تحليل نظم تحكم نظري

٢٦٣٢٦٣אא

א ٢٦٣

FE

،،אא،אW

אא א א א א א אאאא،אאאאאאא

אאאא אאאאאאאWא

אאK אאאאאאא

א ،א אא א א אא א א ، א

א،אאאאאאאאאאאאאאאא

אא،אאאא،אK

א א ? ? ? ?אאאאאאאאK אאאאאאא

،א،אאאאאאאאאK

א אא א WאK

אאאא

א ٢٦٣

FE

אאW אאאW

١K אאאאK ٢K אK ٣K אאאאK ٤K אאK ٥K אאK

אאW

אW ١K אאאאK ٢K אK ٣K אאאאK ٤K אאK ٥K אאK

אאW٣٨K אאW??K

אאאאאא

FFEE

א

א

א

١

א ٢٦٣ אא FE אאא

- ١ -

אאW אW

١K אאאK ٢K אאאאK ٣K אאאאK

١ J١ W אאאאאאאא

אאאKאאאאאאאאאאK

١ J٢ אאאW אאאאאא

אא،אאאאאאאאא

אאאאאאK א،אאא

אאאאאאאאאאאאאKאא

אאאאאאאאאאאאאאאKאא

אאאאאאאאאK

אאאאא،אא،אFאE،אאאאKאאאאW


- ٢ -

אאW אאאFאEאRF

אE،אאאW )()( tRutr =

Rאאu(t)אאאW

⎩⎨⎧

<≥

=0001

)(tfortfor

tu

אאW

sRsR =)(

אF١ J١EאאאאK

F١ J١E،אאאאK

E אאאW אאאאאא،א

אW

⎩⎨⎧

<≥

=000

)(tfortforRt

tr

אאאW

2)(sRsR =

אF١ J٢EאאאאKאאאאאאאK

r(t)

time

R


- ٣ -

F١ J٢E،אאאאאK

E אאW אאאאא،

אאW

⎩⎨⎧

<≥

=000

)(2

tfortforRt

tr

אאW

32)(sRsR =

אF١ J٣EאאאאKאאאאאאאK

F١ J٣E،אאאאK

E אאW אאאאאאא

Wאא،א )()( ttr δ=

δ(t)אאאW

⎪⎩

⎪⎨⎧

>

<≤=ε

εεδ

tfor

tfort0

01)(

r(t)

time

r(t)

time


- ٤ -

εאאאאאאKאאW

1)( =sR אF١ J٤EאאאאK

F١ J٤E،אאאאK

E אאאW אאאא

Tƒ،אאW )sin()( tRtr ω=

Rאאא،ωאאאא\אאאאאW

Hzfπω2

=

.sec1f

T =

אאאאW 0)( 22 >

+= s

ssR

ωω

אF١ J٥EאאאאK

r(t)

time

ε1

2ε

2ε

−


- ٥ -

F١ J٥E،אאאאK

١ J٣ אאאW

אאאאאא

Y(s)،אאאאKאאאאאאאאאאאא

אאKאאאאאאאא،אאא

אK ١ J٣ J١ אאאאW

E אאW אאאאאאאW

١K אאאW אאאאאW

)()()(' tKxtyty =+τ

G(s) Y(s) X(s) x(t) y(t)

r(t)

time

-r(t)

R

T


- ٦ -

W • τWאאK • KWאאאK • x(t)WאK • y(t)WאFאאKE

٢K אאאW אאא،

،א،W )()()0()( sKXsYyssY =+−ττ

אאאאW )()0()()1( sKXysYs +=+ ττ

אאא1+sτאאW

)(1

11

)0()( sXs

Ks

ysY+

++

=ττ

τ

٣K אאאאW אאRאאW

⎩⎨⎧

<≥

=000

)(ttR

tx

אW sRsX =)(

X(s) אאW

)1(1

1)0()(

++

+=

ssKR

sysY

τττ

٤K אאW

אY(s)אW1

)0(+s

yττ

)1(1+ss

KRτ

K

אאאאאW


- ٧ -

τ

ττ t

eys

yL−− =⎟

⎠⎞

⎜⎝⎛

+)0(

1)0(1

אא)1(

1+ss

KRτ

אאW

1)1(1

++=

+ sB

sA

ssKR

ττ

אABאW

KR

sssKRsBKR

sssKRsA τ

τ

ττ

τ−=

−=+

+===+

=1)1(

)1(,0)1(

אABW

1)1( +−=

+ sKR

sKR

ssKR

ττ

τ

⎟⎠⎞

⎜⎝⎛

+−=

11

ssKR

ττ

אW )1(

)1(/1 τ

τteKR

ssKRL −− −=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

אאאW

)1()0()( ττtt

eKReyty−−

−+=


- ٨ -

E אאW אאאאW

• אאτWאאK • אאts Wאא٩٨א٪،אא

W τ4≅st

• אאאyss Wאא،W

KRy

tyy

ss

tss

=

=∞→

)(lim

• אKWאאאאW

RyK ss=

E אאאאW אאKאאא

אW ١K אאאא،אא

W

)1()( τt

eKRty−

−= ٢K אאyssW

KRyss = ٣K אאאW

ssyy 63.0)( =τ ٤K אאtsW

τ4=st ٥K אאאW

sss yty 98.0)( =


- ٩ -

٦K אאאאאF١ J٦E

F١ J٦E،אאאאK

אאא٩٥٪א3τ،א٢٩٨٪4τ،אτאא

٩٨٪א،א٢K٪אאאאFEאאא

א،אאאאK E אאאאאאאW

א،אאאאאאאא،אאאW

)()(Re)(

)1()( /

tyytyKKRty

eKRty

tss

t

t

−=−=

−=

−

−

τ

τ

אאאW

τ 2τ time

0.5yss

yss=KR

y(τ)=0.63yss

3τ ts = 4τ0

y(t)

y(ts)=0.98yss


- ١٠ -

• אאKRyss =W אאאאאא

אK

• אאτt

t Kty−

−= Re)(W אאאאK

F١ J١WE אאאW

0)0()(10)(2)('

==+

ytxtyty

אאאאאKאאK אW

١K אא2y(t)אW )(5)()('5.0 txtyty =+

٢K אאאאאW • אאW5.0=τ • אW5=K

٣K אא،אאאR=1،אW

t

t

etyety

2

5.0

55)()1(5)(

−

−

−=

−=

٤K אאW • אאאW 5=ssy • אאW t

t ety 25)( −= ٥K אW


- ١١ -

١ J٣ Jא ٢אאאW

E אאW אאאאאאאאW

١K אאאW אאאאאW

)()()('2)(" 22 txKtytyty ooo ωωζω =++

)()()('2)(" 22 txKtytyty oo ωωα =++ W • ω0Wא?אK? • ζWא?K? • αWאFoξωα =E?K? • KWאאאK

אאW • 2

oKb ω= • אאאאK

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Step Response

Time (sec)

Ampl

itude

y(τ) = 3.15

τ = 0. 5

yss = 5


- ١٢ -

٢K אאאW אאאא،

،א،W )()()(2)( 2

02 sbXsYssYsYs =++ ωα

אW )()()2( 22 sbXsYss o =++ ωα

22אאא 2 oss ωα ++אW )(

2)( 2

02 sX

ssbsY

ωα ++=

٣K אאאאW אאRאאW

⎩⎨⎧

<≥

=000

)(ttR

tx

אWsRsX =)(

X(s) אאW

)2()( 22

osssbRsY

ωα ++=

אא22 2 oss ωα ++א،אאאאW

222,1 or ωαα −±−=

אr1r2אאW

E אאאW אα>ωoζ>1،א

אW 22

2,1 or ωαα −±−= אאW

))(()(

21 rsrssbRsY

−−=


- ١٣ -

Wאאאאאא

21

)(rs

Crs

BsAsY

−+

−+=

A, B, CW

2210

)(o

bRrr

bRs

ssYAω

===

=

)()()()(

212

2

21111 rr

bRrrrr

bRrs

sYrsBo −

=−

==

−=ω

)()()()(

212

1

21222 rr

bRrrrr

bRrs

sYrsCo −−

=−

−=

=−=

ω

ABCאאאW

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

∗−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

∗−

+=221

1

121

22

111)(rsrr

rrsrr

rs

bRsYoω

אאאאאאW

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

+= trtr

o

err

rerr

rbRty 21

21

1

21

22 1)(

ω

b=Kωo2אאא

אאW

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

+= trtr err

rerr

rKRty 21

21

1

21

21)(

E אאW אα=ωoζ=1،

אאW α−=2,1ror ω−=2,1

אאW

)2()( 22

oosssbRsY

ωω ++=

אאW

2)()(

ossbRsYω+

=


- ١٤ -

אאאאאאאאW

( )to

oetKRty ωω −+−= )1(1)( E אאW

אα<ωoζ<1،אאW

doo jjr ωζωαωα ±−=−±−= 2221,

dωאאאאאאW 222 1 ζωαωω −=−= ood

אאW

)2()( 22

oosssbRsY

ωζω ++=

⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

−= 222 221)(

oo

o

o sss

sbRsY

ωζωζω

ω

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

++−

+++

−= 222221)(

do

o

do

o

o sss

sbRsY

ωζωζω

ωζωζω

ω

אאאאאאW

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−= − tteKRty dd

to ωζ

ζωζω sin1

cos1)(2

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−=

−

φωζ

ζω

d

toeKRty sin1

1)(2

אאW

⎟⎟⎠

⎞⎜⎜⎝

⎛ −= −

ζζφ

21 1tan

אy(t)אאאאאωdK


- ١٥ -

E אאאאW אF١ J٧Eאאאאאאא،

W • אאאWאאאאאאאK • אאWאאאאא،א

אאאאאאאK • אאWאאאאאאא

אאאK

F١ J٧E،אאאK

F١ J٢EW אאW

⎩⎨⎧

<≥

=

===++

0001

)(

,0)0(')0()(2)(2)('3)("

tt

tx

yytxtytyty

W E אא،אK E אK

0

1

ζ<1

ζ>1 א ζ=1

time

C(t)


- ١٦ -

E אאK E אאK

אW E אאאאא

אאW

sec/4.1222 radooo ≅⇒=⇒= ωωω 5.132 =⇒= αα

αωאאאK E אאאאאא

W

12222 2

2 =⇒=⇒=⇒= KKKKo

o ωω

E אאאאאאאאW

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

+= trtr err

rerr

rKRty 21

21

1

21

21)(

r1r2W

( ) ( )

56.3,56.025.45.1,24.45.1

25.45.1

25.15.1

21

21

2,1

222,1

222,1

−≅≅−−=+−=

±−=

+±−=

−±−=

rrrr

r

r

r oωαα

אאאאאW ( )

( ) ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−

−+≅ − tt eety 56.356.0

56.356.056.0

56.356.056.31)(

( )tt eety 56.356.0 13.086.01)( −−−≅ E אאאW


- ١٧ -

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Step Response

Time (sec)

Ampl

itude

E אאאאאאאW

אאאאאאאאאאאא،א

אW sst yyty +=)(

א،אאאאKאאא

אאW )(lim tyy

tss

∞→=

E אאW אאאאא

אאKאאאאאאK


- ١٨ -

F١ J٨E

אאאאאW

١K א tdW אא٥٠א٪אK

٢K א trW אא٩٠א٪אK

٣K ypW אK

٤K אtpW אאא،אאW

20

220 1 ζω

π

αω

π

−=

−=pt

٥K אאtsW אאאאF±٢٪± ٥E٪

٦K MpW 21/ ζζπ −−=−= eyyM sspp

time00

yss

td tr tp ts

±2%Mp

0.5 yss

0.9 yss

yp

y(t)


- ١٩ -

ypyssאאK ٧K אא POW

100×=ss

p

yM

PO

٨K אאא GW

EyG ss=

٩K אאאאאאW 2

022

0 1 ζωαωω −=−=d ١٠K אאאאW

ddT

ωπ2

=

١١K אאאאW

dd T

f 1=


- ٢٠ -

W ١K אאW

⎩⎨⎧

<≥

=

==+

0005

)(

0)0()(10)(10)('

tt

tx

ytxtyty

W E אאK E אK E אאK E אאK

٢K אאאW

⎩⎨⎧

<≥

=

==+

00010

)(

0)0()()()('10

tt

tx

ytxtyty

W E אאK E אK E אאK E אאK

٣K אאW

⎩⎨⎧

<≥

=

===++

0005

)(

0)0(')0()(16)()('4)(''

tt

tx

yytxtytyty


- ٢١ -

W E אא،אK E אK E אאK E אאK

٤K אאW

⎩⎨⎧

<≥

=

===++

0001

)(

,0)0(')0()(10)()('2)("

tt

tx

yytxtytyty


٥K אאW

⎩⎨⎧

<≥

=

===++

0005

)(

0)0(')0()(16)(8)('4)(''

tt

tx

yytxtytyty



- ٢٢ -

٦K אאאאאאW

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Step Response

Time (sec)

Ampl

itude

W

E אtdK E אtrK E אtpK E אאK E MpK E אאK E אאאK

אאאא

FFEE

א

א

٢

א ٢٦٣ אא FE אא

- ٢٣ -

אאW אW

١K אאK ٢K אאK ٣K אאK ٤K אאאK ٥K אאאK ٦K אאאאK


- ٢٤ -

٢ J١ אאW אאאא،אא

אאאKאא،אאאאאאאא

אאאאא،אאאאאאאאK

אאאאאאאאKאF٢ J١EאאK

F٢ J١E،K

٢ J٢ אאW ٢ J٢ J١ אאאW

אאאאאW )()( teKtp p=

KPאא،אאp(t)אe(t)K

٢ J٢ J٢ אאאW אF٢ J٣EאאאK

F٢ J٣E،אאאK

٢ J٢ J٣ אאאW אאאאא

אW

النظام الحاآمy(t) x(t) e(t) p(t)

+-

KpP(s)E(s)


- ٢٥ -

)()( sEKsP p= אאאGc(s)אאW

pc KsEsPsG ==)()()(

٢ J٢ J٤ אאאאאאW אF٢ J٤EאאאאאאK

F٢ J٤E،אאאאאאK

W 1

2

RRK p −=

٢ J٢ J٥ אאאW אאאאKאK

٢ J٣ אאW ٢ J٣ J١ אאאW

אאאאאW ττ deKtp

tI )()(

0∫= KIאK

٢ J٣ J٢ אאאW אF٢ J٥EאאאK

F٢ J٥E،אאאK

٢ J٣ J٣ אאאW אא،אאא

אW

P(s)E(s)

sKI

e(t) R1

R2

p(t)


- ٢٦ -

ssEKsP I)()( =

אאאGc(s)אאW

sK

sEsPsG I

c ==)()()(

٢ J٣ J٤ אאאאאאW אF٢ J٦EאאאאאאK

F٢ J٦E،אאאאאאK

W RC

KI1

−=

٢ J٣ J٥ אאאW א،אאאאאK

٢ J٤ אאW ٢ J٤ J١ אאאW

אאאאאW

dttdeKtp D)()( =

KDאK ٢ J٤ J٢ אאאW

אF٢ J٧EאאאK

F٢ J٧E،אאאK

٢ J٤ J٣ אאאW אא،אאא

אW

P(s)E(s)KDs

e(t) p(t)R

C


- ٢٧ -

)()( ssEKsP D= אאאGc(s)אאW

sKsEsPsG Dc ==)()()(

٢ J٤ J٤ אאאאאאW אF٢ J٨EאאאאאאK

F٢ J٨E،אאאאאאK

W RCKD −= ٢ J٤ J٥ אאאW

אאאאKאאאK

٢ J٥ אאאW ٢ J٥ J١ אאאאW

אאאאאאW

∫+=t

IP deKteKtp0

)()()( ττ ٢ J٥ J٢ אאאאW

אF٢ J٩EאאאאK

F٢ J٩E،אאאאK

٢ J٥ J٣ אאאאW אאאאא،א

אW

sKI

+

+

E(s) P(s)

Kp

e(t) p(t)C

R


- ٢٨ -

ssEKsEKsP IP)()()( +=

אאאאGc(s)אאW

sK

KsEsPsG I

Pc +==)()()(

٢ J٥ J٤ אאאאאאאW אF٢ J١٠EאאאאאאאK

F٢ J١٠E،אאאאאאאK

W RC

KRRK Ip

1,1

2 ==

٢ J٥ J٥ אאאאW אאאאאאאא

אK ٢ J٦ אאאW ٢ J٦ J١ אאאאW

אאאאאאW

dttdeKteKtp DP)()()( +=

٢ J٦ J٢ אאאאW אF٢ J١١EאאאאK

e(t) p(t)

R1

R2

C

R R

R

R


- ٢٩ -

F٢ J١١E،אאאאK

٢ J٦ J٣ אאאאW אא،אאאא

אW )()()( ssEKsEKsP DP +=

אאאאGc(s)אאW

sKKsEsPsG DPc +==)()()(

٢ J٦ J٤ אאאאאאאW אF٢ J١٢EאאאאאאאK

F٢ J١٢E،אאאאאאאK

W RCKRRK Dp == ,

1

2

٢ J٦ J٥ אאאאW אאאאאאאאK

KDs

+

+

E(s) P(s)

Kp

e(t) p(t)

R1

R2

C

R R

RR


- ٣٠ -

٢ J٧ אאאאW אאאאאא،א

F،א،אאEאK ٢ J٧ J١ אאאאW

אאאאאאאW

dttdeKdeKteKtp D

tIP

)()()()(0

++= ∫ ττ

٢ J٧ J٢ אאאאW אF٢ J١٣EאאאאאK

F٢ J١٣E،אאאאאK

٢ J٧ J٣ אאאאאאW אאאא،אאא

אW )()()()( ssEK

ssEKsEKsP DIP ++=

אאאאאGc(s)אאW

sKs

KK

sEsPsG D

IPc ++== +)(

)()(

٢ J٧ J٤ אאאאאאW אF٢ J١٤Eאאאאאא

אK

sKI

+

+

+

E(s) P(s)

KD s

KP


- ٣١ -

F٢ J١٤E،אאאאאאאK

W DDDII

Ip CRKCR

KRRK === ,1,

1

2K

e(t)

p(t)

R2

R1

RI

RD

R

R

R

R

CI

CD


- ٣٢ -

W ١K אאK ٢K אF٢ J٤Eאאאאאא،

R1=10 kΩ R2=5 kΩ W E אאאK E אאK

٣K אF٢ J٤Eאאאאאא،R1=10 kΩאR2אKP=-7؟

٤K אאאא؟ ٥K אF٢ J٦Eאאאאאא،

R=10 kΩ C=1µFW E אאאK E אאK

٦K אF٢ J٨Eא،אאאאאR=5 kΩ C=2µFKאW E אאאK E אאK

٧K אF٢ J١٠Eא،אאאאR=5 kΩ C=2µFR1=20 kΩR2=40 kΩאW

E אאK E אאK

٨K אאאאאאא؟א ٩K אF٢ J١٣Eאאאא

א،א KP=2 KI=5 KD=4W E אאK E אאK E אאאאאאK

אאאאאא

FFEE

א

א

א

٣


- ٣٣ -

אא אW

١K אאאK ٢K אאאאאK ٣K אאאאאK


- ٣٤ -

٣ J١ אאאW אאאאאאאא

א،אאאאאKאאאאאאאK

אאאאאאאאאW?אאאאK?

אLאאאא،אאאL،אאLא

FEאK אאאאאאאאאא

אאאאאאK ٣ J٢ אאאאW

אאאאאאאKאאאאאאאא

אאאאK אאא،אאאא

אאאאאKאאאאK

F٣ J١EW אאאאאאאW

6510

)()(

2 ++=

sssXsY

אW אאאאאאאW

E אאW 0652 =++ ss

E אW 2,3 21 −=−= ss


- ٣٥ -

E אאאאK ٣ J٣ אאאאW

אאאאאאאאאאKאאאאאאא

אW E אאK

١K אאאאאאאאW

)()()(')("....)()( 012)1(

1)( txtyatyatyatyatya n

nn

n =+++++ −−

W y'(t)אאy(t) y"(t)אאy(t)

y(3)אאy(t)אKKK אאאאW

0.... 012

21

1 =+++++ −− asasasasa n

nn

n ٢K אאאאאאא

K E אW

אאאאW ١K אא،א

n+1 ،nאK ٢K אא،אאאאW

.....42 −− nnn aaa ٣K אא،אאאאW

.....531 −−− nnn aaa


- ٣٦ -

٤K אאאאW

sn an an-2 an-4 ……

sn-1 an-1 an-3 an-5 ……

sn-2 C1 c2 …… ……

sn-3 D1 d2 …… ……

….. …… …… …… …… s0 ……

אW

1

5412

1

3211

−

−−−

−

−−−

−=

−=

n

nnnn

n

nnnn

aaaaa

c

aaaaa

c

1

1512

1

21311

3c

caacd

ccaac

d

nn

nn

−−

−−

−=

−=

E אאאW אאאאאאKא

אאא،אאאאK

F٣ J٢EW אאאאאאאW

)(2)(6)(' txtyty =− אW

E אאW s+2=0

E אאאאW • אWn=1אn+1=2K • אאאא

W 01s1

02s0

E אאאאאאאאאאאאאK


- ٣٧ -

F٣ J٣EW אאאאאאאW

)(3)(2)(' txtyty =+ אW

E אאW s-6=0


W 01s1

0-6 s0


F٣ J٤EW אאאאאאאW

)(3)(2)('2" txtytyy =++ אW

E אאW 0222 =++ ss


W 22s2

2 s1

c1 s0

• אc1אW


- ٣٨ -

12

0222

1

1

1

3211

=

×−×=

−=

−

−−−

c

c

aaaaa

cn

nnnn

• אW 22s2

2 s1

1 s0


F٣ J٥EW אאאאאאאW

)(3)(6)('" txtytyy =−+ אW

E אאW 062 =−+ ss


W -61s2

0 1 s1

c1 s0

• אc1אW

61

01)6(1

1

1

1

3211

−=

×−−×=

−=

−

−−−

c

c

aaaaa

cn

nnnn

• אW -61s2

0 1 s1


- ٣٩ -

-6 s0


F٣ J٦EW אאאאאאאW

)()(500)('100)(''4)(''' txtytytyty =+++ אW

E אאW 05001004 23 =+++ sss


W 100 1 s3

5004s2

0 c1 s1

c2 s0

• אc1c2אW

1

5412

1

3211

−

−−−

−

−−−

−=

−=

n

nnnn

n

nnnn

aaaaa

c

aaaaa

c

• אW 100 1 s3

5004s2

0-25s1

500 s0



- ٤٠ -

٣ J٤ אאאאאאאW אאאאאאאא

אאאKאאאW

אK،אאאאאאאW

E אאאK E אאאK E אK E אKאאאאK

אKאאאאK F٣ J٧WE

אKאאאאאאאאW

042 23 =+++ Ksss אW

אאאאW • אWn=3אn+1=4K • אאאא

W 4 1 s3

K2s2

0 2

8 K−s1

K s0

• אאאאאאאאK

אKאאאאאאא

K G(s) Y(s) X(s)

+-

H(s)


- ٤١ -

אאאKאאאW

0

02

8

>

>−

K

K

KאאאאאאW 80 << K


- ٤٢ -

W ١K אאאאאאאאאאW

E 127

10)()(

2 ++=

sssXsY

E 209

3)()(

2 +−=

sssXsY

E 30

1)()(

2 −+=

sssXsY

٢K אאאאאאאאW E )(8)(7)(' txtyty =+

E )(20)(147)(' txtyty =−

E )(8)(4)('6)("2 txtytyty =++

E )(8)(14)('16)(" txtytyty =+−

٣K אאאאאK ٤K אאאאאאאאW

01246 2345 =+++++ sssss

٥K KאאאאאאאאאW

KssssK

sRsC

++++=

23)()(

234

אאאא

FFEE

א

א

٤

א ٢٦٣ אאא FE אא

- ٤٣ -

אאW אW

١K אK ٢K אאאאאK ٣K אאאאאK ٤K אאאאאאK


- ٤٤ -

٤ J١ אאW אאאאאאאא

אאאאאאאאאאאKא

אאאאאאאאK

אאאאאאאאאאKאא

אאאאאאאאאאK אאאאאאא

אא،אאאאאאאKאאאא

אאK

F٤ J١E،K

אF٤ J١E،אאr(t)b(t)،אr(t)אאאb(t)אאאאאc(t)K

אאאאא،אKאאאW

)()()( tbtrte −= אאאsW

)()()( sBsRsE −= אB(s) C(s)אW

G(s) C(s) R(s) +-

H(s)

E(s) e(t) r(t) c(t)

B(s) b(t)


- ٤٥ -

)()()()( sHsCsRsE −= אאאT(s) W

)()(1)(

)()()(

sHsGsG

sRsCsT

+==

אC(s)W )(

)()(1)()( sR

sHsGsGsC

+=

אאC(s)אאR(s)אW

)()()(1

1)( sRsHsG

sE+

=

٤ J٢ אאאW אF٤ J٢E،אאא

אאGp(s)אאאאאGc(s)אאK

F٤ J٢E،אאK

אאGc(s)Gp(s)אאאאאאאאW

G(s)=Gc(s)⋅Gp(s) אאאW

)()()(1)()(

)()(

sHsGsGsGsG

sRsC

pc

pc

+=

אאאW )(

)()()(11)( sR

sHsGsGsE

pc+=

Gc(s) Gp(s) C(s) R(s) +-

H(s)

P(s) E(s)


- ٤٦ -

אאאאאא،אאאאאאאאאW

E אאאW אאאאאאKא

אאאאאאW

ssR 1)( =

E אאW אאאאא

אאאאאW

11)(+

=s

sGp E אאאW

אאאKW 1)( =sH

E אאW אאאK

E אאאW

)()()()(1

1)( sRsHsGsG

sEpc+

=

E אאאW אאאאאW

)(lim0

ssEes

ss→

= אאאאא

אאאאאF٤ J٣KE

F٤ J٣E،K

C(s) s

sR 1)( = +-E(s)

11+s

Gc(s)


- ٤٧ -

٤ J٢ J١ אאאאאW אאאאW

sKsK

sKKsG IpI

pc

+=+=)(

Gc(s)Gp(s)R(s)אאאאW

sss

KsKsEIp

1

111

1)( ∗

+∗

++

=

אאW

Ip KsKsssE

++++

=)1()1()( 2

אאאאאאW

0

)1(1

)(

20

0

lim

lim

=

++++

∗=

=

→

→

ss

Ipsss

sss

e

KsKssse

ssEe

אאאא،אאאאאאאאאאKא

אF٤ J٤KE

F٤ J٤E،אאאאK

0 2 4 6 8 10

12

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Amplitude

Kp=1 ,

Kp=1 ,

Kp=2 ,


- ٤٨ -

٤ J٢ J٢ אאאאאW אאאאW

sKKsG Dpc +=)( Gc(s)Gp(s)R(s)א

אאאW

ss

sKKsE

Dp

1

11)(1

1)( ∗

+∗++

=

אאW

sKsKssE

PD

1)1()1(

)1()( ∗+++

+=

אאאאאאW

11

)1()1(11

)(

lim

lim

0

0

+=

++++

∗∗=

=

→

→

Pss

PDsss

sss

Ke

KsKs

sse

ssEe

אאאאאא،אאאאאאKP K

אאF٤ J٥KE

F٤ J٥E،אאאאK

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

C(t)


- ٤٩ -

٤ J٢ J٣ אאאאאאW אאאאאW

sKsKsK

sKs

KKsG IpDD

Ipc

++=++=

2

)(

Gc(s)Gp(s)R(s)אאאאW

sss

KsKsKsE

IpD

1

111

1)( 2 ∗

+∗

+++

=

אאW

IpD KsKsKssE

+++++

=)1()1(

)1()( 2

אאאאאאW

0

)1()1(1

)(

20

0

lim

lim

=

+++++

∗=

=

→

→

ss

IpDsss

sss

e

KsKsKsse

ssEe

אאאאאאא،אאאאאאאאK

אאF٤ J٦EK

F٤ J٦E،אאאאאK

0 5 10 15 0

1

2,1,1 === DpI KKK

2,1,1 === DpI KKK

2.0,1,2 === DpI KKK

c(t)

time, sec


- ٥٠ -

W ١K אאW

W

E אאאK E אK E אאאK

٢K אאW

W

E אאאK E אK E אאאK

٣K אאא؟KK ٤K אאא؟KK ٥K אK אאאאאאK

C(s) s

sR 1)( = +-

E(s) 210

10+sK

5

C(s) s

sR 10)( = +-

E(s)110

5+s

C(s) s

sR 2)( = +-

E(s) 210

10+s

4s


- ٥١ -

٦K אאאW

W

E אK E אאK E אאאK E אK E אאאessK

٧K אאאW

W

E אK E אאK E אאאK E אK E אאאessK

C(s) s

sR 1)( = +-

E(s) 1

1+s

+ P(s)

+s1

5

s

C(s) s

sR 1)( = +-

E(s) 1

1+s

+ P(s)

+s2

10

אאאא

FFEE

א

א

٥


- ٥٢ -

אא אW

١K אאאאK ٢K K ٣K אאK ٤K אאאאK


- ٥٣ -

٥ J١ אאא،אאאאאK

אאאאאאאאאK

אאKאאאאאאK

٥ J٢ אאW

F٥ J١E،K

אאאאאאאאKאאW

)sin()( θRtr = Rאאאאθאאאאtωωא

אאאאאא (rad/s)fπω 2=fאאא(Hz)Kאאא

אW )sin()( βCtc =

Cאא β אא،אאאאאאאאא

KאאאאאאאF٥ J١EWאאאא

E אאW )()()( sGsRsC ⋅=

E אאאאאאאωאאאsjωאאא

W )()()( ωωω jGjRjC ∗=

G(s) C(s) R(s) r(t) c(t)


- ٥٤ -

E אאאאאW

)Im()()( ωωω jjjRjG e += φωω ∠= )()( jGjG

W ( ) ( )22 )Im()()( ωωω jjRjG e +=

( ))()Im(tan 1

ωωφ

jRj

e

−=

אאאאאאאאW ( ) ( )φωθωβω ∠∗∠=∠ )()()( jGjRjC

E אאW ١K אאW

)()()( ωωω jGjRjC ∗= W

)( ωjGRC ⋅= CאאאRאאאK

٢K אאאאW φθβ ∠+∠=∠

אאאאאאW φωβ += t

אאאאφאאאאK F٥ J١WE

אאאc(t)אאW

אW

אאאאאW )sin()( βCtc =

אאCאאβא

C(s) R(s) r(t)=Rsin(ωt) c(t)1+s

Kτ


- ٥٥ -

אW )( ωjGRC ∗=

φωβ += t אאאאאKאאW

E sאאjωW

1)(

+=

ωτω

jKjG

E אאאW

221)(

τωω

+=

KjG

E אאW )(tan)( 1 ωτωφ −−=∠= jG

אאאאאאאאW

221 τω+=

RKC

אאאאאאאאW )(tan 1 ωτωβ −−= t

אאc(t)אW ( ))(tansin

1)( 1

22ωτω

τω−−

+= tRKtc

٥ J٣ אאאFEW אאאאאא

אKאאאאאאאאאאK

אאאאאאdbאאאω،אאאאK

אאאאאאאאאאאא

אאאK


- ٥٦ -

F٥ J٢WE אאאאW

100)( =sG אW

אאאאKp=100KאאאאאW

E אאאאאsjωאאW

100)( =ωjG E אאאW

100)( =ωjG

E אאאdBW dBjGG

db40100log20)(log20 === ω

E אאאאW o0)

1000(tan 1 == −φ

E א،אF٥ J٢KE

39

39.5

40

40.5

41

Mag

nitu

de (d

B)

100

101

-1

-0.5

0

0.5

1

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٢E،Kp=100K


- ٥٧ -

F٥ J٣WE אאאאW

100)( −=sG אW

אאאאKp=-100KאאאאאW


100)( −=ωjG E אאאW

100)( =ωjG

E אאאdBW dBjGG

db40100log20)(log20 === ω

E אאאאW o180)

1000(tan 1 =

−= −φ

E א،אF٥ J٣KE

39

39.5

40

40.5

41

Mag

nitu

de (d

B)

100

101

179

179.5

180

180.5

181

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٣E،Kp=-100K


- ٥٨ -

F٥ J٤WE אאאאW

1001)( =sG

אW אאאא

1001

=pKKאאאאאW


1001)( =ωjG

E אאאW

1001)( =ωjG

E אאאdBW dBjGG

db40)100log(20)(log20 1 −=== −ω

E אאאאW 011 0)

1000(tan)

10(tan =−= −−φ

E א،אF٥ J٤KE

-41

-40.5

-40

-39.5

-39

Mag

nitu

de (d

B)

100

101

-1

-0.5

0

0.5

1

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٤E،

1001

=pKK


- ٥٩ -

F٥ J٥WE אאאאW

ssG 10)( = אW

אאאא10=DKKאאאאאW


ωω 10)( jjG = E אאאW

ωω 10)( =jG

E אאאdBW dBjGG

db...........)10log(20)(log20 ωω ==

E אאאאW o90)

010(tan 1 == − ωφ

E א،אF٥ J٥KE

15

20

25

30

35

40

Mag

nitu

de (d

B)

100

101

89

89.5

90

90.5

91

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٥E،10=DKK


- ٦٠ -

F٥ J٦WE אאאאW

ssG

101)( =

אW אאאא

101

=IKKאאאאאW


ωω

101)(

jjG =

E אאאW

ωω

101)( =jG

E אאאdBW dBjGG

db............)10log(20)(log20 1−== ωω

E אאאאW o90)

010(tan)

10(tan 11 −=−= −− ωφ

E א،אF٥ J٦KE

-40

-35

-30

-25

-20

-15

Mag

nitu

de (d

B)

100

101

-91

-90.5

-90

-89.5

-89

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٦E،

101

=pKK


- ٦١ -

F٥ J٧WE אאאאW

1)( += ssG τ אW

אאאאאKאאאאאW


τωω jjG +=1)( E אאאW

2)(1)( τωω +=jG

E אאאdBW dBjGG

db...........))(1log(10)(log20 2τωω +==

E אאאאW )(tan 1 τωφ −=

אאאאאdBKאאא،

אW

τω 1

=c אאאW

• אאFω<<ωcWE 1τωτωאW

0)1log(10 =−=dBG • אאFω>>ωcWE

1τωτωW τωτω log20)log(10 2 ==dBG

אω<<ωc0dBω>>ωc20dB/decade0dBω=ωcK


- ٦٢ -

אאτ=10W 1.0

101==cω

אא(ω=ωc)אW

dBjGG cdB3)11log(10)(log20 2 =+== ω

אאאאW 10ωc2ωcωc0.5ωc0.1ωcω 84.3°63.4°45°26.6°5.7° φ

F٥ J٧Eאאτ=10KאאאאK

0

10

20

30

40

Mag

nitu

de (d

B)

10-3

10-2

10-1

100

101

0

45

90

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٧E،אאK

F٥ J٨WE אאאאW

11)(+

=s

sGτ


- ٦٣ -

אW אאאאKא

אאאאW E אאאאאsjωאא

W

τωω

jjG

+=

11)(

E אאאW

2)(11)(τω

ω+

=jG

E אאאdBW dBjGG

db............))(1log(10)(log20 2τωω +−==

E אאאאW )(tan 1 τωφ −−=

אאאא،אאאdBKא

אW

τω 1

=c אאאW

• אאFω<<ωcEW 1τωτωאW

dBG dB 0)1log(10 =−= • אאFω>>ωcEW

1τωτωW dB τωτω log20)log(10 2 −=−=dBG

אω<<ωc0dBω>>ωc-20dB/decade0dBω=ωcK אאτ=10W

1.0101==cω


- ٦٤ -

אא(ω=ωc)אW

dBjGG cdb 32log10)11log(10)(log20 2 −=−=+−== ω אאאאW

10ωc2ωcωc0.5ωc 0.1ωc ω -84.3°-63.4°-45° -26.6° -5.7°φ

F٥ J٨Eאאτ=10KאאKאא

-40

-30

-20

-10

0

Mag

nitu

de (d

B)

10-3

10-2

10-1

100

101

-90

-45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec) F٥ J٨E،אאτ=10K

F٥ J٩WE אאאאW

22

2

2)(

oo

o

sssG

ωζωω

++=

אW אאאאאKאא

אאאW


- ٦٥ -


22

2

)(2)()(

oo

o

jjsG

ωωζωωω

++=

E אאאאW

1)(2)(

1)(2 ++

=

oo

jjjG

ωωζ

ωωω

אאאאאאW

)(2))(1(

1)(2

oo

jjG

ωωζ

ωωω

+−=

E אאאW

222 ))(2())(1(

1)(

oo

jG

ωωζ

ωω

ω+−

=

E אאאdBW

dBjGGoo

db............))(2())(1(log10)(log20 222

⎥⎦

⎤⎢⎣

⎡+−−==

ωωζ

ωωω

E אאאאW

)2(tan 221

ωωζωωφ−

−= −

o

o

אאאW • אאFω<<ωoWE

1⎟⎟⎠

⎞⎜⎜⎝

⎛

oωω⎟⎟

⎠

⎞⎜⎜⎝

⎛

oωωW

dBG dB 0)1log(10 =−= • אאFω>>ωoWE

12

⎟⎟⎠

⎞⎜⎜⎝

⎛

oωω ⎟⎟

⎠

⎞⎜⎜⎝

⎛

oωωζ2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

oωω

אאאאאW

0

22

0log40])[(log10

ωω

ωω

−=−≅dbG


- ٦٦ -

אω<<ωo0dBω>>ωo-40dB/decade0dBω=ω0K

אאאW o180lim −=−=

∞→πφ

ω،0lim

0=

→φ

ω،o90lim

0−=

→φ

ωω

אF٥ J٩Eאאאω0=4،ζ K

אF٥ J٩E،אאK

٥ J٤ אאW אF٥ J١٠EאאW

E אωGW אאא0dBK

E אωpW אאא-180°K

E אmG: אאאא0dBא

Fω=ωpEK

Frequency (rad/sec)

P

hase

(deg

)

Mag

nitu

de (d

B)

-100

-50

0

50

10-2 10 - 10 0 10 1 102 1 3-200

-150

-100

-50

0

ζ=10

ζ=10

ζ=1

ζ=1

ζ=0.1

ζ=0.1


- ٦٧ -

E אmpW אאאא-180°א

Fω=ωGEK 180)( += Gpm ωφ

אF٥ J١٠E،אאK

٥ J٥ אאאאW אאאאאאW

• אאW ?אאאא1dBא?K

• אאW ?אאאא-180 א?K

אאאאאאKאא،אאאאאא

אאא،אאא،אאאא?אאאא

א45°?،אאאK

Frequency (rad/sec)

Pha

se (d

eg)

M

agni

tude

(d

B) -40

-20

0

20

10-1 100 101-300

-200

-100

0

-180

ωp

ωG

mp

mG


- ٦٨ -

F٥ J١٠WE אF٥ J١٠EאאאאאK

אW ١K אאא1אdBא

אK ٢K אאא180-א א

אK ٣K אאא100°،

אא45°אאאא،אאאK


- ٦٩ -

W ١K אאאא؟א ٢K אdB אאW

E 1000)( =sG E 200)( −=sG E

ssG 10)( =

E s

sG 50)( −=

E 11.0

100)(+

=s

sG

E )1)(11.0(

1)(++

=ss

sG

٣K אאאאאא،אKאא

٤K אאאK ٥K אW

)1)(11.0(1)(

++=

sssG

E אK E אK

٦K אאW

)1046.0(7.9)(+

=ss

sG

E אK E אK

٧K אאאW

)101.0)(102.0(100)(

++=

ssssG

E אאK E אאK E אאאK

א ٢٦٣

FE

AC motor

actuator

analog

א armature

א automation

block diagram

bode diagram

cascade

אא characteristic Equation

characteristics

chart recorder

closed loop

compensator

control system

control valve

אאאא controlled variable

controller

critical damping

א cutoff frequency

damping

DC motor

א delay Time

derivative

derivative Controller

design

digital

א ٢٦٣

FE

אא disturbance

، dynamic

error

feedback

feedback path

אא final control element

flow meter

flow rate

forward path

א frequency response

gain

א gain crossover frequency

א gain margin

hydraulic

input

integral

integral Controller

lag compensator

laplace transform

lead compensator

level

magnitude

manual control

matrix

motor

open loop

א oscilloscope

output

א ٢٦٣

FE

א over damping

overshoot

א parallel

א peak time

א performance

אא permanent response

א phase crossover frequency

א phase margin

אא phase shift

،א pneumatic

א polynomial

א potentiometer

process

אא programmablelogiccontrol

proportional

proportional controller

reference input

א resonance frequency

א response

א response curve

א rise time

root

sensor

א series

א،א set point

אא settling time

א signal conversion

א signal processing

א ٢٦٣

FE

simulation

א specification

אא stability

אא stability criteria

א step input

א stepper Motor

summing junction

system

،א tachometer

take off point

אא time constant

א time domain response

transducer

א transfer function

א transient response

אא two position control

underdamping

א unit step

unity feedback

א ٢٦٣ א

אא

1. Johnson, C. D. Process Control Instrumentation Technology, Prentice Hall, 2002 2. Bateson, R. N. Introduction to Control Systems Technology, Prentice Hall, 2002 3. Ogata, K. Modern control Engineering, Prentice Hall, 1997 4. Dorf, R. C. and Bishop, R. H. Modern Control Systems, Addisson Wesley, 1998

5. ،אאא،אא،אאא١٩٩١

א ٢٦٣ א

א ١אאא אאK ١

١ J١ K ١١ J٢ אאאK ١١ J٣ אאאK ٥١ J٣ J١אאאאK ٥١ J٣ J٢אאאאK ١١

K ٢٠٢אא אאK٢٣

٢ J١ אאK ٢٤٢ J٢ אאK ٢٤٢ J٣ אאK ٢٥٢ J٤ אאK ٢٦٢ J٥ אאאK ٢٧٢ J٦ אאאK ٢٨٢ J٧ אאאאK ٣٠

K ٣٢٣אאא

אאK ٣٣٣ J١ אאאK ٣٤٣ J٢ אאאאK ٣٤٣ J٣ אאאKאK ٣٥٣ J٤ אאאאK ٤٠

K ٤٢

א ٢٦٣ א

٤אא אאK ٤٣

٤ J١ אאאK ٤٤٤ J٢ אאאK ٤٥٤ J٢ J١אאאאאאK ٤٧٤ J٢ J٢אאאאאאK ٤٨٤ J٢ J٣אאאאאאאK ٤٩

K ٥٠٥אא

אאK ٥٢ ٥ J١ K ٥٣٥ J٢ אאK ٥٣٥ J٣ אאאFE ٥٥٥ J٤ אאK ٦٦٥ J٥ אאאאK ٦٧

٧٩ א٧٠ אא٧٤

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