- Introduction (prehistory)
- Discrete-time sliding modes
- Observers and estimators
Chattering
problem
- High order sliding modes
Introduction of
Sliding Mode
Control
First Stage – Control in Canonical
Space
Introduction of Sliding Mode Control
■ Concept of Sliding Mode ( Second order relay system )
x  u,
Upper semi-plane :
s  0  u  u0  x  u0
u  u0 sgn( s ), s  cx  x , u0 , c : const
Lower semi-plane :
s  0  u  u0  x  u0
• State trajectories are towards the line switching line s=0
• State trajectories cannot leave and belong to the switching line s=0
• After sliding mode starts, further motion is governed by
s  cx  x  0
: sliding mode
: sliding mode equation
x
m
s  0 , x   u0
In sliding mode,
the system motion is
(1) governed by 1st order
equation (reduced order).
(2) depending only on ‘c’ not
plant dynamics.
x
s  0 , x  u0
n
s  cx  x  0
Mathematical Aspects II
Sliding Mode Existence Conditions
Scalar Control: lim s  0 and lim s  0
s  0
Vector
s =0 be
Control
Trajectories
should
2
s (T )  0, s2 (T )  0
s  0
s=0
1
oriented
towards
the
2
s1 (T ")  0, s2 (T ")  0
switching surface
x  u,
3
1
s1=0
u  u0 sgn( s ), s  cx  x , u0 , c : const
s1  sign s1  2sign Rs2
s2  2sign s1  sign s2 .
[ grad ( s)]T bu  ( x)  [ grad ( s)]T f ( x)  0
s1 (0T)  0
T

[ grad ( s)] bu ( x)  [ grad ( s)] f ( x)  0 s ( x )0
s2 (0)  0
Variable Structure Design
Approaches
 Varying
Structures for
Stabilization
 Use of Singular Trajectories
 SLIDING MODES
Introduction of Sliding Mode Control
■ Concept of Sliding Mode ( Variable Structures System )
x  ax  u,
u  k x sgn( s),
s  cx  x , a, k , c  0
1
If s  0, x  0 or s  0, x  0 then x  ax  kx
2
If s  0, x  0 or s  0, x  0 then x  ax  kx
x
x
x
x
1
x  ax  kx
c0 x  x  0
2
State planes of two unstable structures
x  ax  kx
Introduction of Sliding Mode Control
• If c<c0, the state trajectories are towards the line switching line s=0
• State trajectories cannot leave and belong to the switching line s=0
: sliding mode
• After sliding mode starts, further motion is governed by s  cx  x  0
: sliding mode equation
x
1
s  0, x  0
2
s  0, x  0
In sliding mode,
the system motion is
(1) governed by 1st order
equation (reduced order).
(2) depending only on ‘c’ not
plant dynamics.
x
s  0 or
cx  x  0
2
s  0, x  0
1
s  0, x  0
State planes of Variable Structure System
c0 x  x  0
c  c0
SLIDING MODE CONTROL
0  c  c*
Motion Equation
x  cx  0.
• Order of the motion
equation is reduced
• Motion equation of sliding
mode is linear and
homogenous.
• Sliding mode does not
depend on the plant
dynamics
and
is
determined by parameter
C selected by a designer.
VSS in Canonical Space
x
( n)
 an x
( n 1)
 ...  a2 x  a1 x  bu,
ai , b are plant parameters, u is controlinput.
S.V. Emel’yanov, V.A.Taran, On a class of variable structure control systems, Proc.
of USSR Academy of Sciences, Energy and Automation, No.3, 1962 (In Russian).


The methodology, developed for second-order
systems, was preserved:
sliding mode should exist at any point of switching
plane, then it is called sliding plane.
sliding mode should be stable
the state should reach the plane for any initial
conditions.
VSS in Canonical Space
xi  xi 1
i  1,...,n  1
n
xn   ai xi  bu,
i 1
ai , b are plant parameters, u is controlinput.
u  kx1 ,
n
 k1 if x1 s  0
k
k 2 if x1 s  0
s   ci xi  0, ci  const, c n  1.
i 1
Adaptive VSS
x  b(t )u,
bmin  b(t )  bmax
u  kx,
 k1 if xs  0
k
k2 if xs  0
The rate of decay in
sliding mode may be
increased by varying
the gain C depending
on
b.
s  x  cx  0 bk2  c  bk1
2
Adaptive VSS, State Plane
E.N. Dubrovski, Adaptation principle in VSS, Proceedings of 2nd Bulgarian
Conference on Control, v.1, part 1, Varna, 1967 (In Russian).
While sliding mode exists the
gain C is increased until sliding
mode disappears.
Dubrovnik 1964
IFAC Sensitivity Conference
Dubrovnik 1964
IFAC Sensitivity
Conference
Dubrovnik 1964
IFAC Sensitivity
Conference