chuong 2(mo ta toan hoc)

7/24/2019 Chuong 2(Mo Ta Toan Hoc)

1/101

CHNG 2

MO HNH TOAN HOC CUAHE IEU KHIEN TUYEN TNH

LIEN TUC


2/101

I. GII THIEU:e phan tch mot he thong vat

ly can phai tm ra mot m! "#$"

%&'$ "(). Co the co cac dang a!: "o ta #a$ng % &o 'hoi(

graph tn hie!( % &o ch!yentrang thai.

)ie! die*n #a$ng toan hoc(ha+m tr!yen ,tran-er -!nction(ph/%ng trnh trang thai.


3/101

II. HA*M T+UYEN ,A*A- NG

/. H&0m %1345$ "ot he thong t!yen tnh &%n

#ien( #at #ien theo th%+i gian thph/%ng trnh vi phan co dang:

T1$6 7'c,t: la+ #ien ra.

r,t: la+ #ien va+o.

Cac he o ai, i 0 1( 2 (n va+ #3,4 0 1( .. . (

)(....)()(

)(....)()(

01

1

101

1

1 trbdt

trd

bdt

trd

btcadt

tcd

adt

tcd

a m

m

mm

m

mn

n

nn

n

n ++=+++

5e thongt!yen tnh#at #ien

lien t!c

r,t c,t


4/101

6ay 6aplace hai ve c!a ph/%ng trnh ,7.8 :8&$9$: &$;/9$;/: < : &=>C89> ? 8@m9m: @m;/9m;/: < : @=>+89>

9%i: C, 0 6c,t;( o gi/?a &a! ra va+ &a! va+o@!a phep #ien &oi 6aplace v%i &ie! 'ien&a! #a$ng 1.

0

1-n

1-n

n

n

0

1-m

1-m

m

m

asasa

bsbsb

)(

)()(

+++

+++==

sR

sCsG


5/101

9 d!: =m ha+m tr!yen &at c!a mach &ien a!:

C9

/L9+A89> ++=

A89>

UI B=

C9

/

A89>

U

C9

/IU B

= ==

CssZU

UsG

i )(

1)( 0 ==

C

L

R

UB U


6/101


7/101

2. &'D'$6

&/%c mot lanKm7: =ong c!a tch hai vo+ng 'n 'hongdnh nha!.,&oc lap 'hong dnh ch!ng n!tva+ ha+m tr!yen nhanh

KmM:=ong c!a tch #a vo+ng 'hong dnh


29/101

,W /=m ha+m tr!yen &at c!a he cograph tn hie!.

T& )' 9 75G1&"


30/101

Cac &/%+ng tr!yen th!an:

M/? G/G2GX /?/

M2? G/G

2? /JCo N vo+ng 'n: L/? ;G/G2GX L2?;G/G2H/

LX? ;G2GXH2 L? ;GH2

LZ ? ; G/GKm80 68 O 67 O 6M O 6P O6N0 Q87MQ8758Q7M57QP57 J8P.


31/101

9R 7: =m ha+m tr!yen c!a he coraph tn hie! a!:

"80 y8S7yMSP 80 8 Km80QS7y8J S7yMJ SPyM

Km70 ,Qy8S7,QyMSP KmM0 1

0 8 QKm8O Km7

I3 E0E1E2I1Ei 1z4z2 y3y1

-z2-y1 -y3

4321432321

43210

1

1

zyzyzyzyzy

zyzyM

E

EM 81

i +


32/101

Bi tp


33/101

Bi tp


34/101

T9. Kh/%ng trnh trangthai ,tate pacee@!ation "#ang thai:=rang thai c!a mot he

thong la+ tap h%p nho nhat cac #ien,goi la+ #ien trang thai ma+ ne! #iet giatrD c!a cac #ien na+y tai th%+i &iem t1 va+#iet cac tn hie! va+o % th%+i &iem t U t1(ta hoa+n toa+n co the Bac &Dnh &/%c &ap/ng c!a he thong tai moi th%+i &iem tt1.

5e thong #ac nco n#ien trang thai.Cac #ien trang thai co the chon la+ #ienvat ly hoaVc 'hong phai la+ #ien vat ly.

Vecto# t#ang thai:n#ien trang thai h%ptha+nh vector cot goi la+ vevtor trang thai.

[ ]

T

nxxx 21

=$


35/101


36/101


37/101


38/101


39/101


40/101


41/101


42/101

9iWt K=== tX K=9K:


43/101


44/101


45/101


46/101


47/101

UU ym a t Y B UU Z01 8[Q\ QN;

Z 0

1 8 Q\ QN

UU )01[8;

) 0

1 8

UU C0] 1;

C 0

] 1

UU 0C^,inv,^TQZ^) HHH _nde`ned -!nction or varia#le T.

UU T08 1[1 8;

T 0

8 1 1 8

UU 0C^,inv,^TQZ^) 0 ]E,b7ON^O\


48/101

ym a t Y B Z01 8[Q\ QN; )01[8; C0] 1;

T08 1[1 8; 0C^,inv,^TQZ^)


49/101


50/101


51/101

h h f X


52/101

c nh hm tr!yfn tXphng trnh trng thi


53/101


54/101


55/101


56/101


57/101

,. IEU WIE[N HA*MT+UYEN

,p co the phan tch tha+nh :

=rong &o: Se : nghiem c!a Z,p 0 1.

pi : nghiem c!a ),p 0 1 ,ph/%ng trnh

&aVc tr/ng.

/. , %1Q )D() &0 \41 8%14!$ m&]%"&^$6 "D')>em ha+m tr!yen c!a he t!yentnh lien t!c.

)()(

......)(

01

1

0

1

1

pBpA

apapabpbpbpG

n

n

n

n

m

m

m

m=

=

=

=

=

n

i

i

m

e

e

pp

zp

KpG

1

1

)(

)()(

n

m

a

b

K =


58/101

=hay p 0 4( ta co:

=

=

=

n

i

i

m

e

e

pj

Zj

KjG

1

1

)(

)(

)(

=ren maVt phaLng ph/cma+ tr!c t!ng la+ 4ta&Dnh vD Sero #a$ng'h!yen tro+n ,j va+ c/c#a$ng da! ,

/. , %1Q )D() &0 \41 8%14!$ m&]%"&^$6 "D')>


59/101

Cho vD tr c/c va+ Sero ta co the tnh &/%c #ien&o va+ pha c!a ha+m tr!yen &at tai mot tan

o #at 'y+ nh%+ va+o

=

n

1i

i

m

1e

e

n

1i

i

m

1e

e )pj()zj()j(G

/. , %1Q )D() &0 \41 8%14!$ m&]%"&^$6 "D')>


60/101

)ie! &o c/c la+ anh c!a mot n/a tr!cao d/%ng ang maVt phaLng @!a anh Ba

,4. )ie! &o c/c co the Bac &Dnh#a$ng K 0


61/101

ian &o )ode gom hai&o thD:

o thD logarit #ien &o.

|G8`>|R? 2=F6/=|G8`>|

oc pha theo tan o.

? &16|

G8`

>|

X.GB&_$ 75 R48GB&_$ 75 @B4!$ 7P a "& %"4 F6&1B% %

9J>

1&e#!&e

-40

-20'%

1 10010

0

20

40

&B


62/101


63/101

,I. CAC KHAU C ANT+ONG KT

/. K"&!"4J)" 7&(B

,p 0 I

R9in9o!t

R2

R1

R0

-*

9in9o!t


64/101

K0

ReG

ImG

B4 75

)D()

GB&_$ 75

R4KjG dB lg20)(lg20)( =

20'%K

&B

'%

/. K"&!"4J)" 7&(B


65/101


2. K"&! %Q)""&!$

KjG

jG

)(

2)(

=hay p04

KjjG

0)(

R C-+C

R

p

KpG )(


66/101


2. K"&! %Q)""&!$ian &o )ode

#ien &o:

&B

'%

-20&B+&e#

20'%|K|

= '%20'%20 K

'%20'%20)( =

KjGLdB

-,00

0

ian &o )odegoc pha:


67/101


X. K"&! B"&!$

R

C

-

+

C

R

=

=

KjG

jG

)(

2)(

p-K)p(G =

=hay p04

jKjG

0)(


68/101


X. K"&! B"&!$

lg20lg20lg20lg20)( KKjG dB

&B

'%

*20&B+&e#

20'%|

K|,00

0

ian &o )ode #ien &o

va+ goc pha:


69/101


. K"&! @&P)2

ian &o )ode:

22

2

2 nnn

ppp,

++=

./n s %y n -90o

= n


70/101


+ Kh t!" #$(p) % &-'p

Bin * # $(p) % 1 20 lg$(p) % 0

,ha # !g($(p)) % - '20lg$(p)

lg

$i/n Bd& bin *

!g ($(p))

lg

$i/n Bd& pha


71/101

)10)(1(10)(

++= pppG

V d: V gin Bode ca hm truyn sau:

10

1.1

1.10)( ++= pppG


72/101

10

1.

1

1.10)(

++=

pppG


73/101

c b5c 67 gi/n Bd

B5# 1# 8c 9nh t:n s; gp ?@p th& thA ttCng d:n

h #

=

=

=

n

ii

m

ll

dp

cpKpG

1

1

thD # % cl 6 % di l t:n s; g


74/101

c b5c 67 gi/n Bd

B5# 2#8c 9nh $(E) dBtFi t:n s; nhG nhHt.

-I@ $(p) Jhng cL cc tFi 0 thD ch % 0.

-I@ $(p) cL cc tFi 0thD ?c 9nh Mng tiNm cn cOa$(E)dBJhi 0 (thMng chPn nhG hQn t:n s;

g


75/101

c b5c 67 gi/n Bd

B5# 3#

I@ $(p) Jhng cL cc tFi 0 $i/n Bd& bin * s7 lMng nSm ngang cL * l5n % $(E)dBtFi 0 1ch @nt:n s; g


76/101

c b5c 67 gi/n Bd

B5# 4# I@ tFi t:n s; g


77/101

)10)(1(

10)(

++=

pppG


Gi6 78 B & :


78/101

Z7 gi/n Bd& phabSng phQng php tch !MitYng thnh ph:n !i c*ng lFi.$i/n Bd& pha cOa m*t s; Jh cQ b/n#

+ $ % K K [ 0 thD \ % 0+ $ % K K ] 0 thD \ % -1^0

+ $ % 1Wp thD \ % -_0

+ $ % thD % _0

Gi6n 78 B9&e p:!

8c 9nh bSng cch ?c 9nh hm \#

=

=

n

i i

m

l l darctg

carctg

11

=

=

+

+

=

n

ii

m

ll

dp

cp

KpG

1

1


79/101

1 dec

0o

- 45o

- 90o

1 dec

=a

+ $ % 1W(p+a) (Jh tVch phn)

1 dec 1 dec

90o

45o

0o

=a

+ $ % (p+a) (Jh 6i phn)


80/101

)10)(1(

10)(

++=

pppG



81/101

,Q R( /: 9e? gian &o )ode c!aha+m a!:

=an o ga?y: 8( 81( 811(8111

)1000)(10)(1(

)100(10)(

;

+

+

=

ppp

ppG

oc pha:


82/101

,Q R( /

Gian o Bode bien o.

P1

= /2

Q71

X


83/101

,Q R( /

)ang tnh gocpha:

= / /= /==/=== = ;Z== ;/2= ;/X=;/== ;/b=

Gian o Bode goc pha


84/101

,Q R( 2: 9e? gian &o )ode c!aha+m a!:

=an o ga?y: 8( 81( 811

)100p)(1p(p

)10p(10)p(G

2

=

Ro co c/c tai 1 va+ tan o ga?y nho nhat

la+ 8 nen ta chon tan o nho nhat la+ 1(8

=nh #ien &o tai tano nho nhat.

dBjG 40100lg201001

1010lg20)(lg20

2222

222

==++

+=


85/101

,Q R( 2: 9e? gian &o )ode c!aha+m a!:

oc pha :


86/101

,Q R( 2

Gian o Bode bien o.

= / 2X

71

P1


87/101


88/101

III. M t h thng bng phng trnh tr!ng th"#

1. $h"# n#%&

'

(

)

*

+

+

+ +

r,t


89/101

A. 8h


90/101

8h3 /


91/101

?i@$ &5i &An% p:n% $"n: $"An% $:Di

)$("

!

0

0

)$(


92/101

)()(

)()(

01

1

1 tcadt

tdca

dt

tcda

dt

tcda

n

n

n

n +

)()(

)()(

01

1

1 trbdt

tdrb

dt

trdb

dt

trdb

m

m

m

m +

8B C8&

aVt #ientrang thai

theong!yen

tac:

)(.

)(.

)(

trBxx

trBxx

tcx

nnn ))

))*

)

=

=

=

."Fn% :p .? #:! 7A9 :m #J! n% 9 (m n-1)


93/101

?5i:

'1= bn-1Kan 'A= ,bn-A@ an-1.'1Kan

'L= ,bn-L@ an-1.'A@ an-A'1Kan

'n= ,b0@ an-1'n-1- - a1'1Kan$h# &

)t(!.

B

B

B

)t(?.

aa

...aa

aa

a

a

1...000

0...100

0...010

)t(?

n

2

1

n

1n

n

2

n

1

n

0

+

=

[ ] )t(?0...01c =


94/101

ZV d`# h ,'Z, 6i@t hN phQng t!Dnh m t/ bi@n t!Fng thi

)()(

)()()()(

trdt

tdrtc

dt

tdc

dt

tcd

dt

tcd*+)+)+,-

*

*

.

.+=+++

NOt c"c b#3n tr!ng th"#&

)(.

)(.)(

trBxx

trBxxtcx

**.

))*

)

=

==

8rong & '1

= bn-1

Kan

= 0 K 1 = 0

'A= ,bn-A@ an-1.'1Kan= ,10 @ 5P0K1=10

'L= ,bn-L@ an-1.'A@ an-A'1Kan= ,A0 @ 5 P10 @ QP0= -L0

8haR c"c h 6 /Ba tSnh /


95/101

3 .:n: 'Mp :N p:n% $"n: $"An% $:Di $O s 78 P:i

! Bi@n 7Qi :m $"ySn $:n: .?

ng bi@n i aplac& ngXc U bi@n i sQ Jh;i thnh ,'Z,!i dng ,hQng php e ph:n t!5c U thnh lp m t/ t!Fng thi.

:n% p:Dp $T! 7U p:!

011

1

011

1

---

---

)(

)()(

apapapa

bpbpbpb

pR

pCpM

n

n

n

n

m

m

m

m

=

'Y hm t!=fn#

t bi@n ph` (p) sa ch#C(p)(mpm* m-1pm-1* V* 1p * 0)Y(p)

R(p)(!npn* !n-1pn-1* V* !1p * !0)Y(p)

Bi@n 7Qi L!p'!#e n%# 7W$


96/101

# :n% p:Dp 7W$ i@n $"# $i@p $"Zn s 78 P:i

/

.+p -

*++

p

p

,

)

+

+

p

p

-

* C(p)X1(p)X2(p)

X3(p)

R(p)'a cL#

)p(X;p

2p)p(X 21

=

pX1(p) -;X1(p) * 2X2(p) * pX2(p)

)p(X)p(R4p

3)p(X 32

pX2(p) -4X2(p) - 3X3(p) * 3R(p)

)p(Xp1p

)p(X 13+

+

=

pX3(p) X1(p) - X3 (p) * pX1(p)

'h@ p82(p) e ,'2 6 ,'1 ta cL hN phQng t!Dnh m t/ t!Fng thi.


97/101

4. 8Snh h)-1 BDS dV&8% h


98/101

; \%:iNm #J! p:n% $"n: $"An% $:Di

=

+=

)(-)(

)(-)(-

txCtc

trBtxAx

=

)p(X-C)p(C

)p(R-B)p(X->)0()-1

B R(p) * (pI [ >)-1

(' e#$ n%:iNm #J!p:n% $"n: &e$(^I [ >) 0

p Ang cOa hN th;ng# $

0

&)(RB)$()$(

chuong 2(mo ta toan hoc)

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