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- 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