Solving Engineering System Dynamics Problems With MATLAB

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x 10– 3

Real Axis

x 10– 3 Nyquist Diagram

Imag

inar

y A

xis

"

2 # % # 3!4


,'"-*'','

"-*','"-

.5

,*'%%

')'(')(''

7D+4

4"=*"4

.40

:F .40

7<,,"+"

+((,-F

"F3%1'

+"5"474

+"C+#*1'7

+":"7:

+":"G7'0,':

+"F-7F

'0,'2:"

+""+"G7

.+!

7<,,-+"

+((,-:

":3

+"5"474

+"C+#*'G7

+":"7:

+":"G7'0,':

+"F-7F

F1'/F

+"F-+"G7F

.+!

MATLAB TUTORIAL 191

H,+++":H=IHG+""(

,+

,

7G,""

7D+

,*'%%

')'(')(''

7D+4

4"=*"4

.40

FJJ +"40

G

'0,'

<++(

=++"(

>>>>>>>>>>>>>>>>>>>>>>>>>>

?/?/E

=*"4&'

=-G,"-6"C")D))

50

0

– 50

– 100– 90

– 135

– 180

– 225

– 27010

– 210

– 110

010

1

Frequency (rad/sec)

Bode Diagmram

Mag

nitu

de (

dB)

Pha

se (

deg)

""&'C+#*1'E)':"')'1:"G%)EE

F-%1')'E"+"G)EE


""&'C+#*'G):"')@&1:"G%')1&

F-%)1@F-+"GE&)&'

H,+++":H=IHG+""(

=++"(

>>>>>>>>>>>>>>>>>>>>>>>>>

?/?/E

=*""&'

)'''

F

E)1@

J

E

J

)'&

G

)E'E

=-G,"-6"C")D)

50

0

– 50

– 100– 90

– 135

– 180

– 225

– 27010

– 210

– 110

010

1

Frequency (rad/sec)

Bode Diagmram

Mag

nitu

de

(dB

)P

hase

(d

eg)

'

,'"" 67-

+=

!

" 8

9 # % # 3&

MATLAB TUTORIAL 193

*''')

'0"-

*'0)0

''),'")

. /

7D+

'

7D+4

4"=*"4

=

=.!40

.=

",<,>,+#*+

:F .=

: $:

:

:G'0,':

+((,-:

":3')'

+"G6"-7

.+!

<++(

=++"(

/

>>>>>>>>>>>>>>>>>>>>>>>>

?/?/

=*"4&'


=++"(

&'/''

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

?/?//''

:

)E

:G

')@@

)@

G6"-')1'E

=-G,"-6"C")D)1)

40

20

– 20

– 40

– 60100

135

90

45

0

0

– 4510

– 110

010

110

2

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de (

dB)

Pha

se (

deg)

*''

'-

!

8 3

9 % # ! # 3

. /

7K">++

''

'(')('

,"

,%

+"

MATLAB TUTORIAL 195

",K">++

$,.,L

*,.,K">+"

$)'&)'K">+"

$)')'M

10

8

6

4

2

00 2 4 6 8 10

t Sec

Unit-ramp responseU

nit-

ram

p in

put a

nd o

utpu

t

Unit-ramp input

Output

!

6 % !

$ # 3: # 378

9 # & # 33 # 3: # 37

',

00+,0' -

.;,

''E'

E'

'(')'('

+

,

+"

",K">+

$,.,L

*,.,M


1.6

1.4

1.2

0.8

0.6

0.4

0.2

0

1

0 5 10 15 20t Sec

Unit-step response

Out

put y

(t)

78+"+'7@'7"",,

6-",3')/

6-",3')@/

+"N"%0')'

+"N"

')&''

*$ $

!N"%0')'

!N"

)''

$N+-*$%

$N+-

')@

''6-",')@1;3)'%

,"N"%0')'

,"N"

E)'''

<'''''

'',

8

9 % !

= # 3!

# 3!

# & #

*',

>7-$

MATLAB TUTORIAL 197

7K">++>,"

''

1

'(')('

,"

,%

+"

",K">++

$,.,L

*,.,M

$)'E)K">+"

$E)&)M

10

8

6

4

2

00 2 4 6 8 10

t Sec

Out

put

Unit-ramp input

Output

Unit-ramp response

!"

79$%')

''

1

'(')(

$%')0

,"

,%

+"

",8"$%')

$,.,L


1.2

1

0.8

0.6

0.4

0.2

0

– 0.20 2 4 6 8 10 12

t Sec

Response to input r1 = exp(– 0.7t)In

put a

nd O

utpu

t

Input r1 = exp (– 0.7 )t

Output

+ +

!" #!$

%&

!"#$

"#$

%%$

&$

'(! !$

)*(

(+''( &

,-

( &

MATLAB TUTORIAL 199

14

12

10

8

6

4

2

00 2 4 6 8 10

t Sec

Response to input ( ) = 3 + r t t

Out

put

c(

) a

nd in

put r

()

= 3

+

tt

t

!

& '

( ) *

( ) + +

, $ !

$ ! $ - $ + $ .

()

.

,

++ + + +

+

+ + + +.

/

,

/%&

!"#$

" /#$

,%% $

, % % $

, %%$

,.%%$

,%%/$

,",, ,,.,#$


*- ' !,$

*"//#$('*

)*(

(+- '0()

+('

!)(*('

8

6

4

2

0

– 2

– 4

– 6

– 8– 10 – 5 0 5

Real axis

Root-locus plot of G(s) H(s)

Imag

inar

y ax

is

"#

$0 ' '

( +

,

$ 1 $ 2

"* 3 !

%&

"#$

*'

'

/.&

/.

.

%&

!"#$

MATLAB TUTORIAL 201

"#$

*- ' !$

(''1 *

)*(

(+2- '0(

5

0

– 5– 6 – 4 – 2 0

Root Locus

Real Axis

Imag

Axi

s

$"%

& 0

*

( #! +

4

$ 5! $ 1 $ 4

'63 (

3

"#$

*'

'

.&

.

/

363

!"#$

"#$

1 (' !

*" #$('*$(''1 *

)*(

(41 ('0(


2

1

0

1.5

0.5

– 0.5

– 1

– 1.5

– 2– 2– 3 – 1 0 1 2 3

Real Axis

Nyquist Diagram

Ima

gin

ary

Axi

s

&'()

56**72'(6*()660-(*-!06-*((-

(6-'2''!(' '

* 0

*

( #, $ !

$ 4 $ .

& 63 ( ,#4"45"455

(

,

++ +

/

,

++ +

%&

!"#$

"/#$

'%%$

"*(!'#1 (' !'$

"* (! '#1 ('8 !'$

"*(!'#1 ('8 !'$

*(!* (! *(!

*"#$('*

)*(

(41 ('()*!'

+('

!)(*('

9

9

9

MATLAB TUTORIAL 203

3

2

1

0

– 1

– 2

– 3– 3 – 2 – 1 0 1 2 3

Real Axis

Nyquist diagram

Imag

inar

y ax

is

*'()+

*0*

( #1

$ 4 $ !

& 63

(

( *0

(+ +

. + +

%63 ( 7(

!"#$

".#$

! "#$

".#$

1 (' !

6

: **6

1 (' !

*"#$('*

)*(

/(

(


5

4

3

2

1

0

– 1

– 2

– 3

– 4

– 5– 5 0 5

Real Axis

Nyquist Diagram

Imag

inar

y A

xis

'()#

,8 '89!+:"

#7+;< ! & *

)1s5.0(s

7

+Gc(s)+

–

,-(

;*!;()< /0*6-'2('- &< 6

' !06)-*( ('062' (')(= >

? ?;*6-'2 & < 7?62

*'0*0 -(6*7*'6)0(-(-06)(=2*'0*0 -(

6'(*-'2 &< (')(=

/? ???

56-!'*! '-*( ?-!'*56'(!'0*!0

-!'*('

(,

++

MATLAB TUTORIAL 205

s = –2 + j2 3

jωj2

j1.5

j1

j0.5

–j0.510–1–2–3–4

90º 120º

σ

s = –2 + j 3

jω

–j0.510–1–2–3–4

90º

σ

j2

j1.5

j1

j0.530º

. " ## # - "/

#" 0 1 # 2

,#

07'-6@*06-!'* 6606-!'*

! '-.(**6=6')0?';()< /

A- (',

.

$

$

56)(,('(0*!6-((

. <

,

= − +

++ +

* ,

.

. <

= − +

+

* (

.

++

562*'0*0 -(06-!'''!(')(=

(

+

.

++

.

+

. +

56-'2*'0*0 -(06*()(''!('

.

. + +

56-!'''!'-'2*'0*0 -(('

. + +


BCB5DBE*)*!6 (2'*''- *='06*()(-!2

'''!'(')(=756 (2'*''- *='*'67(;()< /

%&

!" .#$

" .#$

!-" #$

-" . #$

%%$

' !$

' !--$

,-

'

*()(''!

.:!'''!

)*(

(F(2,+'''0*()(,'!:!','!

1.5

0.5

1

00 10.5 1.5 2 2.5 3 3.5 4 4.5 5

Original system

Compensated system

Out

puts

t Sec

Unit-Step Responses of Original System and Compensated System

,

38 '89!+2

#74$<4=

)s5.0(

12Gc(s)

+

– Lead compensator

3

MATLAB TUTORIAL 207

;*'(*-'2&<6)-*( (067

2'6*()((')(=?? ?56*0*6)0(-(-

(')(=

/????

A-6-!'*! '-*( ?

G'--!'*060*!

(,

++

-6'6@*06-!'***(66')

0?606-!'*! '-';()<

jωj2

j1

–j1

10–1–2

90º

σ–3

34 "/ # 0

351 #

56*0* (,

++

76*9! '(0*!6!)( -(('

<

,

= − +

++

* 9

<

= − +

++

56*0*6-!'*-!'

(

++

5600*7**'0*0 -(('

(

.

++


!

"

"

++ + +

!#$%$!#$

&

'(')'*

!+, , -.

!+$ , -.

!+, , " -.

!+$ " -.

!,/,,/$,.

$! 00.

! 00.

0$01200012

134 5 3 6 42

7 142

18 2

70,99016 2

7$0$"9013 2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6 7 8 9 10

Uncompensated system

Compensated systemOut

puts

t Sec

Unit-Step Responses of Uncompensated and Compensated Systems

MATLAB TUTORIAL 209

'(')'*

!+, , , $-.

!+$ , $ ,-.

!+, , , " -.

!+$ " ,-.

!,/,,/$:.

$! 00.

! 00.

00$0120001#2

135 5 3 6 42

7 142

1; 8 2

1201 201#201 2

15

10

5

00 5 10 15

Inpu

t and

Out

puts

Uncompensated SystemCompensated System

t Sec

Unit-Ramp Responses of Uncompensated and Compensated Systems

!"#$#

%#&'(&()&## * !#&#"*

" #*$# *#+(#+#

#&, #*# * # # #

&-# *# #*# **#

# *&


8s4.3s

12 ++s

)1bs)(1as(K +++

–

++

D(s)

! $ $.

+ +

!

" 9 $ $

. + + + + +

! " 9

. . . .+ + + + + +

;<

=0ξ!,:ω!">

!#$,

<

?$,?@,:@"?"!?$,?"?$A

!?$"?:A?$A,

< <$

?"?.?9?.?.?.!,

"?.!$"

9?.?.!:A

.!$A,

!,,AA:0?!,

B;C

!

+ $-.

+ + +!

+ +$A,,,AAA , $

!+ +$,A ":9 $:,

MATLAB TUTORIAL 211

DB;C0

! $" :A $A,

+ + +! $, " $A

+ + +

0 E0

→ ∞

!,

→

!,

→

$, " $A

+ + +$

!,

(')'*

,

7F <G=0

*# #*# *

!+, , $ ,-.

!+$ $" :A $A,-.

!,/,,$/:.

+$0/$0-! 00.

0$

15 34 C ; 2

7 142

18 C ; 2

14

12

10

8

6

4

2

0

– 2

– 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t Sec

Response to Unit-Step Disturbance Inputx 10– 3

Out

put t

o D

istu

rban

ce In

put


0

!

$,A ":9 $:,

$" :A $A,

+ ++ + +

!

$,A "9 $A,

$" :A $A,

+ ++ + +

(')'*

, 7

FH$F0<

*# # *

!+, $,A"9 $A,-.

!+$ $" :A $A,-.

!,/,,$/:.

+0/0-! 00.

0

15 34 5; 2

7 142

18 5; 2

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Response to Unit-Step Reference Input

Out

put t

o R

efer

ence

Inpu

t

t Sec

% "$#%#&'(&(01#

#.##" *#"&

MATLAB TUTORIAL 213

)1s5.1s)(5s)(3s(s

)5s2s(K2

2

++++++

–

+R(s) C(s)

I

!

:

: $: $

.

+ ++ + + +

!

: "

:

: 9 , $:

.

+ ++ + + +

'(')'*

$

H(')'*

!+, , , $ :-.

!+$ : 9 , $: ,-.

0

2!+#9 #: :-.72.71<2

15)B2

5

4

3

2

1

0

– 1

– 2

– 3

– 4

– 5– 8 – 6 – 4 – 2 0 2

Real Axis

Root Locus

Ima

g A

xis

$0 '

ω7


ω70 !ω<

:?:"?9?,?$:?.??:!,

ω:?:ω"?9ω?,?.ω?$:?.ω?:.!,

+:ω"#,?.ω?:.-?+ω:#9ω?$:?.ω-!,< <E0 0

:ω"#,?Iω?:.!, $

ω:#9ω?$:?.ω !,

<

ω!, ω"#9ω?$:?.!,

.!" 9 $:

− ω + ω −"

4 <"<$0

:ω"#+,?J#ω"?9ω#$:-ω#:ω"?F,ω#F:!,

,:ωA#ω"?:F:ω#F:!,

< (')'*

H(')'*

!+,: , # , :F: , #F:-.

-* *#3

!

#"F9A?$$:F#

#"F9A#$$:F#

"F9A?$$:F#

"F9A#$$:F#

,9$::

#,9$::

ω7ω!,9$::

.!" ,9$:: 9 ,9$:: $:

− + × −!$:9"ω!,9$::

.0 .

$:9"KIK,

%"$#%#&'(&(+3

#"

##.##"&

MATLAB TUTORIAL 215

K5s

3s

++

)3s(s

32 +

R(s) C(s)+

–

!I

:

++

+! "

9 $:

.

++ +

'(')'*

H

!+, , , $ -.

!+$ 9 $: , ,-.

0

2!+#A " #: :-.72.71<2

15)B2

5

4

3

2

1

0

– 1

– 2

– 3

– 4

– 5– 6 – 4 – 2 0 2 4

Real Axis

Root Locus

Imag

Axi

s

0I

7

0<

<

"?9?$:?.?.!,


4 !ω<0ω"?9ω?$:ω?.ω?.!,

ω"#$:ω?.?ω#9ω?.!,

< <E0 0

ω"#$:ω?.!, $

ω#9ω?.!,

<

ω!, #9ω?.!,

4 .<$0

ω"#$:ω?@9

ω!,

ω"?ω!,

ω!,ω!%4ω!<ω70 ω!<

.

.!9

ω!

9

@!"

0 .

"K.K,

%"$#%#&'(&((3

#"

#2*.* ##ζ#

#)&4

#

*# *2*#&

)5s)(3s(s

K

+++

–

(')'*

!+, , , $-.

!+$ 9 $: ,-.

0

2!+#A " #: :-.72.71<2

MATLAB TUTORIAL 217

15)B2

5

4

3

2

1

0

– 1

– 2

– 3

– 4

– 5– 6 – 4 – 2 0 2 4

Real Axis

Root Locus

Imag

Axi

s

Dζ ,LζL$θω7

jω

θ

ωn

–ζωnσ

Constant ζ line

0

θ!

ζωω

!ζ

'ζ!,A !#,F:?

,LL∞ . ζ ,A!#,F:?

<

!#,F:? $

??:?.!,


!" #$ ! %$ &" ' ($)'#%

* *

*)

*

+++

*

,"& *

&

+*&*

-)"' # $'& ! '&*#)," !

,")" )) ""

,")#.#$$#) %"&##!/0,102$!

3 45

++

**

6)")#.#$$# #)

++*

," .$$&"%&) &"&##.!/0,102$! ," #! .$$) '"!

3 45

3 45

$

!

#78.-$9$:

;# #7-):

MATLAB TUTORIAL 219

%# #7< $ :

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 5 10 15 20 25 30 35 40

Time (sec)

Step Response

Am

plit

ude

t sec

!"##$"

%&

'(

) * +,)-. * (,

/#$%%

01(!0-2!(-

1)2)(0.6ss(s

1

++R(s) C(s)+

–

=) # !)$ &"&

'β

+β +

+

+β

β>

?&! β

''

+ +

@ "! " )'#)%)


6) %

# →

''

# →

+β +

'

# →

'

# →

+ +

="")$ % &" %. $& ) .

=) $#"2 ! &

'ω

ω ω + ω +

," ! ) ' $" . !#) '&'ω "!*"$#"$" !& A")""" "% #

40

20

0

– 20

– 40– 90

– 135

– 180

– 225

– 27010

– 110

010

1

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de (

dB)

Pha

se (

deg)

!" "

!

," & # !)$ &"$" ) '&"2 ! "& ##AA"$)&$" !)$ &"&) .&"$" ) '-)"& )%)$! $" !&A B"! )'& )%&")$ % #) "' # =)"")& )%ωB-)")& )%& #"! )'& )%"&) "$" ) ' % ##0# A"!'$" ! ## ) )) &"# ! !# )%" # ! )$ ," $" ! A ," $" !# & " )$ $.#$ && )A ω B6))""! )'& )% BC!" ! ) '2"# !)$ !'") % ")"") 2

MATLAB TUTORIAL 221

,"& #!

β

β,"")& )%ωβ,")$"$#&"# !)$ .

β B

6)" && )&"# !)$ !'%

'

++

+

+

-)"! ) #) # β " '

β

,"&")$ '

'

++

,"$.#$ && )&")$ %"&

''

++ + +

," ! $" . !#) '&'ω'ω "!*&

40

20

0

– 20

– 40– 90

– 135

– 180

– 225

– 27010

– 110

010

1

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de(d

B)

Pha

se(d

eg)

100

50

0

– 50

– 100


,"$" !&")$ % A" ' # ,"! ! 2 ))$ #," )'#)%) ," ")$. % &" "" % "# ' #%

=" .$$ . $$&")$ % "! # )$ %,")#.#$ && )&").$ )$ % !'%

3

4

*

++ + + +

3

4

+ + +

$)'#%

0/0,102$! " .$ . $$&")$. )$ %!'#," #! .$$) ' . $$) ' "!* *$)'#%

D 8.$$

3 45

3 45

)3 45

)3 45

EE*5

354$ 5

354$ ))5

$#7:7:

!

#78.-$9$&F$ 8)$ -%:

;# #7-):

%# #7< $ :

;7F$ -%:

;78)$ -%:

6"

3 45

3 45

)3 45

)345

EE5

374$ 5

374$ ))5

$#7:7:7:

MATLAB TUTORIAL 223

1.4

1.2

0.8

0.6

0.4

0.2

0

1

0 5 10 15 20 25 30 35

Out

puts

Unit-Step Responses of Compensated and Uncompensated Systems

Compensated System

Uncompensated System

t sec

!

#78.9 $9$&F$ 8)$ -%:

;# #7-):

%# #7< $ :

;*7F$ -%:

;*78)$ -%:

20

18

16

14

12

10

8

6

4

2

00 2 4 6 8 10 12 14 16 18 20

Uncompensated System

Compensated System

t Sec

Out

put

s

Unit-Ramp Responses of Compensated and Uncompensated Systems


#"##$#$"#&8%&

'

) * ,) * 0,

/'),$%(-1(!

0-2!(-

=) # !.# )$ &"&

'

+ +

β β+ +

," $.#$ & & ) & " )$ % ''

# →

'" )'#)%) " '

%

# →

''

# →

'

+ +

,"&

0/0,102$! $#"2 ! !'# " ! "!

3 45

3 45

40

20

0

– 20

– 40

– 90

– 135

– 180

– 225

– 27010

– 1 100

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de (

dB)

Pha

se (

deg)

– 60

– 80

101

102

##$%&''#(

MATLAB TUTORIAL 225

!"$" !&" )$ %A")""" "% #,! # !.# )$ )" ! )'& )%"$" . !#) '&'ω"$" )'& )%ω B=) #)"! )'& )% B )"" "$" .# !# ω B A0!## !.# )$ ) $'" &$" .# !#

=) &")& )&"$" .# !$&"# !.# )$ F"!")& )%ωB)$!"(&"$" .# !$&")$ ) #"! )'& )% ω B ")& )%ωBβ"' # &β,"' # &β) &") &"# $&")$

," ; $" .# !#φ!'%+αBβ" '!'

φ

β −β +

β)$φ*+A= A$" !) #)β6)

β,"")& )%ωBβ ω," && )&"$" .# !$&"# !.# )$

++

+ +

,"! )'& )%ω B G'G& 2,".&&"# !.# )$ ) 2 ω B""! )'& )% " $# !"#&#$2B) $ !" !""$2 B,")&"# ".2# 2#!'")& ),")& )&"# $ ω* B ω* B6)" && )&"# $&"# !.# )$ !'%

*

*

++

+ +

2%)!" && )&"# ! # $&")$ ) &" && )'&"# !.# )$ !

'*

*

++

++

+ ++ +

,"2 ! &"# !.# )$ ' %"&##!/0,102$! ," #!$#"!&

D

3 45

3 * 45


0

– 5

– 10

– 15

– 2090

45

0

– 45

– 9010

– 310

– 210

– 110

010

110

2

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de (

dB)

Pha

se (

deg)

#)

,"$.#$ && )&")$ %

'' *

*

+ ++ +

+ +

*

+

** +*

+ ++ + + +

," ! $" . !# ) ' & " ! $.#$ & & )'' " "2 ! &!," ! !&##!/0,102$! H" " $#% # !")') &##E

3 * 45

&3 45

)'&

**+*

3 + 45

3 **+*45

MATLAB TUTORIAL 227

100

50

0

– 50

– 100– 45

– 90

– 135

– 180

– 225

– 27010

– 310

– 210

– 110

010

110

2

Frequency (rad/deg)

Bode Diagram

Pha

se (

deg)

Mag

nitu

de (

dB)

#*

!" "$" !! ! )'#).%) ## &

+

H '' *

*

+ ++ +

+ +

3

4

' '

' ' +

*

+

** +* +

+ ++ + + + +

," .$$ %"&##!/0,102$! " .$$) '"!


!""#$ %&

1.2

1

0.8

0.6

0.4

0.2

00 5 10 20 25 30 35 40

Am

litud

e

1.4

15Time (sec)

Step Response

'()""#(%"*)*%(#""+,-'.-/'()""#012"3"(""#01'()"23"*)("+4566

"&

7!)!""#($ %&

8)*'2&

%)*-&

MATLAB TUTORIAL 229

20

18

16

14

12

10

8

6

4

2

00 2 4 6 8 10 12 14 16 18 20

Time (sec)

Unit-Ramp Response of the Designed System

Am

plitu

de

! " " #$

#%&

!$'

+ +

'()))#92% )0)())""# #"(

")"

.(#92%2"""()"#:6;<(),)"#*ω)(#"

∠(ω:6;

'(/"))("+4566=#",-'.-/")


20

0

– 20

– 40

– 90

– 135

– 180

– 225

– 27010

– 1

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de (

dB)

Pha

se (

deg)

– 60

100

101

4"4566=(9())"#;)"22)(#92%ω=)0'()"#(ω)(#92%(:/'() ()#%

" /

" 6

) *)'+,',-.

!$ /#&0 10 /

23

'

K

s(0.1s 1)(s 1)+ +Gc(s)+

–

>"()2")"

α

+α +

+

+α

2 3):+()3

→

α

+α +

'

+ + α

MATLAB TUTORIAL 231

. )#?α ∧'(

∧

'(/"))"#

+ +

6

+ +

"*)*%(#""+,-'.-/")'(/"))("+4566

*"

50

0

– 50

– 100– 90

– 135

– 180

– 225

– 27010

– 1 100

Frequency (rad/sec)

Bode Diagram

Mag

nitu

de (

dB)

Pha

se (

deg)

101

102

4"4566(()))));)@/23%4")())"#;2

φ ;:;1;;'()8()); 2

φ

− α+ α

φ ;

α"*)).2(""α

'"(2"#92ω0)ω0α"#()2")"+"()()8()))φ "22)("2)"#(+"2"


#92"ω0 α '()""#("#2)"()23)

ω0 α "(2""#(10α1(3*%

(

( ω =α

+ ω+ ωα

α

2

α

@=@/

'()"#A(ωA:=@/+(2(2"""ω@)0'(#"+2()(+)2""3#92%ω

α ω B@@

α ω

α

@

=6

" @

=6

++

) ∧

α

C2

@

=6

++

=@

=6@

++

'(""")##2"

=@

=6@

++

+ +

6

6=

=6 =6

++ + +

'(2""")##2"

4

5

6

6=

=6 =6 =

++ + + +

'(#""+,-'.-/")"2("23)("+4566

6=

=6 =6 =

MATLAB TUTORIAL 233

7 !""#>") %&

8)* 2&

%)*D&

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2 2.5 3 3.5

Am

plitu

de

Step Response

t sec Time (sec)

'(#""+,-'.-/")"2()"23)("+4566

6=

=6 =6 =

"

7!)!""#>") %&

8)* 2&

%)*7!)E) %D&


5

4

4.5

3.5

3

2.5

2

1.5

1

0.5

00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t Sec

Unit-Ramp Response of Compensated System

Uni

t-R

amp

Inpu

t and

Sys

tem

Out

put c

(t)

!6 * '7'

6

8

8

8

8

= @ @ 6@

− − − −

6

8

8

8

8

1

6

8

8

8

8

9.'(#""+,-'.-/")%(""#(3%'("23("+4566@

:=:@:@:6@

4

:

4:

MATLAB TUTORIAL 235

8)* 2&

%)*D&

'("("+4566@

350

300

250

200

150

100

50

00 10 20 30 40 50 60 70 80

Time (sec)

Step ResponseA

mp

litud

e

t sec

!

)%+(4:)+"*)(")("+4566@

20

15

10

5

0

– 50 10 20 30 40 50 60 70

Time (sec)

Impulse Response

Am

plitu

de

!

"6 * '

4

5

,

,# " -

" # " ,; " -


-,-'.-/")""*))))2)""#(%3*"+

FF6

FF6=

FF

')##2"

61

G61G16=1

FF0040:

::6=:

4

6

:

4"(,-'.-/"+"*)(#""+))29)"

6

8

8

8

6=

− − −

6

8

8

8

1

6

6

8

8

8

1

#$5 *)'+,'/%

'

)8s()6s()4s()1s(

)5s()3s(10

++++++R(s) C(s)+

–

#$

MATLAB TUTORIAL 237

H <

()3))*#"&

&

IJ:6::::=:@

'&

'#*)2J:

))=&

>""2)"2)#")"&

4444:4

#4

<()3))*#")"&

#

#4

4#44

,")#"&

&

IJ:6::::=:@

&

#*)2J:

))=&

>""2)"2)#"&

4444:4

,")#"&

4:2)"4444:4")&

>""

)

()3))*#"

)

K"0"0)

161

111=1@

)

K"0"0)116

1=1G1@@16


!"

#$ %

#$

&'"'( )

#*$#*$

#*$#*$#*$#*$

MATLAB TUTORIAL 239

#$

&'"'( )

#*$#*$

#*$#*$#+**$

%


!

25ss8s10s7s

502345 +++++

R(s) C(s)

","( -)

" #$%

./"" 0

,

1 23

#,4$

14442 #,4$3

",#$

"#$

",#$

"#$

& '

,

5, )

+*+*+*+**

MATLAB TUTORIAL 241

(#$) *& + & & *&

,

8s9s3s7s

7s5234 ++++

+R(s) C(s)

#$

2345

234

s7s5s9s

8s7s12s3s

+++++++

R(s) C(s)

#$

#$,12

,

12

#,4$

5, )

*

+*+*+**

14442 #,4$3

",#$

"#$


",#$

"#$

,123

123

#,4$

5, )

+*+*+**

+*+*+*+

14442 #,4$3

",#$3

"#$

",#$

"#$

+ & &#$

!

,

-

-

-

. ! .

/ , .

. , 0

!

,

-

-

-

*

. .

!

/ .

&

MATLAB TUTORIAL 243

! . .

. . !

!

,

-

-

-

5, 6 ( !7

#$112

-

− − −

−

8 #$

− + − +

−

99"#$%

997

991323

99#13*3*23

9913323

99: ##$:

1'#+*:*$4 '#+*:*$2

1:'#+*:+*:*$*'#*$4:'#+*:+*:*$2

++ *& '

-

-

2

!

,

− − −

!

,

-

-

3!

&

24! ,5 !

,

-

-

+ & 67 &#$

− − −

1 2

5, #$7

#$#1$1 2

+ − +


1 2

1# $# $ 2 + + +

+ − +

1 2 + +1 2

+ +

+

+ +

1 2

1 2

99"#$%

991323

991323

99123

993

991,42 #444$

,

5, !

& 6728679:67)

-

. ; .

. . ! .

. . . !

/ 0 < /

− −

-3

.

/

;

,

21!0/2-

133323

133323

123

3

"#444$

----

-

-

-

-

MATLAB TUTORIAL 245

&

-

-

-

-

----

+

&

,,

14442 #,4$3

2 6&)+7

5, )

+*+*+**

+*+*+*+

& + +

&#$

-

< !

, .

0 = ,

− − − −

*

!

,

&6 7

24,<>!,5-?-6.72

.

.

.

"#$%

991332

991332

9912


99

991,(4(2 #))))$

,(

*:

(

992 6&)+7

5, )

+*

++*

99"#($

"

: ;&#$

#$

#$ +

)9s)(7s)(5s)(2s(

)6s)(4s(10

++++++

R(s) C(s)+

–

;#$%

#$. !0

.#$0

@A#124124$

.#$0

!<#44$

1,42#4.B0$3

. 0

MATLAB TUTORIAL 247

14442#,4$

",#$3

. !" 0

%B "#$

%B ",#$

%B "#$

.#$%0

.#$0

="<#124124$

.#$0

!<#44$

1,42 #4.B0$3

. 0

14442 #,4$

.%0

144422#4444.0$

",")

#$ !

>

#$

&'"'( )

#*$#*$

#*$#*$#*$#*$

>

#$

&'"'( )

#*$#*$

#*$#*$#+**$

>


!""# $

% %&

% % % %

!""# $

%& %

%& % '%%&

() )

MATLAB TUTORIAL 249

&

&

&

&&&

! "# $

%

& ' (

( ) *

( ( +

− − −

%

, -( * *.%/ %"(# ,

(

(

(


**+,,-,

**+,,-,

**+-,

**,

**0...

%%%

%

%

%

%

%

%

%%%

(//01

**0

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (sec)

Step Response

Am

plitu

de

MATLAB TUTORIAL 251

*

'

%

%

( *

+( 1

− −

%

%

%* *

( *

*

'

*

'

* (

( *

%

%

- *! '.

$

2 311 # / 1/

** 2

**+,-,

**3+,-,

**(+,-,

**4+,-,

**1...

0

– 50

– 1000

– 45– 90

– 135

– 100

0

100

180

0

– 18010

010

110

210

310

010

110

210

3

From : U(1) From : U(2)

Bode Diagram

To :

Y(2

)To

: Y

(2)

To :

Y(1

)To

: Y

(1)

Ph

ase

(deg

) ; M

agn

itu

de

(dB

)

Frequency (rad/sec)

34

*

'

%

%

( *

+( 1

−

%

%

%(

+(

, -* (.

%

%

5 -(.

$

5)6 #/ 1 #//. 768/

) 16/#) 168/... 68/....


**9:2;3#

**+,-,

**3+,-,

**(+-,

**4+-,

**68/.3.(.4

**#1

**<768/=

0.8

0.6

0.4

0.2

0

– 0.2

– 0.4

– 0.6

– 0.8

– 1 – 0.5 0 0.5 1Real Axis

Nyquist Diagram

Imag

inar

y A

xis

!

*

'

%

%

* *

1 (

− −

%

%

%* *

* (

*

'

*

'

* (

( *

%

%

%( (

( (

$

6

*

*

"7 #

"7 #

ωω

! '

*

"7 #

"7 #

ωω

! *

'

"7 #

"7 #

ωω

! '

'

"7 #

"7 #

ωω

34 * ' 8

6 6$

2/11/ 16/#:2;3) 168/

...

**9:2;3#

**+,-,

MATLAB TUTORIAL 253

**+,-,

**+,-,

**+,-,

**68/...

0.5

– 0.5

0

2

0

– 2

– 1 0 1 2 – 1 0 1

From : U(1) From : U(2)Nyquist Diagram

Imag

inar

y A

xis

To :

Y(2

)To

: Y

(1)

Real Axis

!

"9 ! !

*

'

%

%

* *$&

' (

− −

%

%

5 *$&

(

, -* (.

%

%

$

**2:

**+,-,

**+,-,

**+-,

**+-,

**+.%.-... ,

**.

**#1

**<>0=

**? <5)=

**6 <@//=


0.6

0.5

0.4

0.3

0.2

0.1

0

– 0.1

– 0.2

– 0.30 2 4 6 8 10 12

t Sec

Unit-Step response

Out

put

"#$%&

**2:

**+,-,

**+,-,

**+-,

**+-,

**97A #1 1//8/

** +B. ,-,

**+,-,

**+-,

**+-,

**+8.%.-... ,

**%+-C%=,.%...<=

**#1

**<>0 =

**? <5)=

**6 <@// 1/0 /=

**?..<@//=

MATLAB TUTORIAL 255

20

18

16

14

12

10

8

6

4

2

00 5 10 15 20

t Sec

Unit-ramp response

Out

put

and

unit-

ram

p in

put

Output

"#$%&

**2:

**+,-,

**+,-,

**+-,

**+-,

**/...

1.5

0.5

1

0

– 0.5

– 10 2 4 6 8 10 12

Time (sec)

Impulse Response

Am

plitu

de

"#$%&

9 6 $

*

'

%

%

* *

1 (

− −

%

%

%* *

* (

*

'


*

'

* (

( *

%

%

%( (

( (

*

'

**2

**+.,-,

**+,-,

**+,-,

**+,-,

**...

From : U(1) From : U(2)

Bode Diagram

To :

Y(2

)To

: Y

(1)

Am

plit

ud

e

0.4

0.2

– 0.2

– 0.4

0

2

1

0

1.5

0.5

0 2 4 6 8 10 12 0 2 4 6 8 10 12

Time (sec)

9

$

*

'

+

%

%

%

,

& '& &

* ( (

( * (

− − −

*

'

+

%

%

%

5

*

(

(

, -( '& &.

*

'

+

%

%

%

5 -(.

**2

**+,,-,

**+,,-,

**+-,

**+-,

MATLAB TUTORIAL 257

**+.%.-... ,

**.

**#1

**<>0=

**? <5)=

**6 <@// =

1.4

1.2

0.8

0.6

0.4

0.2

0

1

0 1 2 3 4 5 6 7 8 9 10t Sec

Unit-response

Out

put

y(t

)

#$%&

: ;

& '& & (

* ( ( (

( * ( (

( '& & (

− − −

(

(

(

( '& & (

B. ,-

*

( ,

(

( (

+( '& & (-+ (-

**2

**+,,-,

**3+,,-,

**(+-,

**4+-,

**+B. , -,

**+,-,


!"#$ !

%#&'(

)#&*!+#!++

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00 0.5 1 1.5 2 2.5 3 3.5 54.54

t Sec

Unit-ramp response

Inpu

t and

out

put

! "#$%&'

(#$%&

,-

&..

.

MATLAB TUTORIAL 259

#!/&&0.&

#!

,

#!/0.

#!

,

#!/&&0.&

#!

.

1$2#&34& 3 2# 2 ) $ # (!#&&+! ($)& 3#&* ++(!#&

!

$)

* +

! "#$%&

, ) ")

#$%&'

(#$%&

,,.

#!/----0.-

#!

,

,,.

,.

#!/----0.-

#!

,

1$2#&34& 32#2 ) $ & 3#&#!(!#&


!" , . !).

− − − − −

* +

") !) #$%&'

52(!#&)#!& 3#&)62 ) $(#!&&#!&)%#$!"!2#!/(!!6

! "## #′′′′′!

(#$%&

77,7.7.7

&,

#!/&&0.&

#!

,

#!/----0.-

#!

,

8& 32#!/6&&.& ,#!2#!/6 #$# $#! ,2 ) $ ($) #(!#&#!& 3#&

% " ,#$%&'

/

0

1

− − − − −

−

*2+

(#$%& 3

797:7.7;7,-

&7-.

7,

MATLAB TUTORIAL 261

!!3

&!!3&

!!3

$+ !

!!3

!

.-<:.=:777<

7<.-7:.=:77

<77=7

&!!3&

&!

;,=9<..:

;,=97..:

,,.-<

!

7.-,.-<-,:

7.-,.-7-,:

.:=,<

& "" ,#$%&'

4

− −

..797..>6!2$#%#&3

>#(+#2!3#+ 6$#%

#!

.

.<,-.9;

.7,-.9;

'5" ,", !) " "! 6 ) ) 7!" !8 ) 98:, !#$%&'

− − −

4

0

* +


.=7.7-7:>6!2$#%

&.>6!2$#%

>6!2$#%

;

;&>## #" #($

$

$1&#(/;

?!3#+ 6$#

##$>## #" #($

$>$!?!3#+

#!

#!

7;;9.

7,:-<;;;

7,:-7;;;

52( ) $ #&# 2!+$& ##!#342# 2#% (!# ) $+ 6@+2"A+4BC#33#+ # #& 33#+ #+!" #&

(5" ,) " "! 6 ) )8 7! " !8 ) 98:, !#$%&'

/

0

− − −

* +

(#$%&

9 7, 7- 7:

& .

;

;&

$

$6&#(/;

?!3#+ 6$#

MATLAB TUTORIAL 263

##$

$

))

#!

;

.,

.

,7,7-7

.

,

.,

!

!!++ "$$

#!

$

. ,

. 9

, 7, 7: 7;

.

, .

.,

!


8! 3

#!

?!3#+ 6$#

#!

79.

7,---<--,9

7,---7--,9

)* +

)( < #$%&)")

8! ) !" , ,5'=>''

)) ", '

m

k c

x(t)

y(t)

)( ",,"+-./ *

52$#!+626D+!() ! 3!#

EFωE G ... .

.

ω ω − + ζ ω ω

52$#!+6Hω 3!#

EHωE G ..

.

ζω + ω EFωE

42@ω

Hωω

522# #!φ 3!#

φω#!7

,

. .

.

.

ω ζ ω

ω ζω − + ω ω

MATLAB TUTORIAL 265

526D+!()#

ωω

52!!"$! !# ! $#!+ 3!# 2#! $ &)

EH E

ω

.

. .

.

.

ζω + ω

ω ζω − + ω ω

# !2 D+#! I5J ( 4!# 64

*'?'?'?'?'?'?'+?(! )"

*3'3+?(".

" 63 @ A

;@673A.@@@B @6ABA'CA'D@@2'CA'C@B @6ABA'CAA?

)@673A @B @6AB'C72'C@B @6ABA'CA?

!

"@A

) @7;A

@EF DF GEA

@EH@F AHDEA

!

!@EF G'E7EF G'E7EF G'E7EF G'E7EF G'E7EF G0

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MATLAB TUTORIAL 269

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MATLAB TUTORIAL 285

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Solving Engineering System Dynamics Problems With MATLAB

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