Sliding Mode Control with Industrial ApplicationsWu-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 1 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
Prelude
to drive the state to the target.
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The Old Oil-Peddler Ouyang Show(1007~1072A.D.)











http://baike.baidu.com/2008/11/17 3 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
Sliding Mode: an illustration
to fill oil into a bottle through a funnel 2008/11/17 4
Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
A “funnel-like” domain
= − +
2 1x x= −
2x
1x
0s =
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A Physical Example Coulomb Friction
sgn( )
f
, 3, 2
0 3 2 0
0 3 2 0
= = −
> ⇒ = − − < ⇒ ↓
< ⇒ = − > ⇒ ↑
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Contents (1/2)
2( ) . . ( )O T v s O T
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Contents (2/2)
Singular Perturbations Distributed Parameter Systems
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A “funnel-like” domain (cont.)
=
2 1x x= −
2x
1x
0s =
disturbance
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A “funnel-like” domain (cont.)
x x
= = − + + Δ + Δ
2 1x x= −
Parameter variations
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Introduction to Sliding mode -Variable Structure Systems
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(K. D. Young, 1978)1 2
2 2
1 2
1 1
1 1
, if 0 (rigion II)
s cx x
= = −
= + =
> = − <
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2
2
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1 2
2 1 3
2 1 3
= + − + − − = + − + + − −
Q: Can the control law possibly reject the disturbance out of the system?
Introduction to Sliding mode - Invariance Condition
Control subspace: 
Disturbance subspace:
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 14
0 2 1 , 1
−
* The disturbance can be rejected by the control law if  (and only if) the control subspace covers
the disturbance subspace.
In this example,
Will lead to
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 15
2 0 2 1 2 1 1
3 2 1
1 2
0 2 0 2 2 0 2 1 1 ( 2 1 1 ) ( ) 1 ( ) 1 1
= + − + + − − − + − = + − − − −
.
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(Drazenovic, 1969): The system with disturbance
is invariant of in sliding mode iff
In the previous example,
[ ] [ ]| .rank B E rank B=
0 2 2 0 2 1 1 1 1 1 2.
2 1 3 2 1 rank rank
− − = − = − −
=
2 1x x= −
2x
1x
0s =
disturbance
2008/11/17 17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
A “funnel-like” domain (revisit)
x x
= = − + + Δ + Δ
2 1x x= −
Parameter variations
2008/11/17 18 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
VSS Design
2008/11/17 19 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
(K. D. Young, 1978)1 2
2 2
1 2
1 1
1 1
, if 0 (rigion II)
s cx x
= = −
= + =
> = − <
ii. Sliding condition (sufficient, local)
VSS Design
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Sliding Surface: s = 0 u+
u−
s s + −→ →
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sgn( ), 0s sσ σ< − >
s x
(i) Sliding surface
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3 2 2 3 2 1 3 2 1( ) ( ) ( )( ( )), dD a D a D a y t b D b D b u f t D
dt + + + = + + + =
0 1 0 0 0 0 0 1 0 0 ( )
1 1 x x u f t
a a a
= + + − − −
=
3 2 2 1 1( ) 0s x x c x c x= + + = Sliding mode dynamics:
1 2 3 1 1 2 2
2 3
= − − =
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sgn( ), 0s sσ σ< − >
3 2 2 1 1
1 1 2 2 3 3 2 3 1 2( ) s x c x c x
= − − − + + + +
1 1 2 1 2 3 2 3 max( ) ( ) ( )sgn( )u a x a c x a c x f sσ= + − + − − +
max( ) .f t f<
∂ = ∂
1. Choose sliding surface
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( ) 0s x =
f x u t f s
+ +
− −
= >= = <
VSS Design - Sliding Surface
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( )x A x B u D f t= + +
Nonsingular transformation
Canonical form
Sliding surface
TB Tx B x − ×
0 0 ( )
x A A x B D
= + +
[ ]2 1, or ( ) Im mx Kx s x K Tx×= − =
Stabilizing feedback
Lyapunov function
Sliding surface
1( ) 2
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( ) s lid ing su rface: ( ) 0
( )
x A x B u D f t s x C x
= + + = =
= + +
1 1 max( ) ( ) ( )sgn( )u CB CAx CB d sσ− −= − − +
Unit Control (Multi-input case) 1 1
max( ) ( ) ( ) su CB CAx CB d s
σ− −= − − + ⋅
v.s.
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( ) Sliding surface: ( ) ( ) 0
= + + = =
k k
+ = Φ +Γ + = =
0 ( 1)
k kT
τ
However, 2. Non-causal disturbances
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1k k k kx x u d+ = Φ +Γ +
0 0
, (( 1) ) T T
A A ke d B d e Df k T dλ λλ λ λΓ = = + −∫ ∫
range( )kd ∉ Γ
−
− − −
Ultimate achievable accuracy: 2( )O T
Discrete-Time Sliding Mode
2008/11/17 Princeton/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
Discrete-time sliding mode control (Modification of )
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 30
1 1
s C x C u Cd
u C C x d +
−
s C d d O T
− −
+ −
2( ) . . ( )O T v s O T
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Contents (2/2)
Singular Perturbations Distributed Parameter Systems
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Switched Reluctance Motors
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Rotor Stator
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A
A-
1
2
3
4
5
6
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Reluctance force
reluctance force
Magnetic flux
( )L θ
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2 2( )1Torque : 2
a
a'
R
L
7 .5 15 22 .5 30 37 .5 45 52 .5 60 (D egree)
30
10
j j j j j
di dL V L i R i j A B C D
dt dt θ
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A BC D
θ ω θ= − +
Minimum-Time VSC: , ( )
V if i t i − >
= <
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Temperature Control - A Plastic Extrusion Process
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System setup: ITRI (Industrial Technology Research Institute), Taiwan, R.O.C.
Temperature Control - A Plastic Extrusion Process
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Dynamics of Channel temperature: 1dy hA hAy y w
dt cV cV cVρ ρ ρ∞= − + +
Rate of change of energy input: ( , , ), 0
( , , ) ( , , ), 0
+
−
≥ = =
<
ξ ρ
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 41
Single-channel dynamics:
ξ ρ
( ) ( )( ( ) ( ))I P I P desired
= − − = + −
( )P I
k k k k
= + + + ≈ + +
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Switching control law: heat, 0 water, 0
k k
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Rod-less Pneumatic Cylinder Servo
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Equation of motion: sgnL u c TM Y Y Y A Pμ μ= − − + Δ
1 1 2 2 1 2
Pressure dynamics: ( , , , ) ( , , , , )P P P Y X X P P Y Y Xψ ψΔ = +
Rod-less Pneumatic Cylinder Servo
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 45
1 2
d d d u c T T
MP Y k Y Y k Y Y Y Y A A
σ μ μ = Δ − + − + − + +
Wireless Network Power Control
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 46
Zoran Gajic, Plenary Lecture 2007 CACS International Automatic Control Conference National Chung Hsing University, Taichung, Taiwan, November 9-11, 2007
Wireless Network Power Control
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 47
( ) ( ) ( ) ( )( ) , 1,2, , . ( ) ( ) ( ) ( )
i ij j i j i
g t p t g t p tt i N I t g t p t t
γ υ
Control objective: ( ) tar i itγ γ→
,
i i i
= +
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 48
1( 1) ( ) 1 1 ( ( ) ( ))1( 1) ( ) 21 i i
i i
u k u k T T
+ = − + + − −
( ) ( ) 0tar i i is t tγ γ= − =
( 1) ( ) ( ) ( ) ( )i i is k s k Tu k d k v k+ = + + +
= − − − + −
Singular Perturbations Distributed Parameter Systems
2008/11/17 49 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
Singularly Perturbed Systems VSS Quasi-steady state (slow mode) Sliding mode
Fast mode Reaching phase
Continuous control Discontinuous control
Infinite-time reaching (boundary layer)
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 50
Sliding Surface: s = 0
: ( , ) ( , ) ( , )
(0, ) 0 ( , ) ( ) ( )
t xx
Heat Conduction System U x t U x t U x t
Boundary conditions U t U l t Q t f t
α β= +
( , )U x t ( , )w x t (0, ) 0 ( , ) ( ) ( )
t xx
w
w w cw w t w l t Q t f t
α= −
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 52
( ) ( , ) 0 Sliding surface:
x y
λ λ
t xx
w
w w cw w t w l t Q t f t
α= −
2008/11/17 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting 53
0 ( ) = sgn( ( ))
0
( ) ( , ) 0
S t w l t
= =
= − − ∫
parameter variations)
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Conclusions
Filtering methods
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Thank You!
2008/11/17 56 Princeton/Central Jersey Section of IEEE - Circuits and Systems Chapter Meeting
Prelude
Sliding Mode: an illustration
Introduction to Sliding mode - Invariance Condition
Introduction to Sliding mode - Invariance Condition
Introduction to Sliding mode - Invariance Condition
Introduction to Sliding mode - Invariance Condition
A “funnel-like” domain (revisit)
A “funnel-like” domain (revisit)
Rod-less Pneumatic Cylinder Servo
Rod-less Pneumatic Cylinder Servo
Rod-less Pneumatic Cylinder Servo
Wireless Network Power Control
Wireless Network Power Control
Wireless Network Power Control