- 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

n

m

x

x

00 uxs −=> ,

00 uxs =< ,0=+= xcxs

■ Concept of Sliding Mode ( Second order relay system )

const :, , ),sgn(,

00 cuxcxssuuux

+=−== 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

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

>=

ss

2

Scalar Control: 0lim0lim00

and ><−→+→ss

ss

s=0Vector Control

. 2 2

212

211

ssignssignsssignssigns

−−=+−=

Trajectories should be oriented towards the switching surface

const :, , ),sgn(,

00 cuxcxssuuux

+=−==

R

( ) 0

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

T T

T Ts x

grad s bu x grad s f xgrad s bu x grad s f x

+

−=

+ <+ >

Variable Structure DesignApproaches

nn Varying Structures for Varying Structures for StabilizationStabilization

nn Use of Singular TrajectoriesUse of Singular Trajectoriesnn SLIDING MODESSLIDING MODES

0,, , ),sgn( ,

>+=−=

=−

ckaxcxssxkuuaxx

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

1 2 0cc <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()(

ubabuxaxaxax

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

sxksxk

k

.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

<>

=xskxsk

k

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 1964 IFAC Sensitivity Conference

Dubrovnik 1964 IFAC Sensitivity Conference

Dubrovnik 1964 IFAC Sensitivity Conference