lecture_6_7
TRANSCRIPT
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7/21/2019 Lecture_6_7
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Mechanical Engineering DepartmentAutomatic Control
Dr. Talal Mandourah
Lecture 6 & 7Chapter 3
State Space Models
!!!!!
!!!!!
tutDtxtCty
tutBtxtAtx
+=
+=
m
u!t
"!t
ukyybym =++
!!
!!
#
1
tytx
tytx
=
=let
um
ybkym
x
xx
1!1#
#1
+=
=
um
xm
bx
m
kx
xx
1#1#
#1
+=
=
1xy=
u
mx
x
m
b
m
kx
x
+
=
1$1$
#
1
#
1
DuCxy
BuAxx
+=+=
[ ] $%$1%1$
%1$
==
=
= DC
m
B
m
b
m
kA
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#
Mechanical Engineering DepartmentAutomatic Control
Dr. Talal Mandourah
State Space Models
kuubkybymy
uykuybym
Fma
+=++
=
=
!!
kbsms
kbs
sU
sYTF
sUkbssYkbsms
++
+==
+=++
#
#
!
!
!!!!
To get the State Space
m
kbm
bbb
m
kam
baWhere
ubububyayay
um
kum
bym
kym
by
=====
++=++
+=++
#1$#1
#1$#1
%%$%%...
m
"
u
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7/21/2019 Lecture_6_7
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Mechanical Engineering DepartmentAutomatic Control
Dr. Talal Mandourah
State Space Models
Matla Commands
'um()$ $ 1 $*Den()1 1+ ,6 16$*
)A%-%C%D*(t#ss!num%den
)num%den*(ss#t!A%-%C%D
E/ample
[ ]
=
+
=
3
#
1
3
#
1
3
#
1
$$1
1#$
#,
$
,#,,
1$$
$1$
x
x
x
y
u
x
x
x
x
x
x
A()$ 1 $0$ $ 10, #, ,*0
-()$0#,01#$*0
C()1 $ $*0
D()1$*0
)num%den*(ss#t!A%-%C%D
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7/21/2019 Lecture_6_7
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Mechanical Engineering DepartmentAutomatic Control
Dr. Talal Mandourah
State Space ModelsElectrical s"stems
L
C
2
ei eo
=
=++
o
i
eidtC
eidtC
Ridt
diL
1
1
!!11
!!11
!
sEsIsC
sEsIsC
RIsLsI
o
i
=
=++
1
1# ++
=RCsLCsE
E
i
o
State space
iooo eLCeLCeL
Re
11=++
1
#
1
xey
eu
ex
ex
o
i
o
o
==
=
==
u
LCx
x
L
R
LCx
x
+
=
1$
11$
#
1
#
1
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7/21/2019 Lecture_6_7
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Mechanical Engineering DepartmentAutomatic Control
Dr. Talal Mandourah
State Space Models
The mathematical model o an
automoile suspension s"stem is
uite complicated. A 4er" simpliied
4ersion o the suspension s"stem is
sho5n elo5 assuming /i is the input
t the s"stem and /o is the output% ind
the T o8i
m
m
9/o
/i
kxiixbkxooxboxm
or
xixokixoxboxm
+=++
=++
$!!
!!!! # sXikbssXokbsms +=++
kbsms
kbs
sXi
sXo
++
+=
#!
!
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7/21/2019 Lecture_6_7
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Mechanical Engineering DepartmentAutomatic Control
Dr. Talal Mandourah
State Space Models
:ui;1ind the Laplace in4erse o the ollo5ing