Second order sliding mode and adaptive observer for synchronization of a chaotic system: a comparative study

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Second order sliding mode and adaptive observer for synchronization of a chaotic system: a comparative

study

Francisco J. Bejarano, Malek Ghanes, Jean-Pierre Barbot, Leonid Fridman

To cite this version:

Francisco J. Bejarano, Malek Ghanes, Jean-Pierre Barbot, Leonid Fridman. Second order sliding

mode and adaptive observer for synchronization of a chaotic system: a comparative study. IFAC

World Congress, Jul 2008, Séoul, North Korea. �inria-00345027�

observer for synhronization of a haoti

system: a omparative study

Franiso J.Bejarano

∗

Malek Ghanes

∗

Jean-Pierre Barbot

∗

LeonidFridman

∗∗

∗

EquipeCommande desSystèmes(ECS),ENSEA 6Avenue du

Poneau, 95014Cergy-Pontoise, Frane, (e-mail: {Franiso.Bejarano,

Malek.Ghanes, barbot}ensea.fr)

∗∗

NationalAutonomousUniversityof Mexio, EngineeringFaulty,

Divisionof Eletrial Engineering, UNAM04510, Méxio, D.F.,

(e-mail: lfridmanservidor.unam.mx)

Abstrat:

1. INTRODUCTION

2. PROBLEMSTATEMENT

Considerthefollowinghaotisystem



 

 

 

 

 

 

˙

x

1

(t) = a(x

2

(t) − x

1

(t)) + x

2

(t) x

3

(t) + m

1

(t)

˙

x

2

(t) = b (x

1

(t) + x

2

(t)) − x

1

(t) x

3

(t)

˙

x

3

(t) = − cx

3

(t) − ex

4

(t) + x

1

(t) x

2

(t) + m

1

(t)

˙

x

4

(t) = f x

3

(t) − dx

4

(t) + x

1

(t) x

3

(t) + m

2

(t) y

1

(t) = x

1

(t)

y

2

(t) = x

2

(t)

(1)

where

x

i

∈ R

⁽

i = 1, 2, 3, 4

^{)arethe statesof thesystem;}

m

1 ^and

m

2 ^representthe informationto be transmitted, whih for the observation problem are onsidered as the

unknowninputs;

y

1^and

y

2^{aretheoutputsofthesystem,}

these arethesignalstobesentbyapublihannel.

Assumption. 2.1. The signals

m

1 ^and

m

2 ^{are assumed}

to be dierentiable, bounded and with the derivative

bounded.

Thegoalistoreonstrutthestates

x

3^and

x

4^whihallows

to reonstrut the messages

m

1 ^and

m

2^{. For ahieving}

suh a goal two approahes are tested, namely, an ob-

serverbasedonthesuper-twistingalgorithmthatbasially

allows to obtain information from the derivatives of the

outputandonsequentlytoreonstrutthestatesandthe

messages(unknowninputs),and anadaptiveobserver.A

omparisonbetweenthesetwomethods willbedisussed.

Itshouldbenotiethatasingularityappearsinthepoint

x

1

= 0

^{,thatis,fromtheoutputs}

x

1^and

x

2 ^{isnotpossible}

toknowthestates

x

3^and

x

4^andonsequentlyneitherthe messages

m

1^and

m

2^.

3. SUPER-TWISTING OBSERVER

First in ordertogenerateanewoutput,namelythevari-

able

x

3^,weusethesuper-twistingalgorithmfordesigning theobserverfor

x ˆ

3 ^{in thefollowingway:}



 



 



˙

x

a,1

= b (x

1

+ x

2

) + v

1

v

1

= ˆ x

1,3

+ λ

1

| s

1

|

^1/2

sign s ˆ

x

1,3

= α

1

sign s s

1

= x

2

− x

a,1

(2)

Inthisway,thederivativeof

s

1 ^takestheform

˙

s

1

= − x

1

x

3

− v

^{1 (3)}

Choosingthegains

λ

1

≥

^(α1+M_1−θ^1)(1+θ)

q

2 α1+M1

and

α

1

>

M

1

≥

_dt^d

(x

1

x

3

)

⁽

0 < θ < 1

^{), we get, aording to}

(Levant, 93), (Levant, 98), and (Davila, 05), the seond

orderslidingmotion,thatis,

s (t) = 0

^,

s ˙ (t) = 0

^aftersome

nite time

T

1^{. Notie that for}

s = 0

^,

v

¹

= ˆ x

3^{; therefore,}

from(3),weget

ˆ

x

1,3

(t) ≡ − x

1

(t) x

3

(t)

⁽⁴⁾

From (4) we an reonstrut

x

3

(t)

^provided

x

1

(t) 6 = 0

^.

Thus,theobserverfor

x

3 ^{isdesignedin theform}

ˆ x

3

(t) =







− x ˆ

1,3

(t)

x

1

(t)

^if

| x

1

(t) | ≥ ε ˆ

x

3

(t − τ)

^if

| x

1

(t) | < ε

(5)

where

τ

^and

ε

^{areenoughsmallonstants1.Thus,weget}

theidentity

ˆ

x

3

(t) ≡ x

3

(t)

^for

| x

1

(t) | ≥ ε

^.

Reonstrution of

x

4 ^{Now, dening}

y ¯

3

(t) = x

3

(t)

^as

a new output, we an rewrite the dynami equations in

(1)asalinearsystemwithoutputinjetionandunknown

inputs,thatis,

1

The onstant τ is hosen enough small but bigger than the sampling time used during the realization of the observer. The

onstantεshouldbehosensuientlybigtoavoidthesingularity, butalsoshouldbenotiethatanyestimationinthiszone annot

beonsideredasanaeptable estimation.



 

˙ x

1

˙ x

2

˙ x

3

˙ x

4



 

⁼



 

-

a a 0 0 b b 0 0 0 0

^-

c

^-

e 0 0 f

^-

d



 

| {z }

A



  x

1

x

2

x

3

x

4



 

| {z }

x

+



  y

2

y ¯

3

-

y

1

y ¯

3

y

1

y

2

y

1

y ¯

3



 

| {z }

φ(y1,y2,¯y3)

+



  1 0 0 0 1 0 0 1



 

| {z }

D

m

1

m

2

| {z }

w

" y

1

y

2

¯ y

3

#

| {z }

¯ y

=

" 1 0 0 0 0 1 0 0 0 0 1 0

#

| {z } x

C

(6)

Now, let

z

^{be dened by the solution of the following}

dierentialequation

z = Az + φ (y

1

, y

2

, y ¯

3

)

Thus,dening

e

z

= x − z

^{weobtainthedynamiequation}

for

e

z

e

z

(t) = Ae

z

(t) + Dw (t) y

z

= Ce

z

(7)

Hene, the system (7) is basially a linear system with

unknown inputs, just the sort of systems onsidered in

(Bejarano,07).Then,forthereonstrutionof

x

4^wefollow

in essene, but with a little modiation, the algorithm

proposed in (Bejarano, 07). That is, the basi idea pro-

posedin(Bejarano,07)istoobtainanalgebraiexpression

of

e

z ^{in termsofthe output}

y

z ^anditsderivatives.Thus, wegettheequalities

2

y

z

= Ce

z

d

dt (CD)

^⊥

y

z

= (CD)

^⊥

CAe

z

thusweget

3

e

z

(t) =

C

(CD)

^⊥

CA

+

y

z

(t) (CD)

^⊥

y

z

(t)

(8)

Sine wealready know therst3 states,we solve(8) for

e

z,4^{,thefourthstateof}

e

z^.Thus,

e

z,4

= 1

e [ ˙ e

z,1

− e ˙

z,3

+ a (e

z,1

− e

z,2

)

− ce

z,3

+ e

z,1

e

z,2

− e

z,2

e

z,3

]

(9)

3.1 Realization ofthe observerusing super-twisting

To redue the fast dynami and have asmaller gains in

thesuper-twistingalgorithm,wedesignthefollowinglike-

linear estimator

x ˜

^{whose dynamis is governed by the}

followingdierentialequation

˜

x =

− c − e f − d

| {z }

A¯

˜ x +

y

1

y

2

y

1

y ¯

3

+ L (¯ y

3

− y ˜

3

)

˜

y

3

= [ 1 0 ]

| {z }

C¯

˜ x

(10)

The matrix

L

^{is hosen so that the} eigenvalues of the matrix

A ¯ − L C ¯

havenegativerealpart.Inthis waywe

havethatthedynamiequationsfor

¯ e = ˜ x − x

3

x

4

are

2 Y⊥

isafullrowrankmatrixsuhthatY⊥Y = 0.

3

The matrix X⁺ isdened to be the pseudo-inverse of X. The matrixonsideredin(8)belongsto sortofmatriesoffullolumn

rank.Hene,insuhaase,X⁺= XTX

−1

XT

.

¯

e (t) = A ¯ − LC

¯

e (t) + w (t)

where

w

^{isdenedin(6).Henewegetsomeupperbounds}

forthenormof

e ¯

^{andforthenormof}

e ¯

^thatis

k e ¯ (t) k ≤ γ exp ( − λt) k ¯ e (0) k + µ k w (t) k

γ

^,

λ

^,

µ

^{arepositiveonstants}

Thatis,withtheestimator(10),

e ¯

^{isonstrainedtostayed}

inazonedependingontheamplitudeof

m

1 ^and

m

2^;and

thiszone anbemadesmallerbymovingtheeigenvalues

of

A ¯ − LC

moreto theleftin thelefthalf-plane.

Then, following the algebrai expression obtained in (9)

we design the observer for

x

4 ^{using the} super-twisting algorithmasfollows



 

 

 



 

 

 



˙ x

a,2

= 1

e [a (x

2

− x

1

) + c x ˆ

3

− x

1

x

2

+ x

2

x ˆ

3

] + ˜ x

4

+ v

2

v

2

= v

2,1

+ λ

2

| s

2

| sign s

2^,

v ˙

2,1

= α sign s

2

s

2

= 1

e (x

1

− x

3

) − x

a,2

ˆ x

4

(t) =

x ˜

4

+ v

2,1 ^if

| x

1

(t) | ≥ ε ˆ

x

4

(t − τ)

^if

| x

1

(t) | < ε

where

x ˜

4 ^{istheseondomponentof thevetor}

x ˜

^dened

in(10).Then,thetimederivativeof

s

2 ^is

˙

s

2

= x

4

− x ˜

4

+ v

2

Thus, for

λ

2

≥

^(α2+M_1−θ^2)(1+θ)

q

2 α2+M2

and

α

2

> M

2

≥

_dt^d

(x

4

− x ˜

4

)

⁽

0 < θ < 1

^{), after some nite time}

T

2^{, we}

get

s

2

= 0

^and

s ˙

2

= 0

^;therefore,

ˆ

x

4

≡ x

4 ^for

| x

1

(t) | ≥ ε

^.

3.2 Messages reonstrution

Thereonstrution of

m

1^{ismadein thefollowingway}



 

 

 

 

 

 

˙

x

a,3

= a (x

2

− x

1

) + x

2

x ˆ

3

+ v

3

v

3

= v

3,1

+ λ

3

| s

3

| sign s

3

˙

v

3,1

= α

3

sign s

3

ˆ m

1

=

v

3,1 ^if

| x

1

(t) | ≥ ε ˆ

m

1

(t − τ)

^if

| x

1

(t) | < ε s

3

= x

1

− x

³_a

Thus,for

α

3^and

λ

3^satisfying

λ

3

≥

^(α3+M1−³θ^)(1+θ)

q

2 α3+M3

and

α

3

> M

3

≥ | m ˙

1

|

⁽

0 < θ < 1

^{),after somenite time,}

we get the equalities

s

3

= 0

^,

s ˙

3

= 0

^{. Thus, we get the}

equality

ˆ

m

1

≡ m

1^{, for}

| x

1

(t) | ≥ ε

Thereonstrutionof

m

2^{ismadeinasimilarway,thatis,}



 

 

 

 

 

 

˙

x

a,4

= b (x

1

+ x

2

) + f x

3

− dx

4

+ v

4

v

4

= v

4,1

+ λ

4

| s

4

| sign s

4

˙

v

4,1

= α

4

sign s

4

ˆ m

2

=

v

4,1 ^if

| x

1

(t) | ≥ ε ˆ

m

2

(t − τ)

^if

| x

1

(t) | < ε s

4

= x

2

+ x

4

− x

a,4

Thus, taking into aount(1) and the derivativeof

x

a,4^,

withthehoosingof

α

4 ^and

λ

4 ^{satisfyingthe} inequalities

λ

4

≥

^(α4+M_1−θ^4)(1+θ)

q

2 α4+M4

and

α

4

> M

4

≥ | m ˙

2

|

⁽

0 <

θ < 1

^{)(see,(Levant,93),(Levant,98)),weget,aftersome}

nitetime,

ˆ

m

2

≡ m

2 ^for

| x

1

(t) | ≥ ε

mationofthestate

x

3^{,itissuientwithtouse}

x

1

(t) 6 = 0

instead of

| x

1

(t) | ≥ ε

^{, but the justiation is given in}

the following lines. It is known that during the realiza-

tion of the super-twisting algorithm

s

1 ^and

s ˙

1 ^{are not}

exatly zero, hene,instead of having (4) we really have

the equality

x ˆ

1,3

(t) = − x

1

(t) x

3

(t) + ∆ (t)

^{, where}

∆

representstheestimation errorandwhihdoesnottends

tozero.Then,afterdividingover

x

1^{,ityieldstheequality}

¯

x

3

:= −

^ˆxx^1,31

= x

3

−

x^∆1

, whih means that, when

x

1 ^is

verylosetozero,theerrorbetween

x ¯

3 ^and

x

3^isequalto

O (1/x

1

)

^{. Therefore, in a small} neighborhood of

x

1

= 0

^,

theerrorbetween

x ¯

3 ^and

x

3 ^{isextremelybig.Thisjustify}

thestrutureof

x ˆ

3^.

4. ADAPTIVEOBSERVER

Themodelofahaotisystem(1)anberewritteninthe

followinginteronnetedompatform

X ˙

1

= A

1

(y

2

)X

1

+ g

1

(y, X

1

, X

2

) + φ

₁

(t) y

1

= C

1

X

1

(11)

X ˙

2

= A

2

(y

1

)X

2

+ g

2

(y, X

2

, X

1

) + φ

₂

(t) y

2

= C

2

X

2

(12)

where

X

1

= (x

1

, x

3

, x

4

, x

5

)

^{T is the state of the rst}

subsystem with

x

5

:= m

2^,

X

2

= (x

2

, x

3

, x

6

)

^{T is thestate}

oftheseond subsystemwith

x

6

:= m

1^.

y = [x

1

, x

2

]

^{T are}

theoutputofthewholesystem,and

A

1

(y

2

) =



 

0 y

2

0 0 0 0 − c 0 0 0 0 1 0 0 0 0



 

^,

A

2

(y

1

) =

0 − y

1

0 0 0 1 0 0 0

!

g

1

(y, X

2

, X

1

) =



 

a(y

1

− y

2

) + x

6

− ex

4

+ y

1

y

2

+ x

6

f x

3

− dx

4

+ y

1

x

3

0



 

g

2

(y, X

2

, X

1

) =

b(y

1

+ y

2

)

− cx

3

− ex

4

+ y

1

y

2

0

!

φ

_i

(t) =

" 0 0

˙ m

i

#

,

i = 1, 2

C

1

= ( 1 0 0 0 )

^,

C

2

= ( 1 0 0 )

^.

Remark. 4.1. The hoie of the variables of eah subsys-

tem hasbeen onsideredin order to separatein one sub-

systemthemessage

m

1 ^{andintheother onethemessage}

m

2^{. It is lear that other hoie an be onsidered in}

ordertorepresentthesesubsystemsprovidedtheneessary

onditionstodesignanadaptivearesatised.

Next, let us introdue an adaptive observer in order to

estimatethesystem'sstateandtheunknowninputssimul-

taneously. It is basedon interonnetion betweenseveral

subsystemswhihsatises somerequiredproperties, suh

that thepropertyof inputspersisteny((Hammouri, 90),

(Ghanes, 06)).

Atrst,letusintroduethefollowingassumptionsinorder

to establishthe results onerning the adaptiveobserver

1. The signals

X

1 ^and

X

2 ^{are assumed bounded and}

to be regularly persistent ((Hammouri, 90), (Ghanes,

06)) in order to guarantee the observability property of

subsystems(11)and(12),respetively.

2.

A

1

(y

2

)

^and

A

2

(y

1

)

^{areuniformly bounded.}

3.

g

1

(y, X

1

, X

2

)

^{is globally Lipshitz with respet to}

X

2

anduniformlywithrespetto

(y, X

1

)

^.

4.

g

2

(y, X

2

, X

1

)

^{isgloballyLipshitzwithrespet}

X

1 ^and

uniformlywithrespetto

(y, X

2

)

^.

5.Theunknownfuntions

m ˙

i

(t)

⁽

i = 1, 2

^{)areassumedto}

bebounded.

Then,anadaptiveobserverforinteronnetedsubsystems

(11)and(12)estimatingthestateand unknownparame-

tersisgivenby







Z ˙

1

= A

1

(y

2

)Z

1

+ g

1

(y, Z

1

, Z

2

) + S

₁⁻¹

C

₁^T

(y

1

− y ˆ

1

) S ˙

1

= − θ

1

S

1

− A

^T₁

(y

2

)S

1

− S

1

A

1

(y

2

) + C

₁^T

C

1

ˆ

y

1

= C

1

Z

1

(13)







Z ˙

2

= A

2

(y

1

)Z

2

+ g

2

(y, Z

2

, Z

1

) + S

₂⁻¹

C

₂^T

(y

2

− y ˆ

2

) S ˙

2

= − θ

2

S

2

− A

^T₂

(y

1

)S

2

− S

2

A

2

(y

1

) + C

₂^T

C

2

ˆ

y

2

= C

2

Z

2

(14)

where

Z

1

= (ˆ x

1

, x ˆ

3

, x ˆ

4

, x ˆ

5

)

^T;

Z

2

= (ˆ x

2

, x ˆ

3

, x ˆ

6

)

^T

S

i

= S

_i^T

> 0

^,

i = 1, 2

^{. Note that}

S

₁⁻¹

C

1^{T and}

S

₂⁻¹

C

2^{T are the}

gainsoftheobservers(13)and(14),respetively.

Remark. 4.2. Itisworthnotiingthat

k S

1

k

^and

k S

2

k

^are

boundedfor

θ

1^and

θ

2^{largeenoughduetothepersisteny}

ofinputonsideredinassumption4.1.

Now, in order to guarantee the onvergene of the pro-

posedobserver,suientonditionsareestablishedinthe

followingresult.Denotetheestimationerrors:

ǫ

1

= X

1

− Z

1 ^and

ǫ

2

= X

2

− Z

2

whosedynamisaregivenby

ǫ ˙

1

= [A

1

(y

2

) − S

₁⁻¹

C

₁^T

C

1

]ǫ

1

+g

1

(y, X

1

, X

2

) − g

1

(y, Z

1

, Z

2

) + φ

₁

(t)

⁽¹⁵⁾

ǫ ˙

2

= [A

2

(y

1

) − S

₂⁻¹

C

₂^T

C

2

]ǫ

2

+g

2

(y, X

2

, X

1

) − g

2

(y, Z

2

, Z

1

) + φ

₂

(t)

^{. (16)}

Thevaluesof

θ

1^and

θ

2 ^{arehosentosatisfytheinequali-}

ties

δ

1

= (θ

1

− Γη) > 0

^,

δ

2

= (θ

2

− Γ

η ) > 0

⁽¹⁷⁾

where

Γ = ˜ µ

1

+ ˜ µ

2

,

^with

µ ˜

i

= √

^µi

λmin(S1)

√

λmin(S2)

, i = 1, 2;

^and

µ

₁

= k

1

k

2^,

µ

₂

= k

3

k

4^,

η ∈ ]0, 1[

^.

The parameters

k

1

, k

2

, k

3^,

k

4 ^{are positive onstants and}

λ

min

(S

1

), λ

min

(S

2

)

^{aretheminimal}eigenvaluesof

S

1^and

S

2 respetively.

Lemma 4.1. Consider the system (13)-(14) and that as-

sumption4.1holds.Then,thesystem(13)-(14)isaprati-

alexponentialobserverforsystem(11)-(12)for

θ

1^and

θ

2

satisfyingtheinequalities(17).Furthermore,theobserver

onvergesarbitrarilyfastwithaonvergeneratexedby

aparameter

δ

^,

δ = min(δ

1

, δ

2

)

^.

Skethofproof 4.1. ConsiderthefollowingLyapunovfun-

tionandidate:

V

o

= V

1

+ V

2

where

V

1

= ǫ

^T₁

S

1

ǫ

1^and

V

2

= ǫ

^T₂

S

2

ǫ

2^.

k S

1

k ≤ k

1^;

k{ g

1

(y, X

1

, X

2

) − g

1

(y, Z

1

, Z

2

) }k ≤ k

2

k ǫ

2

k

^;

k S

2

k ≤ k

3^;

k{ g

2

(y, X

2

, X

1

) − g

2

(y, Z

2

, Z

1

) }k ≤ k

4

k ǫ

1

k

^.

k φ

₁

k ≤ k

5^.

k φ

₂

k ≤ k

6^.

Computingthetimederivativeof

V

o^{,,byusingtheabove}

inequalitiesitfollowsthat

V ˙

o

≤ − θ

1

ǫ

^T₁

S

1

ǫ

1

+ 2µ

₁

k ǫ

1

k k ǫ

2

k + k φ

₁

k

⁽¹⁸⁾

− θ

2

ǫ

^T₂

S

2

ǫ

2

+ 2µ

₂

k ǫ

2

k k ǫ

1

k + k φ

₂

k

Now,onsider thatthefollowinginequalitiesaresatised

λ min(S

i

) k ǫ

i

k

²

≤ k ǫ

i

k

²Si

≤ λ max(S

i

) k ǫ

i

k

²

, i = 1, 2.

Bywriting(18)intermsoffuntions

V

1 ^and

V

2^,itfollows

that

V ˙

o

≤ − θ

1

V

1

− θ

2

V

2

+ 2(˜ µ

₁

+ ˜ µ

₂

) p V

1

p V

2

+ k

5

+ k

6

wheretheparameters

µ ˜

₁^,

µ ˜

₂^{,aredenedjustbeforelemma}

4.1.

Next,byusingthefollowinginequality

√ V

1

√

V

2

≤

^υ₂

V

1

+

1

2υ

V

2^,

∀ υ ∈ ]0, 1[

^{,one get}

V ˙

o

≤ − (θ

1

− Γ)V

1

− (θ

2

− Γ

υ )V

2

+ k

5

+ k

6

.

where

Γ

^{isdenedjust before lemma4.1.}

Bytaking

δ

^and

r

^suhthat

δ = min(δ

1

, δ

2

)

^and

r = k

5

+k

6

one has

V ˙

o

≤ − δV

o

+ r.

⁽¹⁹⁾

Finally, by hoosing

θ

1 ^and

θ

2 ^{suh that the} inequalities (17)aresatisedandsuientlylarge,theinequality(19)

showsthatarbitrarilyboundedperturbationwillnotresult

in largeerrorestimationdeviations.Thisendstheproof.

5. NUMERICALEXAMPLEANDDISCUSSIONS

Forthesystem(1)weusetheparameters

a = 42.5

^,

b = 24

^,

c = 13

^,

d = 20

^,

e = 50

^,

f = 40

^{.Theparametersused for}

thesuper-twistingobserverare

α

1

= 7 × 10

^7,

λ

1

= 7 × 10

^3,

α

2

= 300

^,

λ

2

= 100

^,

α

3

= 1000

^,

λ

3

= 200, α

4

= 600

^,

λ

4

= 300

^{.Forthe adaptiveobservertheparametersused}

are

θ

1

= θ

2

= 400

^.

Figures1and2showthetrajetoriesofthestates

x

3^and

x

4 ^{as well as the ones of}

x ˆ

3 ^and

x ˆ

4 ^{for both the super-}

twisting and the adaptive observers. We an see in the

guresthatthetrajetoriesofthesuper-twistingobserver

onverge muh faster than the ones of the adaptive ob-

server.

Figures 3 and 4show themessages

m

1 ^and

m

2 ^together

with their estimations

m ˆ

1 ^and

m ˆ

2^, respetively. In this gures we note that the singularity in the point

x

1

= 0

aets more the estimation of the messages made with

the super-twisting than the one made with the adaptive

observer.This is leardue to the explanationof Remark

3.1 and the fat that, in a ball of radio

ε

^{and enter in}

x

1

= 0

^{, it is not done a estimation neither of the states}

norofthemessages.

Inordertotestbothobserverswithrespettoparameters

unertainties,weintrodueavariationof1%in thenomi-

0 2 4 6 8 10

−400

−200 0 200 400 600 800

Time [s]

0 0.02 0.04 0.06 0.08

−1 0 1

Fig.1.

x

3^{(solidline)anditsestimation}

ˆ x

3^{usingthesuper-}

twisting(dot line)andadaptive(dashline)observers

0 2 4 6 8 10

−400

−200 0 200 400 600 800

Time [s]

0 0.05

0 0.5 1 1.5

Fig.2.

x

4^{(solidline)anditsestimation}

ˆ x

4^{usingthesuper-}

twisting(dot line)andadaptive(dashline)observers

unertainties aetsthebehaviorofthe observers.Inthe

ase of the adaptive observerthe parameter unertainty

destroyompletelytheestimationofthemessages.Never-

theless,nomatter what observeis used, in this ase,the

estimationofthemessagesanbeonsiderunaeptable.

Remark. 5.1. Forthesuper-twistingobserverwedosome

simulations with

x ˜ = 0

^{, that is without using a linear}

estimator.Inthesimulations,notshownhere,weobtained

abiggererrorintheestimationof

x

4^{whihhadabigeet}

in the error of the estimation of

m

2^{. It was due to the}

fat that using

x ˜ = 0

^{, the variable to be estimated with}

thesuper-twistingalgorithmbeame muh faster andfor

having an aeptable estimation the sampling step must

beredueonsiderably.

REFERENCES

H. Hammouri,J. De Leon,`Observersynthesis forstate-

anesystems'Pro29thIEEEConferene onDeision

andControl, Honolulu,Hawaii.1990.

M. Ghanes, J. De Leon and A. Glumineau, `Observabil-

ityStudyandObserver-BasedInteronnetedFormfor

Sensorless Indution Motor',Pro 45th IEEE Confer-

eneon DeisionandControl, SanDiego,CA,Deem-

0 2 4 6 8 10

−200

−150

−100

−50 0 50 100 150 200 250

Time [s]

0 0.1 0.2

0 20 40 60

Fig.3.Message

m

1^{(solidline)anditsestimation}

m ˆ

1^using

thesuper-twisting(dot line)andadaptive(dashline)

observers

0 2 4 6 8 10

−150

−100

−50 0 50 100 150

Time [s]

0 0.2 0.4

0 20 40

Fig.4.Message

m

2^{(solidline)anditsestimation}

m ˆ

2^using

thesuper-twisting(dot line)andadaptive(dashline)

observers

0 2 4 6 8 10

−200

−100 0 100 200

0 2 4 6 8 10

−2000 0 2000 4000

Time [s]

Fig.5.Message

m

1^{(solidline)anditsestimation}

m ˆ

1^(dot

line) for the system with

1%

^{of unertainty in the}

parameters.Abovewith thesuper-twistingobserver,

belowwiththeadaptiveobserver

0 2 4 6 8 10

−100 0 100 200

0 2 4 6 8 10

−2

−1 0 1 2x 10⁴

Time [s]

Fig.6.Message

m

2 ^{(solidline)anditsestimation}

m ˆ

2 ^(dot

line) for the system with

1%

^{of unertainty in the}

parameters.Abovewiththe super-twistingobserver,

belowwiththeadaptiveobserver

A. Levant, 1996, `Sliding order and sliding auray in

slidingmodeontrol',InternationalJournalofControl,

vol.58,no.6,pp.1247-1263.

A. Levant, 1998, `Robustexatdierentiation viasliding

modetehnique',InternationalJournalofControl,vol.

34,no.3,pp.379-384.

J.Davila,L.FridmanandA.Levant,2005,`Seond-Order

Sliding-ModeObserverforMehanialSystems',IEEE

TransationsonAutomatiControl,vol.50,no.11,pp.

1785-1789.

F.J.Bejarano,L. Fridmanand A. Poznyak,2007, `Exat

state estimation for linear systems with unknown in-

puts based on hierarhial super-twisting algorithm',

InternationalJournalofRobustandNonlinearControl,

Publishedonline[doi:10.1002/rn.1190℄.

G.Y. Qi, S.Z. Du, G.R. Chen et al., 2005, `On a four-

dimensional haoti system', Chaos, Solutions and

Fratals,vol.23,no.5,pp.1671-1682.