PRINCETON/CENTRAL JERSEY SECTION OF IEEE – CIRCUITS AND SYSTEMS
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.
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
1
Prelude
A Control Engineer as an Archer…
to drive the state to the target.
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
2
The Old Oil-Peddler
賣油翁
宋 歐陽修
3
康肅問曰：﹁汝亦知射乎？吾射不亦精
乎？﹂翁曰：﹁無他，但手熟爾。﹂康肅忿然曰：
﹁爾安敢輕吾射！﹂翁曰：﹁以我酌油知之。﹂
乃取一葫蘆置於地，以錢覆其口，徐以杓酌油瀝
之，自錢孔入，而錢不濕。因曰：﹁我亦無他，
惟手熟爾。﹂康肅笑而遣之。
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
http://baike.baidu.com/ 2008/11/17
陳康肅公堯咨善射，當世無雙，公亦以此
自矜。嘗射於家圃，有賣油翁釋擔而立，睨之，
久而不去。見其發矢十中八九，但微頷之。
Ouyang Show(1007~1072A.D.)
Sliding Mode: an illustration
to fill oil into a bottle through a funnel
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
4
A “funnel-like” domain
⎧ x1 = x2
System: ⎨
⎩ x2 = 2 x1 − 3 x2 + u
Target: ( x1 , x2 ) = (0, 0)
x2
x1
Funnel: x2 = − x1
New target: s = x1 + x2 = 0
2008/11/17
s=0
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
5
A Physical Example
Coulomb Friction
dv
m = − K sgn(v) + f a
dt
f a : applied force
For example, K = 3, f a = −2
dv
= −3 − 2 < 0 ⇒ v ↓
dt
dv
v < 0 ⇒ m = 3− 2 > 0 ⇒ v ↑
dt
v >0⇒m
http://www.physics4kids.com
sliding mode: v ≡ 0 (if f a < K ).
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
6
Contents (1/2)
y
Introduction to Sliding Mode
◦ Variable Structure Systems
◦ Invariance Condition (Matching Condition)
y
Variable Structure Systems (VSS) Design
◦ Sliding Surface Design
◦ Discontinuous Control
y
Discrete-Time Sliding Mode
◦ O(T ) v.s. O(T 2 ) Boundary Layer
◦ Switching v.s. Continuous Control
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
7
Contents (2/2)
Minimum-Time Torque Control (SRM)
y Temperature Control (Plastic Extrusion P.)
y Rod-less Pneumatic Cylinder Servo
y Wireless Network Power Control
y
Singular Perturbations
y Distributed Parameter Systems
y
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
8
A “funnel-like” domain (cont.)
disturbance
⎧⎪ x1 = x2
System:⎨
x2
⎪⎩ x2 = 2 x1 − 3 x2 + u + f (t )
Target: ( x1 , x2 ) = (0, 0)
x1
Funnel: x2 = − x1
New target: s = x1 + x2 = 0
2008/11/17
s=0
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
9
A “funnel-like” domain (cont.)
Parameter
x
=
x
⎧⎪ 1
2
variations
System: ⎨
⎪⎩ x2 = 2 x1 − 3 x2 + u + (Δ1 x1 + Δ 2 x2 )
x2
Target: ( x1 , x2 ) = (0, 0)
Funnel: x2 = − x1
x1
New target: s = x1 + x2 = 0
s=0
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
10
Introduction to Sliding mode
- Variable Structure Systems
⎧ x1 = x2
System : ⎨
⎩ x2 = ax2 − bu
(K. D.Young, 1978)
Switching line : s = cx1 + x2 = 0
(Sliding Surface)
⎧α x1 , if x1s > 0 (region I)
Control : u = ⎨
⎩−α x1 , if x1s < 0 (rigion II)
2008/11/17
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- Circuits and Systems Chapter Meeting
11
Introduction to Sliding mode
- Variable Structure Systems
1⎤
⎡ 0
x
Structure I : x = ⎢
⎥
⎣ −bα a ⎦
s 2 − as + bα = 0 : spiral out
⎡ 0 1⎤
x
Structure II : x = ⎢
⎥
⎣bα a ⎦
s 2 − as − bα = 0 : saddle point
2008/11/17
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- Circuits and Systems Chapter Meeting
12
Introduction to Sliding mode
- Invariance Condition
Q: Can the control law possibly reject
the disturbance out of the system?
⎡0 2⎤
⎡2⎤
x = Ax + ⎢⎢ −1 1 ⎥⎥ u + ⎢⎢ −1⎥⎥ f (t )
⎢⎣ 2 −1⎥⎦
⎢⎣ 3 ⎥⎦
⎡0⎤
⎡2⎤
⎡2⎤
= Ax + ⎢⎢ −1⎥⎥ u1 + ⎢⎢ 1 ⎥⎥ u2 + ⎢⎢ −1⎥⎥ f (t )
⎢⎣ 3 ⎥⎦
⎣⎢ 2 ⎦⎥
⎣⎢ −1⎦⎥
2008/11/17
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- Circuits and Systems Chapter Meeting
13
Introduction to Sliding mode
- Invariance Condition
⎧⎡ 0 ⎤ ⎡ 2 ⎤ ⎫
⎪⎢ ⎥ ⎢ ⎥ ⎪
Control subspace: span ⎨ −1 , 1 ⎬
⎢ ⎥ ⎢ ⎥
⎪ ⎢ 2 ⎥ ⎢ −1⎥ ⎪
⎩⎣ ⎦ ⎣ ⎦ ⎭
⎧⎡ 2 ⎤ ⎫
Disturbance subspace: span ⎪ ⎢ −1⎥ ⎪
⎨⎢ ⎥ ⎬
⎪⎢ 3 ⎥ ⎪
⎩⎣ ⎦ ⎭
* The disturbance can be rejected by the control law if (and only if) the control subspace covers
the disturbance subspace.
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
14
Introduction to Sliding mode
- Invariance Condition
In this example,
.
⎡2⎤
⎡0⎤ ⎡2⎤
⎢ −1⎥ = 2 ⎢ −1⎥ + ⎢ 1 ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢⎣ 3 ⎥⎦
⎢⎣ 2 ⎥⎦ ⎢⎣ −1⎥⎦
d (t )
If is known, then the control law
u1 = v1 − 2 f (t )
u2 = v2 − f (t )
Will lead to
⎡0⎤
⎡2⎤
⎡0⎤ ⎡2⎤
⎡2⎤
⎡0 2⎤
x = Ax + ⎢⎢ −1⎥⎥ v1 + ⎢⎢ 1 ⎥⎥ v2 + ( −2 ⎢⎢ −1⎥⎥ − ⎢⎢ 1 ⎥⎥ ) f (t ) + ⎢⎢ −1⎥⎥ f (t ) = Ax + ⎢⎢ −1 1 ⎥⎥ v
⎢⎣ 2 ⎥⎦
⎢⎣ −1⎥⎦
⎢⎣ 2 ⎥⎦ ⎢⎣ −1⎥⎦
⎢⎣ 3 ⎥⎦
⎢⎣ 2 −1⎥⎦
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
15
Introduction to Sliding mode
- Invariance Condition
(Drazenovic, 1969):
The system with disturbance
x = Ax + Bu + Ef (t )
is invariant of
f (t ) in
sliding mode
s≡0
rank [ B | E ] = rank [ B ] .
iff
In the previous example,
⎡0 2
⎢
rank ⎢ −1 1
⎢⎣ 2 −1
2⎤
⎡0 2⎤
⎥
−1⎥ = rank ⎢⎢ −1 1 ⎥⎥ = 2.
⎢⎣ 2 −1⎥⎦
3 ⎥⎦
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
16
A “funnel-like” domain (revisit)
disturbance
⎧⎪ x1 = x2
System:⎨
x2
⎪⎩ x2 = 2 x1 − 3 x2 + u + f (t )
Target: ( x1 , x2 ) = (0, 0)
x1
Funnel: x2 = − x1
New target: s = x1 + x2 = 0
2008/11/17
s=0
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
17
A “funnel-like” domain (revisit)
Parameter
x
=
x
⎧⎪ 1
2
variations
System: ⎨
⎪⎩ x2 = 2 x1 − 3 x2 + u + (Δ1 x1 + Δ 2 x2 )
x2
Target: ( x1 , x2 ) = (0, 0)
Funnel: x2 = − x1
x1
New target: s = x1 + x2 = 0
s=0
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
18
VSS Design
⎧ x1 = x2
System : ⎨
⎩ x2 = ax2 − bu
(K. D.Young, 1978)
Switching line : s = cx1 + x2 = 0
(Sliding Surface)
⎧α x1 , if x1s > 0 (region I)
Control : u = ⎨
⎩−α x1 , if x1s < 0 (rigion II)
2008/11/17
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- Circuits and Systems Chapter Meeting
19
VSS Design
Existence of sliding mode
i. Reaching condition (sufficient, global)
s < −σ sgn( s), σ > 0
ii. Sliding condition (sufficient, local)
x(0)
lim+ s < 0,
s →0
u
lim− s > 0
s →0
Sliding Surface: s = 0
+
u−
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VSS Design
Finite time reaching: s < −σ sgn( s), σ > 0
Lyapunov function
V = s2
dV
= 2 ss = −2σ s < 0
dt
∴ s( x) → 0
Reaching time
T<
V (0)
σ
=
2008/11/17
s (0)
σ
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VSS Design
A heuristic example
d
( D + a3 D + a2 D + a1 ) y (t ) = (b3 D + b2 D + b1 )(u + f (t )), D =
dt
1
0 ⎤
⎡ 0
⎡0⎤
⎡0⎤
x = ⎢⎢ 0
0
1 ⎥⎥ x + ⎢⎢ 0 ⎥⎥ u + ⎢⎢0 ⎥⎥ f (t )
⎢⎣ − a1 − a2 − a3 ⎥⎦
⎢⎣1 ⎥⎦
⎢⎣1 ⎥⎦
3
2
2
y = [b1 b2
b3 ] x
(i) Sliding surface s ( x ) = x3 + c2 x2 + c1 x1 = 0
Sliding mode dynamics:
⎧ x1 = x2
, with x3 = −c1 x1 − c2 x2 .
⎨
⎩ x2 = x3
2008/11/17
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VSS Design
(ii)
ds ( x) ∂s dx
=
∂x dt
dt
s = x3 + c2 x2 + c1 x1
= −a1 x1 − a2 x2 − a3 x3 + u + f (t ) + c2 x3 + c1 x2
(iii) Discontinuous control
u = a1 x1 + (a2 − c1 ) x2 + (a3 − c2 ) x3 − ( f max + σ ) sgn( s )
- Reaching condition s < −σ sgn( s), σ > 0
satisfied if f (t ) < f max .
2008/11/17
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- Circuits and Systems Chapter Meeting
23
VSS Design
Two steps in VSS design
x = f ( x , u , t )
1.
Choose sliding surface
s( x ) = 0
2.
Design discontinuous control to
render sliding mode
⎧⎪ f ( x , u + , t ) = f + , s > 0
x = ⎨
−
−
=
f
(
x
,
u
,
t
)
f
, s<0
⎪⎩
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24
VSS Design
- Sliding Surface
x = A x + B u + D f ( t )
Eigenspace Approach v.s. Lyapunov Approach
Nonsingular transformation
Stabilizing feedback
x = A x − B K x + D f ( t )
⎡0( n − m )×m ⎤
⎡ x1 ⎤
TB = ⎢
, Tx = ⎢ ⎥
⎥
⎣ B2 ⎦
⎣ x2 ⎦
Lyapunov function
Canonical form
⎡ x1 ⎤ ⎡ A11
⎢ x ⎥ = ⎢ A
⎣ 2 ⎦ ⎣ 21
PAs + AsT P = −Q
A12 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤
⎡0⎤
+
u + ⎢ ⎥ f (t )
A22 ⎥⎦ ⎢⎣ x2 ⎥⎦ ⎢⎣ B2 ⎥⎦
⎣ D2 ⎦
Sliding surface
x2 = − Kx1 , or s( x ) = [ K
I m×m ] Tx
V ( x) =
1 T
x Px
2
Sliding surface
s ( x ) = BT Px
2008/11/17
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- Circuits and Systems Chapter Meeting
25
VSS Design
- Discontinuous Control
x = A x + B u + D f ( t )
slid in g su rfa ce: s ( x ) = C x = 0
s = C A x + C B u + C D f ( t )
Signum function
(Single-input case)
u = −(CB) −1 CAx − (CB) −1 (σ + d max ) sgn( s )
v.s.
Unit Control
(Multi-input case)
u = −(CB) −1 CAx − (CB) −1 (d max + σ ) ⋅
2008/11/17
s
s
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Discrete-Time Sliding Mode
Continuous-time
x = Ax + Bu + Df (t )
Sliding surface: s (t ) = Cx (t ) = 0
Sampled-data
xk +1 = Φxk + Γuk + d k
Sliding surface: sk = Cxk = 0
T
Φ = e , Γ = ∫ e Aτ dτ B,
AT
0
( k +1)T
dk =
∫
kT
T
e A(t −τ ) Df (τ )dτ = ∫ e Aλ Df ((k + 1)T − λ )d λ
0
2008/11/17
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- Circuits and Systems Chapter Meeting
27
Discrete-Time Sliding Mode
T
xk +1 = Φxk + Γuk + d k
T
Γ = ∫ e Aλ d λ B, d k = ∫ e Aλ Df ((k + 1)T − λ )d λ
0
0
Issues on disturbance rejection:
1. Matching condition fails d k ∉ range(Γ)
However, d k = Γf (kT ) + O(T 2 )
2. Non-causal disturbances
However, d k = d k −1 + O(T 2 )
d k −1 = xk − Φxk −1 − Γuk −1
Ultimate achievable accuracy:O(T 2 )
2008/11/17
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- Circuits and Systems Chapter Meeting
28
Discrete-Time Sliding Mode
Quasi-sliding mode sk = O(T ), s (t ) = O(T )
Discontinuous control
2
s
=
0,
s
(
t
)
=
O
(
T
)
Discrete-time sliding mode k
Continuous control
2008/11/17
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- Circuits and Systems Chapter Meeting
29
Discrete-Time Sliding Mode
Ukin’s discrete-time equivalent control
(Continuous control law)
sk +1 = C Φxk + C Γuk + Cd k = 0
−1
u = −(C Γ) C (Φxk + d k )
eq
k
Discrete-time O(T 2 ) sliding mode control
(Modification of ukeq )
uk = −(C Γ ) −1 C (Φxk + d k −1 )
⇒ sk +1 = C ( d k − d k −1 ) = O (T 2 )
2008/11/17
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- Circuits and Systems Chapter Meeting
30
Contents (1/2)
y
Introduction to Sliding Mode
◦ Variable Structure Systems
◦ Invariance Condition (Matching Condition)
y
Variable Structure Systems (VSS) Design
◦ Sliding Surface Design
◦ Discontinuous Control
y
Discrete-Time Sliding Mode
◦ O(T ) v.s. O(T 2 ) Boundary Layer
◦ Switching v.s. Continuous Control
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
31
Contents (2/2)
Minimum-Time Torque Control (SRM)
y Temperature Control (Plastic Extrusion P.)
y Rod-less Pneumatic Cylinder Servo
y Wireless Network Power Control
y
Singular Perturbations
y Distributed Parameter Systems
y
2008/11/17
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- Circuits and Systems Chapter Meeting
32
Switched Reluctance Motors
Rotor
Stator
2008/11/17
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Switched Reluctance Motors
A
A1
2
6
3
5
4
4-phase, 8/6-pole SRM
2008/11/17
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34
Switched Reluctance Motors
料
Stator 導 磁 材Rotor
reluctance
磁 阻force
力
Magnetic
flux
磁力線
阻力
Reluctance磁force
L (θ )
θ
θ
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Switched Reluctance Motors
(m H )
La
30
R
Lb
Lc
Ld
a
L
10
a'
7 .5 1 5 2 2 .5 3 0 3 7 .5 4 5 5 2 .5 6 0
(D eg ree)
V j = L j (θ )
di j
dt
+
dL j (θ )
dt
i j + R j i j , j = A, B, C , D
1 dL j (θ ) 2
Torque ∝ i : T j =
ij
2 dθ
2
2008/11/17
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Minimum-Time Torque Control
Vdc
Q3
Q1
Sensor 1
Sensor 2
A
C
Q2
B
D
System dynamics:
di
= − A(θ , ω )i + B (θ )V (t )
dt
Sliding surface:
Q4
s = i − ir = 0
Drive Circuit
Minimum-Time VSC:
CPLD
XC9536
⎧−Vdc , if i (t ) > ir
V (t ) = ⎨
⎩ Vdc , if i (t ) < ir
Drive
encoder
V * = −K
V* = K
current sensor
i (t 0 )
Control System Setup
2008/11/17
ir
i (t 0 )
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37
Minimum-Time Torque Control
2008/11/17
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Temperature Control
- A Plastic Extrusion Process
System setup: ITRI (Industrial Technology Research Institute), Taiwan, R.O.C.
2008/11/17
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Temperature Control
- A Plastic Extrusion Process
Dynamics of Channel temperature:
dy
hA
hA
1
y+
y∞ +
w
=−
dt
ρ cV
ρ cV
ρ cV
Rate of change of energy input:
⎧a + ( w, u , t ), u ≥ 0
dw
= a ( w, u , t ) = ⎨ −
dt
⎩ a ( w, u , t ), u < 0
Single-channel dynamics:
y = −my + v
a ( w, u , t )
v =
+ξ
ρ cV
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Temperature Control
- A Plastic Extrusion Process
Single-channel dynamics:
y = −my + v
a ( w, u, t )
v =
+ξ
ρ cV
Sliding surface (PI control):
KI
v = − K P e − K I e or V ( s ) = ( K P + )( ydesired ( s ) − Y ( s ))
s
Output feedback sliding mode control
s = e + ( K P + m)e + K I e
sk ≈ a1sk −1 + b0 ek + b1ek −1
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Temperature Control
- A Plastic Extrusion Process
⎧ heat, sk < 0
Switching control law: uk = ⎨
⎩ water, sk > 0
2008/11/17
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Rod-less Pneumatic Cylinder Servo
2008/11/17
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43
Rod-less Pneumatic Cylinder Servo
Equation of motion:
M LY = − μuY − μc sgn Y + AT ΔP
Pressure dynamics:
ΔP = ψ 1 ( P1 , P2 , Y , X ) X +ψ 2 ( P1 , P2 , Y , Y , X )
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Rod-less Pneumatic Cylinder Servo
Sliding surface:
M L 1
⎡⎣Yd + k1 (Yd − Y ) + k2 (Yd − Y ) ⎤⎦ +
⎡⎣ μuY + μc sgn Y ⎤⎦
σ = ΔP −
AT
AT
Variable Structure Control: X = − X v sgn(σ )
2008/11/17
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Wireless Network Power Control
Zoran Gajic, Plenary Lecture 2007 CACS International Automatic Control Conference
National Chung Hsing University, Taichung, Taiwan, November 9-11, 2007
2008/11/17
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- Circuits and Systems Chapter Meeting
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,
Wireless Network Power Control
Signal-to-interference ratio (SIR) for the ith user
gii (t ) pi (t )
gii (t ) pi (t )
γ i (t ) =
=
, i = 1, 2," , N .
I i (t )
∑ gij (t ) p j (t ) + υi (t )
j ≠i
Control objective: γ i (t ) → γ i
tar
Logarithmic scale representation
γ i = gii + pi − I i ( p−i )
System dynamics:
p i (t ) = ui (t ), i = 1, 2," N
γ i (t ) = pi (t ) + fi (t )
2008/11/17
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Wireless Network Power Control
Sliding surface si (t ) = γ i (t ) − γ
Discrete-time representation
tar
i
=0
si (k + 1) = si (k ) + Tui (k ) + d (k ) + v(k )
Discrete-time SMC (continuous)
1
1
ui (k ) = − si (k ) − (d (k − 1) + v(k − 1))
T
T
Stability analysis
⎡ 1
s
(
k
1)
+
⎡ i
⎤ ⎢
⎢u (k + 1) ⎥ = ⎢ 1
⎣ i
⎦ −
⎣ T
⎡z −1
det ⎢ 1
⎢
⎣ T
T⎤
⎥ ⎡ si (k ) ⎤ − ⎡1 ⎤ 1 (d (k ) + v(k ))
⎢
⎥ ⎢ ⎥
−1⎥ ⎣ui (k ) ⎦ ⎣ 2 ⎦ T
⎦
−T ⎤
⎥ = z2 = 0
z + 1⎥
⎦
2008/11/17
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Contents (2/2)
Minimum-Time Torque Control (SRM)
y Temperature Control (Plastic Extrusion P.)
y Rod-less Pneumatic Cylinder Servo
y Wireless Network Power Control
y
Singular Perturbations
y Distributed Parameter Systems
y
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
49
Extended Research Problems
y Singular
Perturbations
x(0)
u+
u
−
Sliding Surface: s = 0
Singularly Perturbed Systems
VSS
Quasi-steady state (slow mode)
Sliding mode
Fast mode
Reaching phase
Continuous control
Discontinuous control
Infinite-time reaching
(boundary layer)
Finite-time reaching
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
50
Extended Research Problems
y Distributed
Parameter Systems
Heat Conduction System :
U t ( x, t ) = αU xx ( x, t ) + βU ( x, t )
Boundary conditions U (0, t ) = 0
U (l , t ) = Q(t ) + f (t )
x=0
x=l
Kernel function
k ( x, y )
wt = α wxx − cw
U ( x, t )
w( x, t )
2008/11/17
w(0, t ) = 0
w(l , t ) = Q(t ) + f w (t )
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
51
Extended Research Problems
y Distributed
Parameter Systems
Sliding surface:
S (t ) = ω x (l , t ) = 0
x=l
x=0
Kernel function : k ( x, y ) = −λ y
I1
(
λ ( x2 − y 2 )
)
λ ( x2 − y 2 )
wt = α wxx − cw
U ( x, t )
w( x, t )
w(0, t ) = 0
w(l , t ) = Q(t ) + f w (t )
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
52
Extended Research Problems
y Distributed
Parameter Systems
Sliding surface: S (t ) = wx (l , t ) = 0
l
S (t ) = U x (l , t ) − k (l , l )U (l , t ) − ∫ k x (l , y )U ( y, t ) dy
0
Sliding Mode Control:
t
Q(t ) = − K m ∫ sgn( S (τ ))dτ
0
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
53
Conclusions
y
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
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
54
Conclusions
y
Chattering Reduction (Elimination)
◦ Discrete-Time Sliding Mode
◦ Filtering methods
y
Interesting Problems
◦ Output feedback
◦ Singular Perturbations
◦ Infinite-dimensional systems
◦ Systems with delays (e.g. wireless network)
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
55
Thank You!
http://www.chneic.sh.cn/
2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting
56