh ri rc vi chui vo {uk} (chui kch) v chui ra {yk} (chui p ng). S ph thuc ca ca tnhiu u ra ca mt h tuyn tnh c c trng bi phng trnh tch chp v bin i z ca n,

0

, ..., 1,0,1, 2,...

( ) ( ) ( ) ( )

= =

= + = + =

= +

k

k m k m k k m m k

m m

y h u v h u v k

Y z H z U z V z(23.62)

trong chui {vk} tng ng vi mt vi u vo ca sai s v nhiu ngoi v vi ( ) { }, ( ) { }, ( ) { }Y z y U z u V z v= = =Z Z Z l

tn hiu u ra v cc tn hiu u vo.Hm trng { }0

( ) k kh kT h

== bng 0 vi km v theo thuyt nhn qu c gi l p ng

xungca h thng s (so snh vi p ng xung ca h thi gian lin tc). p ng xung v bin i z ca n, hm truyn xung:

0

( ) { ( )}

=

= = kkk

H z h kT h zZ (23.63)

xc nh p ng ca h i vi u vo U(z); xem hnh 23.18.

HNH 23.18 S khi c quan h hm truyn c gi thit l H(z) gia u vo U(z), nhiu V(z), hm trung gianX(z), v ura Y(z)

Hm truyn xungH(z) c xc nh bi t s:

( )( )

( )

=X z

H z

U z

(23.64)

v quy nh quan h gia u vo v u ra min tn s ca h thng. c bit s Bode c c lng bi ( )H z v arg

H(z) cho z=exp(ikT) v vi /k n T < = , tc l khiH(z) c c lng bi cc im tn s ti tn s Nyquist ndctheo ng trn n v.

Cc h khng gian - trng thi

Cch biu din quan h vo - ra bng cc hm truyn gi l biu din khng gian - trng thi. Xt phng trnh khng gian -trng thi ri rc, th nguyn hu hn vi mt vector trng thi nkx R , mt u vo

p

ku R v cc quan stm

ky R .

1 0,1...+= +

== +

k k k

k k k

x x uk

y Cx Du

(23.65)

vi hm truyn xung:

1( ) ( )= +H z C zI D (23.66)

v bin u ra:

0

0

( ) ( ) ( )

=

= + k kk

Y z C z x H z U z (23.67)

trong cc hiu ng c th ca iu kin u x0 s xut hin nh s hng u tin. Ch rng cc iu kin u x0 c th thynh cc hiu ng thc ca u vo trong khong thi gian (-, 0).

Cc h thng s c m t bng cc phng trnh sai phn (cc m hnh ARMAX)Mt lp quan trng ca cc qu trnh ngu nhin khng dng l lp m trong mt vi p ng xc nh i vi mt u

vo ngoi v mt qu trnh ngu nhin dng l c tnh cht xp chng. iu l thch hp, v d khi u vo khng th c

26


27/46

Tn hiu v h thngm t mt cch hiu qu bi mt vi phn b xc sut. Mt m hnh ri rc c th c a ra theo dng mt phng trnh saiphn vi mt tn hiu vo {uk} thng c xem nh bit.

1 1 1 1 1 1.... .... = + + + + + + +Lk k n k n k n k n k k n k ny a y a y b u b u w c w c w (23.68)

Bin iz ca phng trnh (23.68) l:

1 1 1( ) ( ) ( ) ( ) ( ) ( ) = +A z Y z B z U z C z W z (23.69)

trong :1 1

1

1 1

1

1 1

1

( ) 1

( ) 1

( ) 1

= + + +

= + + +

= + + +

L

L

L

n

n

n

n

n

n

A z a z a z

B z b z b z

C z c z c z

(23.70)

Cc m hnh ngu nhin bao gm a thc A, theo phng trnh (23.69) v (23.70) c bit n nh cc m hnhautoregressive (AR) v cc m hnh bao gm a thc Cc bit n nh cc m hnh moving-average (MA), trong khi, athcB xc nh cc hiu ng ca u vo ngoi (X). Ch rng s hng moving average yl mt vi li do khng c s hnch cc h s phi cng vo 1 hoc cc h s khng m. Mt cch m t khc l p ng xung hu hn ( finite impulseresponse) hoc b lc tt c cc im khng (allzero filter).

V vy m hnh y ca phng trnh (23.69) l mt m hnh auto regressive moving average vi tn hiu u vo ngoi

(ARMAX) v hm truyn xung ca nH(z)=B(z-1

)/A(z-1

) l n nh nu v ch nu cc cc - tc l cc s phc z1,,znl nghimca phng trnhA(z-1) = 0 l hon ton nm trong ng trn n v, tc l |zi| < 1. Cc im khng ca h thng - tc l ccs phc z1,, zn l nghim ca phng trnhB(z-1) =0 c th nhn gi tr bt k m khng xut hin bt k s mt n nh no,mc d thch hp hn nhn cc im khng nm bn trong ng trn n v, tc l |zi| < 1 (minimum-phase zeros). Do tuyntnh, {yk} c th phn thnh mt qu trnh xc nh hon ton {xk} v mt qu trnh ngu nhin hon ton {vk}:

1 1

1 1

( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

= += = +=

k k ky x vA z X z B z U z

Y z X z V zA z V z C z W z(23.71)

Kiu phn tch ca phng trnh (23.71) tch cc qu trnh xc nh v cc qu trnh ngu nhin c gi l Wolddecomposition.

D bo v khi phc

Xt cc vn d bo trc d bc ca tn hiu ra khi tn hiu u ra {yk} c to bi m hnh ARMA,

1 1( ) ( ) ( ) ( ) =A z Y z C z W z (3.72)

m m hnh ny c iu khin bi n trng trung bnh khng {wk} vi covariance { } 2i j w ijw w =E . Ni cch khc, gi thitrng cc quan st {yk}thu c ti thi im hin ti, cn phi d bo trc d bc u ra nh th no l ti u? Gi thit rnga thc A(z-1) v C(z-1) l nguyn t vi nhau v khng c im khng vi |z| 1. Khai trin a thc C theo phng trnh

Diophantine

1 1 1 1( ) ( ) ( ) ( )dC z A z F z z G z = + (23.73)

trong :

1 11

1 1

0 1

( ) 1 , 1

( ) , max( 1, )

= + + + = = + + + =

L

L

nFnF

nG

nG G A c

F z f z f z nF d

G z g g z g z n n n d (23.74)

Biu din caz-1 nh l ton t dch li (backward shift operator) v s dng cc phng trnh (23.72) v (23.73) cho phpnhn c:

11

1

( )( )

( )

+ + = +k d k d k G z

y F z w yC z

(23.75)

K hiu cc b d bo tuyn tnh dbc lyk+d da trn thng tin thu c ti thi im k. V s hng trungbnh-khng 1( ) k dF z w

+ ca phng trnh (23.75) l khng th d bo c ti thi im k, nn hon ton c th xut b d

bo dbc nh sau:

1

1

( )

( )+

=k d kkG z

y yC z

(23.76)

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28/46

S tay C in t

Sai s d bo tha mn:

1 1 1 1

1 1

1

( )

( ) ( ) ( ) ( )

( ) ( )

( )

+ ++

+

+

=

+=

=

k d k d k d k

d

k k d

k d

y y

G z A z F z z G zy y

C z C z

F z w

(23.77)

K hiu { }kE F l k vng ton hc c iu kini vi thng tin o c thi im k. K vng ton hc v hipphng sai ca vic d bo dbc lin quan n thng tin thu c thi im kl:

1

2 1 2

2

1 1 1 1

2 2 2

1

{ } { ( ) } 0

{( ) } {[ ( ) ] }

{( ) }

(1 ) 0

+ ++

+ ++

+ + +

= =

=

= + + +

= + + + =

L

L

k d k k d k k d k

k d k d k d k

k d k d d k k

nF w

y y F z w

y y F z w

w f w f w

f f

E E

E E

E F

k k

F F

F F(23.78)

Suy ra rng b d bo ca phng trnh (23.76) l khch quan v rng sai s d bo ch ph thuc vo tng lai, cc thnhphn nhiu khng d bo c. D dng ch ra rng b d bo ca phng trnh (23.76) s t c gii hn thp ca phng

trnh (23.78) v rng b d bo ca phng trnh (23.76) l ti u theo ngha phng sai ca sai s d bo l ti thiu.V d 23.1 - B d bo ti u cho m hnh bc nht

Xt m hnh ARMA bc nht

1 1 1 1k k k k y a y w c w+ += + + (23.79)

Phng sai ca b d bo trc mt bc l:

2 2 2

1 1 1 11 1

2 2 2

1 11

{( ) } {( ) } { }

{( ) }

+ ++ +

+

= + +

= + +

k k k k k k k k k k k

k k k w wk k

y y y a y c w w

y a y c w

E F E F E F

E F(23.80)

B d bo ti u tha mn gii hn thp ca phng trnh (23.80) nhn c t phng trnh (23.80) l:

1 11 + = +

o

k kk ky a y c w (23.81)

rt tic l khng th thc hin n bi v wkkhng th o c. V vy, chui n {wk} c thay th bi mtvi hm ca bin quan st c {yk}. Mt b d bo tuyn tnh c chn theo phng trnh (23.76) l:

1

1

1 1

1 1

1

( )

( ) 1+

= =

+k k kkc aG z

y y yC z c z

(23.82)

B lc Kalman

Xt m hnh khng gian trng thi tuyn tnh

1 ,,

+ = + = +

n

k k k k

m

k k k k

x x v xy Cx w y

(23.83)

trong {vk} v {wk} c gi thit l cc qu trnh n trng trung bnh-khng c lp vi cc covariance v wtngng. Gi thit rng {yk} l o c v rng mong mun d bo{xk} t cc php o {yk}. a ra b d bo trng thi:

1 1 1

1

( ),

,

+

=

=

n

k k kk k k k k k

m

k kk k

x x K y y x

y Cx y(23.84)

B d bo ca phng trnh (23.84) c cng ma trn ng lc vi m hnh khng gian trng thi ca phng trnh(101.83), v ngoi ra c mt s hng hiu chnh ( )k k kK y y vi mt h s Kk c chn. Sai s d bo l:

11 1 +

+ +

= % kk k k k x x x (23.85)

Cc ng lc ca sai s d bo l:

1( )

+= + % %

k k k k k k x K C x v K w (23.86)

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29/46

Tn hiu v h thngSai s d bo trung bnh b khng ch bi phng trnh quy:

1{ } ( ) { }+ = % %k k kx K C xE E (23.87)

v sai s trung bnh bnh phng ca sai s d bo b khng ch bi:

1 1{ } {[ ) ][( ) ] }

( ) { }( )+ + = + +

= + +

% % % %

%%

T T

k k k k k k k k k k k k

T T

k k k k v k w k

x x K C x v K w K C x v K w

K C x x K C K K

E E

E(23.88)

Nu chng ta k hiu:{ },= = +%%T Tk k k k w k P x x Q CP C E (23.89)

th phng trnh (23.88) c n gin thnh:

1+ = + +T T T T T

k k k k k k v k k k P P K CP P C K K Q K (23.90)

Bng cch bnh phng ton b cc s hng cha Kkchng ta s nhn c:

1 1 1

1( ) ( ) + = + +

T T T T T T

k k v k k k k k k k k k k P P P C Q CP K P C Q Q K P C Q (23.91)

trong ch s hng cui cng ph thuc vo Kk.Vic ti gin Pk+1 c th c thc hin bng cch chn Kk sao cho nadng ca Kk l s hng ph thuc trong phng trnh (23.91) bin mt. V vy Pk+1 t c gii hn thpca n vi:

1( )= +T Tk k w k K P C CP C (23.92)

v b lc Kalman (hoc b lc KalmanBucy) s c dng:

1 1

1

1

1

1

( )

, ( )

( )

+

+

=

= = +

= + +

k k kk k k k

T T

k k k w k k k

T T T T

k k v k w k k

x x K y y

y Cx K P C CP C

P P P C CP C CP

(23.93)

l b d bo ti u theo ngha sai s bnh phng trung bnh (phng trnh (23.88)) l ti thiu theo mi bc.

V d 23.2B lc Kalman cho mt H bc nht

Xt m hnh khng gian trng thi:1

0.95 ,+ = + = +k k k k k k x x v y x w (23.94)

Trong {vk} v {wk} l cc qu trnh n trng trung bnh-khng vi cc covariance { }2 1kv =E v { }2 1kw =E tng ng.

Cc b lc Kalman s c dng:

1 1 1

2 2

2

1

0.95 ( )

0.95

1

0.950.95 1

1

+

+

=

=+

= + +

k kk k k k k k

kk

k

kk k

k

x x K x y

PK

P

PP P

P

(23.95)

Kt qu ca s thc hin nh vy c ch ra trong hnh 23.19.

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S tay C in t

HNH 23.19 B lc Kalman c dng d bo trc mt bcxk+1 trong phng trnh (23.94). Bin quan st c {yk},

trng thi {xk} v trng thi d bo c { }kx% , variance c tnh c {Pk} v {Kk}, v sai s d bo c ch ra trong 100bc thc hin ca qu trnh ngu nhin. (Source: Johansson, R. 1993. System Modeling and Identification. Prentice-

Hall, Englewood Cliffs, NJ.)

nh ngha cc khi nim M hnh Autoregressive (AR): Mt chui thi gian t li bc n c nh ngha bng

1w

n

k m k m k my a y == + . Chui {wk} thng c gi thit bao gm cc bin ngu nhin phn btrung bnh khng ging nhau wk.

M hnh Autoregressive moving average (ARMA): Mt chui thi gian trung bnh t

li bc n c nh ngha bng1 0

n n

k m k m m k m

m m

y a y c w = =

= + . Chui {wk} thng c gi thit bao gm cc bin ngunhin phn b trung bnh khng ging nhau wk.

Bin i Laplace ri rc: Bin i Laplace ri rc l mt bn sao ca bin i Laplace p dng chocc tn hiu v cc h ri rc. Bin i Laplace ri rc nhn c t bin izbng cch thay th z= exp(sT), trong

Tl chu k trch mu. Qu trnh moving average (MA): Mt chui thi gian trung bnh dch bc n c nh ngha

0

n

k m k mmy c w == . Chui {wk} thng c gi thit bao gm cc bin ngu nhin phn b trungbnh-khng ging nhau wk.

M hnh Rational: AR, MA, ARMA, and ARMAX c gi chung l m hnh hu t.

Chui thi gian: chui ca bin ngu nhin {yk}, trong kthuc tp cc s nguyn dng v m.

Bin i z: Mt hm to c p dng vi cc chui d liu v c c tnh nh mt hm ca bin phc z theo tn s.

Ti liu tham kho

[1] Box, G. E. P. and Jenkins, G. M. 1970. Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, CA.

[2] Hurewicz, W. 1947. Filters and servo systems with pulsed data. In Theory of Servomechanisms, H. M. James, N. B.Nichols, and R. S. Philips, eds., McGraw-Hill, New York.

[3] Jenkins, G. M. and Watts, D. G. 1968. Spectral Analysis and Its Applications. Holden-Day, San Francisco, CA.

[4] Johansson, R. 1993. System Modeling and Identification. Prentice-Hall, Englewood Cliffs, NJ. Jury, E. I. 1956. Synthesisand critical study of sampled-data control systems. AIEE Trans. 75: 141151.

30


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Tn hiu v h thng[5] Kalman, R. E. and Bertram, J. E. 1958. General synthesis procedure for computer control of single and multi-loop linear

systems. Trans. AIEE. 77: 602609.

[6] Kolmogorov, A. N. 1939. Sur linterpolation et extrapolation des suites stationnaires. C. R. Acad. Sci. 208: 20432045.

[7] Ragazzini, J. R. and Zadeh, L. A. 1952. The analysis of sampled-data systems. AIEE Trans. 71:225234. Tsypkin, Y. Z.1950. Theory of discontinuous control. Avtomatika i Telemekhanika. Vol. 5.

[8] Wiener, N. 1949. Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications.John Wiley & Sons, New York.

Thng tin b sungCc kt qu l thuyt gn y lin quan n cc ng dng ca cc h c servo v radar [Hurewicz, 1947]. Tsypkin [1950]

gii thiu bin i Laplace ri rc v nh ngha bin izi xng c gii thiu bi Ragazzini v Zadeh [1952] vi ccpht trin hn na bi Jury [1956]. Nhiu l thuyt d bo c o c pht trin bi Kolmogorov [1939] v Wiener [1949]trong khi cc phng php khng gian - trng thi t c cc tin b bi Kalman and Bertram [1958]. Cc sch gio khoau tin v phn tch chui thi gian v phn tch ph c cung cp bi Box v Jenkins [1970] v Jenkins v Watts [1968].

Cc bo co chi tit v phn tch chui tn hiu thi gian v bin i z v vic ng dng chng x l tn hiu c tm thytrong:

Oppenheim, A. V. and Schafer, R. W. 1989. Discrete-Time Signal Processing. Prentice-Hall, EnglewoodCliffs, NJ.

Proakis, J. G. and Manolakis, D. G. 1989. Introduction to Digital Signal Processing. Maxwell MacMillan Int.

Ed., New York. Cc l thuyt v phn tch chui tn hiu thi gian v vic ng dng n iu khin thi gian ri rc thy c trong:

strm, K. J. and Wittenmark, B. 1990. Computer-Controlled Systems, 2nd ed., Prentice-Hall, EnglewoodCliffs, NJ.

L thuyt v phn tch chui thi gian v phng php lun xc nh v lm cho hp logic cc m hnh thi gianri rc v cc kha cnh khc ca vic nhn dng h thng tm c trong:

Johansson, R. 1993. System Modeling and Identification. Prentice-Hall, Englewood Cliffs, NJ.

Cc ti liu tt kho st cc nghin cu hin thi l:

IEEE Transactions on Automatic Control

IEEE Transactions on Signal Processing

Cc v d cho cc ng dng x l tn hiu l: Cadzow, J. A. 1990. Signal processing via least-squares error modeling. IEEE ASSP Magazine. 7:1231, October.

Schroeder, M. R. 1984. Linear prediction, entropy, and signal analysis. IEEE ASSP Magazine. 1:311, July.

23.3 Cc m hnh khng gian - trng thi thi gian ri rc v lin tc

Kam Leang, Qingze Zou, and Santosh Devasia

Gii thiu

Trong mc ny chng ti s gii thiu vic m hnh ha cc h thi gian lin tc v ri rc s dng phng php khng gian- trng thi. Phng php khng gian trng thi l phng php s dng mt tp cc phng trnh vi phn bc nht biu din

ng xca mt h thng trong min thi gian. Phng php khng gian trng thi c mt thun li hn cc phng php mintn s nh l phng php hm truyn: c th c dng m hnh ha cc h thng tuyn tnh, phi tuyn, bin i theo thigian v a bin, trong phng php hm truyn l ph hp vi cc h tuyn tnh bt bin theo thi gian (LTI) [1, Chapter 3].Ngoi ra, cc m hnh c biu din dng khng gian - trng thi bc nht trong min thi gian c th d dng c giiquyt bng my tnh s hoc b vi x l lm cho phng php ny hon ton c ch thit k v iu khin cc h c in thin i. Hn na, c rt nhiu phn mm my tnh nh MATLAB [2], khai thc cc li th ca dng khng gian - trng thi phn tch v gii quyt cc vn ca bi ton thit k. V vy phng php khng gian - trng thi c th c dng nghin cu ng x v thun tin khi thit k c h thi gian lin tc v ri rc, cc nguyn tc c bn ca n l tiu im camc ny.

Sau y, chng ta bt u bng mt v d: M hnh ha mt c cu chp hnh p gm v dng v d trong sut mc ny.Khi nim trng thi ca h c a ra v chng ta s gii thch phng trnh khng gian trng thi ca cc h tuyn tnh vtrnh by cch gii n. Ch tuyn tnh ha cc h khng tuyn tnh c cp mt cch ngn gn. Mi quan h gia cc mhnh min thi gian v m hnh min tn s c bn lun v mt th tc nhn c m hnh khng gian - trng thi bng

cch dng d liu thc nghim ca min tn s (p ng tn s) cng c trnh by. Mc ny c kt thc bng mt bn lunv vic m hnh ha khng gian - trng thi thi gian ri rc v nhng nhn xt kt lun. Cc lnh ca MATLAB cng c dnra.

31


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S tay C in t

Trng thi v khng gian - trng thi

M hnh v d ca mt c cu chp hnh p gm

Chng ta bt u bng vic m hnh ha c cu chp hnh p gm (piezoceramic actuator) l mt h c in t v d. Khimt in p c t vo mt vt liu p gm th kch thc ca n s thay i. S thay i kch thc ny c th c dng nh v chnh xc mt i tng hoc mt cng c (v d nh mt u o), v vy to ra cc c cu chp hnh p gm ph hpcho nhiu ng dng rng ri khc nhau. V d nh kh nng nh v vi chnh xc nano, cc c cu chp hnh p gm tr

thnh l tng i vi cng ngh nano ang ni ln. c bit, mt c cu chp hnh piezo-tube c dng trong cc knh hin vid qut (scanning probe microscopes - SPMs, xem hnh 23.20) nh v chnh xc mt u d cho c kh chnh xc c nano,sa b mt, v thu thp cc nh ca cc vt nh [3]. Cc u d c th c t trong h ta x,y vz vi mi dch chuync iu khin bi mt ngun in p c lp ( Vx(t), Vy(t), v Vz(t)). Vic qut u d c thc hin song song i vi bmt mu, dc theo trcx vy; S dch chuyn ca trcx cho php u d dch chuyn thng ng vi b mt mu. Mt m hnhton chnh xc ca cc ng lc ca c cu chp hnh piezo-tube c yu cu phn tch v thit k cc h SPM. Mt ngithit k c th khai thc thng tin bit ca h thng t m hnh ca n ci to hoc ti u mt thit k sao cho vic xydng nhanh hn v cc SPMA tin cy hn. V d mt phng php c thc hin thnh cng l phng php iu khin o(inversion-based), tm cc u vo c yu cu t c s hiu chnh chnh xc bng vic o m hnh h thng [3].Phng php ny lm vic tt nht khi cc ng lc ca h c hiu v m t tt. Ni chung, vic phn tch v thit k cc hthng iu khin cng yu cu mt m hnh h thng. V vy phn tch v thit k, n l quyt nh nhn c mt mhnh ton chnh xc m t ng x ca mt h thng. Vic m hnh ha h piezo-tube v d c xt di.

HNH 23.20 Cc thnh phn chnh ca mt knh hin vi qut u d (SPM) c dng phn tch b mt, bao gm c cuchp hnh piezo-tube, u d, v mu. The main components of a scanning probe microscope (SPM) used for surface analysis,which includes the piezo-tube actuator, the probe tip, and the sample. Cu hnh ca u d v mu tng vi cc trc ta (x,

y, vz) c phng to trong hnh.

HNH 23.21: (a) M hnh tp trung n ginca c cu chp hnh piezo-tubedc theo trczbao gm mtkhi lng, mt l so v mt gim chn [4]. Hng dng caz c ch ra bi mi tn v du + . (b) Cc lc tc

dng ln khi lng (s vt t do)

M hnh n gin ca c cu chp hnh Piezo-Tube

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Tn hiu v h thngChng ta s m hnh ha cc ng lc ca c cu chp hnh piezo-tube dc theo trc z trong u vo l in p cp Vz(t)

v u ra ca h thng l dch chuyn ca u dz(t). Chng ta bt u vic m hnh ha bng vic n gin ha h thng nhl mt khi lng c tch ra, mt l xo l tng, v mt gim chn nh ch ra trn hnh 23.21 (a). Ton b khi lng capiezo-tube c dn vo phn t khi lng m, p ng n hi ca piezo-tube c m hnh ha l mt l xo, v s gim chnca cu trc trong piezo-tube c m hnh ha nh mt gim chn hoc mt phn t ma st dnh (cc m hnh nh vy cxem l cc m hnh tp trung(lumped model) [4]). Mt quan h ton hc gia in p cp Vz(t) v dch chuyn ca u dz(t)c th nhn c bng cch dng cc nh lut vt l. p dng nh lut th 2 ca Newton (tng ca cc lc ngoi t ln mtvt bng tch ca khi lng m ca n vi gia tc ( )z t&& ) chng ta c th vit phng trnh chuyn ng nh sau:

( ) ( )= &&ii F t mz t (23.96)

nh ch ra trong hnh23.21(b)(s vt t do), c 3 lc ngoi tc ng ln piezotube. u tin, lc c to bi l xo c githit l tng ng vi dch chuyn ca u d, c ngha l:

( ) ( )= sF t kz t (23.97)

trong kl hng s l xo vi n v [N/m] theo h SI. Th hai, lc gim chn c xem l tng ng vi vn tc cau d ( )z t&& , c ngha l:

( ) ( )= &DF t cz t (23.98)

trong c l h s ma st dnh vi n v l [Ns/m] theo SI. Th ba, l sc cng trong vt liu p gm tng ng vi

in p cp Vz(t) [5], v theo nh lut Hook th ng sut cn bng vi sc cng . V vy, lc Fp(t) (ng sut nhn vidin tch thit din) cn bng vi in p cp Vz(t), tc l:

( ) ( )P z F t bV t = (23.99)

trong b l hng s vi n v l [N/V] theo SI. Vit li phng trnh (23.96) theo ba lc ngoi th phng trnh trthnh:

3

1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )=

= + + = + = & &&i s D P z i

F t F t F t F t kz t cz t bV t mz t (23.100)

c gi l m hnh khi lng- l xo - gim chn. Ch rng quan h gia in p vo Vz(t) v dch chuynz(t) ca ud (tc l m hnh ng lc hc) l phng trnh vi phn bc hai. p ng ca u d (dch chuyn ca khi lng m) i viin p Vz(t) c th nhn c trong min tn s bng cch dng phng php bin i Laplace [6, Chng 2, mc 5]; tuy nhinphng php khng gian - trng thi c th c dng nhn c li gii trc tip trong min thi gian. Trong cc mc cnli, phng php khng gian trng thi m hnh ha s c biu din v m hnh khi lng- l xo - gim chn ca ccu chp hnh piezo-tube s c ly lm v d.

Cc trng thi ca mt h thng

Chng ta bt u bng vic a ra khi nim trng thi l c s cho phng php khng gian - trng thi. Ni chung, mttrng thi c th c nh ngha nh sau:

Trng thix(t0) ca mt h ng ti thi im t0 l mt tp cc bin m cng vi u vo u(t), vi 0t t , xc nh ng xca h vi tt c 0t t [7, Chng 2, mc 1.1].

C s ca nh ngha ny l cho rng trng thi s phn nh cu hnh hin thi ca mt h thng. V vy k c ca mt hng c gi trong cc bin trng thi thi im hin thi t0 (c gi l iu kin u), v ng x trong tng lai ca mt h

ng c xc nh bi iu kin ux(t0) v u vo u(t), vi 0t t . Trng thi ca mt h c th c vit nh sau:

1

2

( )

( )( )

( )

=

M

n

x t

x tx t

x t

(23.101)

trong n l s ca cc trng thi.1Mt tp bt k cc bin tha mn nh ngha trn c th l mt trng thi ng, v vytrng thi khng phi l duy nht [8, Chapter 2, section 2].

V d

Cc bin trng thi m t h khi lng - l xo - gim chn c th c chn l v tr z(t) v vn tc( )z t&

ca khi lng.Chng ta c th vit vc t trng thi l:

1 V tp ti thiu cc trng thi c yu cu m t mt h (s thc hin ti thiu), c th thy trong [7, Chng 7].33


34/46

S tay C in t

1

2

( ) ( )( )

( ) ( )

= =

&x t z t

x tx t z t

(23.102)

Trong s ca cc trng thi l hai (n= 2). Nu v trz(t) v vn tc ( )z t& ca khi lng l bit thi im t0, cng viin p Vz(t) c nh ngha cho 0t t , th ng x trong tng lai ca h thng (tc l trng thi x(t)) c th c xc nhbng cch gii phng trnh vi phn (23.100).

Phng trnh khng gian trng thi tuyn tnh v li giiCho mt h tuyn tnh, s tin trin ca cc trng thi ca h theo thi gian c th c m t bng tp cc phng trnh vi

phn tuyn tnh bc nht sau:

1

1 11 1 1 11 1 1

22 21 1 2 21 1 2

1 1 1 1

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

= = + + + + +

= = + + + + +

= = + + + + +

& L L

& L L

M

& L L

n n p p

n n p p

nn n nn n n np p

dx tx t a t x t a t x t b t u t b t u t

dt

dx tx t a t x t a t x t b t u t b t u t

dt

dx tx t a t x t a t x t b t u t b t u

dt)t

(23.103)

trong n l s trng thi (hoc bc ca h) v p l s u vo.2 nh ngha vct u vo nh sau:

1

2

( )

( )( )

( )

=

M

p

u t

u tu t

u t

(23.104)

v vc t trng thix(t) nh c nh ngha trong phng trnh (23.101), tp cc phng trnh vi phn bc nht nhn cbng phng trnh (23.103) c th c vit li theo dng ma trn nh sau: [8, Chng 2, mc 2]

11 12 111 12 1

21 22 221 22 2

1 2 1 2

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

= +

= +

LL

LL

& M M O M M M O M

L L

pn

pn

n n nn n n np

b t b t b t a t a t a t

b t b t b t a t a t a t

x t x t u t

a t a t a t b t b t b t

A t x t B t u t

(23.105)

trong A(t) l mt ma trn nn vB(t) l mt ma trn np. Cho mt h c xc nh bng q u ra y(t) c gi thit lmt t hp tuyn tnh ca trng thi x(t) v u vo u(t), chng ta c th vit phng trnh ra nh sau:

11 12 111 12 1

21 22 2 21 22 2

1 2 1 2

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

= +

= +

LL

L L&

M M O M M M O M

L L

pn

n p

q q qn q q qp

d t d t d t c t c t c t

c t c t c t d t d t d t x t x t u t

c t c t c t d t d t d t

C t x t D t u t

(23.106)

trong C(t) l mt ma trn q n vD(t) l mt ma trn qp. Ni chung, cc ma trnA(t), B(t), C(t), v D(t) l thay itheo thi gian; tuy nhin, trong chng ny chng ta s ch xt n trng hp bt bin theo thi gian, trong A, B, C, v D lcc ma trn khng i, khi phng trnh. (23.107) v (23.108) c gi l cc phng trnh trng thi v u ra tuyn tnh btbin theo thi gian (LTI),3

( ) ( ) ( )x t Ax t Bu t = +& (23.107)

( ) ( ) ( )y t Cx t Du t = + (23.108)

p ng ca h i vi mt u vo ng dng c th c nh lng bng tin trin ca trng thi x(t) ca h v u ray(t). Phng trnh khng gian trng thi (23.107) l tp cc phng trnh vi phn bc nht dng ma trn c th c gii theo

thi gian nhn c iu kin ux(t0) nh sau [8, Chng 3]

2 Nhn c mt phng trnh vi phn bc cao hn, mt tp cc phng trnh vi phn bc nht c th nhn c bng mt th tc c bit nh vic gimbc nht nh trnh by trong [9].

3 Li gii chi tit ca cc phng trnh tuyn tnh khng n nh theo thi gian c th thy trong [7, Chng 4, mc 5].

34


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Tn hiu v h thng0

0

( ) ( )

0( ) ( ) ( ) = +

tA t t A t

tx t e x t e Bu d (23.109)

Ch rng li gii (23.109) l tng ca hai s hng: s hng u l kt qu ca iu kin u x(t0) v s hng th hai l ktqu ca u vo ng dng u(t) gia 0t t .4 Dng phng trnh ra (23.108) v li gii trng thi nhn c bng (23.109),th u ray(t) s tr thnh:

0

0

( ) ( )

0( ) ( ) ( ) ( ) = + +

tA t t A t

ty t Ce x t Ce Bu d Du t (23.110)

p ng ca hy(t) i vi u vo u(t) c c trng bi cc ma trn ca h (A, B, C, D). V d u ray(t) s b gii hnvi mt u vo gii hn bt k nu h n nh v h n nh nu cc phn thc ca tt c gi tr ring caA nh hn 0 (mhon ton) [8, Chng 4, mc 4].5

V d

Vi h khi lng - l xo - gim chn v d, phng trnh khng gian trng thi c th tm c bng cch ly vi phn trngthix(t) c xc nh trong phng trnh (23.102) v dng phng trnh chuyn ng (23.100) nhn c:

1 2

1 1 2

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

= =

= = + = +

& &

& && &z

x t z t x t

k c b k c bx t z t z t z t V t x t x t u t

m m m m m m

(23.111)

Chng ta chn v tr ca khi lngz(t) l u ra ca h, v vit phng trnh khng gian trng thi v u ra theo dng nhnc bng cc phng trnh (23.107) v (23.108) nh sau:

0 1 0( ) ( ) ( )

( / ) ( / ) /

= +

&x t x t u t k m c m b m

(23.112),

[ ]( ) 1 0 ( )y t x t = (23.113)

Tuyn tnh ha cc h phi tuyn

Mt dng tng qut ca phng trnh khng gian trng thi (cho cc h phi tuyn) l:

( ) ( , )x t g x u=& (23.114),

( ) ( , )y t h x u= (23.115)

trong g v h c th l cc hm phi tuyn.6 ng x ca cc h phi tuyn l vt khi gii hn ca mc ny; tuy nhin mt bnlun chi tit c th tm thy [10]. ng x ca mt h phi tuyn c th c xp x bng mt m hnh tuyn tnh ln cn imcn bng. V vy vic tuyn tnh ha c th n gin ha vic vic phn tch v thit k cc h phi tuyn bi v cc cng c c pht trin cho cc h tuyn tnh c th c dng vi cc iu kin no [10]. x0 v u0 tr thnh im cn bng vu vo cn bng nh [10, Chng 1]

0 0( , ) 0g x u = (23.116),

0 0 0( , )h x u y= (23.117)

Xt cc nhiu nh ca im cn bng 0( ) ( )x t x x t = + , u vo 0( ) ( ) ( )u t u t u t = + , v u ra 0( ) ( )y t y y t = + . Nu nhiu

( )x t l nh vi tt c t, th chng ta nhn c phng trnh di bng cch khai trin (23.114) thnh chui Taylor (b qua ccs hng bc cao ca ( )x t v ( )u t ):

0 0

0 0

0 0 0

0 0

( ) ( ( ), ( ))

( ) ( , ( ) ( )= == =

+ = + +

= + +

&&

& x x x xu u u u

x x t g x x t u u t

g gx t g x u x t u t

x u(23.118)

Chp nhn rng g(x0,x) = 0 chng ta nhn c:

( ) ( ) ( )x t Ax t Bu t = +& (23.119)

trong :

4 Lnh ca MATLAB 1sim m phng p ng thi gian ca cc m hnh LTI vid cc u vo bt k.5 Lnh ca MATLAB eig(A) iu chnh gi tr ring ca ma trn h thng A.6 Lnh ca MATLAB ode45 c th c dng nhn c li gii s cho phng trnh khng gian trng thi tng qut

35


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37/46

Tn hiu v h thngvi cng tn s v mt s dch pha [6, Chng 8]. Bin ca tn hiu ra l bin ca tn hiu vo c cn chnh bi M,gi l h s khuych i cng . H s khuych i cng ny nhn c bng cch ly ln ca G(s) ti s=j, tc:

( )s j

M G s=

=(23.129)

Thng h s khuych i cng M c biu din theo n v(dB), tc M[dB] = 20logM. dch pha l gcca G(s) c ly ti s=j, tc l:

( )s j

G s

=

= (23.130)

vi n v . Biu h s khuych i cng M v dch pha i vi tn s l s biu din bng th p ngtn s (cc biu Bode) ca mt h.9Cc biu ny c th nhn c bng thc nghim bng cch o h s khuych i ln v dch pha gia tn hiu vo v p ng ra ca mt h trn mt di tn s. Ngoi ra, hm truyn ca mt h c th nhnc t d liu ca p ng tn s thc nghim bng cch dng phn mm thch hp. Trong mc "M hnh ha bng thcnghim dng p ng tn s", chng ti trnh by phng php ny xc nh m hnh cho mt h dng d liu ca p ngtn s thc nghim.

Tnh khng gian trng thi t hm truyn

Trong mc Tnh hm truyn t khng gian trng thi, mt m hnh hm truyn nhn c cho mt h theo dng khnggian-trng thi. Sau y, mt phng php thc hin m hnh khng gian trng thi t mt hm truyn G(s) c trnh by. Chomt hm truyn kh thi G(s) ca h SISO c dng:

1

0 1

1

1

( )

+ + +=

+ + +L

L

n n

n

n n

n

b s b s bG s

s a s a(23.131)

dng khng gian trng thi kinh in d iu khin c vit theo cc h s ca G(s) l:

1 2 1 1

1 0 0 0 0

( ) 0 1 0 0 ( ) 0 ( )

0 0 1 0 0

= +

L

L

& L

M M O M M M

L

n na a a a

x t x t u t (23.132)

1 1 0 2 2 0 0 0( ) [( ) ( ) ( )] ( ) [ ] ( )n ny t b a b b a b b a b x t b u t = +L (23.133)

trong s cc trng thi n bng ly tha cao nht ca mu s ca G(s).10Th nguyn nh nht c th thc hinmt h, c gi l thc hin ti thiu, l mt nhn t quan trng cn xem xt khi phn tch v thit k. 11Cc m hnh vi bc tithiu i hi tnh ton t hn khi m phng v thc hin so vi cc m hnh bc cao hn. Cc thng tin v cc dng khng gian -trng thi kinh in tng ng khc tham kho [7, Chng 4, mc 3 v 4].

HNH 23.22 Mt s thc nghim c dng xc nh p ng tn s ca c cu chp hnhpiezo-tube. Mt cm bin cm ng o dch chuyn ca c cu chp hnh dc theo trcx, v d liu ca p ng tn s t

DSA c dng nh gi m hnh h thng

9 Lnh ca MATLAB bode v biu cho s thay i ca h s khuych i cng v pha ca mt h tuyn tnh10 Lnh ca MATLAB tf2ss to ra thc hin dng kinh in d iu khin ca hm truyn G(s). Cc lnh c ch khc bao gm ss2tf, zp2tf, v

tf2zp11 Chi tit v cc thc hin ti thiu cho cc h nhiu u vo, nhiu u ra, c th thy trong [7, Chng 7]

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S tay C in t

HNH 23.23 Biu thc ca h s khuych i v pha theo tn s ca c cu chp hnh piezo-tube c o dc theo trc x vip ng tn s ca m hnh xp chng. ng lin nt biu din d liu thu c bng thc nghim; ng t nt biu din cc

kt qu t m hnh tnh ton.

M hnh ha da vo thc nghim dng p ng tn s

Mt phng php m hnh ha dng d liu ca p ng tn s thu c t thc nghim c biu din trong mc ny.Dng mt b phn tch tn hiu ng (DSA), p ng tn s ca cc tn hiu ng dc theo trc x ca c cu chp hnh piezo-tube c o.12Mt in p u vo dng sin Vx(t) vi s bin i tn s gia 10 Hz v 6 kHz c to ra bi mt DSA vc cp cho h knh hin vi qut u d (SPM) nh ch ra trong hnh 23.22. Bng cch dng mt cm bin cm ng, dchchuynxp(t) ca piezo-tube dc theo trcx-axis c o v c cp li cho DSA tnh p ng tn s (cc biu M v theo tn s ). Hnh 23.23 ch ra cc biu Bode nhn c bng DSA gia in p cp Vx(t) v tn hiu ray(t) cm ng cmbin. Mt c tnh ca m hnh h thng t d liu ca p ng tn s c tm thy bng phn mm MATLAB.13Hm truyngia in p vo Vx(t) v tn hiu ray(t) ca cm bin cm ng c tm thy l:

1

5 4 9 3 15 2 18 23

6 4 5 9 4 13 3 17 2 21 25

( )( )

( )

5.544 10 7.528 10 1.476 10 4.571 10 9.415 10

1.255 10 1.632 10 1.855 10 6.5 10 6.25 10 1.378 10

x

Y sG s

V s

s s s s

s s s s s s

=

+ + = + + + + + + (23.134)

vi n v V/V. Phng trnh (23.135) c cn chnh bng h s khuych i ca cm bin cm ng (30 /V) v hmtruyn gia in p cp Vx(t) v dch chuyn thc ca u d piezo-tubexp(t) nhn c bng:

2

7 4 11 3 16 2 20 25

6 4 5 9 4 13 3 17 2 21 25

( )( )

( )

1.663 10 2.258 10 4.427 10 1.371 10 2.825 10

1.255 10 1.632 10 1.855 10 6.5 10 6.25 10 1.378 10

=

+ + =

+ + + + + +

p

x

X sG s

V s

s s s s

s s s s s s

(23.135)

vi n v /V.

Cn chnh thi gian ca mt m hnh hm truyn

Di y chng ti trnh by mt phng php cn chnh thi gian cho G2(s) t [s] ti [ms]. Chng ti tm tt li tnhcht cn chnh theo thi gian ca hm truyn Laplace transform c trnh by [1, Chng 3, mc 1.4]. F(s) l hmtruyn Laplace ca f(t), tc l :

( ) ( )L

f t F s (23.136)

trong L k hiu ton t ca hm truyn Laplace. By gi, xt mt php cn chnh theo thi gian mi c nhngha l t at= trong a l hng s. Hm truyn Laplace ca ( ) ( )f t f at = nhn c bng :

1 ( ) ( ) ( ) = =

L s f t f at F F s

a a(23.137)

Dng cng thc (23.137), chng ta c th gim bt cc h s ca G2(s) bng cch thay i n v thi gian ca c tn hiuvo u(t) v ra y(t) nh sau:

12 Cc h thng Stanford Research, m hnh SR78513 Lnh ca MATLAB invfreqs a ra cc h s thc ca t s v mu s ca d liu p ng tn s c xc nh bng thc nghim

38


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40/46

S tay C in t

HNH 23.24 S khica mt h thi gian ri rc vi cc tn hiu c ch ra dng th. Ch rng u[k]= u(k.T) vy[k]=y(k.T) , vi k=0,1,2,, v chu k trch mu Tc gi thit l hng s

Tn hiu vo u[k] v tn hiu ray[k] ca h ny l tn hiu ri rc vi u[k]= u(k.T) vy[k]=y(k.T) vi k= 0,1,2, trong Tl chu k trch mu l hng s . Tn hiu vo u[k] c cp cho h lin tc ny l t mt my tnh s hoc b vi x lv c gi khng i trong khong thi gian T(gi bc khng). Mt b trch mu s thu thp tn hiu ra ca h lin tc mikhong thi gian Tto ra tn hiu ra ri rcy[k]. H ri rc gia tn hiu vo u[k] v tn hiu ray[k] [11, Chng 1].14

Biu din tng ng khng gian trng thi ca h thi gian ri rc ca m hnh khng gian trng thi ca h thi gian lin tcnhn c bng cc phng trnh (23.107) v (23.108) l (Chi tit c th tm thy trong [11, Chng 5, mc 5])

[ 1] [ ] [ ]D Dx k A x k B u k + = + (23.142)

[ ] [ ] [ ]D Dy k C x k D u k = + (23.143)

trong

( )0, , = = = =

T AT A

D D D DA e B e d B C C D D (23.144)

v cc ma trn CDand DDkhng b thay i bi vic trch mu.15M hnh ri rc ny (cc phng trnh (23.142) v(23.143)) l biu din ca h trch mu d liu ch ra trn hnh 23.24.

Li gii cho cc phng trnh khng gian - trng thi ca h thi gian ri rc

Li gii cho m hnh ri rc (cc phng trnh (23.142) v (23.143)) l

11

0

[ ] [0] [ ]

=

= + k

k k j

D D D

j

x k A x A B u j (23.145)

11

0

[ ] [0] [ ] [ ]

=

= + +k

k k j

D D D

j

y k CA x C A B u j Du k (23.146)

cho mi bc trch mu k. Chi tit thit lp cng thc c th tm thy [11, Chng 5, mc 3]. p ng trng thi x[k] ivi tn hiu vo u[k] c c trng bi cc ma trn h thng (AD, BD ,CD, DD). c bit, tn hiu ray[k] s b gii hn vi btk tn hiu vo u[k] b gii hn nu h l n nh. Mt h dng phng trnh (23.142) l n nh nu ln ca tt c cc gi

tr ring caADl nh hn n v, tc l, nm trong ng trn n v m tm l gc ca mt phng z [11, Chng 5, mc 6].

Bin i zv mi quan h vi khng gian - trng thi

Quan h vo - ra trong min tn s ca cc h thi gian ri rc c biu din bng mt hm truyn ri rc c gi l biniz, c vit theo cc s hng ca binz[12, Chng 4]. Tng t vi h thi gian lin tc, m hnh mt h ng theo dnghm truyn ri rc c th c dng trong thit k v iu khin cc h [12, Chng 7]. Nu m hnh h thng l c th dnghm truyn ri rc th vic thit lp khng gian trng thi c th c thy nh ch ra di. Cho mt h ri rc c m tbng binzsau:

1

0 1

1

1

( )1

+ + +=

+ + +L

L

n

n

n

n

d d z d z G z

c z c z (23.147)

Dng khng gian trng thi kinh in d kim sot ca G(z) l

1 2 1 1

1 0 0 0 0

[ 1] 0 1 0 0 [ ] 0 [ ]

0 0 1 0 0

+ = +

L

L

L

M M M M M

L

n nc c c c

x k x k u t (23.148)

1 1 0 2 2 0 0 0[ ] [( ) ( ) ( )] [ ] [ ] [ ]= +L n ny k d c d d c d d c d x k d u k (23.149)

S trng thi n tng ng vi ly tha cao nht ca a thc mu s ca G(z). Cc thng tin v cc dng khng gian trngthi kinh in khc tham kho ti [11, Chng 5, mc 2].

V d

14 Chng ta khng bn vic lng t ha v li ca vic lng t ha. Cacs chi tit xem trong [11, Chng 1, mc 3]15 Vi m hnh khng gian - trng thi thi gian lin tc (A, B, C, D), lnh ca MATLAB c2d s a ra tng ng thi gian ri rc cho mt chu k trch

mu T40


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42/46

S tay C in t

vy l mt tn hiu c trng ca h. p ng xung (impulse response) l m t khc ca h thng. Vi mt h c biu dinbng cc phng trnh vi phn tuyn tnh, p ng bc n v l tch phn ca p ng xung n v.

Nu ta biu din vi phn theo thi gian ( /d dt& ) bng ton t o hm theo thi gian p. Khi chng ta c th k hiu ohm theo thi gian ca mt tn hiu ybng py, o hm bc hai ca n l p2y, tch phn ca n theo thi gian bng (1/p)y vtng t. K hiu ton t nh vy lm n gin vic biu din cc phng trnh vi phn. Chng ta s dng cm t cc phngtrnh h thng ch tp hp cc phng trnh vi phn xc nh ton b ng x ca cc bin ph thuc xut hin trong ccphng trnh . Chng ta c th gim mt tp tuyn tnh cc phng trnh ca h thnh mt phng trnh vo-ra n ca h

bng cch loi tr hu ht cc bin ph thuc khi tp . Hm truyn (transfer function) cng vi bin ph thuc l mt biuthc ton hc cha tt c cc thng tin ct yu th hin trong phng trnh vi phn ca h. Bin i Laplace bin i cc tn hiu(cc hm thi gian) thnh cc hm ca mt bin tn s phcs = +j . l mt s tng ng mt mt gia mt tn hiu vbin i Laplace ca n. Chng ta c th tm li c hm thi gian bng php bin i ngc. Bin i Laplace s to ra ccnh c cc tnh cht thun tin hn cc tnh cht ca cc tn hiu gc. c bit, vi phn theo thi gian mt tn hiu s tng ngvi vic nhn bin i Laplace ca n vi bin tn s phc s. V vy, php bin i s bin cc phng trnh vi phn h s hngtuyn tnh thnh cc phng trnh i s tuyn tnh. Vic n gin ha nh vy ca cc php ton min thi gian lm cho bini Laplace tr nn c ch. Bin i Laplace cng bin p ng xung ca mt bin h thng thnh hm truyn cho bin . Vvy chng ta c th thy phng trnh vi phn biu din mt h tuyn tnh nh l mt biu thc ca p ng ca h i vimt tn hiu vo dng xung.

Cc hm truyn

Cc dch chuyn ca ntx1 vx2 v cc lc nnf1 vf2 bn trong cc nhnh ca m hnh tng th hnh 23.25 lin quan timi ci khc bng phng trnh l xo, phng trnh gim chn, v s cn bng ca cc lc nt 2. Phng trnh ca l xo lf2 =k(x1 .x2). Phng trnh cho b gim chn l f1 = b(px1 .px2). Vic cn bng cc lc i hi f1 +f2 = mp2x2. Cc phng trnh nym t y ng x ca h nu l xo v khi lng khng c tip lc. (Nu khi lng ang chuyn ng v hoc l xo angb nn th chng ta c th phi biu din ring r cc trng thi nng lng ban u ca chng m t y cc quan h sauny gia cc bin.)

Loif1 vf2 khi cc phng trnh nhn c phng trnh hot ng:

2

2 1( ) ( )mp bp k x bp k x+ + = + (23.152)

Phng trnh vi phn ny m t y quan h ca trng thi khng gia x1 vx2. Sp xp li phng trnh (23.152) thnhdng t s:

22

1+= + +x bp kx mp bp k (23.153)

Chng ta gi (23.152) l hm truyn tx1 nx2. Hm truyn tp trung ch vo cc php ton hc c trng cho cc quanh ng x hn l vo cc bn cht c th ca cc bin. (Ch rng hm truyn t v1 n v2 trong v1 =px1 v v2 =px2 l ginghm truyn nhn c bi phng trnh (23.153).)

Ni chung, gi thit rngy1 vy2 l hai bin lin quan (theo k hiu ton t) bi phng trnh vi phn tuyn tnh

2 1( )y G p y= (23.154).

Chng ta nh ngha chnh thc hm truyn ty1 tiy2 bng:

2

1

( ) =zs

yG p

y

(23.155)

trong k hiu ZS l trng thi khng. Nuy1 l bin c lp th G(p) l hm truyn vo-ra ca biny2 v miu t y ngx ca n do tn hiu vo y1. Chng ta c th xc nh t hm truyn ng x ca h i vi tn hiu ngun v trng thi banu bt k.

HNH 23.25 M hnh tng th ca mt h c

Bin i Laplace

Bin i Laplace mt v, L, l ton t tch phn bin mt tn hiuf(t) thnh mt hm phcF(s) c dng nh sau:

0

[ ( )] ( ) ( )

=

stf t F s f t e dt L (23.156)

Chng ta gi hm truynF(s) l bin i Laplace ca tn hiuf(t). Hy tng tng gii hn di 0 ca tch phn l mtkhong thi gian ngay trc t=0. Thng thng, dng k hiu ch thng (f) biu din dng tn hiu v ch hoa (F) biu

42


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Tn hiu v h thngdin bin i Laplace ca n. (mc d chng ta ni y l cc tn hiu thi gian, nhng khng c ci g trong phng trnh(23.156) yu cuf(t) l mt hm ca thi gian. Hm truyn c th c dng i vi cc hm ca i lng tno .)

Chng ta phi dng bin i Laplace bin i cc tn hiu ca cc h tuyn tnh bt bin theo thi gian. ng x ca mt hnh vy vi t 0 ch ph thuc vo tn hiu vo vi t 0 v vo trng thi trc ca bin ra ( 0t = ). V vy, khng c ngharng bin i Laplace b qua f(t) vi 0t < . Qu trnh tm hm thi gian f(t) tng ng vi mt bin i Laplace c th F(s)

c gi l bin i Laplace ngc, v c k hiu l L-1. Chng ta cng gif(t) l bin i Laplace ngc caF(s). V bini Laplace mt v b qua 0t < , nnF(s) khng cha thng tin v f(t) vi t + +

s t

t st

t j t

eF s e e dt

s

e es

s s

(23.157)

Chng ta phi yu cu > , trong l phn thc cas, h s m thc hi t ti 0 gii hn trn. ( ln ca h sm phc gi nguyn l 1 vi tt c t). V vy, bin i Laplace ca hm m suy gim ch c nh ngha vi Re[s] > - .S

hn ch trn minFny trong mt phng phcs c th so vi s hn ch 0t

trn minf.Cc c tnh quan trng ca hm tn s phc 1/(s + ) l tnh trng mt cc n v v tr ca cc ,s = - [rad/s]. (Cc ny

xc nh gii hn tri ca min ca mt phng phc s m trn hm truyn 1/(s + ) c xc nh). Cc c tnh quan trngca hm thi gian tng ng l tnh suy gim v tc suy gim vi lu tha - [rad/s]. C s tng ng r rng gia cc ctnh caf(t) v F(s). Chng ta nn ngh ton b hm phc Ftng ng vi ton b tn hiu thi gian f. Nh v d bin i thhai, tf(t) = (t), xung n v vi khong thi gian rt ngn. N tc ng ti thi im t= 0, ch va vi gii hn thp ca tchphn Laplace. N c gi tr 0 0t = . (Bi v chng ta dng 0 nh gii hn di ca tch phn xc nh, nn n khng nhhng g khi m xung dng t = 0 hoc bt u tng t = 0). Xung ch khng l khng vi 0t m 1ste V vy bini Laplace l

0 0( ) ( ) ( )(1) 1sts t e dt t dt

= = (23.158)

BNG 23.11 Cc cp bin i Laplace1( ) [ ( )], 0f t F s t = L ( ) [ ( )]F s f t = L

( )t 1

( )su t1

s

, 1, 2, ...nt n =1

!n

n

s +

te 1

s +

, 1, 2, ...n tt e n =1

!

( )

n

n

s +

+

43


44/46

S tay C in t

0sin( )t

2 2

0

s

s +

0cos( )t

2 2

0

s

s +

sin( )t de t

2 2( )d

ds

+ +

cos( )t de t

2 2( ) d

s

s

++ +

Khng cn phi dn ra bin i Laplace cho mi tn hiu m chng ta s dng khi nghin cu h thng. Bng 23.11 a racc bin i cho mt vi dng tn hiu thng gp cc h ng lc.

Cc thuc tnh ca hm truyn

Mt s thuc tnh c ch ca bin i Laplace c tm tt trong bng 23.12. Theo tnh cht o hm, s nhns tc nghon ton ging ton t o hm thi gian, nhng trong min ca cc tn hiu c bin i Laplace. Khi chng ta bin iLaplace phng trnh cho mt phn t lu gi nng lng nh l khi lng hoc l xo, th tnh cht o hm t ng kt hptrng thi nng lng ban u ca phn t chnh l gi tr ca bin ti 0t = .

BNG 23.12 Cc tnh cht ca bin i Laplace

Khuych i [ ( )] ( )af t aF s=L

Cng1 2 1 2

[ ( ) ( )] ( ) ( )f t f t F s F s+ = +L

o hm [ ( )] ( ) (0 )f t sF s f = &L

Cc o hm 2[ ( )] ( ) (0 ) (0 )f t s F s sf f = && &L

Tch phn0

( )( )

t F sf t dt

s = L

Tch chp1 2 1 2

0( ) ( ) ( ) ( )

t

f f t d F s F s

= L

Gi tr u0

(0 ) lim ( ) lim ( )st

f f t sF s+

+

= =

Gi tr cui0

( ) lim ( ) lim ( )t s

f f t sF s

= =

Tch phn xc nh0 0

( ) lim ( )s

f t dt sF s

=

Suy gim theo hm m [ ( )] ( )te f t F s = +L

Suy gim 00 0 0

[ ( ) ( ) ( ) 0t ssf t t u t t e F s t = L

Nhn thi gian ( )[ ( )]

dF stf t

ds= L

Chia thi gian ( )( )

s

f tF s ds

t

= L

Cn chnh thi gian ( / )[ ( )]

F s af at

a=L

Ti nguyn: Dorny, C,N. 1993. Understanding Dynamic Systems. P413. Prentice-Hall, Englewood Cliffs, NJ. With permission.

44


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Tn hiu v h thngKhi chng ta bin i Laplace phng trnh ca h vo - ra ca mt bin ring bit th tnh cht o hm t ng lin kt ton

b trng thi trc ca h. V vy chng ta c th tm thy li gii cho phng trnh ca h m khng cn xc nh cc iukin u ( 0t += ) - l mt s n gin ng k cho qu trnh gii. V F(s) cha tt c cc thng tin v f(t) vi 0t nn cth tm thy mt vi c trng ca tn hiu f(t) t hm truynF(s) m khng cn thc hin php bin i Laplace ngc. Cc tnhcht 7 9 ca bng 23.12 a ra ba trong s cc c trng , gi l gi tr u ( 0t + ), gi tr cui (t ), v vng didng sng. Cc tnh cht cn li trong bng ch ra nh hng ca cc s thay i khc nhau ca dng tn hiu ln php bin i.Phng php thng dng tm thy cc bin i ngc l s dng mt bng cc cp bin i. Bng phi c lu gitrong ti phn mm nh CC, MATLAB, MAPLE, v.v Bng 23.11 chng minh rng cc bin i ca cc tn hiu in hnh ca

cc h t l vi cc a thc cas. T s ca cc a thc c th c tch thnh tng ca cc phn s ca a thc n gin - qutrnh c gi khai trin tng phn phn s (the partial-fraction expansion). V vy qu trnh bin i ngc c th c honthnh bi mt chng trnh my tnh kt hp vi mt bng tm tt cc bin i.

Bin i v nghim phng trnh h thng

Gi s rng mt ngun ngoi cp mt vn tc c trng v1(t) ti nt 1 ca hnh 23.25. nhn c phng trnh vo - raca h thng lin h vn tc v2 ca nt 2 vi tn hiu vo v1, nhn phng trnh (23.153) vi p v thay th v1 chopx1 v v2 chopx2. Kt qu l :

( )2 2 1( ) ( ) 23.159mp bp k v bp k v+ + = + (23.159)

Hai pha ca phng trnh (23.159) l cc hm ng nht ca thi gian. V vy, cc bin i Laplace ca hai pha ca phngtrnh (23.159) l bng nhau. V bin i Laplace l tuyn tnh (tnh cht 1 v 2 ca bng 23.12) v v cc h s ca phng trnhvi phn l cc hng s nn bin i Laplace c th c p dng mt cch ring bit cho cc s hng ring bit ca mi pha. Ktqu l :

2

2 2 2 2 2 2 1 1 1[ ( ) (0 ) (0 )] [ ( ) (0 )] ( ) [ ( ) (0 )] ( )m s V s sv v b sV s v kV s b sV s v kV s + + = +& (23.160)

trong cc tnh cht o hm ca bin i Laplace (tnh cht 3 v 4 ca bng 23.12) a ra cc gi tr trc 1 (0 ) , 2 (0 )

, v

2(0 ) & ca phng trnh. Theo phng trnh (23.160), xc nh y bin i V 2(s) ca ng x v2(t), chng ta phi ch ra

cc gi tr u v V1(s). c th ch ra rng vic ch ra ba gi tr u l tng ng vi vic ch ra cc trng thi nng lng cal xo v khi lng. Gi thit rng ngun c lp a ra vn tc khng i v1(t) = vc bt u t= 0. Hm truyn tng ng, bimc 2 ca bng 23.11 v tnh cht 1 ca bng 23.12 l V1(s) = vc/s. Thay bin i V1(s) vo phng trnh (23.160) v gii :

2

2 2 1 2

2 2

( ) (0 ) [ (0 ) (0 )] (0 )( )

( )

+ + + +=

+ +

&cbs k v msv bs v v ms vV s

s ms bs k

(23.161)

Chng ta c th tm c dng tn hiu ra v2(t) l mt hm ca cc tham s ca m hnh m, k, b, tham s ca tn hiu ngunvc, v thng tin ca cc trng thi ban u 1 (0 )

, 2 (0 ) , v 2 (0 )

& , nhng biu thc ca li gii ny c th l ln xn. Thay

v vy, chng ta hon thin qu trnh gii cho cc s c th m = 2 kg, b = 4 N s/m, k = 10 N/m, 22 (0 ) 0 /m s =& ,

1(0 ) 0 /m s = , 2 (0 ) 1 /m s

= , v vc= 1 m/s. Khai trin tng phn phn s ca bin i thun v ngc cng c thc hinbi mt chng trnh my tnh, l:

2 2 2

1 2 2( )

( 1) 2

+=

+ +s

V ss s

(23.162)

2( ) 1 2 cos(2 ), 0tv t e t t = (23.163)

Chng ta c th thu c cc bin i Laplace ca cc phng trnh h thng bt k giai on no ca qu trnh pht trin.Thm ch, chng ta c th vit cc phng trnh trc tip theo cc bin c bin i nu chng ta mun. Qu trnh kh ccbin cng c th c thc hin tt trong k hiu ny cng nh k hiu khc. V d, ton t G(p) trong phng trnh (23.154)biu din mt t s ca cc a thc trong ton t o hm thi gianp. V vy, bin i Laplace phng trnh vi phn (23.154) sa ra cc gi tr u ca cc o hm y1 vy2 . Nu cc gi tr u ca tt c cc o hm l khng th phng trnh bin iLaplace l :

2 1( ) ( ) ( )Y s G s Y s= (23.164)

trong ton tp trong phng trnh (23.154) c thay bi bin tn s phcs trong phng trnh (23.164). V vy, s l thchhp xc nh hm truyn trc tip theo cc s hng ca cc tn hiu c bin i Laplace:

2

1 0

( )

( ) ( ) == PV

Y s

G s Y s (23.165)

45


46/46

S tay C in t

trong Y1(s) v Y2(s) l cc bin i Laplace ca cc tn hiu y1(t) v y2(t), v k hiu PV = 0 c ngha rng cc gi tr u (0t = ) cay1(t) vy2(t) v cc o hm c k ra trn trong mi quan h vi phng trnh (23.164) c t ti khng. nh

ngha min tn s, phng trnh (23.165) l tng ng vi nh ngha min thi gian, phng trnh (23.155).

Cho rng tn hiu voy1(t) l xung n v (t). Th tn hiu p ngy2(t) l p ng ca xung n v ca h thng. V bin iLaplace ca xung n v l Y1(s) = (s) = 1, mc 1 ca bng23.11, phng trnh (23.164) ch ra rng bin i Laplace Y2(s) cap ng xung n v l ging ht vi hm truyn ca trng thi khng (c biu din trong min truyn). Hm truyn ca mth tuyn tnh c hai cch th hin. C hai cch th hin u tiu biu cho h thng. Trong min tn s, hm truyn G(s) l s

nhn a ra p ng bng vic nhn hm truyn ca ngun - tn hiu nh trong phng trnh (23.164). Trong min thi gian,chng ta dng mt tn hiu p ng in hnh p ng xung - m t h. Hm truyn G(s) l bin i Laplace ca p ngc trng .

nh ngha cc khi nim

Tn hiu vo (Input): Mt bin c lp.

Phng trnh h thng vo ra (Inputoutput system equation): Mt phng trnh vi phn m t ng x camt bin ph thuc n nh l mt hm thi gian. Bin ph thuc c xem l u ra ca h thng. Bin (cc bin) clp l cc u vo.

Tn hiu ra (Output): Mt bin ph thuc.

Tn hiu (Signal): Mt bin c th quan st; Mt i lng biu l ng x ca mt h thng.

Trng thi (State): Trng thi ca mt h tuyn tnh bc n tng ng vi cc gi tr ca mt bin ph thuc vn-1 o hm thi gian u tin ca n.

Bt bin theo thi gian (Time invariant): Mt h c th c biu din bng cc phng trnh vi phn vicc h s hng.

Trng thi khng (Zero state): l mt trng thi m khi khng c nng lng c lu gi hoc khi tt ccc bin c gi tr bng khng.

Ti liu tham kho

[1] Franklin, G. F., Powell, J. D., and Emami-Naeini, A. 1994. Feedback Control of Dynamic Systems, 3rd ed., AddisonWesley, Reading, MA.

[2] Kuo, B. C. 1991.Automatic Control Systems, 6th ed., Prentice-Hall, Englewood Cliffs, NJ.

[3] Nise, N. S. 1992. Control Systems Engineering, Benjamin Cumming, Redwood City, CA.

Cc thng tin b sung

Vic x l ton hc ca cc bin i Laplace mt cch chi tit c trnh by trong Advanced Engineering Mathematics,biC. Ray Wylie v Louis C. Barrett. Understanding Dynamic Systems,bi C. Nelson Dorny, p dung cc hm truyn v cc khinim lin quan vo cc hon cnh khc nhau. Cc bi bo ca cc tp ch sau s dng cc hm truyn v cc bin i Laplace:

IEEE Transactions on Automatic Control. Published monthly by the Institute of Electrical and Electronics Engineers.

IEEE Transactions on Systems, Man, and Cybernetics. Published bimonthly.

Journal of Dynamic Systems, Measurement, and Control. Published quarterly by the American Society of MechanicalEngineers.

sổ tay cdt chuong 23- signal v system

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