-

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

Introduction of Sliding Mode ControlFirst Stage – Control in Canonical Space

n

m

x

x

00 uxs ,

00 uxs ,

0 xcxs

Slid

ing

Mod

e

■ Concept of Sliding Mode ( Second order relay system )

const :, , ),sgn(

,

00 cuxcxssuu

ux

000 uxuus Upper semi-plane :

000 uxuus Lower semi-plane :

• 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 0 xcxs : sliding mode equation

Introduction of Sliding Mode Control

Sliding Mode Equation

In sliding mode,the system motion is(1) governed by 1st order

equation (reduced order).(2) depending only on ‘c’ not

plant dynamics.

Mathematical Aspects IISliding Mode Existence Conditions

0)( ,0)( 21 TsTs0)"( ,0)"( 21 TsTs

1 s1=03

s2=0

0)0(

0)0(

2

1

s

s

2

Scalar Control: 0lim0lim00

and ss

ss

s=0Vector Control

. 2

2

212

211

ssignssigns

ssignssigns

Trajectories should be oriented towards the switching surface

const :, , ),sgn(

,

00 cuxcxssuu

ux

R

( ) 0

[ ( )] ( ) [ ( )] ( ) 0

[ ( )] ( ) [ ( )] ( ) 0

T T

T T

s x

grad s bu x grad s f x

grad s bu x grad s f x

Variable Structure DesignApproaches

Varying Structures for Varying Structures for StabilizationStabilization

Use of Singular TrajectoriesUse of Singular Trajectories

SLIDING MODESSLIDING MODES

0,, , ),sgn(

,

ckaxcxssxku

uaxx

kxaxx

x

x

kxaxx

x

x

00 xxc

kxaxxxsxs then00or00If ,,1

kxaxxxsxs then00or00If ,,2

1 2

Introduction of Sliding Mode Control

■ Concept of Sliding Mode ( Variable Structures System )

State planes of two unstable structures

In sliding mode,the system motion is(1) governed by 1st order

equation (reduced order).(2) depending only on ‘c’ not

plant dynamics.

• 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

• After sliding mode starts, further motion is governed by 0 xcxs : sliding mode equation

: sliding mode

Introduction of Sliding Mode Control

State planes of Variable Structure System

x

x

00 xs , 00 xs ,

00 xs , 00 xs ,

1 2

120cc

0

or0

xcx

s

00 xxc

SLIDING MODE CONTROL

• 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.

*0 cc

.0 cxxMotion Equation

VSS in Canonical Space

The methodology, developed for second-ordersystems, was preserved:

- sliding mode should exist at any point of switchingplane, then it is called sliding plane.

- sliding mode should be stable- the state should reach the plane for any initial conditions.

input. control is ,parametersplant are ,

,... 12)1()(

uba

buxaxaxax

i

nn

n

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).

VSS in Canonical SpaceVSS in Canonical Space

input. control is ,parametersplant are ,

,

1,...,1

1

1

uba

ubxax

nixx

i

n

iiin

ii

,1kxu

0 if

0 if

12

11

sxk

sxkk

.1 const, ,01

ni

n

iii ccxcs

Adaptive VSS

The rate of decay in sliding mode may be increased by varying

the gain C depending

on b.

,)( utbx

,kxu

0 if

0 if

2

1

xsk

xskk

maxmin )( btbb

12

2 bkcbk 0 cxxs

Adaptive VSS, State PlaneE.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 1964IFAC Sensitivity Conference

Dubrovnik 1964IFAC SensitivityConference

Dubrovnik 1964IFAC SensitivityConference