dktd full 3444

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T RN G I HC B C H K H O A KHOA IN

B MN T NG HA

L thuyt IU KHIN T NG

Lin h : [email protected]

2

MC LC

Phn m u 1 Khi nim.......................................................................................................................5 2 Cc nguyn tc iu khin t ng..................................................................................6

2.1 Nguyn tc gi n nh ...........................................................................................6 2.2 Nguyn tc iu khin theo chng trnh ................................................................6

3 Phn loi h thng KT...............................................................................................6 3.1 Phn loi theo c im ca tn hiu ra....................................................................6 3.2 Phn loi theo s vng kn ......................................................................................6 3.3 Phn loi theo kh nng quan st tn hiu ................................................................7 3.4 Phn loi theo m t ton hc..................................................................................7

4 Biu iu khin t ng trong mt nh my ...............................................................8 5 Php bin i Laplace.....................................................................................................8 Chng 1: M T TON HC CC PHN T V H TH!NG I"U KHI#N T$ %NG 1 Khi nim chung...........................................................................................................10 2 Hm truyn t .............................................................................................................10

2.1 nh ngh&a : ..........................................................................................................10 2.2 Phng php tm hm truyn t...........................................................................10 2.3 Mt s v d' v cch tm hm truyn t ...............................................................11 2.4 Hm truyn t ca mt s thit b in hnh.........................................................13 2.5 i s s khi ..................................................................................................13

3 Phng trnh trng thi .................................................................................................16 3.1 Phng trnh trng thi tng qut ..........................................................................16 3.2 Xy dng phng trnh trng thi t( hm truyn t .............................................18 3.3 Chuyn i t( phng trnh trng thi sang hm truyn ........................................20

Chng 2: )C TNH %NG HC C*A CC KHU V C*A H TH!NG TRONG MI"N TN S! 1 Khi nim chung...........................................................................................................24 2 Phn +ng ca mt khu.................................................................................................24

2.1 Tn hiu tc ng vo mt khu (cc tn hiu tin nh).........................................24 2.2 Phn +ng ca mt khu .........................................................................................24

3 c tnh tn s ca mt khu ........................................................................................25 3.1 Hm truyn t tn s ...........................................................................................25 3.2 c tnh tn s ......................................................................................................26

4 c tnh ng hc ca mt s khu c bn ...................................................................27 4.1 Khu t, l ..............................................................................................................27 4.2 Khu qun tnh b-c 1.............................................................................................27 4.3 Khu dao ng b-c 2.............................................................................................29 4.4 Khu khng n nh b-c 1.....................................................................................31 4.5 Khu vi phn l tng...........................................................................................32 4.6 Khu vi phn b-c 1 ...............................................................................................32 4.7 Khu tch phn l tng........................................................................................33 4.8 Khu ch-m tr........................................................................................................33

Chng 3: TNH /N 0NH C*A H TH!NG I"U KHI#N T$ %NG 1 Khi nim chung...........................................................................................................35 2 Tiu chu1n n nh i s .............................................................................................36

2.1 iu kin cn h thng n nh.........................................................................36 2.2 Tiu chu1n Routh..................................................................................................36 2.3 Tiu chu1n n nh Hurwitz ..................................................................................37

3 Tiu chu1n n nh tn s .............................................................................................37 3.1 Tiu chu1n Nyquist theo c tnh tn s bin pha ..................................................37

3

3.2 Tiu chu1n Nyquist theo c tnh tn s logarit .....................................................37 3.3 Tiu chu1n n nh Mikhailov...............................................................................38

4 Phng php qu2 o nghim s ..................................................................................38 4.1 Phng php xy dng QNS ..............................................................................38

Chng 4: CH3T L4NG C*A QU TRNH I"U KHI#N 1 Khi nim chung...........................................................................................................41

1.1 Ch xc l-p ......................................................................................................41 1.2 Qu trnh qu ...................................................................................................41

2 nh gi ch5t l6ng ch xc l-p............................................................................41 2.1 Khi u(t) = U0.1(t) ..................................................................................................42 2.2 Khi u(t) = U0.t .......................................................................................................42

3 nh gi ch5t l6ng qu trnh qu .........................................................................42 3.1 Phn tch thnh cc biu th+c n gin..................................................................42 3.2 Phng php s Tustin..........................................................................................42 3.3 Gii phng trnh trng thi ..................................................................................44 3.4 S7 d'ng cc hm ca MATAB..............................................................................44

4 nh gi thng qua d tr n nh ...........................................................................45 4.1 d tr bin ..................................................................................................45 4.2 d tr v pha ...................................................................................................45 4.3 Mi lin h gia cc d tr v ch5t l6ng iu khin........................................45

5 Tnh iu khin 6c v quan st 6c ca h thng ....................................................46 5.1 iu khin 6c....................................................................................................46 5.2 Tnh quan st 6c................................................................................................46

Chng 5: NNG CAO CH3T L4NG V T/NG H4P H TH!NG 1 Khi nim chung...........................................................................................................48 2 Cc b iu khin Hiu ch,nh h thng ......................................................................48

2.1 Khi nim .............................................................................................................48 2.2 B iu khin t, l P..............................................................................................48 2.3 B b s8m pha Lead .............................................................................................48 2.4 B b tr. pha Leg..................................................................................................49 2.5 B b tr.-s8m pha Leg -Lead................................................................................50 2.6 B iu khin PI (Proportional Integral Controller) ...............................................51 2.7 B iu khin PD (Proportional Derivative Controller) .........................................51 2.8 B iu khin PID (Proportional Integral Derivative Controller) ...........................52

3 Tng h6p h thng theo cc tiu chu1n ti u ...............................................................53 3.1 Phng php ti u modun ...................................................................................53 3.2 Phng php ti u i x+ng ................................................................................54

Chng 6: H TH!NG I"UKHI#N GIN ON 1 Khi nim chung...........................................................................................................56 2 Php bin i Z.............................................................................................................56

2.1 nh ngh&a ............................................................................................................56 2.2 Mt s tnh ch5t ca bin i Z .............................................................................57 2.3 Bin i Z ng6c ..................................................................................................57

3 L5y m9u v gi m9u .....................................................................................................58 3.1 Khi nim .............................................................................................................58 3.2 L5y m9u................................................................................................................58 3.3 Gi m9u................................................................................................................59

4 Hm truyn t h gin on.........................................................................................60 4.1 Xc nh hm truyn t W(z) t( hm truyn t h lin t'c .................................60 4.2 Xc nh hm truyn t t( phng trnh sai phn.................................................65

5 Tnh n nh ca h gin on ......................................................................................65 5.1 Mi lin h gia mt ph:ng p v mt ph:ng z........................................................65 5.2 Php bin i tng ng ...................................................................................65

Ph' l'c: CONTROL SYSTEM TOOLBOX & SIMULINK TRONG MATLAB

4

1 Control System Toolbox ...............................................................................................66 1.1 nh ngh&a mt h thng tuyn tnh ......................................................................66 1.2 Bin i s tng ng ..................................................................................68 1.3 Phn tch h thng.................................................................................................69 1.4 V d' tng h6p ......................................................................................................71

2 SIMULINK ..................................................................................................................73 2.1 Khi ng Simulink..............................................................................................73 2.2 To mt s n gin.........................................................................................74 2.3 Mt s khi th;ng dng ......................................................................................75 2.4 V d'.....................................................................................................................76 2.5 LTI Viewer ...........................................................................................................77

Phn m u

5

iu khin hc l khoa hc nghin cu nhng qu trnh iu khin v thng tin trong cc my mc sinh vt. Trong iu khin hc, i tng iu khin l cc thit b, cc h thng k thut, cc c c sinh vt

iu khin hc nghin cu qu trnh iu khin cc i tng k thut c gi l iu khin hc k thut. Trong iu khin t ng l c s l thuyt ca iu khin hc k thut.

Khi nghin cu cc qui lut iu khin ca cc h thng k thut khc nhau, ngi ta s dng cc m hnh ton thay th cho cc i tng kho st. Cch lm ny cho php chng ta m rng phm vi nghin cu v tng qut bi ton iu khin trn nhiu i tng c m t ton hc ging nhau.

Mn hc iu khin t ng cung cp cho sinh vin cc kin thc c bn v xy dng m hnh ton hc ca mt i tng v ca c h thng. Trn c s , sinh vin c kh nng phn tch, nh gi cht lng ca h thng iu khin. Ngoi ra, bng cc phng php ton hc, sinh vin c th tng hp cc b iu khin thch hp h thng t c cc ch tiu cht lng ra.

1 Khi nim Mt h thng KT 6c xy dng t( 3 b ph-n ch yu theo s sau :

Trong : - O : i t6ng iu khin - C : b iu khin, hiu ch,nh - M : c c5u o l;ng

Cc loi tn hiu c trong h thng gm : - u : tn hiu ch o (cn gi l tn hiu vo, tn hiu iu khin) - y : tn hiu ra - f : cc tc ng t( bn ngoi - z : tn hiu phn hi - e : sai lch iu khin

V d v mt h thng iu khin n gin

C O

M

u

f

y e

z

h

l

Qi

Q0

Phn m u

6

2 Cc nguyn tc iu khin t ng 2.1 Nguyn tc gi n nh Nguyn tc ny gi tn hiu ra b

Phn m u

7

3.3 Phn loi theo kh nng quan st tn hiu

3.3.1 H thng lin tc Quan st 6c t5t c cc trng thi ca h thng theo th;i gian. M t ton hc : phng trnh i s, phng trnh vi phn, hm truyn

3.3.2 H thng khng lin tc Quan st 6c mt phn cc trng thi ca h thng. Nguyn nhn: - Do khng th t 6c t5t c cc cm bin. - Do khng cn thit phi t cc cm bin. Trong h thng khng lin t'c, ng;i ta chia lm 2 loi: a) H thng gin on (S. discret) L h thng m ta c th quan st cc trng thi ca h thng theo chu k? (T). V bn ch5t, h thng ny l mt dng ca h thng lin t'c. b) H thng vi cc s kin gin on (S vnement discret) - c trng bi cc s kin khng chu k? - Quan tm n cc s kin/ tc ng

V d v h thng lin tc, gin on, h thng vi cc s kin gin on

3.4 Phn loi theo m t ton hc - H tuyn tnh: c tnh t&nh ca t5t c cc phn t7 c trong h thng l tuyn tnh. c

im c bn: xp chng. - H phi tuyn: c t nh5t mt c tnh t&nh ca mt phn t7 l mt hm phi tuyn. - H thng tuyn tnh ha: tuyn tnh ha t(ng phn ca h phi tuyn v8i mt s iu

kin cho tr8c 6c h tuyn tnh gn ng.

Bng chuyn 2

Piston 3 2

Piston 1

Bng chuyn 3

Bng chuyn 1

Phn m u

8

4 Biu iu khin t ng trong mt nh my

5 Php bin i Laplace Gi s7 c hm f(t) lin t'c, kh tch. nh Laplace ca f(t) qua php bin i laplace, k

hiu l F(p) 6c tnh theo nh ngh&a:

0

( ) ( ) ptF p f t e dt

=

- p: bin laplace - f(t): hm gc - F(p): hm nh

Mt s tnh cht ca php bin i laplace

1. Tnh tuyn tnh { }1 2 1 2( ) ( ) ( ) ( )L af t bf t aF p bF p+ = +

2. nh laplace ca o hm hm gc { }'( ) ( ) (0)L f t pF p f=

Nu cc iu kin u b

Phn m u

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3. nh laplace ca tch phn hm gc

0

( )( )t F pL f d

p

=

4. nh laplace ca hm gc c tr. { }( ) ( )pL f t e F p =

5. Hm nh c tr. { }( ) ( )atL e f t F p a = +

6. Gi tr u ca hm gc (0) lim ( )

pf pF p

=

7. Gi tr cui ca hm gc

0( ) lim ( )

pf pF p

=

NH LAPLACE V NH Z CA MT S HM THNG DNG

f(t) F(p) F(z) (t) 1 1 1 1

p

1z

z

t 2

1p

( )21Tz

z

21

2t 3

1p

( )( )

2

3

12 1

T z z

z

+

e-at 1p a+

aT

z

z e

1-e-at ( )

a

p p a+

( )( )( )

1

1

aT

aT

e z

z z e

sinat 2 2

a

p a+ 2

sin2 cos 1z aT

z z aT +

cosat 2 2

pp a+

2

2cos

2 cos 1z z aT

z z aT

+

Chng 1 M t ton hc

10

M T TON HC CC PHN T

V H THNG IU KHIN T NG 1 Khi nim chung

- phn tch mt h thng, ta phi bit nguyn tc lm vic ca cc phn t7 trong s , bn ch5t v-t l, cc quan h v-t l,

- Cc tnh ch5t ca cc phn t7/h thng 6c biu di.n qua cc phng trnh ng hc, th;ng l phng trnh vi phn.

- thu-n l6i hn trong vic phn tch, gii quyt cc bi ton iu khin, ng;i ta m t ton hc cc phn t7 v h thng b

Chng 1 M t ton hc

11

2.3 Mt s v d v cch tm hm truyn t Nguyn tc chung :

- Thnh l-p phng trnh vi phn ; - S7 d'ng php bin i laplace a v dng hm truyn t theo nh ngh&a.

V d 1 : Khuch i lc b

Chng 1 M t ton hc

12

2

2 ei i i

LJ d RJ LB d RBu K

K dt K dt K

+

= + + +

V-y ( )22 2 0( ) ( )U p a p a p a p= + + v8i 2 1 0; ; e

i i i

LJ RJ LB RBa a a K

K K K

+

= = = +

Hm truyn t ca ng c in mt chiu l:

22 2 0

( ) 1( ) ( )pW p

U p a p a p a

= =

+ +

V d 3: Tm hm truyn t ca mch in t7 dng KTT, gi thit khuch i thu-t ton l l tng.

Ta c:

22

ii

V V dV dVC V V R CR dt dt

= = + (1.5)

Xt dng in qua V+ 0

01 1

2i iV V V V V V V

R R

+ ++

= = + (1.6)

Mt khc, do gi thit KTT l l tng nn V- = V+. T( (1.5) v (1.6)

02 0 2

ii

dV dVR C V R C Vdt dt

+ = 0 2

2

( ) 1( ) ( ) 1iV p R CpW pV p R Cp

= =

+

V d 4:

Vi V0

R1

R1

R2

C

+Vcc

-Vcc

y(t)

u(t)

r h

Chng 1 M t ton hc

13

Trong : u(t): lu l6ng ch5t lAng vo; y(t) l lu l6ng ch5t lAng ra; A l din tch y ca b ch5t lAng. Gi p(t) l p su5t ca ch5t lAng ti y b, bit cc quan h sau:

( )( ) p ty tr

= (r l h s)

( ) ( )p t h t= Tm hm truyn t ca b ch5t lAng. Gii Theo cc quan h trong gi thit, ta c:

( )( ) p ty t hr r

= = (1.7)

gia tng chiu cao ct ch5t lAng l:

( ) ( )dh u t y tdt A

= (1.8) T( (1.7) v (1.8), suy ra:

( ) ( )dy u t y tdt r A

=

( ) ( )dyrA y t u tdt

+ =

Hm truyn t ca b ch5t lAng trn l:

( )( ) ( ) 1 1Y p KW pU p rAp Tp

= = =

+ +

2.4 Hm truyn t ca mt s thit b in hnh - Cc thit b o l;ng v bin i tn hiu: W(p) = K - ng c in mt chiu: 2

1 2 2

KW(p)=T T 1p T p+ +

- ng c khng ng b 3 pha KW(p)=T 1p +

- L nhit KW(p)=T 1p +

- Bng ti -W(p)= pKe

2.5 i s s khi i s s khi l bin i mt s ph+c tp v dng n gin hn thu-n tin cho vic tnh ton.

2.5.1 Mc ni tip 1 2W(p)= . ... nW W W

2.5.2 Mc song song 1 2W(p)= ... nW W W

2.5.3 Mc phn hi

1

1 2

W(p)=1

WWW

W1

W2

-

+

U(p) Y(p)

Chng 1 M t ton hc

14

2.5.4 Chuyn tn hiu vo t trc ra sau mt khi

2.5.5 Chuyn tn hiu ra t sau ra trc mt khi

V d 1: I"U KHI#N M$C CH3T LBNG TRONG B# CHCA Cho mt h thng iu khin t ng mc ch5t lAng trong b ch+a nh hnh vD, bit r

Chng 1 M t ton hc

15

Ti

T

T Ta

Qe

=

+==

11

)()()(

pTpNpQpG

V

eV v8i Tv=4

Yu cu : 1. Thnh l-p s iu khin ca h thng. 2. Tm cc hm truyn t

0( ), ( ), ( )

aHU HQ HQW p W p W p 3. Gi s7 cha c b iu khin C(p) = 1. Tm gi tr xc l-p ca ct n8c ng ra nu u(t)= 5.1(t) v Qa = 2.1(t).

S

V d 2 : Cho m hnh ca mt b iu ha nhit ch5t lAng nh hnh vD

Trong : - Ti : nhit ch5t lAng vo b - T : nhit ch5t lAng trong b - Ta : nhit mi tr;ng

Bit r

Chng 1 M t ton hc

16

ai e

T TdTC VHT Q VHTdt R

= +

1 1

i e adTC VH T VHT Q Tdt R R

+ + = + +

( )1 0 0 0( ) ( ) ( ) ( )i e aa p a T p b T p Q p c T p+ = + + [ ]0 0

1 0

1( ) ( ) ( ) ( )i e aT p b T p Q p c T pa p a

= + ++

M hnh iu khin l :

Ngoi phng php i s s khi, chng ta cn c th dng phng php Graph tn hiu tm hm truyn t tng ng ca mt h thng ph+c tp.

3 Phng trnh trng thi 3.1 Phng trnh trng thi tng qut

3.1.1 Khi nim - i v8i mt h thng, ngoi tn hiu vo v tn hiu ra cn phi xc nh, i khi ta cn quan st cc trng thi khc. V d' i v8i ng c in l dng in, gia tc ng c, tn hao, v.v - Khc v8i tn hiu ra phi o l;ng 6c b

Chng 1 M t ton hc

17

- m tn hiu vo: u1(t), u2(t), , um(t), vit 1

...

m

u

Uu

=

, mU

- r tn hiu ra: y1(t), y2(t), , yr(t), vit 1

...

r

yY

y

=

, rY

- n bin trng thi : x1(t), x2(t), , xn(t), vit 1

...

n

x

Xx

=

, nX

Phng trnh trng thi dng tng qut ca h thng 6c biu di.n d8i dng :

X AX BUY CX DU = +

= +

V8i , , ,nxn nxm rxn rxmA B C D A, B, C, D gi l cc ma tr-n trng thi, nu khng ph' thuc vo th;i gian gi l h thng d(ng.

Nhn xt : - Phng trnh trng thi m t ton hc ca h thng v mt th;i gian d8i dng cc phng trnh vi phn. - H thng 6c biu di.n d8i dng cc phng trnh vi phn b-c nh5t.

3.1.3 V d thnh lp phng trnh trng thi V d 1 Xy dng phng trnh trng thi ca mt h thng cho d8i dng phng trnh vi phn nh sau :

2

22 5d y dy y udt dt

+ + =

Gii H c mt tn hiu vo v mt tn hiu ra.

t 1

2

x ydy

x ydt

=

= =

T( phng trnh trn, ta c : 2 2 12 5x x x u+ + = Nh v-y :

1 2

2 1 25 1 12 2 2

x y x

x x x u

= =

= +

[ ]

1 1

2 2

1

2

0 1 05 1 12 2 2

0 1

x xu

x x

xy

x

= +

=

Chng 1 M t ton hc

18

t A, B, C, D l cc ma tr-n tng +ng, suy ra X AX BUY CX DU = +

= +

V d 2 Cho mch in c s nh hnh vD sau, hy thnh l-p phng trnh trng thi cho

mch in ny v8i u1 l tn hiu vo, u2 l tn hiu ra.

Gii Gi s7 mch h ti v cc iu kin u b

Chng 1 M t ton hc

19

( )1( ) 1( ) ( )

n

i i

Y pW p KU p p p

=

= =

t cc bin trung gian nh hnh vD, ta c :

1 1 1

2 2 2 1

1

...

n n n n

x p x Kux p x x

x p x x

= +

= + = +

v y = xn

Suy ra phng trnh trng thi l :

[ ][ ]

1 1

2 2

1 2

1 0

0 1 0

0 0 1n n

Tn

x p Kx p

u

x p

y x x x

= +

=

3.2.2 Khai trin thnh tng cc phn thc n gin Nu hm truyn t 6c khai trin d8i dng :

1

( )( ) ( )n

i

i i

K Y pW pp p U p

=

= =

1

( ) ( )n

i

i i

KY p U pp p

=

=

S c5u trc nh sau :

Nh v-y : i i ipX p X U= + i i ix p x u= +

1

1p p

2

1p p

1np p

U

X1

X2

Xn

K1

K2

Kn

Y1

Y2

Yn

Y

1

Kp p 2

1p p

1np p

U Y x1 x2 xn

Chng 1 M t ton hc

20

Hay

[ ][ ]

1 1

2 2

1 2 1 2

111

0 1n nT

n n

x px p

u

x p

y K K K x x x

= +

=

3.2.3 S dng m hnh tch phn c bn Tr;ng h6p hm truyn t c dng

1 0

( )( ) ( ) ...nnY p KW pU p a p a p a

= =

+ + +

t ( 1) ( )1 2 1 3 2, , ,..., ,n n

n nx y x x y x x y x y x y

= = = = = = =

Suy ra :

1 2

2 3

111

...

...

nn n

n n n

x x

x x

aa Kx x x u

a a a

=

=

= +

3.3 Chuyn i t phng trnh trng thi sang hm truyn 1( ) ( )W p C pI A B D= +

MT S BI TP CH !NG 1 Bi tp 1 I"U KHI#N LU L4NG CH3T LBNG TRONG !NG DEN Cho s iu khin mc lu l6ng ca mt ;ng ng d9n ch5t lAng nh hnh vD

Bit hm truyn ca c c5u chuyn i t( dng in sang p su5t + van LV + ;ng ng + b

chuyn i t( lu l6ng sang dng in l 12.2)(

)()(+

==

pe

pXpYpH

p

Hy thnh l-p m hnh iu khin ca h thng.

Bi tp 2

I"U CHFNH NHI T % C*A MY LOI KH CHO NGI HHI

FE

FT

FIC FY

Y

X

FE : o lu l6ng FT : chuyn i lu l6ng/ dng in FIC : b iu khin lu l6ng FY : chuyn i dng in/p su5t LV

Chng 1 M t ton hc

21

N8c tr8c khi 6c a vo l hi cn phi qua my loi kh nh

Chng 1 M t ton hc

22

Yu cu iu khin l gi cho nhit ra T2 ca ch5t lAng cn lm nng khng i v8i mi lu l6ng Qf. Mt tn hiu iu khin X a n van sD khng ch nhit T2 ca ch5t lAng, nhit ny 6c th hin qua tn hiu o l;ng Y. Hm truyn ca van TV + b trao i nhit + b o

TT l ( )3124.1

)()()(

+==

ppXpYpH . Mt khc, nu gi tn hiu iu khin X khng i nhng

lu l6ng Qf ca ch5t lAng cn lm nng thay i cIng lm nh hng n nhit ra T2. nh hng ca Qf n T2 6c cho bi hm truyn ( )215.0

2)(

)()(+

==

ppQpYpD

f


Bi tp 4 I"U KHI#N NHI T % C*A M%T MY HA LBNG GA (liqufacteur) S khi ca mt my ha lAng ga 6c cho trong hnh sau :

Trong : TT : b chuyn i nhit TIC : b iu ch,nh nhit FT1 : b chuyn i lu l6ng (in t() FT2 : b chuyn i lu l6ng v8i o l;ng tuyn tnh

M

FT1

TIC

FT2

TT

Q2, T1

Q2, T2 Q1, T3

Q1, T4

Ga cn ha lAng

Ga lAng Ch5t lm lnh

Y X

FIC X1

TT

TIC

TV

FT

Qf,T1

Qf,T2 Qc,2

Qc,1 Ch5t lAng cn lm nng

Ch5t lAng mang nhit

Y

X

TT : b chuyn i nhit TV : van iu ch,nh nhit TIC : b iu ch,nh nhit FT : b chuyn i lu l6ng

Chng 1 M t ton hc

23

iu khin nhit ca ga 6c ha lAng, ng;i ta i lu l6ng Q1 ca ch5t lm lnh bi b iu khin TIC. Ga tr8c khi ha lAng c nhit T1, sau khi 6c ha lAng sD c nhit T2. Hm truyn ca cc khu trong s 6c nh ngh&a nh sau :

peK

pQpTpH

p

1

1

1

21 1)(

)()(1

+==

)()()(

2

22 pQ

pTpH = )()()(

3

23 pT

pTpH =

)()()(

1

24 pT

pTpH = 1)()()(

25 == pT

pYpH 1)()()( 16 == pX

pQpH

V8i K1=2, 1=1 min, 1=4 min.


Chng 2 c tnh ng hc

24

"C TNH NG HC CA CC KHU

V CA H THNG TRONG MIN TN S 1 Khi nim chung - Nhim v' ca chng : xy dng c tnh ng hc ca khu/h thng trong min tn s. M'c ch : + Kho st tnh n tnh + Phn tch tnh ch5t + Tng h6p b iu khin - Khu ng hc : nhng i t6ng khc nhau c m t ton hc nh nhau 6c gi l khu ng hc. C mt s khu ng hc khng c phn t7 v-t l no tng +ng, v d' ( ) 1W p Tp= + hay

( ) 1W p Tp= .

2 Phn ng ca mt khu 2.1 Tn hiu tc ng vo mt khu (cc tn hiu tin nh) 2.1.1 Tn hiu bc thang n v

1 0( ) 1( )0 0

tu t t

t

= =

Chng 2 c tnh ng hc

25

nh ngh&a: Phn ng ca mt khu (h thng) i vi mt tn hiu vo xc nh chnh l c tnh qu hay c tnh thi gian ca khu .

2.2.1 Hm qu ca mt khu Hm qu ca mt khu l phn ng ca khu i vi tn hiu vo 1(t). K hiu : h(t) Biu th+c : 1 ( )( ) W ph t L

p

=

2.2.2 Hm trng lng ca mt khu Hm trng lng ca mt khu l phn ng ca khu i vi tn hiu vo (t). K hiu : (t) Biu th+c : { }1( ) W(p)t L = hay ( )( ) dh tt

dt =

V d : Cho mt khu c hm truyn t l

5( )2 1

W pp

=

+

Tm phn +ng ca khu i v8i tn hiu u(t) = 2.1(t-2)-2.1(t-7).

3 c tnh tn s ca mt khu 3.1 Hm truyn t tn s

3.1.1 nh ngha: Hm truyn t tn s ca mt khu, k hiu l W(j), l t s gia tn hiu ra vi tn

hiu vo trng thi xc lp khi tn hiu vo bin thin theo qui lut iu ha ( ) sinmu t U t= .

- J trng thi xc l-p (nu h thng n nh): yxl(t)= Ymsin(t + ) - Biu di.n d8i dng s ph+c :

( )( ) j tu t e

( )( ) j tmy t Y e +

- Theo nh ngh&a : ( )

( )( )( ) ( )

j tjxl m m

j tmm

y t Y e YW j eu t UU e

+

= = =

Nhn xt: Hm truyn t tn s - L mt s ph+c - Ph' thuc vo tn s tn hiu.

Do W(j) l s ph+c nn c th biu di.n n nh sau :

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

jW j A eW j P jQ

=

= +

3.1.2 Cch tm hm truyn t tn s t hm truyn t ca mt khu C th ch+ng minh 6c hm truyn t tn s 6c tm 6c t( hm truyn t ca mt

khu (h thng) theo quan h sau : ( ) ( )

p jW j W p ==

V d : Tm hm truyn t tn s ca khu c hm truyn 5( )2 1

W pp

=

+.

ngha ca W(j)

Chng 2 c tnh ng hc

26

- Xc nh 6c h s khuch i / gc lch pha i v8i tn hiu xoay chiu - Xc nh 6c phng trnh ca tn hiu ra trng thi xc l-p.

3.2 c tnh tn s

3.2.1 c tnh tn s bin pha (Nyquist) Xu5t pht t( cch biu di.n hm truyn t tn s ( ) ( ) ( )W j P jQ = +

- Xy dng h tr'c v8i tr'c honh P, tr'c tung Q. - Khi bin thin, vD nn c tnh tn s bin pha.

nh ngh!a : c tnh tn s bin pha (TBP) l qu o ca hm truyn t tn s W(j) trn mt phng phc khi bin thin t - n .

c im : - TBP i x+ng qua tr'c honh nn ch, cn xy dng

c tnh khi bin thin t( 0 n v l5y i x+ng qua tr'c honh 6c ton b c tnh.

- C th xc nh 6c mdun A, gc pha t( TBP

3.2.2 c tnh tn s logarit (Bode) Quan st s bin thin ca bin v gc pha theo tn s Xy dng h gm 2 c tnh :

* #c tnh tn s bin logarit TBL - Honh l hay log [dec] - Tung L [dB]. Hm L 6c xc nh 20log ( )L A = TBL biu di.n bin thin ca h s khuch i tn hiu theo tn s tn hiu vo.

* #c tnh tn s pha logarit TPL - Honh l hay log [dec] - Tung [rad], 6c xc nh trong W(j). TPL biu di.n bin thin ca gc pha theo tn s tn hiu vo.

* c im ca c tnh logarit Khi h thng c n khu ni tip :

log

L

log

P

jQ

A

Chng 2 c tnh ng hc

27

1 2

1 2

...

...

n

n

L L L L

= + + +

= + + +

4 c tnh ng hc ca mt s khu c bn 4.1 Khu t l

W(p) = K

4.1.1 Hm truyn t tn s

4.1.2 c tnh Nyquist P = K Q = 0

4.1.3 c tnh Bode 20 lg0

L K

=

=

4.1.4 Hm qu ( ) .1( )h t K t=

4.2 Khu qun tnh bc 1

( ) 1

KW pTp

=

+


2 2 2 2

2 2

,

1 1

,

1

K KTP QT T

KA arctg TT

= =

+ +

= =

+

4.2.2 c tnh Nyquist

Chng 2 c tnh ng hc

28

-2 0 2 4 6 8 10-5

-4

-3

-2

-1

0

1

2

3

4

5Nyquist Diagram

Real Axis

Imaginary Axis

c tnh Nyquist ca khu qun tnh b-c 1 (K = 10, T = 0.1)

4.2.3 c tnh Bode 2 220lg 20lg 1L K T = +

arctg T =

-20

-10

0

10

20

30

40

Magnitude

(dB)

10-1 100 101 102 103-90

-45

0

45

Phase (deg)

Bode Diagram

Frequency (rad/sec)

c tnh Bode ca khu qun tnh b-c 1 (K = 10, T = 0.1)

Trn h tr'c logarit, c th vD c tnh bin pha gn ng ca khu qun tnh b-c nh5t nh sau : * c tnh bin logarit - 0 : L L1 = 20lgK; - : L L2 = 20lgK 20lg; - = g = 1/T: L1(g) = L2(g)

* c tnh pha logarit - 0 : 0;

Chng 2 c tnh ng hc

29

- : -pi/2; - = g = 1/T: (g) = -pi/4

Ch : sai lch gia c tnh gn ng v c tnh chnh xc khng 6c l8n hn 3dB. 4.2.4 Hm qu

( )/( ) 1 t Th t K e=

0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

10

12Step Response

Time (sec)

Amplitude

c tnh qu ca khu qun tnh b-c 1 (K = 10, T = 0.1)

4.3 Khu dao ng bc 2 20

2 20 0

( ) 2

W p Kp p

= + + v8i

Chng 2 c tnh ng hc

30

4.3.2 c tnh Nyquist

-2 0 2 4 6 8 10-8

-6

-4

-2

0

2

4

6

8Nyquist Diagram

Real Axis

Imaginary Axis

c tnh Nyquist ca khu dao ng b-c 2 (K = 10, 0 = 0.5, = 0.9)

4.3.3 c tnh Bode

( )22 2 2 2 2 20 0 020 lg 20 lg 4L K = +

-80

-60

-40

-20

0

20

40

Magnitude

(dB)

10-2 10-1 100 101 102-180

-135

-90

-45

0

45

Phase

(deg)

Bode Diagram

Frequency (rad/sec)

c tnh Bode ca khu dao ng b-c 2 (K = 10, 0 = 0.5, = 0.9)

Cch vD c tnh bin pha gn ng : * c tnh bin logarit - 0 : L L1 = 20lgK; - : L L2 = 20lgK02 40lg; - = g = 0: L1(g) = L2(g).

Chng 2 c tnh ng hc

31

0 6c gi l tn s dao ng t nhin

* c tnh pha logarit - 0 : 0; - : -pi; - = g = 0: (g) = -pi/2

4.3.4 Hm qu

( )0 2021( ) 1 sin 1 arccos1 th t K e t

= +

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14Step Response

Time (sec)

Amplitude

c tnh qu ca khu dao ng b-c 2 v8i cc h s khc nhau

4.4 Khu khng n nh bc 1

( ) 1

KW pTp

=


2 2 2 2

2 2

,

1 1

,

1

K KTP QT T

KA arctg TT

pi

= =

+ +

= =

+

4.4.2 c tnh Nyquist

4.4.3 c tnh Bode 2 220 lg 20lg 1L K T = +

arctg T pi=

4.4.4 Hm qu ( )/( ) 1t Th t K e=

Chng 2 c tnh ng hc

32

4.5 Khu vi phn l tng ( ) W p Kp=

4.5.1 Hm truyn t tn s 0,

,

2

P Q KA K

pi

= =

= =

4.5.2 c tnh Nyquist

4.5.3 c tnh Bode 20 lg 20 lgL K = +

4.6 Khu vi phn bc 1 ( )( ) 1W p K Tp= +


2 2

,

1,

P K Q TKA K T arctgT

= =

= + =

4.6.2 c tnh Nyquist

-2 0 2 4 6 8 10 12-200

-150

-100

-50

0

50

100

150

200Nyquist Diagram

Real Axis

Imaginary Axis

c tnh Nyquist ca khu vi phn b-c nh5t

4.6.3 c tnh Bode 2 220 log 20 log 1

1g

L K T

T

= + +

=

Chng 2 c tnh ng hc

33

10-1 100 101 102 1030

45

90

135

Phase (deg)

0

10

20

30

40

50

60

Magnitude

(dB)

Bode Diagram

Frequency (rad/sec)

c tnh Bode ca khu vi phn b-c 1 (K = 10, T = 0.1)

4.7 Khu tch phn l tng ( ) KW p

p=


0,

,

2

KP QKA

pi

= =

= =

4.7.2 c tnh Nyquist

4.7.3 c tnh Bode 20 lg 20lgL K =

4.8 Khu chm tr -( ) pW p e =

4.8.1 Hm truyn t tn s ( )

1,

jW j eA

=

= =

4.8.2 c tnh Nyquist

4.8.3 c tnh Bode 0L

=

=

Chng 2 c tnh ng hc

34

10-1 100 101 102 103-180

-135

-90

-45

0

45

Phase (deg)

-20

-10

0

10

20

30

40

Magnitude

(dB)

Bode Diagram

Frequency (rad/sec)

c tnh Bode ca khu qun tnh b-c 1 (xanh blue) v khu qun tnh b-c nh5t c tr. 0.5s (xanh verte)

Cc lnh thc hin vD c tnh trn trong MATLAB : num=10 den=[0.1 1] W1=tf(num,den) W2=W1; set(W2,IODelay,0.5); W2 bode(W1); hold on bode(W2);

Chng 3 Tnh n nh ca h thng

35

TNH $N %NH CA H THNG IU KHIN T& NG

1 Khi nim chung Kho st mt h thng iu khin t ng 6c m t ton hc d8i dng hm truyn t :

1 0

1 0

... ( )( )... ( )

m

m

n

n

b p b p b Y pW pa p a p a U p

+ + += =

+ + + (3.1)

Phng trnh vi phn tng +ng ca h thng l :

1 0 1 0... ...

n m

n mn m

d y dy d u dua a a y b b b u

dt dt dt dt+ + + = + + + (3.2)

Nghim ca phng trnh vi phn (3.2) c dng nh sau : 0( ) ( ) ( )qdy t y t y t= + (3.3) Trong : y0(t) l nghim ring ca phng trnh (3.2) c v phi, c trng cho qu trnh xc lp. yqd(t) l nghim tng qut ca (3.2), c trng cho qu trnh qu .

Tnh !n nh ca mt h thng ch ph thuc vo qu trnh qu , cn qu trnh xc lp l mt qu trnh !n nh.

nh ngha : a) Mt h thng KT n nh nu qu trnh qu tt dn theo th;i gian.

lim ( ) 0qdt

y t

=

b) Mt h thng KT khng n nh nu qu trnh qu tng dn theo th;i gian. lim ( )qd

ty t

=

c) Mt h thng KT bin gi8i n nh nu qu trnh qu khng i hay dao ng khng tt dn.

Xt nghim yqd(t) trong (3.3), dng tng qut ca nghim qu nh sau :

,

1 1( ) i

n np t

qd i qd ii i

y t C e y= =

= = (3.4) v8i n l b-c v pi l nghim ca phng trnh c tnh 1 0( ) ... 0nnN p a p a p a= + + + = (3.5) Ci l cc h


36

Kt lun : 1) H thng iu khin t ng n nh nu tt c cc nghim ca phng trnh c tnh c

ph"n thc m. 2) H thng iu khin t ng khng n nh nu c t nht mt nghim ca phng trnh c

tnh c ph"n thc dng. 3) H thng iu khin t ng bin gi8i n nh nu c t nh5t mt nghim ca phng trnh

c tnh c ph"n thc bng 0, cc nghim cn li c ph"n thc m.

2 Tiu chun n nh i s 2.1 iu kin cn h thng n nh

Xt mt h thng iu khin t ng c phng trnh c tnh tng qut nh sau : 1 0( ) ... 0nnN p a p a p a= + + + =

Pht biu : iu kin cn mt h thng KT tuyn tnh !n nh l t"t c cc h s ca phng trnh c tnh dng

2.2 Tiu chu n Routh

2.2.1 Cch thnh lp bng Routh pn an an-2 an-4 a0 pn-1 an-1 an-3 an-5 (a0) pn-2 cn-2,1 cn-2,2

p2 c2,1 c2,2 p1 c1,1 c1,2 p0 c0,1

V8i :

2 4

1 3 1 52,1 2,2

1 1

;

n n n n

n n n n

n n

n n

a a a a

a a a ac c

a a

= = ;

2,1 2,2

1,1 2,30,1

1,1

c c

c cc

c=

Quy t*c : M=i s hng trong bng Routh l mt t, s, trong :

- T7 s l nh th+c b-c 2, mang d5u m. Ct th+ nh5t ca nh th+c l ct th+ nh5t ca 2 hng +ng st trn hng c s hng ang tnh ; ct th+ hai ca nh th+c l ct +ng st bn phi s hng ang tnh cIng ca 2 hng trn.

- M9u s : T5t c cc s hng trn cng mt hng c cng m9u s l s hng ct t+ nh5t ca hng st trn hng c s hng ang tnh.

2.2.2 Pht biu tiu chun Routh iu kin cn v h thng tuyn tnh !n nh l t"t c cc s hng trong ct th

nh"t ca bng Routh phi dng.

2.2.3 Cc tnh cht ca bng Routh - C th nhn hoc chia t5t c cc s hng trn cng mt hng ca bng Routh v8i mt s

dng. - S ln i d5u ca cc s hng trong ct th+ nh5t ca bng Routh b


37

- Nu trong ct th+ nh5t ca bng Routh c mt s hng b0. 3.2.2 p dng tiu chun - Trong c tnh logarit


38

+ C+ giao im dng : l giao ca () v8i ;ng th:ng -pi, c chiu theo chiu tng ca . + C- giao im m : l giao ca () v8i ;ng th:ng -pi, c chiu theo chiu tng ca .

- Tiu chu1n ch, p d'ng cho h kn phn hi -1, h h n nh.

3.3 Tiu chu n n nh Mikhailov

3.3.1 Pht biu iu kin cn v h thng tuyn tnh !n nh l biu # vect a thc c tnh

A(j) xu"t pht t trc thc dng quay n gc phn t ngc chiu kim #ng h# khi t%ng t 0 n .

3.3.2 p dng tiu chun - Tiu chu1n ny 6c p d'ng xt n nh cho h b5t k? (h/kn) - a th+c c tnh l a th+c t7 s ca hm truyn t.

4 Phng php qu o nghim s Phng php qu2 o nghim s (QNS) th;ng dng cho h thng c mt thng s bin i

tuyn tnh. V8i m=i gi tr ca thng s, phng trnh c tnh ca h thng sD c mt t-p nghim, m=i nghim 6c biu di.n b


39

- m qu2 o xu5t pht t( 'ip v kt thc ''

jp ;

- (n m) qu2 o xu5t pht t( 'ip v tin ra v cng. Khi phng trnh N0(p) = 0 c nghim ph+c lin h6p th cp qu2 o tng t+ng ca n sD i x+ng qua tr'c thc.

4.1.4 Xc nh cc "ng tim cn C (n-m) ;ng th:ng tim c-n cho cc qu2 o tin ra v cng.

- Tm tim c-n : ' ''01 1

1 n mi j

i jR p p

n m= =

=

- Gc to bi cc ;ng tim c-n v tr'c honh : 2 1kk

n m pi

+=

, k = 0,1,,n-m-1

4.1.5 Xc nh im tch kh#i trc th$c v hng dch chuyn ca qu o

- Kho st hm s 00

( )( ) ( )N pf pM p

= xc nh h8ng di chuyn ca qu2 o

- Cc nghim ca phng trnh ( ) 0df pdp

= chnh l cc im tch khAi tr'c thc ca QNS.

4.1.6 Xc nh giao im ca trc o vi QNS Gi jc l im ca QNS v8i tr'c o. Thay p = jc vo phng trnh c tnh N(p) = 0, c 6c xc nh t( h phng trnh :

Re ( ( )) 0Im( ( )) 0

c

c

al N jN j

=

=

V d' : VD QNS ca mt h thng c phng trnh c tnh c thng s K bin thin nh sau :

3 2( ) 3 ( 2) 10 0N p p p K p K= + + + + = Gii : Tr8c tin, ta bin i phng trnh trn v dng 3.6 nh sau : ( )3 2( ) 3 2 ( 10) 0N p p p p K p= + + + + = Nh v-y : ( )3 20 ( ) 3 2N p p p p= + + v 0 ( ) ( 10)M p p= + - Cc im xu5t pht ca QNS :

' ' '

0 1 2 3( ) 0 0; 1; 2;N p p p p= - Cc im kt thc ca QNS :

''

0 1( ) 0 10M p p= - V-y c 3 im xu5t pht, 1 im kt thc nn sD c 2 qu2 o tin ra v cng (tng +ng v8i 2 tim c-n) - Tm tim c-n : R0 = 7

- Gc cc tim c-n so v8i tr'c honh : 3(2 1) ;2 2 2k

k pi pi pi = + =

- Giao im v8i tr'c o : 207c

= ti K = 6/7.


40

-10 -8 -6 -4 -2 0 2 4-30

-20

-10

0

10

20

30Root Locus

Real Axis

Imag

inar

y Ax

is

Hnh vD trn biu di.n Qu2 o nghim s ca h thng trong v d' trn (6c vD b

Chng 4 Cht lng ca qu trnh iu khin

41

CH+T L ,NG CA QU TRNH IU KHIN

1 Khi nim chung Ch5t l6ng ca mt h thng iu khin t ng 6c nh gi qua 2 ch : ch xc l-p

v qu trnh qu .

1.1 Ch xc lp Ch5t l6ng iu khin 6c nh gi qua sai lch t&nh (hay cn gi l sai s xc l-p) Sai lch tnh (St) l sai lch khng !i sau khi qu trnh qu kt thc.

1.2 Qu trnh qu Ch5t l6ng ca h thng 6c nh gi qua 2 ch, tiu chnh : a) qu iu chnh ln nht max : l sai lch cc i trong qu trnh qu so v8i gi tr xc l-p, tnh theo n v phn trm.

max *100%maxy y

y

= (4.1)

b) Thi gian qu ln nht Tmax : V mt l thuyt, qu trnh qu kt thc khi t . Trong iu khin t ng, ta c th xem qu trnh qu kt thc khi sai lch ca tn hiu 6c iu khin v8i gi tr xc l-p ca n khng v6t qu 5% (mt s ti liu chn bin l 2%). Khong th;i gian gi l Tmax.

Thc t iu khin cho th5y : khi gim max th Tmax tng v ng6c li. Thng th;ng, qui nh cho mt h thng iu khin :

max = (20 30)% Tmax = 2 n 3 chu k? dao ng quanh gi tr xc l-p

c) Thi gian tng tm : l th;i gian t( 0 n lc tn hiu iu khin t 6c 90% gi tr xc l-p ln u tin.

2 nh gi cht lng ch xc l p Xt mt h thng kn phn hi -1.

Wh(p) U(p) Y(p) E(p)

max

Tmax tm

t

y


42

Theo nh ngh&a, ta c :

0lim ( ) lim ( )tt p

S e t pE p

= =

Theo s khi trn, ta c : ( )( )1 ( )h

U pE pW p

=

+

V-y 0

( )lim ( ) lim1 ( )t t p h

U pS e t pW p

= =

+ (4.2)

Trng hp h thng kn bt k&, ta chuyn v h thng kn phn h#i 1 tng ng v p dng cng thc tnh sai lch t!nh cho h tng ng ny.

Nhn xt : sai lch t&nh St ph' thuc - Hm truyn t ca h h - Tn hiu kch thch.

Hm truyn t ca h h c dng tng qut nh sau :

' '

10'

... 1( ) ( )... 1

m

mh n

n

b p b pK KW p W pp a p p

+ + += =

+ +

l b-c tch phn

2.1 Khi u(t) = U0.1(t) 1( )U pp

= 0

0

1lim1 ( )

t pS K W p

p

=

+

- V8i = 0 : 01tUS

K=

+

- V8i = 1,2,.. St = 0

2.2 Khi u(t) = U0.t 02( )

UU pp

= 00

0

lim1 ( )

t p

USKp W pp

=

+

- V8i = 0 : tS =

- V8i = 1: 0tUSK

=

- V8i = 2,3,.. St = 0

3 nh gi cht lng qu trnh qu Phi vD 6c p +ng qu y(t) ca h thng

3.1 Phn tch thnh cc biu thc n gin Trong phng php ny, tn hiu ra Y(p) 6c phn tch thnh tng ca cc thnh phn n

gin. S7 d'ng bng tra Laplace hay hm ilaplace trong MATLAB tm hm gc y(t).

3.2 Phng php s Tustin

3.2.1 Ni dung phng php S ha tn hiu lin t'c thnh tn hiu gin on tm p +ng th;i gian, ngh&a l : chuyn hm truyn t t( h lin t'c sang h gin on. - Trong h gin on, quan tm n y(kT) - Bin i ton hc trong h gin on l Y(z)


43

kT (k+1)T

- c im : y(kT) -> Y(z) y(k+m)T -> zmY(z)

Xc -nh mi lin h gi.a h lin tc v h gin on Xt mt quan h gia Y(p) v U(p) d8i dng hm truyn t :

( ) 1( ) ( )Y pW pU p p

= = (4.3) Phng trnh vi phn tng +ng l :

0

( ) ( )t

y t u t dt= (gi thit cc iu kin

u b


44

Cc h s ai, bj 6c xc nh t( phng trnh trn. Gi thit bit tr8c cc gi tr u y(0), y(1), y(2), y(3), ta c th tnh ln l6t cc gi tr cn li ca tn hiu ra y(kT).

3.3 Gii phng trnh trng thi Nghim ca phng trnh trng thi :

X=AX+BUY=CX+DU

(4.6)

c dng sau :

( )

0

( ) (0) ( )t

At A tX t e X e BU d = + (4.7)

( )

0

( (0) ( )t

At A tY t C e X e BU d DU

= + +

(4.8) Trong : ( ){ }11Ate L pI A = Ghi ch :

1

det( )adjAAA

= v8i Aadj l ma tr-n c cc phn t7 ( 1) det( )i jij jia A+= trong Aji l ma

tr-n c 6c b


45

Cu lnh: LSIM(sys,u,t) V8i: + sys l tn ca hm truyn t 6c nh ngh&a tr8c + u l vect tn hiu vo + t l vect th;i gian. V d': t = 0:0.01:2*pi; u = sin(t); lsim(W1,u,t);

4 nh gi thng qua d tr! n nh 4.1 d tr bin

( )L L pi =

4.2 d tr v pha 180 ( )c = +

C th xc nh cc d tr v bin , v pha b


46

5 Tnh iu khin c v quan st c ca h thng 5.1 iu khin #c

5.1.1 nh ngha Xt mt h thng 6c m t ton hc d8i dng phng trnh trng thi :

X AX BUY CX DU = +

= +

V8i , , ,nxn nxu rxn rxmA B C D

Mt h thng c gi l iu khin c nu t$ mt vect ban "u X0 bt k&, ta lun c th tm c vect tn hiu Ud chuyn h thng t$ trng thi X0 n trng thi Xd mong mun.

5.1.2 iu kin Xy dng ma tr-n iu khin

P = [B, AB, A2B,, An-1B] iu kin cn v mt h thng m t ton hc di dng phng trnh trng thi

iu khin c l rank(P) = n.

Nhn xt : - Tnh iu khin 6c ch, ph' thuc vo cc ma tr-n trng thi A, B. - Lin quan n vic chn cc bin trng thi

V d' : Cho h thng c m t ton hc d8i dng hm truyn t nh sau :

220( )

2 4W p

p p=

+ +

Gi s7 t cc bin trng thi l :

1

1 2

x yx x

=

=

Xc nh tnh iu khin 6c ca h thng. Gii Ta c :

1 2

2 1 22 0.5 10x x

x x x u

=

= +

hay 1 1

2 2

0 1 02 0.5 10

x xu

x x

= +

Ma tr-n P

[ ] 0 0 1 0 0 10,10 2 0.5 10 10 5

P B AB

= = =

det(P) = -100 0 nn rank(P) = 2. V-y h thng v8i cch t bin trng thi nh trn l iu khin 6c.

5.2 Tnh quan st #c

5.2.1 nh ngha Mt h thng c gi l quan st c nu t$ cc vect U v Y c, ta c th xc nh

c cc bin trng thi X ca h thng.

5.2.2 iu kin Xy dng ma tr-n quan st

L = [C, AC, (A)2C,, (A)n-1C]


47

iu kin cn v mt h thng m t ton hc di dng phng trnh trng thi quan st c l rank(L) = n.

Nhn xt : - Tnh iu khin 6c ch, ph' thuc vo cc ma tr-n trng thi A, C.

V d' : Xt trong v d' trn, ma tr-n trng thi C sD l :

C = [1 0] Ma tr-n quan st

[ ] 1 0 2 1 1 0' ' ' 0 1 0.5 0 0 1L C A C

= = =

Do rank(L) = 2 nn h trn quan st 6c.

Chng 6 H thng iu khin gin on

48

NNG CAO CH+T L ,NG V T$NG H,P H THNG

1 Khi nim chung Trong mt h thng iu khin t ng, vai tr ca b iu khin C l :

- /n nh ha h thng - Nng cao ch5t l6ng iu khin.

2 Cc b iu khin Hiu ch"nh h thng 2.1 Khi nim

- C nhiu loi b iu khin (khc nhau v c5u to, m t tan hc, tc d'ng iu khin,) - M'c ch l nh

+

2.3.2 c tnh tn s logarit = arctg(aT) - arctg(T)

max

1

1sin 0

1

max T aa

a

=

= >+

Wh(p) U(p) Y(p) E(p)

Wc(p)


49

-2

0

2

4

6

8

10

12

14

16

18

20

Magnitude

(dB)

10-1 100 101 102 1030

45

90

Phase

(deg)

Bode Diagram

Frequency (rad/sec)

c tnh logarit ca b b s8m pha (K=1, T=0.1, a = 5)

Nhn xt : - c tnh bin lm tng h s khuch i vng tn s cao - Gy ra s v6t pha vng tn s trung bnh.

2.3.3 Tc dng hiu ch%nh Ty thuc vo cch chn h s khuch i K, cc thng s a, T m tc d'ng hiu ch,nh r5t

khc nhau. Nn t-n d'ng s v6t pha tn s trung bnh lm tng d tr v pha ca h thng.

2.4 B b tr pha Leg

2.4.1 Hm truyn t 1( ) , 1

1aTpW p K aTp

+= T2)

Hay

( )1 21

1( ) 1 1 ( )* ( )PI PDW p KT T p W p W pT p

= + + =

2.8.2 c tnh tn s logarit Nhn xt :

- L s kt h6p ca b iu khin PI v PD


53

2.8.3 Tc dng hiu ch%nh - PI : gim b-c sai lch t&nh - PD : tng

3 Tng hp h thng theo cc tiu chun ti u 3.1 Phng php ti u modun

- Kho st h kn phn hi -1. Hm truyn h kn l k( )* ( )W ( )

1 ( )* ( )c h

c h

W p W ppW p W p

=

+

- Mt trong nhng tiu chu1n chn b iu khin Wc(p) l tn hiu ra lun bm theo tn hiu vo, ngh&a l Y(p) = X(p) hay ( ) 1,kW p = . - Thc t, vic t 6c tiu chu1n ny l v cng kh khn do : bn thn h thng c qun tnh, dao ng, tr., Tuy nhin nhng h thng thc t li c mt c im t nhin h6p l l suy gim mnh tn s cao, nh; v-y m n tn ti v8i nhi.u. - thAa thu-n gia yu cu l tng v iu kin thc t, yu cu l tng h6p h thng sao cho

' ( ) 1kW j (*) trong mt di tn s cng rng cng tt.

hay ni cch khc 20lg 0k kL A= . Di tn s lm Lk = 0 cng l8n th ch5t l6ng h thng kn cng cao.

Phng php ny hin nay ch, m8i 6c p d'ng cho mt s h h c bit d8i y. Tr;ng h6p cc h tng qut, ta a v cc h c bit nh; phng php gn ng.

3.1.1 H h& l khu qun tnh bc nht

- H h : ( )1h

KW pTp

=

+

- B iu khin ( ) Pci

KW pT p

=

- H h v8i b iu khin : ( )' ( )

1h RKW p

T Tp=

+ v8i ( ) iR

P

TT pK

=

- Hm truyn h kn v8i b iu khin

( )' ( )

1k RKW p

T p Tp K=

+ +

( ) ( )'

2 22( )k

R R

KW pK T T T

=

+

Do 22

'

2 2 2 2 2 4( ) ( 2 )k R R RKW p

K T KT T T T =

+ +

iu kin (*) thAa mn trong di tn s cng rng cng tt, ta c th chn TR sao cho :

lg

L

Lk


54

2 2 0 2iR R RP

TT KT T T KTK

= = =

3.1.2 H h& l khu qun tnh bc 2

- H h : ( )( )1 2( ) 1 1hKW p

T p T p=

+ +

- B iu khin 1( ) 1c Pi

W p KT p

= +

- Tr8c tin chn TI = T1 b m9u s (T1p + 1). Thc hin tng t phn cn li, ta sD 6c : 1

22

22

iR P

P

T TT KT KK KT

= = =

3.1.3 H h& l khu qun tnh bc 3

- H h : ( )( )( )1 2 3( ) 1 1 1hKW p

T p T p T p=

+ + +

- B iu khin ( )( )' '1 21 11( ) 1c P d

i R

T p T pW p K T p

T p T p

+ + = + + =

v8i ( ) iR

P

TT pK

=

trong : ' '

1 2' '

1 2

i

i d

T T T

T T TT

+ =

=

- u tin, ta chn ' '1 1 2 2;T T T T= =

Sau n gin cc biu th+c v thc hin nh trn, ta 6c 1 232

PT TK

KT+

= .

3.2 Phng php ti u i xng - Nh6c im ca tng h6p ti u modun trn l h h phi n nh, hm qu h(t) c dng tip xc v8i tr'c honh ti gc 0. - Xt h kn phn hi -1, ta c :

' '

' '

' '1 1h k

k hh k

W WW WW W

= =+

- T( phng php ti u modun, thay v ' ( ) 1kW j , ta phi xc nh b iu khin sao cho

' ( ) 1hW j (**) - c tnh tn s logarit mong mun l :

c

i

1


55

c tnh xy dng c 3 phn + Tn s th5p : L cc l8n sai lch t&nh b


56

H THNG IUKHIN GIN O/N

(H xung s) 1 Khi nim chung - Trong iu khin, ng;i ta phn thnh 2 loi h thng : h lin t'c v h khng lin t'c. Trong h khng lin t'c li c 2 loi chnh l : h gin on (h xung s) v h thng v8i cc s kin gin on. V c im ca h gin on l ta ch, c th quan st cc trng thi ca h thng mt cch gin on nhng c chu k? (T). - Nguyn nhn hnh thnh cc h thng gin on l :

o S hnh thnh ca cc b iu khin s : linh hot, d. dng thay i v khng ch cc thng s.

o Gim st cc tn hiu b


57

0( ) ( ) i

iF z f iT z

=

= (6 .4) F(z) 6c gi l bin i Z ca hm gin on f(iT). K hiu l : F(z) = Z{f(iT)} Hay f(iT) = Z-1{F(Z)}

Nhn xt : - Bin i Z l dng bin i laplace. - Ch, c bin i Z ca hm gin on ch+ khng c bin i Z ca hm lin t'c.

V d : Cho hm f(t) = e-at. Tm bin i Z ca hm f(iT). Gii Ta c f(t) = e-at nn f(iT) = e-aiT. Theo nh ngh&a

1 2 2

0

1

( ) ( ) 1 ...

1( )1

i aT a T

i

aT aT

F z f iT z e z e z

zF ze z z e

=

= = + + +

= =

v8i iu kin e-aTz-1


58

V d :

2( ) 3 2zF z

z z=

+

Phn tch hm F(z) trn ta 6c :

1 2 3 4( ) 3 7 15 ...F z z z z z = + + + + V-y f(iT) = 2i -1.

3 Ly m#u v gi! m#u 3.1 Khi nim

c th a b iu khin s vo h thng, cn c qu trnh l5y m9u v gi m9u. - L5y m9u l chuyn tn hiu lin t'c thnh tn hiu gin on. - Gi m9u l qu trnh chuyn tn hiu gin on thnh tn hiu lin t'c.

Kho st mt qu trnh l5y m9u v gi m9u n gin nh hnh vD sau, trong tn hiu gin on khng qua b5t k? mt khu bin i no.

c im th;i gian ca cc tn hiu trn nh sau :

Nhn xt : ( )e t l tn hiu lin t'c t(ng on. Sau qu trnh bin i (l5y m9u v gi m9u), ( )e t khc v8i e(t)

ban u. Khi tn s l5y m9u l8n cng l8n (T b) th ( )e t cng gn ging dng ca e(t).

3.2 L"y m%u Phng trnh ca tn hiu e*(t) sau khi 6c l5y m9u l :

t

e

a) t

e*(t)

b) T 2T 3T iT

t

e(t)

c) T 2T 3T iT

K s L5y m9u Gi m9u e(t) e*(t) e*(t) e(t)

E(p) E*(p) E*(p) E(p)

K s Wh(p) L5y m9u Gi m9u u y


59

*

0( ) ( ) ( )

ie t e iT t iT

=

= (6 .5) Do :

*

0( ) ( ) ipT

iE p e iT e

=

= (6.6)

3.2.1 nh ngha Mt b l5y m9u 6c gi l l tng nu sau khi l5y m9u, nh laplace ca tn hiu l5y m9u

c biu th+c nh trong 6.6.

S thay th ca b l5y m9u l tng nh sau :

Nu bit nh laplace ca tn hiu c l5y m9u E(p), ta c th tm 6c nh laplace ca tn hiu 6c l5y m9u l tng theo biu th+c sau :

* 1 2 (0)( )2n

eE p E p jnT T

pi

=

= + +

(6.7)

Ghi ch : c kh nng nhiu tn hiu khc nhau sau khi 6c l5y m9u sD c phng trnh ton hc nh nhau.

3.2.2 nh l ly m)u (nh l Shannon) Mt tn hiu lin t'c theo th;i gian e(t) ch, c th ph'c hi sau qu trnh l5y m9u nu thAa

mn iu kin : ax2 mf f (6.8)

Trong : - f l tn s l5y m9u (f = 1/T) - fmax l tn s cc di ca tn hiu cn l5y m9u

3.2.3 Tnh cht ca tn hiu E*(p) Tnh cht 1

Hm E*(p) tun hon trong mt ph:ng p v8i chu k? jp trong 2p Tpi

= (T l chu k? l5y m9u)

Tnh cht 2 Nu E(p) c mt cc ti p = p1 th E*(p) phi c cc ti p = p1 + jp v8i m = 0, 1, 2, 3.3 Gi m%u

3.3.1 B gi* m)u bc 0 c im ca b gi m9u b-c 0 l tn hiu 6c gi m9u khng i gia 2 ly l5y m9u v b


60

2

2

0

1 1 1 1( ) (0) ( ) ...

1 (0) ( ) (2 ) ...

1 ( )

pT pT pT

pTpT pT

pTipT

i

E p e e e T e ep p p p

ee e T e e T e

p

ee iT e

p

=

= + +

= + + +

=

Kt h6p v8i 6.6, ta 6c

*1( ) ( )pTeE p E p

p

=

(6.8)

Nh v-y, m t ton hc ca b gi m9u b-c 0 (Zero Order Hold) l :

Hm truyn t ca b gi m9u b-c 0 l :

1( )pT

ZOHeW pp

= (6.9)

3.3.2 B gi* m)u bc 1 Tn hiu gi m9u gia 2 ln l5y m9u lin tip nT v (n+1)T l

( ) ( ) '( )( )ne t e nT e nT t nT= + , ( 1)nT t n T < +

v8i [ ]( ) ( 1)'( ) e nT e n Te nTT

=

Ch+ng minh tng t, ta tm 6c hm truyn t ca b gi m9u b-c nh5t (First Order Hold) l :

21 1( )

pT

FOHpT eW p

T p

+ =

Nh v-y, s thay th ca b l5y m9u v gi m9u l :

Ch : B l5y m9u v gi m9u trong s trn khng th l m hnh ton hc cho mt thit b c' th no trong thc t. Tuy nhin, s kt h6p gia b l5y m9u v gi m9u li l m hnh chnh xc ca b chuyn i ADC va DAC.

4 Hm truyn t h gin on -nh ngha

Hm truyn t h gin on, k hiu l W(z), l t, s gia tn hiu ra v8i tn hiu vo d8i dng ton t7 z.

( )( ) ( )Y zW zU z

= (6.10)

4.1 Xc nh hm truyn t W(z) t hm truyn t h lin tc 4.1.1 Mi lin h gi*a E*(p) v E(z)

Theo cng th+c (6.6), ta c nh laplace ca tn hiu lin t'c e(t) sau khi 6c l6ng t7 ha l :

1 pTep

E*(p) ( )E pT E(p)

1 pTep

E*(p) ( )E p


61

*

0( ) ( ) ipT

iE p e iT e

=

=

CIng tn hiu lin t'c e(t), sau khi 6c lng t7 ha v thc hin bin i Z, theo cng thc (6.4), ta c :

0( ) ( ) i

iE z e iT z

=

=

T( 2 cng th+c trn, c th th5y rn, ta c :

( ) ( )( )

( )( )2

2 2( )

T T

T T T T

z e ez zE zz e z e z e z e

= =

( )

( )( )2

*

2( )

pT T T

pT T pT T

e e eE p

e e e e

=

Ch : chng ta sD dng k hiu sau biu di.n nh laplace ca tn hiu 6c l6ng t7 ha { }**( ) ( )E p E p= (6.13)

Tnh cht ca php bin i *(p) Nu ta c quan h F(p) = H(p).E*(p) (6.14) th F*(p) = H*(p).E*(p) (6.15)

4.1.2 Hm truyn t h h& Xt mt h h gin on c s khi nh hnh vD


62

Hm truyn t phn lin t'c quy i l : ( ) ( ) ( )LTQD LG hW p W p W p= Tn hiu ra l :

* *( ) ( ) ( ) ( ) ( ) ( )LTQD LG hY p W p U p W p W p U p= = Thc hin bin i *(p) 2 v phng trnh trn, ta 6c { }** *( ) ( ) ( ) ( )LG hY p W p W p U p= Bit r


63

Bin i Z-1, ta 6c y(iT) = 1 - e-iT

Ch : V8i h thng gin on, ta ch, c th bit 6c gin tr ca tn hiu ng ra ti nhng th(oi im l5y m9u. J gia cc khong l5y m9u, ta khng th bit 6c gi tr chnh xc ca tn hiu.

4.1.3 H h& c b iu khin s Xt h h c b iu khin s nh sau :

Trong b iu khin s c hm truyn l :

( )( ) ( )cM zW zU z

= hay ( ) ( ) ( )cM z W z U z= Ta c :

*( ) ( ). ( ) ( ). ( ) ( )h h LGY p W p M p W p W p M p= = { } { }* * ** * *( ) ( ). ( ) . ( ) ( ). ( ) . ( ). ( )h LG h LG cY p W p W p M p W p W p W p U p= = { }( ) ( ). ( ) . ( ). ( )h LG cY z Z W p W p W z U z= { }( )( ) ( ). ( ) . ( )( ) h LG c

Y zW z Z W p W p W zU z

= =

4.1.4 H kn Xt h kn gin on c s khi nh sau :

Ta c :

* *( ) ( ). ( ) ( ). ( ). ( ) ( ). ( )h h LG LTQDY p W p E p W p W p E p W p E p= = =

{ }** *( ) ( ) . ( )LTQDY p W p E p= Mt khc :

* * *( ) ( ) ( ) ( ) ( ) ( )E p U p Y p E p U p Y p= = { }** * *( ) ( ) ( ) ( )LTQDY p W p U p Y p =

{ }{ }

*

* *

*

( )( ) ( )1 ( )

LTQD

LTQD

W pY p U p

W p=

+

hay { }

{ }( )( ) ( )

1 ( )LTQD

LTQD

Z W pY z U z

Z W p=

+

Wh(p) WLG(p) U(p) E*(p) ( )E p Y(p)

Wh(p) y(t) u(t) u(kT)

K s

( )m kT

U(p) U*(p) ( )M p Y(p) AD DA

m(kT)

M*(p)


64

( )( )1 ( )

hk

h

W zW zW z

=

+

4.1.5 H kn c b iu khin s

Ch+ng minh tng t, ta 6c :

( ) ( )( )1 ( ). ( )

h ck

h c

W z W zW zW z W z

=

+ v8i { }( )( ) ( ) ( )( )h LG h

Y zW z Z W p W pU z

= =

4.1.6 H gin on iu khin t my tnh S khi ca h thng nh sau :

Ta c :

*

1 1( ) ( ) ( ) ( ). ( ). ( )LGY p W p M p W p W p M p= = { }** *1( ) ( ). ( ) . ( )LGY p W p W p M p= hay { }1( ) ( ). ( ) . ( )LGY z Z W p W p M z= Theo s th :

* * * * * *( ) ( ) ( ) ( ) ( ) ( )c cM p W p E p W p U p R p = = hay [ ]( ) ( ) ( ) ( )cM z W z U z R z= Ngoi ra do :

*

2 1 2( ) ( ). ( ) ( ). ( ). ( ). ( )LGR p W p Y p W p W p W p M p= = nn { }1 2( ) ( ). ( ). ( ) ( )LGR z Z W p W p W p M z= Suy ra { }1 2( ) ( ) ( ) ( ). ( ). ( ) ( )c LGM z W z U z Z W p W p W p M z = Hay { }1 1

( ). ( )( )1 ( ). ( ). ( ). ( )

c

c LG

W z U zM zW z Z W p W p W p

=

+

Thay vo cng th+c ca Y(z), ta 6c :

W1(p) DA u(kT) e(kT) ( )m t y(t)

Wc(z) m(kT)

W2(p) AD r(t) r(kT)

W1(p) WLG(p)U*(p) E*(p) ( )M p Y(p)

Wc(z) M*(p)

W2(p) R(p) R*(p)

Wh(p) WLG(p)U(p) E*(p) ( )M p Y(p)

Wc(z) M*(p)


65

{ }{ }

1

1 2

( ). ( ). ( )( ) ( )1 ( ). ( ). ( ). ( )

c LG

c LG

W z Z W p W pY z U z

W z Z W p W p W p=

+

Hay { }{ }1

1 2

( ). ( ). ( )( )( ) ( ) 1 ( ). ( ). ( ). ( )c LG

c LG

W z Z W p W pY zW zU z W z Z W p W p W p

= =

+

V d :

Cho h iu khin gin on kn phn hi -1 trong 2 1( )czW zz

= v 1( )

1pW z

p=

+. Tm

hm truyn t ca h thng.

4.2 Xc nh hm truyn t t phng trnh sai phn Mt h thng gin on c th 6c cho d8i dng phng trnh sai phn tng qut nh sau :

[ ] [ ] [ ] [ ]1 0 1 0( ) ... ( 1) ( ) ( ) ... ( 1) ( )n ma y i n T a y i T a y iT b u i m T b u i T b u iT+ + + + + = + + + + + Gi s7 cc iu kin u b

Control System Toolbox & Simulink

66

Ph'l'c CONTROL SYSTEM TOOLBOX & SIMULINK TRONG MATLAB 'ng dng phn tch, thit k v m ph(ng cc h thng tuyn tnh

GIKI THI U MATLAB, tn vit tt ca t( ting Anh MATrix LABoratory, l mt mi tr;ng mnh dnh cho cc tnh ton khoa hoc. N tch h6p cc php tnh ma tr-n v phn tch s da trn cc hm c bn. Hn na, c5u trc ha h8ng i t6ng ca Matlab cho php to ra cc hnh vD ch5t l6ng cao. Ngy nay, Matlab tr thnh mt ngn ng chu1n 6c s7 d'ng rng ri trong nhiu ngnh v nhiu quc gia trn th gi8i. V mt c5u trc, Matlab gm mt c7a s chnh v r5t nhiu hm vit s>n khc nhau. Cc hm trn cng l&nh vc +ng d'ng 6c xp chung vo mt th vin, iu ny gip ng;i s7 d'ng d. dng tm 6c hm cn quan tm. C th k ra mt s th vin trong Matlab nh sau :

- Control System (dnh cho iu khin t ng) - Finacial Toolbox (l&nh vc kinh t) - Fuzzy Logic (iu khin m;) - Signal Processing (x7 l tn hiu) - Statistics (ton hc v thng k) - Symbolic (tnh ton theo biu th+c) - System Identification (nh-n dng) -

Mt tnh ch5t r5t mnh ca Matlab l n c th lin kt v8i cc ngn ng khc. Matlab c th gi cc hm vit b


67

T=0.5;sys2=tf(num,den,T)

H thng nhiu tn hiu vo/ra

Cu lnh : G11=tf(num11,den11,T); G12=tf(num12,den12,T);...; G1n=tf(num1n,den1n,T); G21=tf(num21,den21,T); G22=tf(num22,den22,T);...; G2n=tf(num2n,den2n,T);

Gp1=tf(nump1,denp1,T); G12=tf(nump2,denp2,T);...; Gpn=tf(numpn,denpn,T); sys=[G11,G12,...,G1n;G21;G22;...;G2n;...;Gp1,Gp2,...,Gpn];

1.1.2 nh ngha b+ng zero v c$c H thng mt tn hiu vo/ra

Cu lnh: sys=zpk(Z,P,K,T)

- Z,P l cc vect hng ch+a danh sch cc im zer v cc ca h thng. - K l h s khuch i

Ch : nu h thng khng c im zer (cc) th ta t l []

V d':

)5(2)(

+

+=

ppppF Z=-2;P=[0 -5];K=1;sys=zpk(Z,P,K);

H thng nhiu tn hiu vo/ra Cu lnh :

G11=zpk(Z11,P11,T); G12=zpk(Z12,P12,T);...; G1n=zpk(Z1n,P1n,T); G21=zpk(Z21,P21,T); G22=zpk(Z22,P22,T);...; G2n=zpk(Z2n,P2n,T);

Gp1=zpk(Zp1,Pp1,T); G12=zpk(Zp2,Pp2,T);...; Gpn=zpk(Zpn,Ppn,T); sys=[G11,G12,...,G1n;G21;G22;...;G2n;...;Gp1,Gp2,...,Gpn];

1.1.3 Phng trnh trng thi

Cu lnh: sys=ss(A,B,C,D,T)

- A,B,C,D l cc ma tr-n trng thi nh ngh&a h thng - T l chu k? l5y m9u.

Chuyn i gi*a cc dng biu di,n - Chuyn t( phng trnh trng thi sang hm truyn

[num,den] = ss2tf(A,B,C,D) - Chuyn t( dng zero/cc sang hm truyn

[num,den] = zp2tf(Z,P,K) - Chuyn t( hm truyn sang phng trnh trng thi

[A,B,C,D]=tf2ss(num,den)

G(r) U1

Un

Y1

Yn

=

)()()(...

)()()()(...)()(

)(

21

22221

11211

rGrGrG

rGrGrGrGrGrG

rG

pnpp

n

n


68

1.1.4 Chuyn i gi*a h lin tc v gin on S ha mt h thng lin tc

Cu lnh: sys_dis=c2d(sys,T,method)

- sys, sys_dis h thng lin t'c v h thng gin on tng +ng - Ts th;i gian l5y m9u - method phng php l5y m9u: zoh l5y m9u b-c 0, foh l5y m9u b-c 1, tustin phng

php Tustin

V d': chuyn mt khu lin t'c c hm truyn 15.0

2)(+

=

ppG sang khu gin on b


69

1.3 Phn tch h thng

1.3.1 Trong min th"i gian Hm qu h(t) Cu lnh: step(sys)

VD hm qu ca h thng tuyn tnh sys. Khong th;i gian vD v b8c th;i gian do Matlab t chn.

Mt s tr;ng h6p khc - step(sys,t_end): vD hm qu t( th;i im t=0 n th;i im t_end. - step(sys,T): vD hm qu trong khong th;i gian T. T 6c nh ngh&a nh sau

T=Ti:dt:Tf. i v8i h lin t'c, dt l b8c vD, i v8i h gin on, dt=Ts l chu k? l5y m9u.

- step(sys1,sys2,sys3,) : vD hm h(t) cho nhiu h thng ng th;i. - [y,t]=step(sys): tnh p +ng h(t) v lu vo cc bin y v t tng +ng

Hm tr0ng l)ng (t) Cu lnh: impulse(sys)

1.3.2 Trong min tn s #c tnh bode Cu lnh: bode(sys)

VD c tnh tn s Bode ca h thng tuyn tnh sys. Di tn s vD do Matlab t chn.

Mt s tr;ng h6p khc - bode(sys,{w_start,w_end}): vD c tnh bode t( tn s w_start n tn s w_end. - bode(sys,w) vD c tnh bode theo vect tn s w. Vect tn s w 6c nh ngh&a b


70

200

2

20

2)(

++=

pppG v8i 0=1rad/s v =0,5

w0=1 ;xi=0.5 ;num=w0^2 ;den=[1 2*xi*w0^2 w0^2] ;G=tf(num,den); w=logspace(-2,2,100) ; bode(G,w) ; % vD c tnh bode trong di tn s w nichols(G); % vD c tnh nichols trong di tn s t chn ca Matlab nyquist(G); % vD c tnh nyquist

1.3.3 Mt s hm phn tch Hm margin

- margin(sys) vD c tnh Bode ca h thng SISO v ch, ra d tr bin , d tr pha ti cc tn s tng +ng.

- [delta_L,delta_phi,w_L,w_phi]=margin(sys) tnh v lu d tr bin vo bin delta_L ti tn s w_L, lu d tr v pha vo bin delta_phi ti tn s w_phi.

Hm pole vec_pol=pole(sys) tnh cc im cc ca h thng v lu vo bin vec_pol.

Hm tzero vec_zer=tzero(sys) tnh cc im zero ca h thng v lu vo bin vec_zer.

Hm pzmap - [vec_pol,vec_zer]=pzmap(sys) tnh cc im cc v zero ca h thng v lu vo cc bin

tng +ng. - pzmap(sys) tnh cc im cc, zero v biu di.n trn mt ph:ng ph+c.

Hm dcgain G0=dcgain(sys) tnh h s khuch i t&nh ca h thng v lu vo bin G0.

1.3.4 Mt s hm c bit trong khng gian trng thi Hm ctrl Cu lnh: C_com=ctrl(A,B) C_com=ctrl(sys)

Tnh ma tr-n iu khin c C ca mt h thng. Ma tr-n C 6c nh ngh&a nh sau: C=[B AB A2B An-1B] v8i Anxn

Hm obsv Cu lnh: O_obs=obsv(A,C) O_obs=obsv(sys)

Tnh ma tr-n quan st c O ca mt h thng. Ma tr-n O 6c nh ngh&a nh sau: O=[C CA CA2 CAn-1]

Hm ctrbf Cu lnh: [Ab,Bb,Cb,T,k]=ctrbf(A,B,C)

Chuyn v dng chu1n (canonique) iu khin 6c ca mt h thng biu di.n d8i dng phng trnh trng thi. Trong : Ab=TAT-1, Bb=TB, Cb=CT-1, T l ma tr-n chuyn i.

Hm obsvf Cu lnh: [Ab,Bb,Cb,T,k]=obsvf(A,B,C)


71

Chuyn v dng chu1n quan st 6c ca mt h thng biu di.n d8i dng phng trnh trng thi. Trong : Ab=TAT-1, Bb=TB, Cb=CT-1, T l ma tr-n chuyn i.

1.4 V d tng h#p Cho mt h thng kn phn hi -1, trong hm truyn ca h h l

200

2

20

2*)1()(

+++=

ppppKpG v8i K=1, =10s, 0=1rad/s v =0.5

1. VD c tnh tn s Nyquist. Ch+ng tA r>K=1;to=10;w0=1;xi=0.5; >>num1=K;den1=[to 1 0]; >>num2=w0^2;den2=[1 2*xi*w0 w0^2] ; >>G=tf(num1,den1)*tf(num2,den2) Transfer function: 1 ----------------------------

10 s^4 + 11 s^3 + 11 s^2 + s >>w=logspace(-3,2,100) ; % to vect tn s vD cc c tnh tn s >>nyquist(G,w); c tnh 6c biu din trn hnh 6.1

xt tnh n nh ca h kn dng tiu chu1n Nyquist, tr8c tin ta xt tnh n nh ca h h. Nghim ca phng trnh c tnh ca h h 6c xc nh : >>pole(G) ans = 0 -0.5000 + 0.8660i -0.5000 - 0.8660i -0.1000 H h c 1 nghim b


72

Quan st c tnh tn s Nyquist ca h h trn hnh 6.1 (phn zoom bn phi), ta th5y c tnh Nyquist bao im (-1,j0), v do h h bin gi8i n nh nn theo tiu chu1n Nyquist, h thng kn s1 khng n -nh. Cu 2 >>G_loop=feedback(G,1,-1) ; % hm truyn h kn >>step(G_loop) ;

Cu 3 >>K=0.111 ;num1=K ; % thay i h s khuch i K >>GK=tf(num1,den1)*tf(num2,den2) Transfer function: 0.111 ----------------------------

10 s^4 + 11 s^3 + 11 s^2 + s >>margin(GK) c tnh tn s Bode ca h h hiu ch,nh 6c biu di.n trn hnh 6.3. T( c tnh ny, ta c th xc nh 6c L=18.34dB ; = 44.78 ; c=0.085rad/s

Time (sec.)

Amplitude

Step Response

0 50 100 150 200 250 300 350 400 450 500-10

-5

0

5

10

15From: U(1)

To: Y(1)

Hnh 6.2 : p +ng qu h kn

Frequency (rad/sec)

Phase

(deg); Magnitude

(dB)

Bode Diagrams

-150

-100

-50

0

50Gm=18.344 dB (at 0.30151 rad/sec), Pm=44.775 deg. (at 0.084915 rad/sec)

10-3 10-2 10-1 100 101-400

-350

-300

-250

-200

-150

-100

-50

0

Hnh 6.3 : c tnh tn s Bode ca h h hiu ch,nh


73

Hnh 6.4 p +ng qu h kn hiu ch,nh

Hnh 6.5 C7a s chnh ca Simulink

Time (sec.)

Amplitude

Step Response

0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

1.4From: U(1)

To: Y(1)

Cu 4 >>GK_loop=feedback(GK,1,-1) ; >>step(GK_loop);

S7 d'ng con trA chut v kch vo cc im cn tm trn c tnh, ta xc nh 6c max=23%; Tmax= 70.7s

2 SIMULINK Simulink 6c tch h6p vo Matlab (vo khong u nhng nm 1990) nh mt cng c' m

phAng h thng, gip ng;i s7 d'ng phn tch v tng h6p h thng mt cch trc quan. Trong Simulink, h thng khng 6c m t d8i dng dng lnh theo kiu truyn thng m d8i dng s khi. V8i dng s khi ny, ta c th quan st cc p +ng th;i gian ca h thng v8i nhiu tn hiu vo khc nhau nh : tn hiu b-c thang, tn hiu sinus, xung ch nh-t, tn hiu ng9u nhin b


74

- Continuous : h thng tuyn tnh v lin t'c - Discrete : h thng tuyn tnh gin on - Nonliear : m hnh ha nhng phn t7 phi tuyn nh rle, phn t7 bo ha - Source : cc khi ngun tn hiu - Sinks : cc khi thu nh-n tn hiu - Function & Table : cc hm b-c cao ca Matlab - Math : cc khi ca simulink v8i cc hm ton hc tng +ng ca Matlab - Signals & System : cc khi lin h tn hiu, h thng con

2.2 To mt s n gin lm quen v8i Simulink, ta bt u b


75

xem ng th;i tn hiu vo v ra trn cng mt Scope, ta to s m phAng nh hnh 6.8. Kt qu m phAng biu di.n trn hnh 6.9.

2.3 Mt s khi th'ng dng

Th vin Sources Step To ra tn hiu b-c thang lin t'c hay gin on. Ramp To tn hiu dc tuyn tnh (rampe) lin t'c. Sine Wave To tn hiu sinus lin t'c hay gin on. Constant To tn hiu khng i theo th;i gian. Clock Cung c5p ng h ch, th;i gian m phAng. C th xem 6c ng h ny khi

ang thc hin m phAng. Ch : Mun khi clock ch, ng th;i im ang m phAng, tham s Sample time 6c t nh sau

0 : h lin t'c >0 : h gin on, clock lc ny sD ch, s chu k? l5y m9u t trong Sample time.

Th vin Sinks Scope Hin th cc tn hiu 6c to ra trong m phAng. XY Graph VD quan h gia 2 tn hiu theo dng XY. Khi ny cn phi c 2 tn hiu

vo, tn hiu th+ nh5t tng +ng v8i tr'c X, tn hiu vo th+ hai tng +ng v8i tr'c Y.

To Workspace T5t cc cc tn hiu ni vo khi ny sD 6c chuyn sang khng gian tham s ca Matlab khi thc hin m phAng. Tn ca bin chuyn vo Matlab do ng;i s7 d'ng chn.

2.3.1 Th vin Continuous Transfer Fcn M t hm truyn ca mt h thng lin t'c d8i dng a thc t s/a thc

m)u s. Cc h s ca a th+c t7 s v m9u s do ng;i s7 d'ng nh-p vo, theo b-c gim dn ca ton t7 Laplace. V d' nh-p vo hm truyn c

dng 1

122 ++

+

ss

s, ta nh-p vo nh sau :Numerator [2 1], Denominator [1 1 1].

State Space M t hm truyn ca mt h thng lin t'c d8i dng phng trnh trng thi. Cc ma tr-n trng thi A, B, C, D 6c nh-p vo theo qui 8c ma tr-n ca Matlab.

Integrator Khu tch phn. sDerivative Khu o hm Transport Delay Khu to tr.

Hnh 6.8

Hnh 6.9


76

Th vin Discrete Discrete Transfer Fcn M t hm truyn ca mt h thng gin on d8i dng a thc t

s/a thc m)u s. Cc h s ca a th+c t7 s v m9u s do ng;i s7 d'ng nh-p vo, theo b-c gim dn ca ton t7 z.

Discrete State Space M t hm truyn ca mt h thng gin on d8i dng phng trnh trng thi. Ng;i s7 d'ng phi nh-p vo cc ma tr-n trng thi A,B,C,D v chu k? l5y m9u.

Discrete-Time Integrator Khu tch phn ca h thng gin on. First-Order Hold Khu gi m9u b-c 1. Ng;i s7 d'ng phi nh-p vo chu k? l5y m9u. Zero-Order Hold Khu gi m9u b-c 0. Ng;i s7 d'ng phi nh-p vo chu k? l5y m9u.

Th vin Signal&Systems Mux Chuyn nhiu tn hiu vo (v h8ng hay vect) thnh mt tn hiu ra

duy nh5t dng vect. Vect ng ra c kch th8c b


77

2.5 LTI Viewer Nh ta bit, khi thc hin m phAng trn Simulink, ta ch, c th quan st 6c cc c tnh

th;i gian ca h thng. c th phn tch ton din mt h thng, ta cn cc c tnh tn s nh c tnh Bode, c tnh Nyquist, qu2 o nghim s v.v

LTI Viewer l mt giao din ha cho php quan st p +ng ca mt h thng tuyn tnh, trong l&nh vc tn s cIng nh th;i gian, m khng cn g li lnh hay l-p trnh theo t(ng dng lnh nh trong Control System Toolbox. N s7 d'ng trc tip s khi trong Simulink.

2.5.1 Kh&i ng LTI Viewer khi ng LTI Viewer t( Simulink, ta chn menu Tool -> Linear Analysis. Lc ny, Matlab sD m 2 c7a s m8i: - C7a s LTI Viewer (hnh 6.13) c 2 phn chnh:

o Phn c7a s ha dng biu di.n cc ;ng c tnh. o Thanh cng c' pha d8i ch, d9n cch s7 d'ng LTI Viewer

- C7a s ch+a cc im input v output (hnh 6.14). Cc im ny 6c dng xc nh im vo/ra trn s Simulink cn phn tch.

2.5.2 Thit lp cc im vo/ra cho LTI Viewer Dng chut ko r cc im input point, output point trn c7a s hnh 6.14 v t ln cc v tr tng +ng trn s Similink.

Hnh 6.11 : p +ng qu (K=1) Hnh 6.12 : p +ng qu (K=0.111)

Hnh 6.13 Hnh 6.14


78

Ch : Vic chn cc im t input, output phi ph hp yu c"u phn tch. LTI Viewer tnh hm truyn bng cch tuyn tnh ha h thng vi 2 im input/output c nh ngh!a. Khi v* cc c tnh t"n s c+ng nh thi gian, LTI s dng cc h thng c tuyn tnh ha ny.

2.5.3 Tuyn tnh ha mt m hnh tm m hnh gia 2 im input/output nh ngh&a, ta thc hin nh sau: Chn c7a s LTI Viewer (hnh 6.13) Chn memu Simulink Get linearized model Lc ny, trong phn ha ca c7a s LTI Viewer sD xu5t hin t tnh qu ca m hnh tuyn tnh ha tm 6c. xem cc c tnh khc trn LTI Viewer, ta ch, vic kch chut phi vo phn ha, chn menu Plot Type chn loi c tnh cn quan st.

Ghi ch: - C+ m=i ln thc hin tuyn tnh ha mt m hnh (Simulink Get linearized model) th LTI

Viewer sD np m hnh hin hnh ti ca s Simulink vo khng gian ca n. Nu gia 2 ln thc hin tuyn tnh ha, m hnh khng c s thay i (c5u trc hay thng s) th 2 m hnh tm 6c tng +ng sD ging nhau.

- C th b-t/tt c tnh ca mt hay nhiu m hnh tm 6c trong LTI Viewer b


79

Sau 4 ln tuyn tnh ha trong LTI Viewer, ta 6c 4 h thng ln l6t l baitap1_simulink_1 n baitap1_simulink_4 (s trong Simulink c tn l baitap1_simulink).

Trn c7a s ha lc ny sD hin th ng th;i c tnh qu ca c 4 m hnh trn. - xem c tnh Nyquist ca h h tr8c v sau hiu ch,nh:

o Kch chut phi vo phn ha, chn Systems, chn 2 m hnh 1 v 2. o Tip t'c kch chut phi vo phn ha, chn Plot Type Nyquist.

Trn c7a s ha sD xu5t hin 2 c tnh Nyquist v8i 2 mu phn bit. - xem c tnh qu ca h kn tr8c v sau hiu ch,nh:

o Kch chut phi vo phn ha, chn Systems, chn 2 m hnh 3 v 4. o Tip t'c kch chut phi vo phn ha, chn Plot Type Step.

Cc c tnh khc 6c tin hnh mt cch tng t.

a)

b)

c)

d)

Hnh 6.15 : S v c5u trc tuyn tnh ha

dktd full 3444

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