Chuong 5_LTDKNC.pdf

MnMn hchc

L THUYT IU KHIN NNG CAOL THUYT IU KHIN NNG CAO

Ging vin: PGS TS Hunh Thi HongGing vin: PGS. TS. Hunh Thi HongB mn iu Khin T ng

Khoa in in T i h B h Kh TP HCMi hc Bch Khoa TP.HCM

Email: [email protected]: http://www4.hcmut.edu.vn/~hthoang/

15 January 2014 H. T. Hong - www4.hcmut.edu.vn/~hthoang/ 1

p g p g

ChngChng 55gg

IU KHIN BN VNGIU KHIN BN VNG


Gii thi

NiNi dung dung chngchng 55

Gii thiu Chun ca tn hiu v h thng

Tnh n nh bn vng Cht lng bn vng Thit k h thng iu khin bn vng dng

phng php chnh li vng (loop-shaping) Thit k h thng iu khin ti u bn vng (SV

t c thm)


Feedback Control Theory J Doyle B Francis andTi liu tham khoTi liu tham kho

Feedback Control Theory, J.Doyle, B. Francis, and A. Tannenbaum, Macmillan Publishing Co. 1990.

Linear Robust Control M Green and D J N Linear Robust Control, M. Green and D. J.N. Limebeer, Prentice Hall, 1994.

Robust and Optimal Control, K. Zhou, J.C. Doyle Robust and Optimal Control, K. Zhou, J.C. Doyle and K. Glover, Prentice Hall.


GII THIUGII THIU


nh ngha iu khin bn vngnh ngha iu khin bn vng

H thng iu khin bn vng l h thng c thit k H thng iu khin bn vng l h thng c thit k sao cho tnh n nh v cht lng iu khin c m bo khi cc thnh phn khng chc chn (sai s m hnh p g (ha, nhiu lon,) nm trong mt tp hp cho trc.

y(t)

u(t)u(t) y(t) y(t)

G ++u(t)

Gu(t) y(t)

i t K ki h i i t K b G: m hnh danh nh : thnh phn khng chc chn

i tng K kinh in i tng K bn vng


: thnh phn khng chc chn

Cc thnh phn khng chc chnCc thnh phn khng chc chn

Cc yu t khng chc chn c th lm gim cht Cc yu t khng chc chn c th lm gim cht lng iu khin, thm ch c th lm h thng tr nn mt n nh.nn mt n nh.

Cc yu t khng chc chn xut hin khi m hnh ha h thng vt l. g

Cc yu t khng chc chc c th phn lm hai loi: M hnh khng chc chn M hnh khng chc chn Nhiu t mi trng bn ngoi


M hnh khng chc chnM hnh khng chc chn

M hnh khng chc chn do s khng chnh xc M hnh khng chc chn do s khng chnh xc hoc s xp x trong khi m hnh ha: Nhn dng h thng ch thu c m hnh gn Nhn dng h thng ch thu c m hnh gn ng: m hnh c chn thng c bc thp v cc thng s khng th xc nh chnh xcg g

B qua tnh tr hoc khng xc nh chnh xc tr

B qua tnh phi tuyn hoc khng bit chnh xc cc yu t phi tuyn

Cc thnh phn bin i theo thi gian c th c xp x thnh khng bin i theo thi gian hoc s


bin i theo thi gian khng th bit chnh xc.

Nhiu lon t bn ngoiNhiu lon t bn ngoi

Cc tn hiu nhiu xut hin t mi trng bn ngoi Cc tn hiu nhiu xut hin t mi trng bn ngoi, th d nh ngun in khng n nh nh ngun in khng n nh nhit , m, ma st, thay i nhiu o lng nhiu o lng


Th d: H thng khng bn vngTh d: H thng khng bn vng

i tng tht: 3)(~G i tng tht: 2)11.0)(1()( sssG

M hnh b qua c tnh tn s cao: 3)( sGi tng tht

M hnh b qua c tnh tn s cao:)1(

)( ssG

M hnh

Biu Bode ca i t thti tng tht v m hnh trng nhau gmin tn s thp, sai lch min tn s cao


tn s cao

Th d: H thng khng bn vng (tt)Th d: H thng khng bn vng (tt)y(t)r(t) y(t)r(t)

K G

B iu khin thit k da vo m hnh sssK )1(10)( s H kn khi thit k c cc ti 30, cht lng p ng tt.


Th d: H thng khng bn vng (tt)Th d: H thng khng bn vng (tt)y(t)r(t) y(t)r(t)

K G~

S dng b K thit k cho i tng tht: c tnh ng hc min tn s cao b qua khi thit k lm h


ng hc min tn s cao b qua khi thit k lm h thng khng n nh H thng khng n nh bn vng

Th d: H thng c cht lng bn vngTh d: H thng c cht lng bn vng

i t tht )(~ kG k i tng tht:1

)( TsksG

M hnh danh nh: 4)( sG53 k %)30( 5.0 T

M hnh danh nh:)15.0(

)( ssG

M h h d h h20

Bode Diagram

M hnh danh nhi tng tht

-10

0

10

M

a

g

n

i

t

u

d

e

(

d

B

)

Biu Bode ca m hnh danh nh v

-30

-20M

0

) danh nh v m hnh tht khi thng s thay i

-45

P

h

a

s

e

(

d

e

g

)


10-1

100

101

102

-90

Frequency (rad/sec)

Th d: H thng c cht lng bn vng (tt)Th d: H thng c cht lng bn vng (tt)

y(t)u(t) y(t)G

u(t)

4

5Plant response (20 samples)

2

3

A

m

p

l

i

t

u

d

e

0

1

p ng ca h h khi tn hiu vo l hm nc: b

0 0.5 1 1.5 2 2.5 3 3.5 40

Time (sec)


p ng ca h h khi tn hiu vo l hm nc: b nh hng nhiu khi thng s ca i tng thay i

Th d: H thng c cht lng bn vng (tt)Th d: H thng c cht lng bn vng (tt)y(t)r(t)

B i khi

y(t)( ) K G~

B iu khin:

1 4Closed-loop response (20 samples)sK 1)(

1

1.2

1.4s

sK4

)(

p ng ca h kn: h thng n nh

0.6

0.8

A

m

p

l

i

t

u

d

e

h thng n nh, cht lng thay i khng ng k khi

0

0.2

0.4thng s i tng thay i cht lng bn vng


0 1 2 3 4 5 6 7 8 9 100

Time (sec)

lng bn vng

M phng HT c thng s khng chc chn dng MatlabM phng HT c thng s khng chc chn dng Matlab

% Khu qun tnh bc nht vi thi hng v h s khuch i khng chc chn% Khu qun tnh bc nht vi thi hng v h s khuch i khng chc chn

>> T = ureal('T',0.5,'Percentage',30); % T = 0.5 (30%), T0=0.5>> k = ureal('k' 4 'range' [3 5]); % 3k5 k0=4>> k = ureal( k ,4, range ,[3 5]); % 3k5, k0=4>> G = tf(k,[T 1])>> figure(1); bode(usample(G,20)) % Biu Bode h khng chc chn>> figure(2); bode(tf(G nominal)) % Biu Bode i tng danh nh>> figure(2); bode(tf(G.nominal)) % Biu Bode i tng danh nh

% B iu khin>> KI 1/(2*T N i l*k N i l)>> KI = 1/(2*T.Nominal*k.Nominal);>> Gc = tf(KI,[1 0]); % B iu khin Gc(s)=KI/s>> Gk = feedback(G*Gc,1) % Hm truyn h kn

% M phng h h v h kn>> figure(3); step(usample(G,20)), title('Plant response (20 samples)')

fi ( ) ( l ( k )) i l ( l d l ( l ) )


>> figure(4); step(usample(Gk,20)), title('Closed-loop response (20 samples)')

Cc phng php thit k HTK bn vngCc phng php thit k HTK bn vng

Cc phng php phn tch v tng hp h thng Cc phng php phn tch v tng hp h thng iu khin bn vng: Phng php trong min tn s Phng php trong min tn s Phng php trong khng gian trng thi


S lc lch s pht trin LTK bn vngS lc lch s pht trin LTK bn vng

(1980 ): iu khin bn vng hin i (1980-): iu khin bn vng hin i u thp nin 1980: Phn tch ( analysis) Gia thp nin 1980: iu khin H v cc phin Gia thp nin 1980: iu khin H v cc phin

bn Gia thp nin 1980: nh l Kharitonov Gia thp nin 1980: nh l Kharitonov Cui 1980 n 1990: Ti u li nng cao, c bit

l ti u LMI (Linear Matrix Inequality)l ti u LMI (Linear Matrix Inequality) Thp nin 1990: Cc phng php LMI trong iu

khinkhin


CHUN CA CHUN CA TN HIU V H THNGTN HIU V H THNG


nh ngha chun ca vectornh ngha chun ca vector

Cho X l khng gian vector Mt hm gi tr thc || || Cho X l khng gian vector. Mt hm gi tr thc ||.||xc nh trn X c gi l chun (norm) trn X nu hm tha mn cc tn cht sau:hm tha mn cc tn cht sau:

0x00 xx

axaax , axaax ,yxyx

ngha: chun ca vector l i lng o di ca vector


ca vector

Cc chun vector thng dngCc chun vector thng dng

Cho nTxxx ][xCho nxxx ],...,,[ 21xp

npx:x Chun bc p:

n

p

iip

x

1

:x Chun bc p:

n

iix

11

:x Chun bc 1:

n

iix

1

22

:x Chun bc 2:

inix

1max:x Chun v cng:


Tnh chun vector Tnh chun vector Th d 1 Th d 1

Cho T]2031[Cho

41 ixx Chun bc 1:T]2031[ x

62031 11 i

ixx

4 2 Chun bc 2:62031

1420)3(1 222

1

22

iixx Chun bc 2:

Ch

1420)3(1 222

ii x41max x Chun v cng: 32,0,3,1max


nh ngha chun ma trnnh ngha chun ma trn

Cho ma trn A=[a ]Cmn Chun ca ma trn A l:Cho ma trn A=[aij]Cmn. Chun ca ma trn A l:p

AxA sup: Chun bc p:

pp xx 0

p

p

Chun bc 1: m amax:A (tng theo ct) Ch b 2 )( * AAA

Chun bc 1:

i

ijnja

111

max:A (tng theo ct)

Chun bc 2: )(max:12

AAA ini trong A* l ma trn chuyn v lin hp ca A,

l cc tr ring ca . )( * AAi AA* Chun v cng: n amax:A (tng theo hng)


Chun v cng:

j

ijmia

11max:A (tng theo hng)

Tnh cht ca chun ma trnTnh cht ca chun ma trn

nn CAA ,0

nn CC AAA00 AA

nn CC AAA ,,. nn CBABABA CBA,BABA ,

nn CBA,BAAB ,


Tnh chun ma trn Tnh chun ma trn Th d 1Th d 1

Ch t 2jACho ma trn

202: jA

Chun bc 1: 2max aA 4|)2||2(||)0||(|max j Chun bc 1:

121

1max

iijj

aA

Chun bc 2:*

4|)2||2(||),0||(|max j

)(max: *212

AAA ii

8221

202

220* jjjAA 822022 jAA

0)det()()( *** AAIAAAA soleig 5311.8 4689.021

Chun v cng: 2A 2 9208.25311.8,4689.0max:

12

niA

3|)2||0(||)2||(|15 January 2014 H. T. Hong - www4.hcmut.edu.vn/~hthoang/ 32

Chun v cng:

121

max:j

ijiaA 3|)2||0(||),2||(|max j

Tnh chun ma trn Tnh chun ma trn Th d 2Th d 2

1jCho ma trn

321

:j

jA

Tnh chun : , , 1A 2A A


Chun ca tn hiuChun ca tn hiu

Chun ca t/hiu x(t) [ +] c nh ngha l:Chun ca t/hiu x(t) [,+] c nh ngha l:p

p dttxtx )(:)( Chun l : dttt )()(Ch l

p

tp

dttxtx

)(:)( Chun lp:

t

dttxtx )(:)(1 Chun l1:

( b 2

Chun l2:

t

dttxtx )(:)( 22

(cn bc 2 ca nng lng ca tn hiu)

)(sup:)( txtxt

Chun l : ngha: Chun ca tn hiu l i lng o ln

(gi tr cc i ca t/h)


ngha: Chun ca tn hiu l i lng o ln ca tn hiu

Tnh chun ca tn hiu Tnh chun ca tn hiu Th d 1Th d 1

1/1 ttCho tn hiu:

10

1/1)( ttttx

t

dttxtx )()(1

Chun l1:

1

1ln1

t

tdttt

2/1

2 1t

112/12/1

Chun l2 : 22 )()(

tdttxtx 111

112

tdttt

)(sup)( txtxt

Chun l : 11sup1

tt


t 1 tt

Tnh chun ca tn hiu Tnh chun ca tn hiu Th d 2Th d 2

Ch t hi 3tCho tn hiu:

Tnh chun l1, l2 , l

)(.)( 3 tuetx t1 2


Chun ca h thng Chun ca h thng Cho h thng tuyn tnh c hm truyn G(s)Cho h thng tuyn tnh c hm truyn G(s).

Chun bc 2:21

2)(1:)(

djGjG

2)(

2:)( djGjG

Ch do nh l Parseval ta c:Ch do nh l Parseval, ta c:21

221

2

2)()(

21:)(

dttgdjGjG 2 2

trong g(t) l p ng xung ca h thng.

Chun v cng: )(sup:)( jGjG


)(p)(

jj

Biu din chun v cng trn biu Biu din chun v cng trn biu 1

Nyquist Diagram

1

0

1

0

20Bode Diagram

-2

-1

m

a

g

i

n

a

r

y

A

x

i

s

)( jG -40-20

a

g

n

i

t

u

d

e

(

d

B

)

)(lg20 jG

-4

-3I m )( jG

100

101

102

-80

-60

M

-3 -2 -1 0 1 2 3-5

Real AxisFrequency (rad/s)

10 10 10

Chun v cng bng khong cch t gc ta ca Chun v cng bng khong cch t gc ta ca mt phng phc n im xa nht trn ng cong Nyquist ca G(j), hoc bng nh cng hng trn


yq (j ), g g gbiu Bode bin |G(j)|

Cch tnh chun bc 2Cch tnh chun bc 2 Nu G(s) c bc t s bc mu s : )( jG Nu G(s) c bc t s < bc mu s v tt c cc cc u nm bn tri mp phc. Ta c:

Nu G(s) c bc t s bc mu s : 2)( jGp p

djGjG 222 )(21)(

j

dssGsGj

)()(21 dssGsGj )()(21 jj2 j2

trong l ng cong kn gm trc o v na ng trn bn knh v hn bao na tri mt phng phctrn bn knh v hn bao na tri mt phng phc.Theo /l thng d: )()()(lim)( 2

2sGsGpsjG

iips i


(pi l cc bn tri mt phng phc ca G(s)G(s))

Th d tnh chun bc 2 ca h thngTh d tnh chun bc 2 ca h thng

)1(10 sCho . Tnh)5)(3(

)1(10)( ss

ssG 2G

Gii Gii)()()(lim2

2sGsGpsG

iips i

)5)(3(

)1(10)5)(3(

)1(10)3(lim3

2

2

sss

ssssG

s

)5)(3()1(10

)5)(3()1(10)5(lim

)5)(3()5)(3(

5

32

sss

sssss

6667615252 G 5822G

)5)(3()5)(3(5 sssss


6667.61532

G 582.22G

Cch tnh chun v cngCch tnh chun v cng

)( jGd Cch 1: tm cc i ca

bng cch tm nghim phng trnh:

)(

0)(

2 jGd

djGd

)( jGg g p g

0)(

2

djGd

C h 2 h d bi B d Cch 2: tnh gn ng da vo biu Bode20

Bode Diagram

-20

0

t

u

d

e

(

d

B

) )(lg20 jG

80

-60

-40

M

a

g

n

i

t


Frequency (rad/s)10

010

110

2-80

Th d tnh chun v cng ca h thngTh d tnh chun v cng ca h thng

Ch T h)1(10)( sG GCho . Tnh)5)(3(

)1(10)( ss

ssG G

Gii Cch 1: Gii phng trnh tm cc i (SV t lm) Cch 2: Dng biu Bode

Da vo biu Bode, ta c05

Bode Diagram

,

dBjG 23.2)(lg20 -10

-5

g

n

i

t

u

d

e

(

d

B

)

)(lg20 jG

2927.1)( jG-20

-15

M

a


10-1

100

101

102

20

F ( d/ )Frequency (rad/s)

Tnh chun dng MatlabTnh chun dng Matlab

Chun ca vector hoc ma trn: Chun ca vector hoc ma trn:>> norm(X,1) % chun bc 1 ca vector hoc ma trn X

(X 2) % h b 2 t h t X>> norm(X,2) % chun bc 2 ca vector hoc ma trn X>> norm(X,inf) % chun v cng ca vector hoc ma trn X

Chun ca h thng:h2(G) % h b 2 h h G>> normh2(G) % chun bc 2 ca h thng G

>> normhinf(G) % chun v cng ca h thng G% Ch : G phi c khai bo bng lnh tf (transfer

% function) hoc ss (state-space model)


Quan h vo Quan h vo ra ra

Cho h tuyn tnh c h/truyn G(s) p ng xung l g(t) Cho h tuyn tnh c h/truyn G(s), p ng xung l g(t).

y(t)Gu(t) Vn t ra l xc nh ln ca

t/hi (t) khi bit l t/hi G

B 1 Ch t hi B 2 l i h th

t/hiu ra y(t) khi bit ln ca t/hiu vo u(t)

u(t) = (t) u(t) = sin(t) ||u||2 ||u||Bng 1: Chun ca tn hiu ra Bng 2: li ca h thng

||y||2 ||G||2 ||y|| ||g|| |G(j)|

||y||2 ||G|| ||y|| ||G||2 ||g||1

ng dng: Bng 1&2 thng c s dng nh gi: Sai s ca h thng khi bit tn hiu vo, hoc


g , nh hng ca nhiu n tn hiu ra ca h thng

Th d: nh gi sai sTh d: nh gi sai sd(t)

y(t)G++

r(t) K

d(t)e(t)

Cho h thng iu khin hi tip m n v, trong 2)( sG 4)( sK

2)( ssG 4)( sK

Xt trng hp nhiu bng 0. Tnh gi tr cc i g g gca sai s trong cc trng hp:(a) Tn hiu vo l r(t)=sin(3t)

15 January 2014 H. T. Hong - HCMUT 46

( ) ( ) ( )(b) Tn hiu vo r(t) bt k c bin nh hn 1

Th d: Kho st nh hng ca nhiu (tt)Th d: Kho st nh hng ca nhiu (tt)

Gii Gii:y(t)

G++r(t)

K

d(t)e(t)

Hm truyn tng t r(t) n e(t)

)()(11)(

sGsKsGre 241

1

)()(

2)( ssG2

41 s


10)( ssGre


( ) T h (t) i (3t) e(t)r(t)(a) Trng hp r(t)=sin(3t) e(t)Grer(t)

Gi tr cc i ca sai s khi tn hiu vo hnh sin Gi tr cc i ca sai s khi tn hiu vo hnh sin theo bng 1 l:

)()( jGte re42 432

Bng 1: Chun ca tn hiu 100

4)(2

jGre 3453.01003

43)3(2

jGre

u(t) = (t) u(t) = sin(t)||y||2 ||G||2

ra3453.0)3()( jGte re||y|| ||g|| |G(j)|



(b) Trng hp r(t) bt k c bin e(t)r(t)(b) Trng hp r(t) bt k c bin nh hn 1

e(t)Gre

r(t)

Gi t i i th b 2 l Gi tr cc i ca sai s theo bng 2 l:

)()( 1 trgte re trere ets

ssGtg 1011 8)(102)()(

LL

Bng 2: li ca h

dttgtg rere )()( 1 8.110818)(

0

10

dtedtt t||u||2 ||u||

||y||2 ||G|| thng18.1)()()(

1 trtgte re

81)( ||y|| ||G||2 ||g||115 January 2014 H. T. Hong - HCMUT 49

8.1)( te

Th d: Kho st nh hng ca nhiuTh d: Kho st nh hng ca nhiu

d(t)y(t)

G++r(t)

Kd(t)

Cho h thng iu khin hi tip m n v, trong

22)( sG 4)( sK

2)( s )(

Xt trng hp tn hiu vo bng 0. Tnh nng lng v gi tr cc i ca tn hiu ra trong cc trng hp:tr cc i ca tn hiu ra trong cc trng hp:

(a) Nhiu d(t) l xung dirac(b) Nhiu d(t) l tn hiu ngu nhin bt k c nng lng nh


(b) Nhiu d(t) l tn hiu ngu nhin bt k c nng lng nh hn 0.4


Gii: Gii:y(t)

G++r(t)

K

d(t)

Hm truyn tng t d(t) n y(t)2

)()(1)()(

sGsKsGsGdy 241

22

s

102)( ssGdy

241 s


10s


(a) Trng hp d(t) l xung dirac y(t)d(t)(a) Trng hp d(t) l xung dirac y(t)Gdyd(t)

Nng lng ca tn hiu ra theo bng 1 l:22)( Gt22

)( dyGty )()()(lim

2

2sGsGpsG dy

idyipsdy i

2.0)10( 2)10( 2)10(lim10 ssssi )()( 2.0)( 2

2

2

2 dyGty

Gi tr cc i ca tn hiu ra theo bng 1 l:

Bng 1: Chun ca tn hiu


)()( tgty dy2 rau(t) = (t) u(t) = sin(t)

||y||2 ||G||2 tdyyd essGtg 1011 2102)()( LL

2)()( tgty


||y|| ||g|| |G(j)|2)()( tgty dy


(b) Trng hp d(t) l nhiu c y(t)d(t)40)( 2 td(b) Trng hp d(t) l nhiu c y(t)Gdyd(t)

Nng lng ca tn hiu ra theo bng 2 l:)()( tdGty

4.0)(2td

22)()( tdGty dy

2.0ydG (xc nh c d dng da vo biu Bode)

016.04.0)2.0()()( 222

22

2 tdGty dy


Bng 2: li ca h


22)()( tdGty dy

||u||2 ||u||||y||2 ||G||

thng

2830404470)()( tdGty

447.02dyG (xem cch tnh cu a)

||y|| ||G||2 ||g||1


283.04.0447.0)()(22

tdGty dy

M HNH KHNG CHC CHNM HNH KHNG CHC CHNM HNH KHNG CHC CHNM HNH KHNG CHC CHN


M hnh khng chc chnM hnh khng chc chn

M h h t h kh th t h t h h M hnh ton hc khng th m t hon ton chnh xc h thng vt l cn quan tm n nh hng ca sai s m hnh n cht lng iu khinca sai s m hnh n cht lng iu khin

Phng php c bn xt n yu t khng chc chn l m hnh ha h thng thuc v mt tp hp m hnh M.

Hai dng m hnh khng chc chn: M hnh khng chc chn c cu trc (cn gi l

m hnh tham s khng chc chn) M hnh khng chc chn khng cu trc


M hnh khng chc chn c cu trcM hnh khng chc chn c cu trc

M hnh khng chc chn c cu trc: h thng M hnh khng chc chn c cu trc: h thng m t bi hm truyn hoc PTTT trong mt hoc nhiu thng s ca hm truyn hoc PTTT thay i g y ytrong min xc nh trc.

Mt s th d: m hnh bc 2 khng chc chn (nh h xe-l xo

-gim chn hoc h RLC)

maxmin2 :1

8 aaaass

M

m hnh c tr khng chc chn (nh l nhit)

e s

M


maxmin:15 sM

Th d m hnh c tham s khng chc chnTh d m hnh c tham s khng chc chn

Cho h thng gim sc m t bi PTVP bc 2: Cho h thng gim sc m t bi PTVP bc 2:

)()()()(22

tftKydt

tdyBdt

tydM M: khi lng tc ng ln bnh xe,B h s ma st, K cng l xo

dtdt

s a st, c g of(t): lc do sc: tn hiu voy(t): dch chuyn ca thn xe: tn hiu ra

)()(2 dd

Gi s khng bit chnh xc thng s ca h thng, PT trn c th biu din li di dng

)()()()()()()( 0022

0 tftykdttdyb

dttydm kbm

trong : m b k l cc thng s danh nh;


trong : m0, b0, k0 l cc thng s danh nh; m, b, k biu din s thay i ca cc thng s

Th d m hnh tham s khng chc chnTh d m hnh tham s khng chc chn t cc bin trng thi: )()()()( tytxtytx t cc bin trng thi: )()(),()( 21 tytxtytx Phng trnh trng thi m t i tng:

21 xx 2010

02

21

)()(1 fxbxkm

x bkm

1xy

1

S khi:

mm 01

bb 0

k


kk 0

Th d m hnh tham s khng chc chnTh d m hnh tham s khng chc chn

Bi i khi Bin i s khi:

m

0b

0k

b

k



t cc bin z z z d d d nh trn s khi t cc bin z1, z2, z3, d1, d2, d3 nh trn s khi.

Phng trnh trng thi ca h thng c thng s khng chc chn c th biu din li di dng:chn c th biu din li di dng:.

fddxbkx

1

1001 1000010 f

mddxmmx

03

22

0

0

0

02 111

100 bkf

ddd

xx

zzz

3

2

1

2

1

3

2

1

000000111

00

1

0110

00

0

0

0

mmb

mk

101 xxy

33 001


2x


t M l ma trn hm truyn ca h thng S t M l ma trn hm truyn ca h thng. S khi h thng c th biu din di dng:

b

m 0

kb

0


M hnh khng chc chn khng cu trcM hnh khng chc chn khng cu trc

M hnh khng chc chn khng cu trc: m t M hnh khng chc chn khng cu trc: m t yu t khng chc chn dng chun h thng.

M hnh khng chc chn khng cu trc thng M hnh khng chc chn khng cu trc thng dng hn v 2 l do: Tt c cc m hnh dng trong thit k h thng Tt c cc m hnh dng trong thit k h thng iu khin u cha ng trong cc yu t khng chc chn khng cu trc bao hm ckhng chc chn khng cu trc bao hm c tnh ng hc khng m hnh ha, c bit l min tn s cao.

S dng m hnh khng chc chn khng cu trc c th d dng hn trong vic xy dng cc


phng php v phn tch thit k HTK bn vng.

Cc dng MH khng chc chn khng cu trcCc dng MH khng chc chn khng cu trc Bn MH khng chc chn khng cu trc thng dng: Bn MH khng chc chn khng cu trc thng dng: 1:)1(~ GWG mM 1~ WGGM

(M hnh nhiu nhn)

(M hnh nhiu cng) 1: mWGGM (M hnh nhiu cng)

1:1

~GW

GGM (M hnh nhiu cng ngc) 1 GWm

1:1

~W

GGM (M hnh nhiu nhn ngc) 1 mW Trong :

G gi l m hnh danh nh (nominal model)g ( ) l m hnh khng chc chn : l hm truyn n nh, thay i bt k tha mn ||||1

dng m t yu t khng chc chn khng cu trc

G~


dng m t yu t khng chc chn khng cu trc. Wm: hm truyn n nh, ng vai tr l hm trng s

M hnh nhiu nhnM hnh nhiu nhn

G~

Wm

y(t)G ++

u(t)

Biu thc m hnh nhiu nhn: 1:)1(~ GWG m

Thng dng m t cc yu t khng chc chn: Thng dng m t cc yu t khng chc chn: c tnh tn s cao ca i tng Z kh h h


Zero khng chc chn

M hnh nhiu cngM hnh nhiu cng

G~

Wm

y(t)G ++

u(t)

Biu thc m hnh nhiu cng: ~ 1:~ mWGG

Thng dng m t cc yu t khng chc chn: Thng dng m t cc yu t khng chc chn: c tnh tn s cao ca i tng Zero khng chc chn


Zero khng chc chn

M hnh nhiu cng ngcM hnh nhiu cng ngc

WmG~

y(t)G+u(t)

Biu thc m hnh nhiu cng ngc:G 1:

1~ GW

GGm

Thng dng m t cc yu t khng chc chn: Thng dng m t cc yu t khng chc chn: c tnh khng chc chn min tn s thp Cc khng chc chn


Cc khng chc chn

M hnh nhiu nhn ngcM hnh nhiu nhn ngc

WmG~

y(t)G +u(t)

Biu thc m hnh sai s nhn ngc:~ G 1:

1~ mW

GG

Thng dng m t cc yu t khng chc chn: Thng dng m t cc yu t khng chc chn: c tnh khng chc chn min tn s thp Cc khng chc chn


Cc khng chc chn

Xy dng m hnh khng chn chn Xy dng m hnh khng chn chn Cch 1Cch 1 Bc 1: Xy dng m hnh danh nh G dng phng Bc 1: Xy dng m hnh danh nh G dng phng

php m hnh ha thng thng vi b thng s danh nh ca i tng.

Bc 2: Xc nh hm truyn trng s Wm, ty theo tng m hnh, hm truyn trng s cn chn tha mn /kin:

M hnh nhiu nhn:

)(~ jG

1:)1(~ mWGG

,1)()()(

jGjGjWm

M hnh nhiu cng:

)()(~)( jGjGjW1:~ mWGG


,)()()( jGjGjWm

Xy dng m hnh khng chc chn (tt)Xy dng m hnh khng chc chn (tt)

M h h hi 1~ GG M hnh nhiu cng ngc 1:1

GWGG

m

11)( jW ,)()(~)( jGjGjWm

~ G M hnh nhiu nhn ngc 1:1

mWGG

1)()( jGjW ,1)(~)()(

jGjjWm

Bc 3: xc nh biu thc hm truyn trng s tha

Ch : thng thng W c bin tng dn theo tn

Bc 3: xc nh biu thc hm truyn trng s tha iu kin bc 2 da vo biu Bode


Ch : thng thng Wm c bin tng dn theo tn s, do min tn s cng cao bt nh cng ln

Chng minh iu kin hm trng sChng minh iu kin hm trng s M hnh nhiu nhn: M hnh nhiu nhn:

1:)1(~ mWGG)(~ jG

1)(~

)()( jGW)()()()(1

jGjGjWj m

1)(~

)()( jGjWj

1)()()()(

jGjGjWj m

1)(~

)( jGjW

1)()()()( jG

jjWj m

,1)()()(

jGjjWm

CM theo cch tng t cho m hnh nhiu cng, m hnh


CM theo cch tng t cho m hnh nhiu cng, m hnh nhiu s cng ngc v m hnh nhiu nhn ngc.

Xy dng m hnh khng chn chn Xy dng m hnh khng chn chn Cch 2Cch 2Ch p dng trong trng hp hm truyn i tng tht G~Ch p dng trong trng hp hm truyn i tng tht ch c 1 tham s khng chc chn, chng hn: maxmin

G

Bc 1: t , trong : 10 , g10 2/)( maxmin0 2/)( minmax1 11

Bc 2: Thay vo hm truyn v thc hin G~ Bc 2: Thay vo hm truyn v thc hin bin i rt ra G v Wm t m hnh:

10 G

M hnh nhiu nhn: 1:)1(~ mWGG )( m M hnh nhiu cng: 1:~ mWGG

M h h hi ~ G M hnh nhiu cng ngc: 1:1

GWGG

m

M hnh nhiu nhn ngc: 1:~ GG


M hnh nhiu nhn ngc: 1:1

mWG

Th d 1: H thng c li khng chc chnTh d 1: H thng c li khng chc chn

Bi ton: Cho HT m t bi hm truyn thc: ~ kG Bi ton: Cho HT m t bi hm truyn thc:)1( ssG

trong li k nm trong khong 0.1 k 10 Xy dng m hnh nhiu nhn m t h thng trn.

Gii:

Chn m hnh danh nh:

M hnh nhiu nhn: 1:)1(~ GWG m

)1(0

sskG

Chn m hnh danh nh:

)1( ss05.5

2101.0

2maxmin

0 kkk


22

Th d 1: H thng c li khng chc chnTh d 1: H thng c li khng chc chn Cn chn W tha mn iu kin: Cn chn Wm tha mn iu kin:

,1

)()(~)(

jGjGjWm )( jG

,1)(kkjWm )101.0( k

0k

05595.41max)(

0101.0

k

kjWkm

981.0)( jWm05.50k Kt lun: m hnh nhiu nhn tm c l:

1:)1(~ GWG 1:)1( GWG mtrong : 981.0)( sW05.5G


g 981.0)(sWm)1( ssG

Th d 2: H thng thi hng khng chc chnTh d 2: H thng thi hng khng chc chn

Bi ton: Cho HT c hm truyn thc l: )1(8~ sG Bi ton: Cho HT c hm truyn thc l:)110)(12(

)( ssG

trong nm trong khong 0.2 5.0 Xy dng MH nhiu nhn m t HT khng chc chn trn

Gii:

)16.2(8 sG Chn m hnh danh nh: M hnh nhiu nhn: 1:)1(~ GWG m

)110)(12( ssG Chn m hnh danh nh: Cn chn Wm tha mn iu kin:

,1

)()(~)(

jGjGjWm

,1

16.21)(

jjjWm


Chn Wm tha mn /kin trn vi 0.2 5.0 dng b/ Bode

10

Th d 2: H thng c thi hng khng chc chn (tt)Th d 2: H thng c thi hng khng chc chn (tt)

0

10

)(log20 jWm

-10

-30

-20

(

d

B

)

-40

60

-50T=0.2T=1.3T=2.0T=2.5

=0.2 =1.3 =2.0 =2.5

10-2

10-1

100

101

-60


(rad)0.3


Ks Da vo b/ Bode, c th chn Wm c dng: 1)( Ts

KssWm

D thy:

(sec)33.33.0

11 g

T

y

33.3)( ssW)(0lg20 dB

TK 33.3K

133.3)( ssWm

Kt lun: m hnh nhiu nhn tm c l: 1:)1(~ GWG 1:)1( GWG m

trong : 33.3)( ssW)16.2(8 sG


g133.3

)( ssWm)110)(12( ssG


Bi di h h hi h d M tl bBi di h h hi h d M tl b

% i tng c thi hng khng chn chn

Biu din m hnh nhiu nhn dng MatlabBiu din m hnh nhiu nhn dng Matlab

% i tng c thi hng khng chn chn>> tau = ureal('tau',2.6,'range',[0.2 5]);>> G =tf(8*[tau 1],[20 12 1]); %Hm truyn c tham s khng chn chn>> figure(1)g ( )>> bode(usample(G,10),{0.01,100}) %Biu Bode ca i tng kg chc chn

% M hnh sai s nhn (Multiplicative Uncertainty Model)>> Gnom=tf(8*[2.6 1],[20 12 1]); % M hnh danh nh>> Wm=tf([3.33 0],[3.33 1]); % Hm truyn trng s>> Delta = ultidyn('Delta',[1 1]);>> G G *(1+W*D lt ) % M h h i h>> G = Gnom*(1+W*Delta) ; % M hnh sai s nhn>> figure(2)>> bode(usample(G,10),{0.01,100}) % Biu Bode m hnh nhiu nhn


Th d 2: H thng c thi hng khng chc chn (tt)Th d 2: H thng c thi hng khng chc chn (tt)Bode Diagram Bode Diagram

-20

0

20

n

i

t

u

d

e

(

d

B

)

-20

0

20

n

i

t

u

d

e

(

d

B

)

g

-60

-40

M

a

g

n

45

0-60

-40

M

a

g

n

45

0

-180

-135

-90

-45

P

h

a

s

e

(

d

e

g

)

-180

-135

-90

-45

P

h

a

s

e

(

d

e

g

)

1:)1(~ GWG m)110)(12(

)1(8~ sG 10

-210

-110

010

110

2-180

Frequency (rad/sec)10

-210

-110

010

110

2-180

Frequency (rad/sec)

)110)(12()16.2(8

ss

sG133.3

33.3)( sssWm

)110)(12( ss0.52.0

Bi Bode ca i t ng Bi Bode m


Biu Bode ca i tng c thi hng khng chc chn

Biu Bode m hnh nhiu nhn

Th d 3: H thng c tr khng chc chnTh d 3: H thng c tr khng chc chn

Bi ton: Cho h/thng m t bi h/truyn thc: 15~eG

s Bi ton: Cho h/thng m t bi h/truyn thc:

12.0 sGtrong thi gian tr nm trong khong 0 0.1

Xy dng MH nhiu nhn m t HT khng chc chn trn

Gii:

15G Chn m hnh danh nh: M hnh nhiu nhn: 1:)1(~ GWG m

12.0 sG Chn m hnh danh nh: Cn chn Wm tha mn iu kin:

,1

)()(~)(

jGjGjWm ,1)( jm ejW


Chn Wm tha mn iu kin trn da vo biu Bode

Th d 3: H thng c tr khng chc chn (tt)Th d 3: H thng c tr khng chc chn (tt)20

10

20

7)(log20 jWm

-10

0

-20

(

d

B

)

-40

-30

60

-50 )(01.0),(1.0,1log20 greenbluee j


10-1 100 101 102 103 104-60 (rad)

Th d 3: H thng c tr khng chc chn (tt)Th d 3: H thng c tr khng chc chn (tt)

Ks Da vo b/ Bode, c th chn Wm c dng: 1)( Ts

KssWm

D thy:

(sec)1.01011

g

T

y

224.0)( ssW)(7lg20 dB

TK 224.0K

11.0)( ssWm

Kt lun: m hnh nhiu nhn tm c l: 1:)1(~ GWG 1:)1( GWG m

trong : 224.0)( ssW15G


g11.0

)( ssWm12.0 sG

Th d 4: H thng c cc khng chc chnTh d 4: H thng c cc khng chc chn

Bi ton: Cho h/thng m t bi h/truyn thc: 5~G Bi ton: Cho h/thng m t bi h/truyn thc :12 assG

trong thng s a nm trong khong 0.1 a 1.7Xy dng m hnh nhiu cng ngc m t h thng trnXy dng m hnh nhiu cng ngc m t h thng trn Gii: C th biu din a nh sau: 8.09.0a 11

5~ G

Thay a vo :G~

5 )19.0(5

2 ss1)8.09.0(2 ssG sss 8.0)19.0( 2

)19.0(516.01 2

ss

s

)(~ sP)()(1

)(sPsW

sPGm

t 5)(G s160


trong 19.0

5)( 2 sssG ssssWm 16.010001.0

16.0)(

Th d 4: H thng c cc khng chc chn (tt)Th d 4: H thng c cc khng chc chn (tt)

Bi di h h hi d M tl bBi di h h hi d M tl b

% i tng c cc khng chn chn

Biu din m hnh nhiu cng ngc dng MatlabBiu din m hnh nhiu cng ngc dng Matlab

% i tng c cc khng chn chn>> a = ureal(a',0.9,'range',[0.1 1.7]);>> G =tf(5,[1 a 1]); %Hm truyn c tham s khng chn chn>> figure(1)g ( )>> bode(usample(G,20),{0.1,10}) %Biu Bode ca i tng kg chc chn

% M hnh sai s cng ngc (Inverse Additive Uncertainty Model)>> Gnom=tf(5,[1 0.9 1]); % M hnh danh nh>> Wm=tf(0.16*[1 0],[0.0001 1]); % Hm truyn trng s>> Delta = ultidyn('Delta',[1 1]);>> G G /(1+W*D lt *G ) % M h h i >> G = Gnom/(1+W*Delta*Gnom) ; % M hnh sai s cng ngc>> figure(2)>> bode(usample(G,20),{0.01,100}) % Biu Bode m hnh nhiu cng ngc


Th d 4: H thng c cc khng chc chn (tt)Th d 4: H thng c cc khng chc chn (tt)30

Bode Diagram30

Bode Diagram

-10

0

10

20

30

g

n

i

t

u

d

e

(

d

B

)

-10

0

10

20

g

n

i

t

u

d

e

(

d

B

)

-30

-20

-10

M

a

g

45

0-30

-20

-10

M

a

g

45

0

180

-135

-90

-45

P

h

a

s

e

(

d

e

g

)

180

-135

-90

-45

P

h

a

s

e

(

d

e

g

)

1:)1/(~ GWGG m5~ 2G10

-110

010

1-180

Frequency (rad/sec)10

-110

010

1-180

Frequency (rad/sec)

19.05

2 ssG 11016.0)( 4 s

ssWm12 ass

7.11.0 aBi Bode ca i t ng Bi Bode m hnh


Biu Bode ca i tng c cc khng chc chn

Biu Bode m hnh nhiu cng ngc

Cu trc MCu trc M-- H thng iu khin vng kn bt k vi thnh phn khng H thng iu khin vng kn bt k vi thnh phn khng

chc chn c th bin i v cu trc chun Mwz w0z0

M

C b bi i HTK th h t h M Cc bc bin i HTK thnh cu trc chun M Xc nh tn hiu vo ca M (t/hiu ra ca ), k hiu l w0. Xc nh tn hiu ra ca M (tn hiu vo ca ) k hiu l z0 Xc nh tn hiu ra ca M (tn hiu vo ca ), k hiu l z0 Tch thnh phn khng chc chn ra khi s Tm hm truyn M t w0 n z0


y 0 0

Th d: Cu trc MTh d: Cu trc M-- Hy bin i h thng di y v cu trc chun M Hy bin i h thng di y v cu trc chun M

WmM

y(t)G ++

r(t) K

H


Th d: Cu trc MTh d: Cu trc M-- Gii Gii

WmM

z0 w0

y(t)G ++

r(t) K

H

Hm truyn t w0 n z0: w0z0

)()()(1)()()()()(

sHsGsKsHsGsKsWsM m

M

00


M

TNH N NH NITNH N NH NITNH N NH NI TNH N NH NI


H thng iu khin vng knH thng iu khin vng knd(t)

y(t)r(t) GK ++

d(t)

u(t)x1(t) x2(t)

n(t)H ++v(t) x3(t)H +

r(t): tn hiu t y(t): tn hiu ra ca i tng u(t): tn hiu ra ca b iu khin

v(t): tn hiu ra ca cm bin d(t): nhiu h thng (t) hi l


n(t): nhiu o lng

Cc hm truynCc hm truynd

yr GK ++

d

ux1 x2

nH ++vx3H +

rxH 101

nd

xx

GK

3

2

1001

dr

HKKHGH

GHKxx

11

11

2

1


nGGKGHKx 113

nh ngha n nh ni nh ngha n nh ni d(t)

y(t)r(t) GK ++

d(t)

u(t)x1(t) x2(t)

n(t)H ++v(t)x3(t)H +

Nhc li khi nim n nh BIBO: H thng c Nhc li khi nim n nh BIBO: H thng cgi l n nh nu tn hiu vo b chn th tn hiu ra bchn (Bounded Input Bounded Output)

H thng c gi l n nh ni (Internal Stability) nu tn hiu vo b chn th tn hiu ra v tt c cc tn


hiu bn trong h thng u b chn.

nh l n nh ninh l n nh ni

H th h i khi h khi h i i ki H thng n nh ni khi v ch khi hai iu kin sau y c tha mn: Hm truyn (1+GHK) khng c zero nm bn phi Hm truyn (1+GHK) khng c zero nm bn phi

mt phng phc Khng c trit tiu cc zero bn phi mt phng Khng c trit tiu cczero bn phi mt phng

phc khi tnh tch cc hm truyn GHK.


Hm truyn kn v hm nhyHm truyn kn v hm nhyd(t)

y(t)r(t) GK ++

d(t)

u(t)e(t)

n(t)+++

KGKGT 1

Hm truyn kn:

Hm nhy: nh lng nhy ca T i vi s thay ica G:

GdTTT /

KG1

TG

dGdT

GGTTS

G.

//lim:

0

KGS 11


1 ST Ch : T cn c gi l hm b nhy

N NH BN VNGN NH BN VNGN NH BN VNGN NH BN VNG


nh ngha n nh bn vngnh ngha n nh bn vngd(t)

y(t)r(t) K ++

d(t)

G~

n(t)++

H thng c gi l n nh bn vng nu h thng

+

n nh ni vi mi i tng thuc lp m hnhkhng chc chn cho trc.G~g

nh gi tnh n nh bn vng nh l Kharitonov


nh l Kharitonov nh l li b

nhnh ll KharitonovKharitonov Cho h thng iu khin c phng trnh c trng l: Cho h thng iu khin c phng trnh c trng l:

0...665

54

43

32

21

10 nnnnnnn sasasasasasasatrong cc h s ca PTT nm trong min cho trc:

),...,1,0( , niaaa iii trong cc h s ca PTT nm trong min cho trc:

nh l Kharitonov: HT n nh bn vng vi minu v ch nu bn a thc di y u l a thc Hurwitz (tc l a thc c tt c cc nghim nm bn tri mp phc)

iii aaa

(tc l a thc c tt c cc nghim nm bn tri mp phc). ...)( 66

55

44

33

22

1101 nnnnnnn sasasasasasasas

)( 654321 nnnnnnn ...)( 66554433221102 nnnnnnn sasasasasasasas...)( 66

55

44

33

22



...)( 665

54

43

32

21


nhnh ll Kharitonov Kharitonov Th d 1Th d 1y(t)r(t) y(t)r(t)

G

Cho h thng /khin hi tip m vi:)(

)( 2 kbsmssKsG P

62;85;31;101 PKkbmtrong : nh gi tnh n nh bn vng ca h thng.

Gii: Phng trnh c trng: 1 ( ) 0G s g g ( )

0)(

1 2 kbsmssKP


023 PKksbsms

nhnh ll Kharitonov Kharitonov Th d 1 (tt)Th d 1 (tt)

Xt cc a thc Kharitonov: Xt cc a thc Kharitonov:

681)( 231 ssss Do nn 1(s) l a thc Hurwitz. 06181

283)( 232 ssss Do nn 2(s) l a thc Hurwitz. 02183

6510)( 233 ssss ( ) Do nn 3(s) khng phi l a thc Hurwitz. 010551

(khng cn xt 4(s)) Kt lun: Theo nh l Kharitonov, h thng khng n nh bn

vng.


g

nh l li nh (Small Gain Theorem)nh l li nh (Small Gain Theorem)y(t)r(t) y(t)r(t)

G

nh l li nh: Cho h h G(s) n nh. H kn n nh nu 1)( jG 1)( jGnu 1)( jG ,1)( jG

Im Chng minh: D dng h i h d ti h

Re1

chng minh dng tiu chun n nh Nyquist

Ch : nh l li nh l iu G(j) Ch : nh l li nh l iukin nh gi n nh

H thng khng tha nh l


li nh vn c th n nh

nh l n nh bn vngnh l n nh bn vng

nh l n nh bn vng: Cho h thng nh l n nh bn vng: Cho h thng iu khin vng kn nh hnh v, trong M(s) l hm truyn n nh v l (s) hm

M(s) l hm truyn n nh v l (s) hm truyn n nh bt k tha ||(j)||1 . H thng kn n nh khi v ch khi:

M

g

1)( jMCh i h

() S dng nh l li nh Chng minh:

1)()( jMj

() Phn chng. Gi s h kn khng n nh v 1)( jM 1)( j (tri gi thit)


1)( jM

1)( j (tri gi thit)

iu kin n nh bn vng m hnh nhiu nhniu kin n nh bn vng m hnh nhiu nhn

y(t)G +

Wmr(t)

K G ++( )

K

nh l: H thng iu khin m hnh nhiu nhn n nh bn vng vi mi nu v ch nu h thng n nh danh1vng vi mi nu v ch nu h thng n nh danh nh, ng thi b iu khin K tha mn iu kin:

1TW ][0lg20 dBTW

1

1TWm

trong : KGLST 1 (hm nhy b)

][0lg20 dBTWm


trong :KGL

ST 111 (hm nhy b)

iu kin n nh bn vng m hnh nhiu nhn (tt)iu kin n nh bn vng m hnh nhiu nhn (tt) Chng minh: Chng minh:

M

y(t)+

Wm

r(t)

M

y( )G ++

r(t) K

Bin i tng ng h thng v dng vng M-, trong :TW

KGKGWM mm 1

g g g g g g


Sau p dng nh l n nh bn vng.

iu kin n nh bn vng m hnh nhiu nhn (tt)iu kin n nh bn vng m hnh nhiu nhn (tt)

Biu din hnh hc: Ch :

1TW

Biu din hnh hc:

1TWm

,1)(1)()(

jLjLjWm

I )(1 jL ,)(1)()( jLjLjWm

Im

Ti mi tn s, im ti hn (1, j0) phi nm ngoi hnh trn tm L(j)

ReL(j)

1

ngoi hnh trn tm L(j), bn knh |Wm(j)L(j)| |WmL|


iu kin n nh bn vng m hnh nhiu cngiu kin n nh bn vng m hnh nhiu cng

Wmy(t)

G ++r(t)

K

nh l: H thng iu khin m hnh nhiu cng n nh bn i i h h th h d h1vng vi mi nu v ch nu h thng n nh danh nh, ng thi b iu khin K tha mn iu kin:

1

1KSWm

t S11

(h h )

][0lg20 dBKSWm


trong :KGL

S 11 (hm nhy)

iu kin n nh bn vng m hnh nhiu cng (tt)iu kin n nh bn vng m hnh nhiu cng (tt) Chng minh: Chng minh:

M

y(t)+

Wm

r(t)

M

y( )G ++

r(t) K

Bin i tng ng h thng v dng vng M-, trong :KSW

KGKWM mm 1

g g g g g g


Sau p dng nh l n nh bn vng.

iu kin n nh bn vng m hnh nhiu cng (tt)iu kin n nh bn vng m hnh nhiu cng (tt)


1KSW

Biu din hnh hc:

1KSWm

,1)(1)()(

jLjKjWm

I )(1 jL ,)(1)()( jLjKjWm

Im

Ti mi tn s, im ti hn (1, j0) phi nm ngoi hnh trn tm L(j)

ReL(j)

1

ngoi hnh trn tm L(j), bn knh |Wm(j)K(j)| |WmK|


iu kin n nh bn vng MH nhiu cng/nhn ngciu kin n nh bn vng MH nhiu cng/nhn ngc

y(t)r(t) y(t)r(t) K G~

Cho h thng iu khin hi tip m n v (xem hnh). Nu i tng m t bi m hnh nhiu cng ngc: g g g

1:1

~ GWGG

m

1GSWmth iu kin n nh bn vng l: Nu i tng m t bi m hnh nhiu nhn ngc: g g

1:1

~ mWGG


1SWmth iu kin n nh bn vng l:

nh gi tnh n nh bn vng nh gi tnh n nh bn vng Th d 1Th d 1

y(t)G +

Wmr(t)

K G ++( )

K

Bi ton: Cho h thng iu khin c s khi nh hnh v, i tng khng chc chn m t bi m hnh nhiu nhn, trong :

133333.3)( s

ssWm)162)(12(1

ssG 1 133.3 s)16.2)(12( ssnh gi tnh n nh bn vng ca HT trong 2 trng hp:

10 10


ssK 1.03)(

ssK 1.030)(

nh gi tnh n nh bn vng nh gi tnh n nh bn vng Th d 1 (tt) Th d 1 (tt) Gii: Gii:

Trng hp 1:

)16.2)(12(

11.03133.3

33.3

1ssss

s

KGKGWTW mm

)16.2)(12(11.031

1sss

KGm

0057.02502.0035.1185.10192.05769.0

234

2

ssssssTWm

Xt biu Bode K(j)G(j) v Wm(j)T(j)


nh gi tnh n nh bn vng nh gi tnh n nh bn vng Th d 1 (tt) Th d 1 (tt) Bode DiagramBiu Bode K(j)G(j)

0

50(

d

B

)

gu ode (j)G(j)

-50

M

a

g

n

i

t

u

d

e

(

-45

0-100

-135

-90

-45

P

h

a

s

e

(

d

e

g

)

10-3

10-2

10-1

100

101

102

-180

Frequency (rad/sec)


Do GM > 0 v M > 0 nn h danh nh n nh

nh gi tnh n nh bn vng nh gi tnh n nh bn vng Th d 1 (tt) Th d 1 (tt)

-20

0Bode DiagramBiu Bode bin |Wm(j)T(j)|

-40

20

a

g

n

i

t

u

d

e

(

d

B

)

3 2 1 0 1 2-80

-60

M

a

Da vo biu Bode bin |Wm(j)T(j)|, ta xc nh c: 10

-310

-210

-110

010

110

2

F ( d/ )

][0][85.1lg20 dBdBTWm Do h thng danh nh n nh ng thi |Wm(j)T(j)|


Trng hp 2:

)162)(12(

11.0301333

33.3 sKGW

Trng hp 2:

)162)(12(

11.0301

)16.2)(12(133.31

ssssKGKGWTW mm

)16.2)(12( sss0192.0769.5

234

2 ssTWm 0057.0809.1227.6185.1 234 ssssTWmXt biu Bode K(j)G(j) v Wm(j)T(j)


Bode Diagram

nh gi tnh n nh bn vng nh gi tnh n nh bn vng Th d 1 (tt) Th d 1 (tt) Biu Bode K(j)G(j)

50

100(

d

B

)

gu ode (j)G(j)

-50

0

M

a

g

n

i

t

u

d

e

(

-100

-45

0

-135

-90

-45

P

h

a

s

e

(

d

e

g

)

10-4

10-3

10-2

10-1

100

101

102

-180

Frequency (rad/sec)


Do GM > 0 v M > 0 nn h danh nh n nh

BIU DIN CHT LNG BIU DIN CHT LNG DANH NH DNG HM NHYDANH NH DNG HM NHY


Nhc li: Hm truyn kn v hm nhyNhc li: Hm truyn kn v hm nhyd(t)

y(t)r(t) GK ++

d(t)

u(t)e(t)

n(t)+++

KGKGT 1 Hm truyn kn:

Hm nhy: nh lng nhy ca T i vi s thay i ca G: GdTTT /

KG1

1TG

dGdT

GGTTS

G.

//lim:

0

1 ST Ch :

KGS 1

1


1 STHm truyn tng t r(t) n e(t) chnh bng hm nhy

Th d hm truyn kn v hm nhyTh d hm truyn kn v hm nhyd(t)

y(t)r(t) GK ++

d(t)

u(t)e(t)

n(t)+++

2)101.0)(14.0(4)( sssG i tng:

)6(4 sKG

B iu khin:s

sK 61)(

)6(4)101.0)(14.0()6(4

1 2 ssss

sKG

KGT Hm truyn kn:

)1010)(140(1 2 sss


Hm nhy:)6(4)101.0)(14.0(

)101.0)(14.0(1

12 ssss

sssKG

S

Th d hm truyn kn v hm nhyTh d hm truyn kn v hm nhyBiu Bode hm

50Bode Diagram Bode Diagram

Biu Bode h hBiu Bode hm

nhy v hm b nhy

-50

0

t

u

d

e

(

d

B

)

-40

-20

0

t

u

d

e

(

d

B

)

ST

C

-150

-100Ma

g

n

i

t

-1 0 1 2 3

-80

-60

M

a

g

n

i

B

-180

-135

-90

e

(

d

e

g

)

K*G10

110

010

110

210

3

Tn s ct bin ca h h b th h k

10-1

100

101

102

103

104

-270

-225Ph

a

s

e

BC xp x bng thng h kn


10 10 10 10 10 10Frequency (rad/sec)

Cht lng iu khinCht lng iu khin

d(t)y(t)r(t)

GK ++d(t)

u(t)e(t)

n(t)++

Sai s: SrrKG

e 11

Nhc li mt s kt lun trong mn CST: Nu r l hm nc: exl=0 nu KG c t nht 1 khu TPLT

N l h d 0 KG t ht 2 kh TPLT Nu r l hm dc: exl=0 nu KG c t nht 2 khu TPLT

Ch tiu cht lng nu r thuc v mt tp tn hiu c


chun b chn?

Biu din cht lng danh nh dng hm nhyBiu din cht lng danh nh dng hm nhy

Trng hp 1: Xt trng hp r l tn hiu hnh sin Trng hp 1: Xt trng hp r l tn hiu hnh sin c tn s bt k v bin bng 1. Yu cu cht lng l bin sai s nh hn . lng l bin sai s nh hn .

Do SrrKG

e 11KG1

S Ch tiu cht lng c th biu din nh sau:

/1)( sWp tS

pu(t) = (t) u(t) = sin(t)

||y||2 ||G||2 Ch tiu cht lng c th vit li di dng:

1SW15 January 2014 H. T. Hong - www4.hcmut.edu.vn/~hthoang/ 120

||y|| ||g|| |G(j)|1SWp


rr

fF rWr Trng hp 2: Tn hiu vo r c dng trongr

WFrpf

Ch i

rpf l tn hiu hnh sin tn s bt k c bin bng 1. pfF rWr Trng hp 2: Tn hiu vo r c dng trong

SWChun v cng ca sai s: SWe FGi s yu cu cht lng l: ey gt /Fp WW

e

u(t) = (t) u(t) = sin(t)||y||2 ||G||2 1SW

Yu cu cht lng tng ng iu kin:e


||y|| ||g|| |G(j)|1SWp


rr

Trng hp 3: Tn hiu vo r l tn hiu rpf c nng

rWF

rpf

Trng hp 3: Tn hiu vo r l tn hiu rpf c nng lng bng 1 i qua mt b lc WF 1,: pfpfF rrWrr 1,: 2 pfpfF rrWrr

SWe F2Chun bc 2 ca sai s:Gi s yu cu cht lng l:

/Fp WW

2e

t ||u||2 ||u||

||y||2 ||G|| 1SW

Fp

Yu cu cht lng tng ng iu kin:2

e


||y|| ||G||2 ||g||11SWp


Trng hp 4: Trong mt s ng dng ngi thit k da Trng hp 4: Trong mt s ng dng, ngi thit k da vo kinh nghim bit rng t cht lng tt, biu Bode bin ca hm nhy phi nm di mt ng

cong no . tng thit k ny c th vit di dng:

)()( 1jWjS 1SW ,)()( jWjS p 1SWp10

Bode Diagram

-10

0

10

e

(

d

B

)

)( jS

-40

-30

-20

M

a

g

n

i

t

u

d

e

)(1 jWp


10-3

10-2

10-1

100

101

102

103

-50


Tm liTm li: ty theo ng dng c th v ty theo lp tn Tm liTm li: ty theo ng dng c th v ty theo lp tn hiu vo, bng cch chn b lc trng s cht lng W (s) thch hp ta c th biu din ch tiu chtWp(s) thch hp, ta c th biu din ch tiu cht lng di dng:

1SW 1WS1SWp ,1pWS


B lc trng s cht lng thng dngB lc trng s cht lng thng dng Hm truyn trng Bi B d 1

s Hm truyn trng

s cht lng:0

10

B

)

Bode Diagram

20lgB

Biu Bode )(lg20 1 jWp

B

Bp s

ssW

)(

-30

-20

-10

M

a

g

n

i

t

u

d

e

(

d

B

20lg

B

Bp s

ssW

)(1

10-3

10-2

10-1

100

101

102

103

-50

-40

M

20lg

ngha ch tiu cht lng danh nh vi trng s cht lng trn l:

S i l i i t hi l h h h

1SWp Sai s xc lp i vi tn hiu vo l hm nc nh hn Sai s bm theo tn hiu hnh sin c bin bng 1, tn s

bt k nh hn


Bng thng ca h thng xp x B

Biu din hnh hc ch tiu cht lngBiu din hnh hc ch tiu cht lng Ch rng:

,1)(1

)(jL

jWp Ch rng:

1SWp (vi ))()()( jGjKjL ,)(1)( jLjWp

iu kin h thng tha cht lng l N i t L(j ) h h hi i

1|||| SWpng cong Nyquist L(j) ca h h phi nm ngoi vng trn tm 1, bn knh |Wp(j)|

Re|Wp|

Im

Re|Wp|

Im

L(j)1L(j)

1


nh gi cht lng danh nh nh gi cht lng danh nh Th d 1Th d 1d(t)

y(t)r(t) K ++

d(t)

G

n(t)++ Cho h thng, trong :

15

+

)3(8 )1(

15)( ssG )5()3(8)(

sssK

10 Xt hm trng s cht lng:2.05.0

10)( s

ssWp


H thng c tha mn cht lng danh nh hay khng?

nh gi cht lng danh nh nh gi cht lng danh nh Th d 1Th d 1 Gii: Gii: Hm nhy:

365126)1)(5(

)()(11

2 ss

sssGsK

S

)365126)(2.05.0(

)1)(5)(10(2

sss

sssSWp

V Biu Bode

)()( jSjW 510

d

B

)

Bode Diagram )()(lg20 jSjWp

)()( jSjWp-10

-5

0

M

a

g

n

i

t

u

d

e

(

d

10-2

10-1

100

101

102

103

104

-15

Da vo biu ta thy (v )1SW 06lg20 dBSW


Da vo biu , ta thy (v ) do h thng khng tha mn cht lng danh nh.

1SWp 06lg20 dBSWp

nh gi cht lng danh nh nh gi cht lng danh nh Th d 2Th d 2d(t)

y(t)r(t) K ++

d(t)

G

n(t)++ Cho h thng, trong :

5

+

20)10)(2(

5)( sssG ssK205)(

1 Xt hm trng s cht lng:s

ssWp 5.11)(

H th th ht l d h h h kh ?


H thng c tha mn cht lng danh nh hay khng?

nh gi cht lng danh nh nh gi cht lng danh nh Th d 2Th d 2 Gii: Gii: Hm nhy:

1004512)10)(2(

)()(11

23 sss

ssssGsK

S

)1004512(5.1

)10)(2)(1(23

sss

sssSWp

V biu Bode bin : -5

0

d

B

)

Bode Diagram )()(lg20 jSjWp

)()( jSjWp-15

-10

M

a

g

n

i

t

u

d

e

(

d

10-2

10-1

100

101

102

-20

Theo b Bode ta thy (v )1SW 080lg20 dBSW


Theo b. Bode, ta thy (v ) do h thng tha mn cht lng danh nh.

1SWp 08.0lg20 dBSWp

CHT LNG BN VNGCHT LNG BN VNGCHT LNG BN VNGCHT LNG BN VNG


nh ngha cht lng bn vngnh ngha cht lng bn vngd(t)

y(t)r(t) K ++

d(t)

G~

n(t)++

G

H thng c gi l c cht lng bn vng nu

+

h thng n nh ni v tha mn ch tiu cht lng mong mun vi mi i tng thuc lp m hnh

~khng chc chn cho trc.G

~


Cht lng bn vng m hnh nhiu nhnCht lng bn vng m hnh nhiu nhn

Wy(t)

G ++

Wmr(t)

K

Xt hm trng s cht lng )(sW Hm nhy ca m hnh nhiu nhn )1(~ mWGG

SS 11~

Xt hm trng s cht lng )(sWp

iu kin t cht lng bn vng:TWWKGGK

Smm

1)1(1~1

1~1

SW

TWm1,

1

1

SW

TW

p

m

1,


1SWp

11 TWm

nh l cht lng bn vng m hnh nhiu nhnnh l cht lng bn vng m hnh nhiu nhn

y(t)G ++

Wmr(t)

K G + K

nh l: iu kin cn v h thng iu khin m hnh nhiu nhn t cht lng bn vng l:1

1

TWSW mp

Chng minh: Tham kho Feedback Control Theory, trang 47-48


nh l cht lng bn vng m hnh nhiu nhn (tt)nh l cht lng bn vng m hnh nhiu nhn (tt)


1)()()( jLjWjW mp

Biu din hnh hc:

1 TWSW ,1)(1)()(

)(1)(

jLjj

jLj mp

)(1)()()( jLjLjWjW

I

1

TWSW mp

,)(1)()()( jLjLjWjW mp Ti mi tn s, vng trn

Im

, gtm (1, j0), bn knh |Wp(j)| khng c ct t t L(j ) b

ReL(j)

1|Wp|

vng trn tm L(j), bn knh |Wm(j)L(j)| |WmL|


Cht lng bn vng m hnh nhiu cngCht lng bn vng m hnh nhiu cng

Wy(t)

G ++

Wmr(t)

K

Xt hm trng s cht lng )(sW Hm nhy ca m hnh nhiu cng

mWGG ~SS 11~

Xt hm trng s cht lng )(sWp

KSWWGKGKS

mm 1)(1~1

iu kin t cht lng bn vng:

1~

1

SW

KSWm1,

1

1

SW

KSW

p

m

1,


1SWp

11 KSWm

nh l cht lng bn vng m hnh nhiu cngnh l cht lng bn vng m hnh nhiu cng

Wy(t)

G ++

Wmr(t)

K

nh l: iu kin cn v h thng iu khin m hnh nhiu cng t cht lng bn vng l:1

1

KSWSW mp

Chng minh: Tham kho Feedback Control Theory.


nh gi cht lng bn vng nh gi cht lng bn vng Th d 1 Th d 1

y(t)G ++

Wmr(t)

K

Bi ton: Cho HTK c s khi nh hnh v i tng

G

Bi ton: Cho HTK c s khi nh hnh v, i tng khng chc chn m t bi m hnh nhiu nhn, trong :

92.005.0 s26800G K 8.181)(11064.0

92.005.0)(

sssWm)60)(250(

6800 ssG

Hm trng s cht lng l:

ssK 8.18.1)(

01.05.0)( sWHm trng s cht lng l:0001.0

)( ssWp(a) H thng c tha cht lng danh nh ?1SWp


p(b) H thng c tha cht lng bn vng ?1

TWSW mp

nh gi cht lng bn vng nh gi cht lng bn vng Th d 1 (tt)Th d 1 (tt)

Gii Gii:

01050 Kim tra iu kin cht lng danh nh

26800810001.0

01.05.0

1

s

s

KGW

SW pp

)60)(250(268008.18.111

sssKG

234

824.4482506324031015075031555.0

234

234

ssssssssSWp

V biu : )()( jSjWp


nh gi cht lng bn vng nh gi cht lng bn vng Th d 1 (tt)Th d 1 (tt)100

10-110

2

Theo biu :10-1 100 101 102 103

10-2

16207.0 SWp


p H thng tha iu kin cht lng danh nh

nh gi cht lng bn vng nh gi cht lng bn vng Th d 1 (tt)Th d 1 (tt)

Kim tra iu kin cht lng bn vng

268008.18.192.005.0 s Kim tra iu kin cht lng bn vng

268008.18.11)60)(250(11064.0

1ssss

KGKGWTW mm

)60)(250( sss41710043980022670 2 ssTW

453400642600661504.319 234 ssssTWm

bi V biu : )()()()( jTjWjSjW mp


nh gi cht lng bn vng nh gi cht lng bn vng Th d 1 (tt)Th d 1 (tt)11

1 0 1 2 30.5

10-1 100 101 102 103

Theo biu : 19383.0 TWSW mp


H thng tha iu kin cht lng bn vng

THIT K B IU KHIN BN VNG DNG THIT K B IU KHIN BN VNG DNG PHNG PHP CHNH LI VNGPHNG PHP CHNH LI VNGPHNG PHP CHNH LI VNGPHNG PHP CHNH LI VNG

(Loopshaping)(Loopshaping)( p p g)( p p g)


tng thit k dng phng php chnh li vng tng thit k dng phng php chnh li vng

y(t)G ++

Wmr(t)

K

Bi ton: Cho i tng khng chc chn m t bi MH nhiu

G

tng thit k:

Bi ton: Cho i tng khng chc chn m t bi MH nhiu nhn. TK b K K(s) sao cho h kn t cht lng bn vng

g Chnh li vng |L(j)| tha t cht lng bn vng:

1 TWSW 1 LWW mp1 TWSW mp

Sau tnh hm truyn b iu khin: )()( jLjK

111

LL


Sau tnh hm truyn b iu khin:)()()( jG

jjK

Cc rng bucCc rng buc Rng buc i vi S v T: Rng buc i vi S v T:

S v T cn tha mn ng thc: , Trng hp ring, ti tn s bt k S v T khng th

1TSng thi nh hn 1/2

Rng buc i vi W v W : Rng buc i vi Wp v Wm: K cn h thng t cht lng bn vng l: 1)()(i jWjW ,1)(,)(min jWjW mp

Ngha l ti mi tn s, |Wp| hoc |Wm| phi nh hn 1

Thng thng |Wp| n iu gim sai s bm nh trong min tn s thp v |Wm| n iu tng v bt


nh tng min tn s cao.

C s ton hc ca phng php chnh li vng C s ton hc ca phng php chnh li vng

t: )()()()()( jTjWjSjWj t: )()()()()( jTjWjSjWj mp )()()()( jLjWjWj mp

)(1)(1)( jLjLj

iu kin cht lng bn vng tng ng vi:

,1)( j T biu thc nh ngha (j) suy ra cc bt ng thc: T biu thc nh ngha (j), suy ra cc bt ng thc:

LLWW

LLWW mpmp

11 LL 11 Do rng buc nn ti mi tn

,1)(,)(min jWjW mp


s ta phi c hoc 1)( jWp 1)( jWm

C s ton hc ca PP chnh li vng (tt) C s ton hc ca PP chnh li vng (tt)

Xt trng hp WW 11 p

WW

L

11

Xt trng hp pm WW 1

mW11 pWL 11

mWL 1

Nu th v phi 2 bt .thc trn gn bng1pW pWW

1p mW1 min tn s thp tha , iu kin

h thng t cht lng bn vng l:mp WW 1

h thng t cht lng bn vng l:

pWL 15 January 2014 H. T. Hong - www4.hcmut.edu.vn/~hthoang/ 147

mWL 1

C s ton hc ca phng php chnh li vng (tt) C s ton hc ca phng php chnh li vng (tt)

Xt trng hp WW 11

11

p

WW

L

Xt trng hp mp WW 1

1mW1 1 pWL1

1 mWL

Nu th v phi 2 bt .thc trn gn bng1mW pWW1mW

min tn s cao tha , iu kin h thng t cht lng bn vng l:

mp WW 1thng t cht lng bn vng l:

pWL 1


mWL

Trnh t thit k dng PP chnh li vng Trnh t thit k dng PP chnh li vng

y(t)G ++

Wmr(t)

K

Bi ton: Cho i tng K m t bi m hnh nhiu nhn

G

Bi ton: Cho i tng K m t bi m hnh nhiu nhn. Thit k b K K(s) sao cho h kn t cht lng bn vng

1 TWSW mp Bc 1: V hai biu Bode bin

min t/s thp tha : v biu (1)pW

WW 1

mp

min t/s thp tha : v biu (1) mW1

i t/ th bi (2)WW 1 pW1

mp WW 1


min t/s cao tha : v biu (2) mp WW 1m

p

W

Trnh t thit k dng PP chnh li vng Trnh t thit k dng PP chnh li vng Bc 2: V biu Bode bin |L(j)| sao cho: Bc 2: V biu Bode bin |L(j)| sao cho:

min tn s thp: |L(j)| nm pha trn biu Bode (1), ng thi |L(j)| >>1. min tn s cao: |L(j)| nm pha di biu Bode (2), ng thi |L(j)|

Phng php chnh li vng Phng php chnh li vng Th d 1 Th d 1

y(t)G ++

Wmr(t)

K

G + K

Bi ton: Cho TK m t bi m hnh nhiu nhn:

)(1)( ssWm10)( sG 1

Mc tiu iu khin l tn hiu ra y(t) bm theo tn hiu chun (t) d h h i t bt k t i 0 1 d/

)101.0(20)( sm)13()( ssG

r(t) c dng hnh sin, tn s bt k nm trong min 0 1 rad/s vi sai s nh hn 2%. Yu cu: Thit k b iu khin K(s) sao cho h kn t cht


Yu cu: Thit k b iu khin K(s) sao cho h kn t cht lng bn vng.


Gii: Gii: Chn hm trng s cht lng:

1050

101.01050

)(

neu neu

jWp

Hm trng s cht lng c chn nh trn tn hiu ra ca i tng bm theo t/hiu chun hnh sinhiu ra ca i tng bm theo t/hiu chun hnh sin trong min 0 1 (rad/s) vi sai s nh hn 2%.Xt bi B d bi )( jW )( jW Xt biu Bode bin : v )( jWp )( jWm


Phng php chnh li vng Phng php chnh li vng Th d 1 Th d 1 40

Bode Diagram

)( jWp20

40

e

(

d

B

)

34

p

)( jWm-20

0

M

a

g

n

i

t

u

d

e

Bc 1: Da vo biu Bode trn, ta thy:10

-210

-110

010

110

210

310

4-40

y

Trong min : 10 Trong min : 210WW 1

1s

101.01050

)(

neuneu

jWpmpWW 1 mp WW 1

pW V biu V biu pW1)101.0(20

1)(

sssWm


mW1 V biu V biu

mW

Phng php chnh li vng Phng php chnh li vng Th d 1 Th d 1 Bode Diagram

20

40

60

34.3

m

p

WW1

48.5

-20

0

20

M

a

g

n

i

t

u

d

e

(

d

B

)

-14.06

m

m

p

WW1

10-2

10-1

100

101

102

103

104

-60

-40

m

Bc 2: Chnh li vng:10 10 10 10 10 10 10

Min :10 pWL Min :

Min :

10

210mW

L 1pWL

12

1

2

)1()1()(

sT

sTKsL


Min : 10mW

L 1


Bc 3: Biu thc L(s) Bc 3: Biu thc L(s)5.48log20 K 266K

5.01 21 T32 33.02 T

2)12()133.0(266)(

s

ssL

Bc 4: Tnh hm truyn b iu khin)1330(266

ss

sGsLsK 10

)12()133.0(266

)()()(

2

s

sssK 2)12()13)(133.0(6.26)(


ssG

)13()(

s )12(

Phng php chnh li vng Phng php chnh li vng Th d 1 Th d 1 Bc 5: Kim tra li iu kin cht lng bn vng g g

V biu TWSW mp

100

10-1

A

m

p

l

i

t

u

d

e

10-1 100 101 102 103 10410-2

10 10 10 10 10 10Frequency (rad/s)

19558.0)max(

TWSWTWSW mpmp


Kt lun: HT thit k tha mn .kin cht lng bn vng


y(t)G ++

Wmr(t)

K

G + K

Bi ton: Cho i tng K m t bi m hnh nhiu nhn:

10501.0)( ssWm2)010(

1)( sG 1

Mc tiu iu khin l tn hiu ra y(t) bm theo tn hiu chun (t) d h h i t bt k t i 0 1 d/

105.0)( sm2)01.0()( s

r(t) c dng hnh sin, tn s bt k nm trong min 0 1 rad/s vi sai s nh hn 10%. Yu cu: Thit k b iu khin K(s) sao cho h kn t cht


Yu cu: Thit k b iu khin K(s) sao cho h kn t cht lng bn vng.

Phng php chnh li vng Phng php chnh li vng Th d 2 Th d 2 Gii: Gii: tn sai s bm theo tn hiu chun hnh sin trong min 0 1 (rad/s) vi sai s nh hn 10%, chn hm trng s cht lng l b lc Butterworth c li bng 10. Trong th d ny, ta chn Wp(s) l b lc Butterworth bc 3:

10122

10)( 23 ssssWp

Xt biu Bode bin : v )( jWp )( jWm


Phng php chnh li vng Phng php chnh li vng Th d 2 Th d 2 105

)( jWp10010

)( jWm10-5

Bc 1: Da vo biu Bode ta thy:10-1 100 101 102 103

10-10

10

Trong min : Bc 1: Da vo biu Bode, ta thy:

10 )(1)( jWjW

Trong min : 50)(1)( jWjW

1.0)( ssW122

10)( 23 ssssWp)(1)( jWjW mp )(1)( jWjW mp

pW V biu V biu pW1105.0

)( ssWm15 January 2014 H. T. Hong - www4.hcmut.edu.vn/~hthoang/ 159

mW1 mW


30Bode diagram

20

30B

)

m

p

WW1

27

0

10

M

a

g

n

i

t

u

d

e

(

d

m

m

p

WW1

10-1 100 101 102 103-20

-10m

40dB/dec

Bc2: Chnh li vng: Min :10 pWL Min : 10

mWL 1

Min : 50 pWL 1 )1)(1()(

21

sTsTKsL


mWL


B 3 Bi th L( ) Bc 3: Biu thc L(s)27log20 K 38.22K

6.01 66.11 T302 033.02 T

)1033.0)(166.1(38.22)( sssL

Bc 4: Tnh hm truyn b iu khin

3822ss

sGsLsK 1

)1033.0)(166.1(38.22

)()()(

ssssK

)10330)(1661()01.0(38.22)(

2


ssG

2)01.0()(

ss )1033.0)(166.1(


Bc 5: Kim tra li iu kin cht lng bn vng Bc 5: Kim tra li iu kin cht lng bn vng V biu TWSW mp

10-1

100

3

10-2

10-1 100 101 102 10310-4

10-3

19785.0)max(

TWSWTWSW mpmp

10 10 10 10 10


Kt lun: H thng thit k tha mn .kin cht lng bn vng

Nhn xt phng php chnh li vng Nhn xt phng php chnh li vng

u im: u im: n gin, s dng k thut v biu Bode quen thuc l

thuyt iu khin kinh in d i d d h h h b h p dng tng i d dng trong trng hp h thng bc thp

Khuyt im: y l phng php gn ng trong nhiu trng hp phi y l phng php gn ng, trong nhiu trng hp phi

chnh li vng (bc 2) nhiu ln mi tha mn c iu kin cht lng bn vng (bc 5). p dng kh kh khn trong trng hp h bc cao nu phi v cc biu Bode bng tay

Phng php chnh li vng khng nu ln c iu kin g p p g g cn v tn ti li gii ca bi ton thit k

Li gii tm c khng phi l li gii ti u


Phng php thit k ti u H

THIT K H THNG THIT K H THNG IU KHIN TI U BN VNGIU KHIN TI U BN VNG


Cu trc chun PCu trc chun P--KK

w(t): tn hiu vo t bn ngoiz(t)Pw(t)

w(t): tn hiu vo t bn ngoi (bao gm tn hiu t, nhiu,)

z(t): tn hiu ra bn ngoi

y(t)K

u(t)

z(t): tn hiu ra bn ngoiu(t): tn hiu ra ca b iu khiny(t): tn hiu vo ca b iu khin

wPPz 1211H h wPz

C th biu din h thng iu khin di dng chun cu trc P-K:

u

P

PPy 22211211 H h:

Lut iu khin: Kyu

uwPy

z

H kn: wKPKPIPPz 211221211 Lut iu khin: Kyu


Hm truyn kn t w(t) n z(t): 211221211 KPKPIPPTzw

Cc bc bin i h thng thnh cu trc PCc bc bin i h thng thnh cu trc P--KK

Bc 1: Xc nh cc vector tn hiu vo ra ca cu trc P K: Bc 1: Xc nh cc vector tn hiu vo ra ca cu trc P-K: z gm tt c cc tn hiu dng nh gi cht lng iu khin.

hi b i w gm tt c cc tn hiu t bn ngoi y gm tt c cc tn hiu c a vo b iu khin K

u gm tt c cc tn hiu ra ca K Bc 2: Tch K ra khi s khi h thng Bc 3: Vit cc biu thc z v y theo w v u: Bc 4: Xc nh ma trn P tha:

uwPy

z


Bin i h thng thnh cu trc PBin i h thng thnh cu trc P--K K Th d 1Th d 1

Hy biu din h thng di y di dng cu trc chun PK bit

WeF (t)

Hy biu din h thng di y di dng cu trc chun PK, bit rng tn hiu ra dng nh gi cht lng iu khin l eF(t)

y(t)G

Wp

r(t)K u (t) G Ke (t)

Gii: Bc 1: Tn hiu vo ra ca cu

trc PKP

)()( trtw )()( tetz Frw

Fez ey

K


yuu Ku(t) )()( tety

Bin i h thng thnh cu trc PBin i h thng thnh cu trc P--K K Th d 1 (tt)Th d 1 (tt)

e (t) Bc 2: Tch K ra khi s :

y(t)

Wp

r(t)

eF (t)

u (t)

Bc 2: Tch K ra khi s :

y(t)G

r(t) e (t)

u (t)

Bc 3: Quan h vo ra:

)( GurWeWez ppF GuWwWz pp

)()( trtw

pp pp

Gurey Guwy B 4 X h P

)()( trtw

)()( tetz F)()( tety

Bc 4: Xc nh P:

wG

GWWz pp1

GGWW

P pp1


)()(y)()( tutu uGy 1 G1

Bin i h thng thnh cu trc PBin i h thng thnh cu trc P--K K Th d 2Th d 2

Hy biu din h thng di y di dng cu trc chun PK bit

WeF(t)

Hy biu din h thng di y di dng cu trc chun PK, bit rng tn hiu dng nh gi cht lng iu khin l eF(t) v yF(t)

y(t)G

Wp

r(t)K

d(t)

WyF(t)++ G Ke (t) u (t) Wm+

Gii: Bc 1: Tn hiu vo ra ca cu

trc PKTd ][ P

)(tw )(tz

Tdrw ][T

FF yez ][ey

K


yuu Ku(t) )(ty


Bc 2: Tch K ra khi s :Tdrw ][ Bc 2: Tch K ra khi s :

WpeF(t)

][T

FF yez ][ey

y(t)G

p

r(t)

d(t)

e (t) u (t)Wm

yF(t)++

uu

B 3 Q h

e (t) u (t)

Bc 3: Quan h vo ra:

)(1 GuGdrWeWez ppF GuWGwWwWz ppp 211

GuGdrey

)(2 GuGdWyz mF )( 22 GuGwWz m GuGwwy


GuGdrey GuGwwy 21


Bc 4: Xc nh P: Bc 4: Xc nh P:

wGWGWWz ppp 11 wPPz 1211

u

wGGGWGW

yz mm 22

10

uPPy 2221

1211

GWGWWPP ppp1211

)()( trtw

GGGWGW

PPPP

P mm10

2221

1211

)()( trtw )()( tetz F

)()( tety GuWGwWwWz ppp 211

)( 22 GuGwWz m


)()(y)()( tutu

)( 22 GuGwWmGuGwwy 21

Bi ton thit k ti u HBi ton thit k ti u H22z(t)w(t) Cho h thng iu khin biu din

P( ) Cho h thng iu khin biu din

di dng cu trc P-K. M hnh ton hc ca i tng l

y(t)K

u(t)

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

)()()()(

121

21

tuDtxCtztuBtwBtAxtx

)()()( 212 twDtxCty

BABBA 21 DBAsIC

DCBA

DCDCsP

1212

121 ][0

0:)(

Bi ton ti u H2: Tm b iu khin K hp thc n nh ni P, ng thi ti thiu chun H2 ca hm truyn Tzw t w(t) n z(t)


2min)( zwKopt TsK gstabilizin

iu kin tn ti li gii bi ton ti u Hiu kin tn ti li gii bi ton ti u H22z(t)w(t)

0:)( 12121

DCBBA

sP

z(t)P

w(t)

0

)(

212

121

DCy(t)

Ku(t)( )

Gi tht: 1. n nh c v pht hin c;),( 2BA ),( 2 AC p ;

2. v 012*121 DDR

),( 2

BIjA

),( 20*21212 DDR

3. l ma trn hng y ct vi mi

4 l t h h i i

121

2

DCBIjA

1BIjA


4. l ma trn hng y hng vi mi

212

1

DCj

Li gii bi ton ti u HLi gii bi ton ti u H22 Li gii bi ton ti u H2 lin quan n hai ma trn Hamilton: Li gii bi ton ti u H2 lin quan n hai ma trn Hamilton:

*1

*12

1121

*12

1112

*1

*2

1121

*12

112

)()( CDRBACDRDICBRBCDRBA

H 112121121121 )()( CDRBACDRDIC

)()()(

1**1*2

12

*2

*2

12

*211

CRDBABDRDIBCRCCRDBA

J )()( 222111212211 CRDBABDRDIB

t: v 0)( JY Ric0)( HX Ric

nh l: Li gii duy nht ca bi ton ti u H2 l:

)(

)(1

2*211

*2 RDBYCAK K

i

0)(

)()(

1*12

*2

11

22112

CDXBRsK Kopt

1****1 )()( CRDBYCCDXBRBAA


vi 21

2211211221

12 )()( CRDBYCCDXBRBAAK

Li gii bi ton cn ti u HLi gii bi ton cn ti u H n ginn gin

Li gii bi ton cn ti u H lin quan n hai ma trn Hamilton: Li gii bi ton cn ti u H lin quan n hai ma trn Hamilton:

*

1*1

*22

*11

2

ACCBBBBA

H

ABBCCCCA

J *11

2*21

*1

2* 11 11

nh l: Tn ti b iu khin n nh sao cho nu v ch nu 3 iu kin di y ng thi c tha mn:

zwT1. v ;2 . v ;

)(RicdomH )(HX Ric)(RicdomJ )(JY Ric

23. ( l bn knh ph ca A)Mt b iu khin tha l :

2)( XY )()( max AXY zwT

0)(

)( *2

*2

12

XBYCYXIA

sK Ksubopt


vi 2*2

12*22

*11

2 )( CYCYXIXBBXBBAAK

Bi ton thit k ti u HBi ton thit k ti u Hz(t)w(t) Pht bi bi t Cho h thng z(t)

Pw(t) Pht biu bi ton: Cho h thng

iu khin biu din di dng cu trc P-K. Thit k b iu khin K

y(t)K

u(t)

n nh h thng, ng thi tn hiu ra z(t) l ti thiu vi mi tn hiu vo w(t) c nng lng nh hn y( )u(t)vo w(t) c nng lng nh hn hoc bng 1.

Bi ton trn tng ng vi tm b iu khin K sao cho ti thiu h h ( ) ( ) i i Hchun H ca hm truyn t w(t) n z(t) Bi ton ti u H

zwK T gstabilizin min 211221211min PKPIKPPK gstabilizing g

Bi ton cn ti u H : tm b iu khin K sao cho chun H ca

Bi ton ti u H khng gii c trong trng hp tng qut


Bi ton cn ti u H: tm b iu khin K sao cho chun H ca hm truyn t w(t) n z(t) nh hn h s >0 cho trc.

Bi ton thit k cn ti u HBi ton thit k cn ti u H n ginn ginz(t)w(t) Bi ton cn ti u H n gin:

P( ) Bi ton cn ti u Hn gin:

tm b iu khin K sao cho chun H ca hm truyn t w(t) n z(t)h h h >0 h t t

y(t)K

u(t)

nh hn h s >0 cho trc trong trng hp i tng tng qut c m t bi PTTT:

)()()(

)()()()( 21tuDtxCtz

tuBtwBtAxtx

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

212

121

twDtxCtytuDtxCtz

DBAsICDCBA

DCBBA

sP

1121

21

][0:)(


DCDC 212 0

Phng trnh i s RicattiPhng trnh i s Ricatti

Phng trnh i s Ricatti (ARE - Algeraic Ricatti Equation): Phng trnh i s Ricatti (ARE Algeraic Ricatti Equation):

0* QXRXXAXA trong : *RR *QQ Phng trnh Ricatti c v s li gii. X c gi l li gii n nh

nu A+RX n nh. Li gii n nh ca phng trnh Ricatti l duy nhtnht.

Tng ng vi mi phng trnh Ricatti, c th thnh lp ma trn Hamilton:

nnAQRA

H22

*

Hamilton:

B : Cc tr ring ca H i xng qua trc o


Li gii phng trnh RicattiLi gii phng trnh Ricatti

X 1 Gi s H khng c tr ring nm trn trc o. t l

c s ca khng gian bt bin n chiu n nh.

T l i t h

nnXX

T

22

1

THT Tc l vi ma trn n nh B : Nu th l nghim n nh ca 0)det( 1 X 112 XXX

THT nn

gphng trnh Ricatti

1 12

Nghim n nh nghim ca phng trnh Ricatti tng ng vi ma g g p g g gtrn Hamilton H c k hiu l:

)(HX Ric K hiu: nu cc gi thit H1 v H2 tha mn;

l nghim n nh ca phng trnh Ricatti.

)(0 RicdomH)( 0HX Ric


g p g)( 0HX Ric

B gi tr thc b chn (Bounded Real Lemma)B gi tr thc b chn (Bounded Real Lemma)

Gi s trong n nh c phtBAICG 1][)( )( CBA Gi s trong n nh c v pht hin c. t ma trn Hamilton:

BAsICsG 1][)( ),,( CBA

*BBA

**0 ACCBBA

H

nh l: Gi s . Cc pht biu di y l tng ng:1 ;

RHG1G1. ;

2. khng c tr ring trn trc o v

3 T t i hi h h t h Ri tti

1G0H )(0 RicdomH

3. Tn ti nghim n nh ca phng trnh Ricatti: 0*** CCXXBBXAXA


iu kin tn ti li gii bi ton cn ti u Hiu kin tn ti li gii bi ton cn ti u H n ginn gin

z(t)w(t)

0:)( 12121

DCBBA

sP

z(t)P

w(t)

0

)(

212

121

DCy(t)

Ku(t)( )

Gi tht: 1 iu khin c v quan st c;)( BA )( AC1. iu khin c v quan st c;

2. n nh c v pht hin c;

3

),( 1BA

]0[][* IDCD

),( 2BA

),( 1 AC

),( 2 AC

3.

4.

]0[][ 12112 IDCD

I

DDB 0*

211


ID21

Li gii bi ton cn ti u HLi gii bi ton cn ti u H n ginn gin

Li gii bi ton cn ti u H lin quan n hai ma trn Hamilton: Li gii bi ton cn ti u H lin quan n hai ma trn Hamilton:

*

1*1

*22

*11

2

ACCBBBBA

H

ABBCCCCA

J *11

2*21

*1

2* 11 11

nh l: Tn ti b iu khin n nh sao cho nu v ch nu 3 iu kin di y ng thi c tha mn:

zwT1. v ;2 . v ;

)(RicdomH )(HX Ric)(RicdomJ )(JY Ric

23. ( l bn knh ph ca A)Mt b iu khin tha l :

2)( XY )()( max AXY zwT

0)(

)( *2

*2

12

XBYCYXIA

sK Ksubopt


vi 2*2

12*22

*11

2 )( CYCYXIXBBXBBAAK

Gii bi ton thit k ti u bn vng dng MatlabGii bi ton thit k ti u bn vng dng Matlab

z(t)w(t)

0:)( 12121

DCBBA

sP

z(t)P

w(t)

0

)(

212

121

DCy(t)

Ku(t)( )

Bc 1: Bin i h thng v cu trc chun P-K. Tm cc ma trn trng thi m t i tng tng qut P.g g g q

Bc 2: Tm li gii bi ton thit k ti u bn vng dng Matlab Bi ton ti u H2: to t u :

>> [Kopt,Tzw] = h2syn(P,ny,nu) Bi ton cn ti u H:

>> [Ksubopt Tzw ]=hinfsyn(G ny

Documents

Chuong 5_LTDKNC.pdf