- Sliding mode control is a variable structure control method where the control signal is switched between two values based on the sign of a switching function (s). This causes the system trajectories to slide along the switching surface (s=0). - In sliding mode, the system motion is governed by a reduced order sliding mode equation that depends only on the designer-selected parameter c, not the plant dynamics. - For sliding mode to exist, the system trajectories must be oriented towards the switching surface. Adaptive sliding mode control can vary the parameter c depending on system parameters to increase the rate of decay in sliding mode.

2. - Introduction (prehistory)
- Discrete-time sliding modes
- Observers and estimators
- Chattering problem
- High order sliding modes

10. IntroductIon of
SlIdIng Mode
control
First Stage – Control in Canonical
Space

11. Introduction of Sliding Mode Control
■ Concept of Sliding Mode ( Second order relay system )
 = u ,
x Upper semi-plane : s > 0 → u = −u0 →  = −u0
x
u = −u0 sgn( s ), s = cx + x, u0 , c : const
 Lower semi-plane : s < 0 → u = u0 →  = u0
x
• 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
e
od
x

s > 0 ,  = − u0
x M g
m In sliding mode,
din
the system motion is
Sli
(1) governed by 1st order
x equation (reduced order).
(2) depending only on ‘c’ not
plant dynamics.
s = cx + x = 0

n
s < 0 ,  = u0
x Slid
in
E qu g M o d
atio e
n

12. Mathematical Aspects II
Sliding Mode Existence Conditions
Scalar Control:
Vector
Control s =0
Trajectories should be 2 s=0
oriented towards the
2
switching surface
 = u ,
x 3 1 s1=0
u = −u0 sgn( s ), s = cx + x, u0 , c : const

R [ grad ( s)]T bu + ( x) + [ grad ( s )]T f ( x) < 0
[ grad ( s)]T bu − ( x) + [ grad ( s)]T f ( x ) > 0 s ( x )= 0

14. Variable Structure Design
Approaches
Varying Structures for
Stabilization
Use of Singular Trajectories
SLIDING MODES

16. Introduction of Sliding Mode Control
■ Concept of Sliding Mode ( Variable Structures System )
 − ax = u,
x 1 If s > 0, x < 0 or s < 0, x > 0 then  − ax = kx
x
u = − k x sgn( s), s = cx + x, a, k , c > 0

2 If s > 0, x > 0 or s < 0, x < 0 then  − ax = − kx
x
x
 x

x x
c0 x + x = 0

1  − ax = kx
x 2  − ax = − kx
x
State planes of two unstable structures

17. 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
x equation (reduced order).
(2) depending only on ‘c’ not
plant dynamics.
s = 0 or
cx + x = 0

s < 0, x > 0
c < c0
2 s < 0, x < 0 1
c0 x + x = 0

State planes of Variable Structure System

18. SLIDING MODE CONTROL
• Order of the motion
equation is reduced
• Motion equation of sliding
mode is linear and
homogenous.
• Sliding mode does not
Motion Equation depend on the plant
dynamics and is
determined by parameter
C selected by a designer.

20. VSS in Canonical Space
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.

21. VSS in Canonical Space

22. Adaptive VSS
The rate of decay in
sliding mode may be
increased by varying
the gain C depending
on b.

23. 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.

25. Dubrovnik 1964
IFAC Sensitivity Conference

26. Dubrovnik 1964
IFAC Sensitivity
Conference

27. Dubrovnik 1964
IFAC Sensitivity
Conference