IntroductIon of SlIdIng Mode control Second Stage – Multidimensional Sliding Modes

Sliding mode in discontinuity surfaces and their intersection

c a

d

b

0=)(xs j

0=)(xsi

SECOND STAGEFinite Dimensional MIMO Systems

PlantController

Disturbance

Output

ControlReferenceInput

),,( utxfx = nRx∈ mRu∈

=<>

=−

+

mixstxuxsxu

uii

iii ,...,1 ,

0)( if ),(0)( if t),(

),(t),( txuxu ii−+ ≠

>++=−=

+=−=

+=++=

=

0,)sgn(esdisturbancboundedunknownare,)sgn(

where)(

)(213212222

21211111

223

1132

21

MMxxxssMuffxxssMu

utfxutfxx

xx

22211132

3212

)()( signsMtfsignsMtfxxxxxs

−+−++=++=

signsdifferent haveand)()(If

22

121322

ssMtftfxxM

++++>

213

3212 0

xxx

xxxs

−−=

→++=

( )21111111212

21 )sgn()(

xxssMuwhereutfxxx

xx+=−=

++−−==

1111

211

)( signsMtfxxxs

−+−=+=

signs different haveand)(If

11

111

sstfxM

+−>

12

212 0

xx

xxs

−=

→+=

{ 11 xx −=

■ Two dimensional Sliding Mode

Introduction of Sliding Mode Control

In two dimensional sliding mode, the system motion is (1) of order which is by two less

than the order of the original system.

(2) not depending on the disturbances f1 and f2 .

Two dimensional sliding mode

02 =S

01 =S

3x

2x

1x

11 xx −=

Introduction of Sliding Mode Control

■ Multi-dimensional Sliding Mode

Sliding mode in discontinuity surfaces and their intersection

c a

d

b

0=)(xs j

0=)(xsi

=<>

=

∈∈+=

−

+

mxstxuxstxuu

RuRxutxBtxfx

ii

iii

mn

,...,1i 0)( if ),(

0)( if ),(

,,),(),(

• In general case,

1

( ) 0 ( 1,.., ),

( ) 0, ( ,..., ).i

Tm

s x i m or

in manifold s x s s s

= =

= =

• Sliding mode occurs in the intersection of all m surfaces,

• The order of motion equation is by ‘m’ less than the order of original system

■ Outline of Sliding Mode Methodology

• Order reduction of the system

• Insensitivity with respect to parameter variations and disturbances

Enables a designer to simplify and decouple the design procedure.

s u = u0sign(s) s

u

u+

u-

s u u+

u-

Uav

)( 0→∞≈= ss

UK aveq

Introduction of Sliding Mode Control

Attained by finite control actions, unlike continuous high gain controls in conventional system.

Multidimensional Sliding Modes(Late Sixtieth)

mnm

T sxsxsxsxs −ℜ∈=== )0 ( )],(),...,([)( ,0)( 1

MatheMatIcal ProbleMS:

• Sliding Mode Equations • Existence Conditions

Mathematical Aspects ISliding Mode Equations

Sliding Mode Equation on s=0 ??? ),( , 0 cxxsssignuuux +=−== 0=+ cxx WHY ???

01 =b 111

22 xccx −−=).()( 111

1212111 tfdxccaax +−= −

Yu. Kornilov (1950),A. Popovski (1950)

. ),( 2211 xcxcssMsignu +=−=)()(

222221122

112121111

tfdubxaxaxtfdubxaxax

+++=+++=

Ya. Dolgolenko (1955), (0,0,...,0,1).Tx Ax bu b= + =

FILIPPOV”s

METHOD (1958):

( , ), ( , ) [ ( , , )]

sm

smu

x f x tf x t f x t uconv=

∈

fIlIPPov Method 1958

MatheMatIcal aSPectS I SlIdIng Mode equatIonS (cont).

A.F. Filippov, Application of the theory of differential equations with discontinuous right-hand sides to non-linear problems of automatic control, Proceedings of 1st IFAC Congress in Moscow, 1960, Butterworths, London, 1961.

grad s xn

x1

s(x)=0

fsm

−f

<>

==−

+

0)( if 0)( if

)( ),(xsfxsf

xfxfxConvex Hull fsm belongs to convex hull

Yu.I. Neimark, Note on A. Filippov’s paper, 1st IFAC Congress. dx/dt=Ax+bu+dv, u=-sign(s), v=-sign(s), s=cx

Nonuniqueness !?

+f

dx/dt=fsm

Professor N.

Equivalent Control MethodEquivalent Control Method

( , , )x f x u t=Control system

Discontinuous control u enforces sliding mode inmanifold s=(s1,…,sm )=0. }.{)( xsxG ∂∂=Then solution to ( , , ) 0 = ( , )eq eqs Gf x t u u u x t= = ⇒is substituted to system equation

( , , )eqx f x t u=s(x)=0 x2=s0(x1) , 02

msx ℜ∈ m-n1 ℜ∈x :

. )],(,,[ 110111mnfxstxfx −ℜ∈=

non-unIqueneSS, vector caSe

RegularizationA.A. Andronov (1954): Ideal relay is replaced by elementwith delay and then the value of delay is tended to zero.J. Andre and P. Seibert (1956): both delay and hysteresic.

u M

-M

a-as

1

2

3

s=-a

s=as=0

Lim[x(a,t)]=solution to Filippov’s equationwith parameter ‘a’ tending to zero.

c a

d b

0=)(xs j

0=)(xsi

RegularizationBoundary Layer

trajectorystate 0)( =xsmanifold

0)( =xsm

0)(1 =xs

0)( if )(0)( if )(

)(

<>

=−

+

xsxuxsxu

xuii

iii

2 ( , ) ( , ), ( ) , s ( ).Tu x x t s x s s∆ ⇒ ∆ ≤ ∆ =

)(),(~ 0

txtxmil =∆→∆

,)( ,, ),,(

mn xufxuxfx

ℜ∈ℜ∈

=

∆≤)(

xslayerboundary

Equivalent Control MethodEquivalent Control Method

utxBtxfx ),(),( +=Affine system

Discontinuous control u enforces sliding mode inmanifold s=(s1,…,sm )=0. ,0)det( ≠GB }.{)( xsxG ∂∂=Then solution to .)-(= ,0 -1GfGBuGBuGfs eq=+=is substituted to system equation

GfGBBfx -1)(= −s(x)=0 x2=s0(x1) , 02

msx ℜ∈ m-n1 ℜ∈x :

. )],(,,[ 110111mnfxstxfx −ℜ∈=

E.C.M. is substantiated by boundary layer regularization

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

−−=+−=

c a

d b

0=)(xs j

0=)(xsi

( ) ( )[ ]xsxsxs mT ,,)( 1 =

matrix is nmxsG ×

∂∂

=

Stability of the origin in subspace S

Motion Equation:

GBuGfs +=

( ) ( )( ) ( )

. ,

0 if0 if

mm RuRs

xsxuxsxu

u

∈∈

<>

=−

+Component-wise

PIece-vISe lyaPunov functIonS I

Motion Equations:

)()(2),(2)(

212

211

ssignssignsssignssigns

−−=+−=

Lyapunov Function:

=

==

)()(

)(

),,( ),(

2

1

21

ssignssign

ssign

sssssignsV TT

Time Derivative:

222

11

−=∂∂

+∂∂

= ssVs

sVV

Sliding Mode Exists in s=0.

0)( ,0)( 21 =′>′ TsTs0)"( ,0)"( 21 <= TsTs

1 s1=0 3

s2=0

0)0(0)0(

2

1

>=

ss

2

PIece-vISe lyaPunov functIonS II

Lyapunov Function:

Time Derivative:

)()(2),()(2

212

211

ssignssignsssignssigns

+−=−−=

Motion Equations:

.1004

,)()(

)(

),,( ),(

2

1

21

=

=

==

Pssignssign

ssign

ssssPsignsV TT

.0)(67 2122

11

<−−=∂∂

+∂∂

= sssignssVs

sVV

Existence ConditionsLyapunov ApproachStability in subspace s: GBuGfs +=Finite-Time Convergence !!!Equivalent to Equation:

)](),...,([)]([ 1 mT ssignssignssign =

)()()( ssignxDxds α−=

Theorem. If 0>+ TDD then there exists such thatsliding mode exists in manifold

00 >α

0αα >0=s for .The statement of the theorem may be proven using sum of absolute values of

asa Lyapunov function.

is0)( >= ssignsV T

Sliding Mode Control DesignSliding Mode Control DesignSliding Mode Equation- is of reduced order- does not depend on control- depends on the equation of switching surfaces.

Design is decoupled into two independent subproblems:- design of the desired dynamics for a system of the (n-m)-thorder by a proper choice of a sliding manifold s=0;- enforcing sliding motion in this manifold which is equivalent to stability problem of the m-th order system.

. )],(,,[ 110111mnfxstxfx −ℜ∈=

Sliding Mode Control DesignSliding Mode Control DesignInvarianceInvariance

),(),(),( txhutxBtxfx ++=),( txh is disturbance vector

)(),( Brangetxh ∈Matching Condition

B. Drazenovic,The invariance conditions in variable structure systems, Automatica, v.5, No.3, Pergamon Press, 1969.

Design, Regular Form

.0)det( , ,),,(),,(

),,,(

222122122

12111

≠∈+=

∈= −

BRxutxxBtxxfx

Rxtxxfxm

mn

)( 102 xsx −= - psuedo control

0)(),( 10221 =+= xsxxxs - sliding manifold

]),(,[ 10111 txsxfx −= - sliding mode equation

depends neither nor),,( 212 txxf ),,( 212 txxB .

Reducing the System to the Regular Form

■ Regular Form

)( 102 xsx −=

+==

utxxBtxxfxtxxfx

),,(),,(),,(

2122122

2111

matrixmmr nonsingula :,,where

2

21

×ℜ∈ℜ∈ −

Buxx mmn

the first block does not depend on control

dimension second block = dimension of control

• Regular Form consists of two blocks

• In the Regular Form, the design procedure of sliding mode becomes much simpler.

▪ State x2 in the first block is handled as ‘intermediate control’ and selected as a function of the state x1 according to some performance criterion

0)()( 102 →+= xsxxsEquivalent to

▪ Discontinuous control u is selected such that the function have different signs and enforce sliding mode in the manifold s(x)=0

ss and

( )txsxfx ),(, 10111 −=

▪ After sliding mode occurs, further system motion is governed by the sliding mode equation

( ) ( ) 0 ,sgn 21 =+=−= xsxSSMu

( )( )txsxfx ,, 2222 =

( )21 xsx −=

( )XTxx

=

1

2

Regular Form

( )( ) ( )

( ) 0det, ,

,,,,,,

12

1211211

1222

≠∈∈

+==

− BRuRxutxxBtxxfx

txxfx

mmn

( ) ( ) mBRuRXuXBtXfX mn =∈∈+= Rank , , ,),(

Design, Frobenius TheoremReducing to Regular Form

.0)det( , ,),,(),,(

,),,(),,(

222122122

12112111

≠∈+=

∈+= −

BRxutxxBtxxfx

RxutxxBtxxfxm

mn

Coordinate Transformation: ),(1 xy ϕ= , ,1mny −ℜ∈ϕ

tBu

xf

xy

∂∂

+∂∂

+∂∂

=ϕϕϕ

1 ,

.0=∂∂ B

xϕ Frobenius Theorem

(Louk’yanov, 1981)

• To find a coordinate transformation for this system, suppose a system of ordinary differential equations,

0)(and),..,1(,)()( ≠=+= xbniuxbxfx niii

0)(),,..,1()( ≠== xbnixbx nii

)1,..,1(,)()(

−=== nixbxb

dxdx

xx

n

i

n

i

n

i

then we may have an (n-1)th order auxiliary system,

This implies that xi should be represented as

reducIng the SySteM to the regular forM

■ Nonlinear Coordinate Transformation for the Regular Form • Consider a nonlinear system with a scalar control,

constants)nintegratio (),,..,1(,),..,,,( 121 === − innii cnicccxx δ

)1,..,1(),,..,,( 21 −== nixxxc nii φand solve these equations with respect to constants ci , then

Differentiating both sides of equation w.r.t. xn yields

)1,..,1(,0...)()(

)()( and 0... 2

2

1

1

2

2

1

1−==

∂∂

++⋅∂∂

+⋅∂∂

=∂∂

++∂∂

+∂∂ ni

xxbxb

xxbxb

xdxdx

xdxdx

xdxdx

x n

i

n

i

n

i

n

n

n

i

n

i

n

i φφφφφφ

(2.1)

)1,..,1(,)( −== nixy ii φthen new state vector (y1 , . . , yn-1 , xn) is expressed as

( ) ( ) ( )

( )

−=+==

+==∂∂

++∂∂

+∂∂

==

)1,..,1()()()(grad

)(grad)(gradgrad...0

22

11

niuxbxfxxf

uxbxfxxx

xx

xx

y

nnn

Ti

Ti

Ti

Tin

n

iiii

φ

φφφφφφ

reducIng the SySteM to the regular forM

Finally, we have a function such that )(xiφ

( ) )1,..,1(0)(grad or 0)(...)()( T2

21

1−===

∂∂

++∂∂

+∂∂ nixbxb

xxb

xxb

x inn

iii φφφφ

• Let us introduce this function as a nonlinear coordinate transformation for the system (2.1)

)(xiφ

By substituting the (x1 , . . , xn-1) as a function of (y1 , . . , yn-1 , xn), we have the ‘Regular Form’

+=

=

order 1:),(),(

order 1-n : ),(st**

th*

uxybxyfx

xyfy

nnnnn

n

Enforcing Sliding Modes

enforces sliding mode0)( >+ TGBGB

)( ),( ssigntxMu −=

).( equuGBs −=),( txM

.)(s* *),(),( sGBssigntxMu T=−=equtxM >),(

Finite time convergence in the subspace s:

If then there exists scalar functionsuch that control in manifold s=0.General case: Transformation of discontinuity surfaces

Only upper estimate of equivalent control is needed

Integral SlIdIng Mode

Can stabilize the system in the desired way

System with disturbances

ProbleM StateMent

Integral SlIdIng Mode deSIgn PrIncIPle

Sliding manifold: