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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
1/101
CHNG 2
MO HNH TOAN HOC CUAHE IEU KHIEN TUYEN TNH
LIEN TUC
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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.
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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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
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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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
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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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
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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2. &'D'$6
&/%c mot lanKm7: =ong c!a tch hai vo+ng 'n 'hongdnh nha!.,&oc lap 'hong dnh ch!ng n!tva+ ha+m tr!yen nhanh
KmM:=ong c!a tch #a vo+ng 'hong dnh
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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,W /=m ha+m tr!yen &at c!a he cograph tn hie!.
T& )' 9 75G1&"
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Cac &/%+ng tr!yen th!an:
M/? G/G2GX /?/
M2? G/G
2? /JCo N vo+ng 'n: L/? ;G/G2GX L2?;G/G2H/
LX? ;G2GXH2 L? ;GH2
LZ ? ; G/GKm80 68 O 67 O 6M O 6P O6N0 Q87MQ8758Q7M57QP57 J8P.
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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9R 7: =m ha+m tr!yen c!a he coraph tn hie! a!:
"80 y8S7yMSP 80 8 Km80QS7y8J S7yMJ SPyM
Km70 ,Qy8S7,QyMSP KmM0 1
0 8 QKm8O Km7
I3 E0E1E2I1Ei 1z4z2 y3y1
-z2-y1 -y3
4321432321
43210
1
1
zyzyzyzyzy
zyzyM
E
EM 81
i +
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
32/101
Bi tp
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
33/101
Bi tp
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
34/101
T9. Kh/%ng trnh trangthai ,tate pacee@!ation "#ang thai:=rang thai c!a mot he
thong la+ tap h%p nho nhat cac #ien,goi la+ #ien trang thai ma+ ne! #iet giatrD c!a cac #ien na+y tai th%+i &iem t1 va+#iet cac tn hie! va+o % th%+i &iem t U t1(ta hoa+n toa+n co the Bac &Dnh &/%c &ap/ng c!a he thong tai moi th%+i &iem tt1.
5e thong #ac nco n#ien trang thai.Cac #ien trang thai co the chon la+ #ienvat ly hoaVc 'hong phai la+ #ien vat ly.
Vecto# t#ang thai:n#ien trang thai h%ptha+nh vector cot goi la+ vevtor trang thai.
[ ]
T
nxxx 21
=$
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
39/101
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
40/101
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
41/101
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9iWt K=== tX K=9K:
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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\
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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ym a t Y B Z01 8[Q\ QN; )01[8; C0] 1;
T08 1[1 8; 0C^,inv,^TQZ^)
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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
50/101
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51/101
h h f X
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c nh hm tr!yfn tXphng trnh trng thi
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,. 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&]%"&^$6 "D')>em ha+m tr!yen c!a he t!yentnh lien t!c.
)()(
......)(
01
1
0
1
1
pBpA
apapabpbpbpG
n
n
n
n
m
m
m
m=
=
=
=
=
n
i
i
m
e
e
pp
zp
KpG
1
1
)(
)()(
n
m
a
b
K =
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=hay p 0 4( ta co:
=
=
=
n
i
i
m
e
e
pj
Zj
KjG
1
1
)(
)(
)(
=ren maVt phaLng ph/cma+ tr!c t!ng la+ 4ta&Dnh vD Sero #a$ng'h!yen tro+n ,j va+ c/c#a$ng da! ,
/. , %1Q )D() &0 \41 8%14!$ m&]%"&^$6 "D')>
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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Cho vD tr c/c va+ Sero ta co the tnh &/%c #ien&o va+ pha c!a ha+m tr!yen &at tai mot tan
o #at 'y+ nh%+ va+o
=
n
1i
i
m
1e
e
n
1i
i
m
1e
e )pj()zj()j(G
/. , %1Q )D() &0 \41 8%14!$ m&]%"&^$6 "D')>
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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)ie! &o c/c la+ anh c!a mot n/a tr!cao d/%ng ang maVt phaLng @!a anh Ba
,4. )ie! &o c/c co the Bac &Dnh#a$ng K 0
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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
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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,I. CAC KHAU C ANT+ONG KT
/. K"&!"4J)" 7&(B
,p 0 I
R9in9o!t
R2
R1
R0
-*
9in9o!t
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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K0
ReG
ImG
B4 75
)D()
GB&_$ 75
R4KjG dB lg20)(lg20)( =
20'%K
&B
'%
/. K"&!"4J)" 7&(B
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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,I. CAC KHAU C ANT+ONG KT
2. K"&! %Q)""&!$
KjG
jG
)(
2)(
=hay p04
KjjG
0)(
R C-+C
R
p
KpG )(
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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,I. CAC KHAU C ANT+ONG KT
2. K"&! %Q)""&!$ian &o )ode
#ien &o:
&B
'%
-20&B+&e#
20'%|K|
= '%20'%20 K
'%20'%20)( =
KjGLdB
-,00
0
ian &o )odegoc pha:
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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,I. CAC KHAU C ANT+ONG KT
X. K"&! B"&!$
R
C
-
+
C
R
=
=
KjG
jG
)(
2)(
p-K)p(G =
=hay p04
jKjG
0)(
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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,I. CAC KHAU C ANT+ONG KT
X. K"&! B"&!$
lg20lg20lg20lg20)( KKjG dB
&B
'%
*20&B+&e#
20'%|
K|,00
0
ian &o )ode #ien &o
va+ goc pha:
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
69/101
,I. CAC KHAU C ANT+ONG KT
. K"&! @&P)2
ian &o )ode:
22
2
2 nnn
ppp,
++=
./n s %y n -90o
= n
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,I. CAC KHAU C ANT+ONG KT
+ 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
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)10)(1(10)(
++= pppG
V d: V gin Bode ca hm truyn sau:
10
1.1
1.10)( ++= pppG
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10
1.
1
1.10)(
++=
pppG
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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
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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
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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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 % $(E)dBtFi 0 1ch @nt:n s; g
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c b5c 67 gi/n Bd
B5# 4# I@ tFi t:n s; g
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)10)(1(
10)(
++=
pppG
V d: V gin Bode ca hm truyn sau:
Gi6 78 B & :
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Z7 gi/n Bd& phabSng phQng php tch !MitYng thnh ph:n !i c*ng lFi.$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
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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)
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)10)(1(
10)(
++=
pppG
V d: V gin Bode ca hm truyn sau:
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,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:
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,Q R( /
Gian o Bode bien o.
P1
= /2
Q71
X
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,Q R( /
)ang tnh gocpha:
= / /= /==/=== = ;Z== ;/2= ;/X=;/== ;/b=
Gian o Bode goc pha
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,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
==++
+=
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,Q R( 2: 9e? gian &o )ode c!aha+m a!:
oc pha :
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,Q R( 2
Gian o Bode bien o.
= / 2X
71
P1
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III. M t h thng bng phng trnh tr!ng th"#
1. $h"# n#%&
'
(
)
*
+
+
+ +
r,t
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A. 8h
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8h3 /
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?i@$ &5i &An% p:n% $"n: $"An% $:Di
)$("
!
0
0
)$(
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)()(
)()(
01
1
1 tcadt
tdca
dt
tcda
dt
tcda
n
n
n
n +
)()(
)()(
01
1
1 trbdt
tdrb
dt
trdb
dt
trdb
m
m
m
m +
8B C8&
aVt #ientrang thai
theong!yen
tac:
)(.
)(.
)(
trBxx
trBxx
tcx
nnn ))
))*
)
=
=
=
."Fn% :p .? #:! 7A9 :m #J! n% 9 (m n-1)
7/24/2019 Chuong 2(Mo Ta Toan Hoc)
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?5i:
'1= bn-1Kan 'A= ,bn-A@ an-1.'1Kan
'L= ,bn-L@ an-1.'A@ an-A'1Kan
'n= ,b0@ an-1'n-1- - a1'1Kan$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 =
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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 /
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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 $T! 7U p:!
011
1
011
1
---
---
)(
)()(
apapapa
bpbpbpb
pR
pCpM
n
n
n
n
m
m
m
m
=
'Y hm t!=fn#
t bi@n ph` (p) sa ch#C(p)(mpm* m-1pm-1* V* 1p * 0)Y(p)
R(p)(!npn* !n-1pn-1* V* !1p * !0)Y(p)
Bi@n 7Qi L!p'!#e n%# 7W$
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# :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.
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4. 8Snh h)-1 BDS dV&8% h
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; \%: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)$()$(