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Sliding Mode Control with Industrial Applications
Wu-Chung Su, Ph. D.Department of Electrical Engineering
National Chung Hsing University,Taiwan, R. O. C.
Department of Electrical and Computer Engineering
Rutgers, The State University of New Jersey, U. S. A.
PRINCETON/CENTRAL JERSEY SECTION OF IEEE CIRCUITS AND SYSTEMS
2008/11/17 1Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Prelude
A Control Engineer as an Archer
to drive the state to the target.
2008/11/17 2Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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The Old Oil-PeddlerOuyang Show(1007~1072A.D.)
http://baike.baidu.com/2008/11/17 3Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Sliding Mode: an illustration
to fill oil into a bottle through a funnel2008/11/17 4
Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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A funnel-like domain
System:
Target:
Funnel:
New target:
1 2
2 1 22 3x xx x x u=
= +
1 2( , ) (0, 0)x x =
2 1x x=
1 2 0s x x= + =
2x
1x
0s =
2008/11/17 5Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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A Physical ExampleCoulomb Friction
sgn( )
: applied force
a
a
dvm K v fdt
f
= +
http://www.physics4kids.com
sliding mode: 0 (if ).av f K = <
< = >
2008/11/17 6Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Contents (1/2)
Introduction to Sliding Mode Variable Structure Systems Invariance Condition (Matching Condition)
Variable Structure Systems (VSS) Design Sliding Surface Design Discontinuous Control
Discrete-Time Sliding Mode Boundary Layer Switching v.s. Continuous Control
2( ) . . ( )O T v s O T
2008/11/17 7Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Contents (2/2)
Minimum-Time Torque Control (SRM)Temperature Control (Plastic Extrusion P.)Rod-less Pneumatic Cylinder ServoWireless Network Power Control
Singular PerturbationsDistributed Parameter Systems
2008/11/17 8Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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A funnel-like domain (cont.)
System:
Target:
Funnel:
New target:
1 2
2 1 22 3 ( )
x x
x x x u f t
=
= + +1 2( , ) (0, 0)x x =
2 1x x=
1 2 0s x x= + =
2x
1x
0s =
disturbance
2008/11/17 9Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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A funnel-like domain (cont.)
System:
Target:
Funnel:
New target:
1 2
2 1 2 1 1 2 22 3 ( )
x x
x x x u x x
= = + + +
2x
1x
0s =
1 2( , ) (0, 0)x x =
2 1x x=
1 2 0s x x= + =
Parameter variations
2008/11/17 10Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Introduction to Sliding mode-Variable Structure Systems
2008/11/17 11Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
(K. D. Young, 1978)1 2
2 2
1 2
1 1
1 1
System :
Switching line : 0(Sliding Surface)
, if 0 (region I)Control :
, if 0 (rigion II)
x xx ax bu
s cx x
x x su
x x s
= =
= + =
>=
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Introduction to Sliding mode-Variable Structure Systems
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 12
2
2
0 1Structure I :
0 : spiral out
0 1Structure II :
0 : saddle point
x xb a
s as b
x xb a
s as b
=
+ =
=
=
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Introduction to Sliding mode- Invariance Condition
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 13
1 2
0 2 21 1 1 ( )
2 1 3
0 2 21 1 1 ( )
2 1 3
x Ax u f t
Ax u u f t
= + + = + + +
Q: Can the control law possibly reject the disturbance out of the system?
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Introduction to Sliding mode- Invariance Condition
Controlsubspace:
Disturbancesubspace:
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 14
0 21 , 1
2 1span
21
3span
*Thedisturbancecanberejectedbythecontrollawif(andonlyif)the control subspace covers
the disturbance subspace.
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Inthisexample,
Ifisknown,thenthecontrollaw
Willleadto
Introduction to Sliding mode- Invariance Condition
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 15
2 0 21 2 1 1
3 2 1
= +
( )d t
1 1
2 2
2 ( )( )
u v f tu v f t= =
1 2
0 2 0 2 2 0 21 1 ( 2 1 1 ) ( ) 1 ( ) 1 1
2 1 2 1 3 2 1x Ax v v f t f t Ax v
= + + + + = +
.
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Introduction to Sliding mode- Invariance Condition
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 16
(Drazenovic, 1969): The system with disturbance
is invariant of in sliding mode iff
In the previous example,
( )x Ax Bu Ef t= + +( )f t 0s
[ ] [ ]| .rank B E rank B=
0 2 2 0 21 1 1 1 1 2.
2 1 3 2 1rank rank
= =
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A funnel-like domain (revisit)
System:
Target:
Funnel:
New target:
1 2
2 1 22 3 ( )
x x
x x x u f t
=
= + +1 2( , ) (0, 0)x x =
2 1x x=
1 2 0s x x= + =
2x
1x
0s =
disturbance
2008/11/17 17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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A funnel-like domain (revisit)
System:
Target:
Funnel:
New target:
1 2
2 1 2 1 1 2 22 3 ( )
x x
x x x u x x
= = + + +
2x
1x
0s =
1 2( , ) (0, 0)x x =
2 1x x=
1 2 0s x x= + =
Parameter variations
2008/11/17 18Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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VSS Design
2008/11/17 19Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
(K. D. Young, 1978)1 2
2 2
1 2
1 1
1 1
System :
Switching line : 0(Sliding Surface)
, if 0 (region I)Control :
, if 0 (rigion II)
x xx ax bu
s cx x
x x su
x x s
= =
= + =
>=
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Existence of sliding modei. Reaching condition (sufficient, global)
ii. Sliding condition (sufficient, local)
VSS Design
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 20
Sliding Surface: s = 0u+
u
0 0lim 0, lim 0s s
s s+
< >
sgn( ), 0s s < >
x(0)
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VSS Design
Finite time reaching:
Reaching time
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 21
sgn( ), 0s s < >
Lyapunov function2
2 2 0
( ) 0
V sdV ss sdt
s x
=
= =
3 2 2 1 1
1 1 2 2 3 3 2 3 1 2( )s x c x c x
a x a x a x u f t c x c x= + +
= + + + +
1 1 2 1 2 3 2 3 max( ) ( ) ( )sgn( )u a x a c x a c x f s= + + +
max( ) .f t f= =
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VSS Design- Sliding Surface
Eigenspace Approach v.s. Lyapunov Approach
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 25
( )x A x B u D f t= + +
Nonsingular transformation
Canonical form
Sliding surface
( ) 1
2 2
0,n m m
xTB Tx
B x = =
1 11 12 1
2 21 22 2 2 2
0 0( )
x A A xu f t
x A A x B D
= + +
[ ]2 1, or ( ) Im mx Kx s x K Tx= =
Stabilizing feedback
Lyapunov function
Sliding surface
( )x A x B K x D f t= +
1( )2
Ts s
T
PA A P Q
V x x Px
+ =
=
( ) Ts x B Px=
-
VSS Design- Discontinuous Control
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 26
( )s lid ing su rface: ( ) 0
( )
x A x B u D f ts x C x
s C A x C B u C D f t
= + += =
= + +
Signum function (Single-input case)
1 1max( ) ( ) ( )sgn( )u CB CAx CB d s
= +
Unit Control(Multi-input case) 1 1
max( ) ( ) ( )su CB CAx CB ds
= +
v.s.
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Discrete-Time Sliding Mode
Continuous-time
Sampled-data
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 27
( )Sliding surface: ( ) ( ) 0
x Ax Bu Df ts t Cx t
= + += =
1
Sliding surface: 0k k k k
k k
x x u ds Cx
+ = + += =
0( 1)
( )
0
, ,
( ) (( 1) )
TAT A
k T TA t A
kkT
e e d B
d e Df d e Df k T d
+
= =
= = +
-
Discrete-Time Sliding Mode
Issues on disturbance rejection:1. Matching condition fails
However, 2. Non-causal disturbances
However,
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 28
1k k k kx x u d+ = + +
0 0
, (( 1) )T T
A Ake d B d e Df k T d
= = +
range( )kd
21
1 1 1
( )k kk k k k
d d O Td x x u
= +=
2( ) ( )kd f kT O T= +
Ultimate achievable accuracy: 2( )O T
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Discrete-Time Sliding Mode
Quasi-sliding mode
Discrete-time sliding mode
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 29
20, ( ) ( )ks s t O T= =
( ), ( ) ( )ks O T s t O T= =
Discontinuous control
Continuous control
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Discrete-Time Sliding Mode
Ukins discrete-time equivalent control(Continuous control law)
Discrete-time sliding mode control(Modification of )
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 30
11
0
( ) ( )k k k keqk k k
s C x C u Cd
u C C x d+
= + + =
= +
eqku
11
21 1
( ) ( )
( ) ( )k k k
k k k
u C C x d
s C d d O T
+
= +
= =
2( )O T
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Contents (1/2)
Introduction to Sliding Mode Variable Structure Systems Invariance Condition (Matching Condition)
Variable Structure Systems (VSS) Design Sliding Surface Design Discontinuous Control
Discrete-Time Sliding Mode Boundary Layer Switching v.s. Continuous Control
2( ) . . ( )O T v s O T
2008/11/17 31Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Contents (2/2)
Minimum-Time Torque Control (SRM)Temperature Control (Plastic Extrusion P.)Rod-less Pneumatic Cylinder ServoWireless Network Power Control
Singular PerturbationsDistributed Parameter Systems
2008/11/17 32Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Switched Reluctance Motors
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 33
Rotor Stator
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Switched Reluctance Motors
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 34
A
A-
1
2
3
4
5
6
4-phase, 8/6-pole SRM
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Switched Reluctance Motors
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 35
Stator Rotor
Reluctance force
reluctance force
Magnetic flux
( )L
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Switched Reluctance Motors
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 36
2 2( )1Torque :2
jj j
dLi T i
d
=
a
a'
R
L
7 .5 15 22 .5 30 37 .5 45 52 .5 60(D egree)
30
10
(m H )
La Lb Lc Ld
( )( ) , , , ,j jj j j j j
di dLV L i R i j A B C D
dt dt
= + + =
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Minimum-Time Torque Control
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 37
A BC D
Sensor 1 Sensor 2
Q1
Q2
Q3
Q4
Vdc
DriveCPLD
XC9536
encoder
current sensor
Drive Circuit
Control System Setupri
KV =* KV =*
i)( 0ti )( 0ti
System dynamics:
( , ) ( ) ( )di A i B V tdt
= +
Sliding surface:0rs i i= =
Minimum-Time VSC:, ( )
( ), ( )
dc r
dc r
V if i t iV t
V if i t i >
=
-
Minimum-Time Torque Control
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 38
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Temperature Control - A Plastic Extrusion Process
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 39
System setup: ITRI (Industrial Technology Research Institute), Taiwan, R.O.C.
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Temperature Control - A Plastic Extrusion Process
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 40
Dynamics of Channel temperature:1dy hA hAy y w
dt cV cV cV = + +
Rate of change of energy input:( , , ), 0
( , , )( , , ), 0
a w u t udw a w u tdt a w u t u
+
= =
-
Temperature Control - A Plastic Extrusion Process
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 41
Single-channel dynamics:
( , , )y my v
a w u tvcV
= +
= +
Sliding surface (PI control):
( ) ( )( ( ) ( ))IP I P desiredKv K e K e or V s K y s Y ss
= = +
Output feedback sliding mode control
1 1 0 1 1
( )P Ik k k k
s e K m e K es a s b e b e
= + + + + +
-
Temperature Control - A Plastic Extrusion Process
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 42
Switching control law:heat, 0water, 0
kk
k
su
s
-
Rod-less Pneumatic Cylinder Servo
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 43
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Rod-less Pneumatic Cylinder Servo
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 44
Equation of motion:sgnL u c TM Y Y Y A P = +
1 1 2 2 1 2
Pressure dynamics:( , , , ) ( , , , , )P P P Y X X P P Y Y X = +
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Rod-less Pneumatic Cylinder Servo
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 45
1 2
Sliding surface:1( ) ( ) sgnL d d d u c
T T
MP Y k Y Y k Y Y Y YA A
= + + + +
Variable Structure Control: sgn( )vX X =
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Wireless Network Power Control
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 46
Zoran Gajic, Plenary Lecture 2007 CACS International Automatic Control ConferenceNational Chung Hsing University, Taichung, Taiwan, November 9-11, 2007
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Wireless Network Power Control
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 47
( ) ( ) ( ) ( )( ) , 1,2, , .( ) ( ) ( ) ( )
ii i ii ii
i ij j ij i
g t p t g t p tt i NI t g t p t t
= = =+
Signal-to-interference ratio (SIR) for the ith user
Control objective: ( ) tari it
( )i ii i i ig p I p = + Logarithmic scale representation
,
( ) ( ), 1, 2,( ) ( ) ( )
i i
i i i
p t u t i Nt p t f t= =
= +
System dynamics:
-
Wireless Network Power Control
Sliding surfaceDiscrete-time representation
Discrete-time SMC (continuous)
Stability analysis
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 48
1( 1) ( ) 1 1 ( ( ) ( ))1( 1) ( ) 21i i
i i
Ts k s kd k v k
u k u k TT
+ = + +
( ) ( ) 0tari i is t t = =
( 1) ( ) ( ) ( ) ( )i i is k s k Tu k d k v k+ = + + +
1 1( ) ( ) ( ( 1) ( 1))i iu k s k d k v kT T= +
21
det 01 1
z Tz
zT
= = +
-
Contents (2/2)
Minimum-Time Torque Control (SRM)Temperature Control (Plastic Extrusion P.)Rod-less Pneumatic Cylinder ServoWireless Network Power Control
Singular PerturbationsDistributed Parameter Systems
2008/11/17 49Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Singularly Perturbed Systems VSSQuasi-steady state (slow mode) Sliding mode
Fast mode Reaching phase
Continuous control Discontinuous control
Infinite-time reaching(boundary layer)
Finite-time reaching
Extended Research Problems Singular Perturbations
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 50
Sliding Surface: s = 0
u+
u
x(0)
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Extended Research Problems
Distributed Parameter Systems
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 51
:( , ) ( , ) ( , )
(0, ) 0( , ) ( ) ( )
t xx
Heat Conduction SystemU x t U x t U x t
Boundary conditions U tU l t Q t f t
= +== +
Kernel function( , )k x y
( , )U x t ( , )w x t (0, ) 0( , ) ( ) ( )
t xx
w
w w cww tw l t Q t f t
= == +
0x = x l=
-
Extended Research Problems
Distributed Parameter Systems
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 52
( ) ( , ) 0Sliding surface:
xS t l t= =
( )2 212 2
( )Kernel function : ( , )
( )
I x yk x y y
x y
=
( , )U x t ( , )w x t (0, ) 0( , ) ( ) ( )
t xx
w
w w cww tw l t Q t f t
= == +
0x = x l=
-
Extended Research Problems
Distributed Parameter Systems
2008/11/17Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 53
0( ) = sgn( ( ))
Sliding Mode Control:t
mQ t K S d
0
( ) ( , ) 0
( ) ( , ) ( , ) ( , ) ( , ) ( , )
Sliding surface: xl
x x
S t w l t
S t U l t k l l U l t k l y U y t dy
= =
=
-
Conclusions
Merits of VSS
Robustness (against disturbances and system
parameter variations)
Reduced-order system design
Simple control structure
Fits switching power electronics control
2008/11/17 54Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Conclusions
Chattering Reduction (Elimination) Discrete-Time Sliding Mode
Filtering methods
Interesting Problems Output feedback
Singular Perturbations
Infinite-dimensional systems
Systems with delays (e.g. wireless network)
2008/11/17 55Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
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Thank You!
http://www.chneic.sh.cn/
2008/11/17 56Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
http://tw.wrs.yahoo.com/_ylt=A3eg8qx0hCBJ6aYAlQxt1gt./SIG=1k5n82qhu/EXP=1226954228/**http:/tw.info.search.yahoo.com/search/images/view?back=http://tw.info.search.yahoo.com/search/images?p=%E6%BC%8F%E6%96%97&ei=UTF-8&fr=yfp&x=wrt&w=335&h=365&imgurl=static.flickr.com/3212/2454169225_9ca81eabfc.jpg&rurl=http://www.flickr.com/photos/g-economy/2454169225/&size=26.6kB&name=%E6%BC%8F%E6%96%970&p=%E6%BC%8F%E6%96%97&type=jpeg&no=2&tt=8,651&oid=9d25d034aa1fe444&ei=UTF-8&fusr=G-economy.com&hurl=http:/www.flickr.com/photos/g-economy/&src=flickr
Sliding Mode Control with Industrial ApplicationsPreludeThe Old Oil-Peddler Ouyang Show(1007~1072A.D.)Sliding Mode: an illustrationA funnel-like domainA Physical ExampleContents (1/2)Contents (2/2)A funnel-like domain (cont.)A funnel-like domain (cont.)Introduction to Sliding mode - Variable Structure SystemsIntroduction to Sliding mode - Variable Structure SystemsIntroduction to Sliding mode - Invariance ConditionIntroduction to Sliding mode - Invariance ConditionIntroduction to Sliding mode - Invariance ConditionIntroduction to Sliding mode - Invariance ConditionA funnel-like domain (revisit)A funnel-like domain (revisit)VSS DesignVSS DesignVSS DesignVSS DesignVSS DesignVSS DesignVSS Design - Sliding Surface VSS Design - Discontinuous ControlDiscrete-Time Sliding ModeDiscrete-Time Sliding ModeDiscrete-Time Sliding ModeDiscrete-Time Sliding ModeContents (1/2)Contents (2/2)Switched Reluctance MotorsSwitched Reluctance MotorsSwitched Reluctance MotorsSwitched Reluctance MotorsMinimum-Time Torque ControlMinimum-Time Torque ControlTemperature Control - A Plastic Extrusion ProcessTemperature Control - A Plastic Extrusion ProcessTemperature Control - A Plastic Extrusion ProcessTemperature Control - A Plastic Extrusion ProcessRod-less Pneumatic Cylinder ServoRod-less Pneumatic Cylinder ServoRod-less Pneumatic Cylinder ServoWireless Network Power ControlWireless Network Power ControlWireless Network Power ControlContents (2/2)Extended Research Problems Extended Research Problems Extended Research Problems Extended Research Problems ConclusionsConclusionsThank You!