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.

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 1
Prelude
A Control Engineer as an Archer…

to drive the state to the target.

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 2
 3

賣油翁 宋 歐陽修
Ouyang Show(1007~1072A.D.)

Princeton/Central Jersey Section of IEEE

- Circuits and Systems Chapter Meeting

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。
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。﹂
The Old Oil-Peddler

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，

http://baike.baidu.com/ 2008/11/17
惟手熟爾。﹂康肅笑而遣之。
Sliding Mode: an illustration

to fill oil into a bottle through a funnel

Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 4
A “funnel-like” domain

⎧ x1 = x2
System: ⎨ x2
⎩ x2 = 2 x1 − 3 x2 + u
Target: ( x1 , x2 ) = (0, 0)
x1
Funnel: x2 = − x1

New target: s = x1 + x2 = 0 s=0

Princeton/Central Jersey Section of IEEE
2008/11/17 - 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
v >0⇒m = −3 − 2 < 0 ⇒ v ↓
dt
dv http://www.physics4kids.com
v < 0 ⇒ m = 3− 2 > 0 ⇒ v ↑
dt

sliding mode: v ≡ 0 (if f a < K ).

Princeton/Central Jersey Section of IEEE

2008/11/17 - 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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 7
Contents (2/2)
y 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

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 8
 A “funnel-like” domain (cont.)
⎧⎪ x1 = x2 disturbance
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 s=0

Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 9
 A “funnel-like” domain (cont.)
 = Parameter
⎧⎪ 1x x2
System: ⎨ variations
⎪⎩ 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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 10
Introduction to Sliding mode
- Variable Structure Systems
⎧ x1 = x2 (K. D.Young, 1978)
System : ⎨
⎩ x2 = ax2 − bu

Switching line : s = cx1 + x2 = 0

(Sliding Surface)

⎧α x1 , if x1s > 0 (region I)

Control : u = ⎨
⎩−α x1 , if x1s < 0 (rigion II)
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 11
Introduction to Sliding mode
- Variable Structure Systems
⎡ 0 1⎤
Structure I : x = ⎢ ⎥ x
⎣ −bα a ⎦
s 2 − as + bα = 0 : spiral out

⎡ 0 1⎤
Structure II : x = ⎢ ⎥ x
⎣bα a ⎦
s 2 − as − bα = 0 : saddle point

Princeton/Central Jersey Section of IEEE

2008/11/17 - 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 )
⎣⎢ 2 ⎦⎥ ⎣⎢ −1⎦⎥ ⎢⎣ 3 ⎥⎦
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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.
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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⎥⎦
.
If       is known, then the control law
d (t )
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⎥⎦
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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 iff
rank [ B | E ] = rank [ B ] .
In the previous example,
⎡0 2 2⎤ ⎡0 2⎤
⎢ ⎥
rank ⎢ −1 1 −1⎥ = rank ⎢⎢ −1 1 ⎥⎥ = 2.
⎢⎣ 2 −1 3 ⎥⎦ ⎢⎣ 2 −1⎥⎦
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 16
 A “funnel-like” domain (revisit)
⎧⎪ x1 = x2 disturbance
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 s=0

Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 17
 A “funnel-like” domain (revisit)
 = Parameter
⎧⎪ 1x x2
System: ⎨ variations
⎪⎩ 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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 18
VSS Design
⎧ x1 = x2 (K. D.Young, 1978)
System : ⎨
⎩ x2 = ax2 − bu

Switching line : s = cx1 + x2 = 0

(Sliding Surface)

⎧α x1 , if x1s > 0 (region I)

Control : u = ⎨
⎩−α x1 , if x1s < 0 (rigion II)
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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, lim− s > 0
s →0 s →0

+ Sliding Surface: s = 0
u

u− 2008/11/17
Princeton/Central Jersey Section of IEEE
- Circuits and Systems Chapter Meeting 20
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

V (0) s (0)
Reaching time T< =
σ σ

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 21
 VSS Design
A heuristic example
d
( D + a3 D + a2 D + a1 ) y (t ) = (b3 D + b2 D + b1 )(u + f (t )), D =
3 2 2

dt
⎡ 0 1 0 ⎤ ⎡0⎤ ⎡0⎤
x = ⎢⎢ 0 0 1 ⎥⎥ x + ⎢⎢ 0 ⎥⎥ u + ⎢⎢0 ⎥⎥ f (t )
⎢⎣ − a1 − a2 − a3 ⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣1 ⎥⎦
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 Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 22
 VSS Design
ds ( x) ∂s dx
(ii) =
dt ∂x 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 .
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 24
 VSS Design
- Sliding Surface
x = A x + B u + D f ( t )
Eigenspace Approach v.s. Lyapunov Approach
Nonsingular transformation Stabilizing feedback
⎡0( n − m )×m ⎤ ⎡ x1 ⎤ x = A x − B K x + D f ( t )
TB = ⎢ ⎥ , Tx = ⎢ ⎥
⎣ B2 ⎦ ⎣ x2 ⎦ Lyapunov function
Canonical form PAs + AsT P = −Q
⎡ x1 ⎤ ⎡ A11 A12 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤ ⎡0⎤ 1 T
⎢ x ⎥ = ⎢ A +
A22 ⎥⎦ ⎢⎣ x2 ⎥⎦ ⎢⎣ B2 ⎥⎦
u + ⎢ ⎥ f (t )
V ( x) = x Px
⎣ 2 ⎦ ⎣ 21 ⎣ D2 ⎦
2
Sliding surface Sliding surface
x2 = − Kx1 , or s( x ) = [ K I m×m ] Tx s ( x ) = BT Px
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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) s
u = −(CB) −1 CAx − (CB) −1 (d max + σ ) ⋅
s
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 26
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 T
dk = ∫
kT
e A(t −τ ) Df (τ )dτ = ∫ e Aλ Df ((k + 1)T − λ )d λ
0
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 27
 Discrete-Time Sliding Mode
xk +1 = Φxk + Γuk + d k
T 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 )
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 28
 Discrete-Time Sliding Mode
Quasi-sliding mode sk = O(T ), s (t ) = O(T )

Discontinuous control

Discrete-time sliding mode k

s = 0, s (t ) = O (T 2
)

Continuous control

Princeton/Central Jersey Section of IEEE

2008/11/17 - 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 )
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 31
Contents (2/2)
y 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

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 32
Switched Reluctance Motors

Rotor Stator
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2008/11/17 - Circuits and Systems Chapter Meeting 33
Switched Reluctance Motors
A
A-

1
2 6

3 5
4

4-phase, 8/6-pole SRM

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 34
 Switched Reluctance Motors
Stator 導 磁 材Rotor
料

reluctance
磁 阻force
力

Magnetic
磁力線
flux

Reluctance磁force
阻力

L (θ )

θ
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 35
Switched Reluctance Motors
(m H )
R La Lb Lc Ld
30
a

a' 10
7 .5 1 5 2 2 .5 3 0 3 7 .5 4 5 5 2 .5 6 0
(D eg ree)

di j dL j (θ )
V j = L j (θ ) + i j + R j i j , j = A, B, C , D
dt dt

1 dL j (θ ) 2
Torque ∝ i : T j = 2
ij
2 dθ

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 36
 Minimum-Time Torque Control
Vdc
System dynamics:
Q1 Q3
di
Sensor 1 Sensor 2 = − A(θ , ω )i + B (θ )V (t )
A C B D
dt
Q2 Q4
Sliding surface:
s = i − ir = 0
Drive Circuit
Minimum-Time VSC:
CPLD ⎧−Vdc , if i (t ) > ir
XC9536 Drive V (t ) = ⎨
⎩ Vdc , if i (t ) < ir
encoder
V* = K V * = −K
current sensor i
i (t 0 ) ir i (t 0 )

Control System Setup

Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 37
Minimum-Time Torque Control

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 38
Temperature Control
- A Plastic Extrusion Process

System setup: ITRI (Industrial Technology Research Institute), Taiwan, R.O.C.

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 39
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:
dw ⎧a + ( w, u , t ), u ≥ 0
= 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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 40
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
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 41
Temperature Control
- A Plastic Extrusion Process
⎧ heat, sk < 0
Switching control law: uk = ⎨
⎩ water, sk > 0

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 42
Rod-less Pneumatic Cylinder Servo

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2008/11/17 - Circuits and Systems Chapter Meeting 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 )

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 44
 Rod-less Pneumatic Cylinder Servo
Sliding surface:
M L    1
σ = ΔP − ⎡⎣Yd + k1 (Yd − Y ) + k2 (Yd − Y ) ⎤⎦ + ⎡⎣ μuY + μc sgn Y ⎤⎦
AT AT
Variable Structure Control: X = − X v sgn(σ )

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 45
 Wireless Network Power Control

Zoran Gajic, Plenary Lecture 2007 CACS International Automatic Control Conference
National Chung Hsing University, Taichung, Taiwan, November 9-11, 2007
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 46
, 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 )
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 47
Wireless Network Power Control
Sliding surface si (t ) = γ i (t ) − γ =0
tar
i

Discrete-time representation
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 T⎤
⎡ i
s ( k 1) ⎤ ⎢ ⎥ ⎡ si (k ) ⎤ − ⎡1 ⎤ 1 (d (k ) + v(k ))
⎢u (k + 1) ⎥ = ⎢ 1 ⎢ ⎥ ⎢ ⎥
⎣ i ⎦ − −1⎥ ⎣ui (k ) ⎦ ⎣ 2 ⎦ T
⎣ T ⎦
⎡z −1 −T ⎤
det ⎢ 1 ⎥ = z2 = 0
⎢ z + 1⎥ Princeton/Central Jersey Section of IEEE
⎣ T ⎦ 2008/11/17 - Circuits and Systems Chapter Meeting 48
Contents (2/2)
y 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

Princeton/Central Jersey Section of IEEE

2008/11/17 - Circuits and Systems Chapter Meeting 49
Extended Research Problems
y Singular Perturbations
x(0)

u+

−
Sliding Surface: s = 0
u
Singularly Perturbed Systems VSS
Quasi-steady state (slow mode) Sliding mode
Fast mode Reaching phase
Continuous control Discontinuous control
Infinite-time reaching Finite-time reaching
(boundary layer)
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2008/11/17 - 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 )
x=l

Boundary conditions U (0, t ) = 0

x=0

U (l , t ) = Q(t ) + f (t )
Kernel function
k ( x, y )
wt = α wxx − cw
U ( x, t ) w( x, t ) w(0, t ) = 0
w(l , t ) = Q(t ) + f w (t )
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 51
Extended Research Problems
y Distributed Parameter Systems
Sliding surface:
S (t ) = ω x (l , t ) = 0 x=0 x=l

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 )
Princeton/Central Jersey Section of IEEE
2008/11/17 - 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

Princeton/Central Jersey Section of IEEE

2008/11/17 - 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

Princeton/Central Jersey Section of IEEE

2008/11/17 - 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)
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 55
Thank You!

http://www.chneic.sh.cn/
Princeton/Central Jersey Section of IEEE
2008/11/17 - Circuits and Systems Chapter Meeting 56