Chattering-free digital sliding-mode control with state observer and disturbance rejection

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observer and disturbance rejection

Vincent Acary, Bernard Brogliato, Yury Orlov

To cite this version:

Vincent Acary, Bernard Brogliato, Yury Orlov. Chattering-free digital sliding-mode control with state

observer and disturbance rejection. [Research Report] RR-7326, INRIA. 2010. �inria-00494417v2�

a p p o r t

d e r e c h e r c h e

N 0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 7 3 2 6 -- F R + E N G

Domaine 1

Chattering-free digital sliding-mode control with state observer and disturbance rejection

Vincent Acary — Bernard Brogliato — Yury Orlov

N° 7326

June 2010

Centre de recherche INRIA Grenoble – Rhône-Alpes

Vinent Aary

∗

, Bernard Brogliato

†

, YuryOrlov

‡

Domaine: Mathématiquesappliquées,alul etsimulation

Équipes-ProjetsBipop

Rapportdereherhe n° 7326June201032pages

Abstrat: Inthis paperanoveldisrete-timeimplementationofsliding-mode

ontrolsystemsisproposed,whihfullyexploitsthemultivaluednessofthedy-

namis onthesliding surfae. It isshown to guarantee asmoothstabilization

on thedisretesliding surfaein the disturbane-freease, heneavoidingthe

hattering eets due to the time-disretization. In addition when a distur-

bane ats on thesystem, the ontrollerattenuates thedisturbane eets on

the sliding surfaeby a fator

h

^(where

h

^{is the sampling period). Most im-}

portantly this holds even for large

h

^{. The ontroller is based on an impliit}

Euler method and is veryeasy to implement with projetionson the interval

[−1, 1]

^{(or asthesolutionofaquadratiprogram). The}zero-order-hold(ZOH) method is also investigated. First and seond order perturbed systems (with

a disturbanesatisfying themathing ondition) withoutand with dynamial

disturbaneompensationareanalyzed,withlassialandtwistedsliding-mode

ontrollers.

Key-words: Nokeywords

∗

INRIA,655avenuedel'Europe,Inovallée,38334Saint-Ismier,Frane

†

INRIA,655avenuedel'Europe,Inovallée,38334Saint-Ismier,Frane

‡

CICESE, Departamento de Eletronia y Teleomuniaiones, Km.107, Carretera

Tijuana-Ensenada,22860Mexio

Résumé: Cetravailonernelaommandeparmodesglissantsentempsdis-

ret. L'aspetmultivaluéestpleinementexploité,equipermetdesupprimerle

hatteringdûàladisrétisationentemps,etd'autrepartd'atténuerlespertur-

bationsparunfateur

h

⁽

h

^lepasd'intégrationoulapérioded'éhantillonnage) ouparunfateur

h ²

^{. Lesméthodesd'EulerimpliiteetZOH(zeroorderholder)}

sontétudiées.

Mots-lés : sliding-mode,bakwardEuler method, zero-order-holdmethod,

disrete-timeslidingmode,disturbaneompensation,twistingontroller

1 Introdution

Sliding-mode ontrol is animportant eld of feedbakontrol,with manyap-

pliations, see e.g. [6, 15, 19, 22, 27, 28℄. The issue related to the digi-

tal denition and implementation of sliding mode systems, has been the ob-

jet of many works sine the publiation of pioneering works [9, 20℄, see e.g.

[4,12,17,27,31,28,24,25℄. Itappearshoweverthatsuhontrolmethodsare

notyetfullyunderstoodandtheirimplementationisstillpronetoseriousprob-

lems likenumerialhattering[13,30, 14,18, 16,29, 28,32, 5℄. Theobjetive

of thispaperisthreefold: a)to show thatan impliitEuler ontrollerpermits

to numerially implement the multivalued part of disontinuous sliding-mode

ontrollers andonsequentlysuppress the numerial hattering that is present

in the expliitimplementations, b) to extend it to the ase when one part of

the stateis observed, ) to show that when adisturbane ats onthe system

(full-stateor partial-statefeedbak)thenumerialhatteringisstillsuppressed

andthedisturbaneisrejeted. Bydisturbanerejetionitismeantthatinthe

ideal (analytial) ontinuous-time system, the disturbane is exatlyrejeted,

while inthe digitalimplementationit isattenuatedbyafator

h

^where

h > 0

is the sampling time. The major features of the impliit ausal disrete-time

input are on one hand that the ontinuous-time system sliding surfae (that

maybeofodimensionlargerthanone)isnothangedafterthedisretization,

ontheotherhand anite samplingfrequenyissuienttoassure thesliding

motion ofthe disrete-timesystem, andnally the hatteringeets observed

onexpliitontrollers(namedthenumerialhattering)aresuppressed.

Denition1 The numerial hattering orresponds to the osillations (limit

yles) whiharesolely duetothedigital implementationofthe ontroller.

Denition2 The disturbane hattering orresponds to the osillations that

an appearduetoahigh frequeny disturbaneating onthe system.

A rstfundamental stepis to eliminate the numerialhattering with the

appliationofasuitableimpliitdisrete-timeontroller. Thedisturbanehat-

teringwillnotbeeliminatedinthesystem'sstatearoundtheslidingsurfae,but

thedisturbaneisattenuatedbyafator

h

^(ofafator

h ²

^{onthesystem'spo-}

sitionforanorder-twosystem). Inpratieitisexpetedthatthisorresponds

to ahighompensation ofthedisturbane. Theontrolinputobtainedbythe

impliitmethodisnotofthebang-bangtypewhenthestateevolvesontheslid-

ing surfae. On theontrary itis aontinuousinput whih evolvesinside the

multivaluedpartofthesignmultifuntion(themultivaluedpartorrespondsin

theFilippovasetothesetrepresentingthelosedonvexlosureofthevetor

elds on theswithing surfae,whih is asegmentif theodimension isequal

toone).

Denition3 Let

h = t k+1 − t k > 0

^{be the sampling period,}

k > 0

^{. An}

m

^-

disrete-timeslidingsurfae

Σ d

^{isaodimension}

m

^{subspaeofthe statespae,}

suh that the disrete state vetor

x _k = ^∆ x(t k )

^satises

x _k ∈ Σ d

^{for all}

k min 6 k 6 k max

^,

k min < k max − 1

^,

k min > 0

^{. Moreoverthis holdswhatever}

h > 0

^.

Averyattrativefeatureof thedigitalmethodbasedonthe impliitEuler

methodisthatthenumerialslidingsurfae

Σ d

^andtheontinuous-timesliding

surfae

Σ c

^satisfy

Σ d = Σ c

^{: the}disretizationdoesnotmodifytheslidingsurfae [1℄. If, forinstane,

Σ c = {x ∈ IR ⁿ | Cx + D = 0}

^,

C ∈ IR ^m×n

^,

D ∈ IR ^m

^{, then}

Σ d = {x k ∈ IR ⁿ | Cx k + D = 0}

^{. The ontrollers whih are designedin this}

paperonsistofthe stabilizationofanunperturbednominalplant,oupledto

theplant's dynamis(see gure1). Both thenominal andthe realplanthave

tobedisretizedwiththesamemethod(impliit Eulerorzero-orderholder).

nominalsystem

τ _k+1

−τ _k+1

x _k

−

disrete-timeplant

ϕ _k+1

solver equation

generalized

Figure1: Thedisrete-timelosed-loopsystem.

Thepaper isorganizedasfollows: setion2is dediated to theanalysis of

a simple rst-order system, without and with disturbaneompensation. An

extensiontohigher-ordersystemsisalsopresented,withtheEulerandtheZOH

methods. In setion 3 seond-order systems are treated and several types of

ontrollersareanalyzed. Inallasestheontinuous-timesystemisintrodued,

thenitstime-disretizationisstudied,andnallysimulationresultsareshown.

Conlusionsend thepaper.

Notation: Inthesequelsgn

(x)

^{is the}multivaluedsign funtion: sgn

(x) =







+1

^if

x > 0

−1

^if

x > 0

[-1,1℄ if

x = 0

. Let

K ⊂ IR ⁿ

^{be a losed non empty onvex set. The}

normal one to

K

^at

x ∈ K ⊂ IR ⁿ

^is

N K (x) = {z ∈ IR ⁿ | z ^T (y − x) >

0

^forall

y ∈ K}

^{. Let}

M = M ^T > 0

^bean

n × n

^{matrix. Forany}

x ∈ IR ⁿ

^and

y ∈ IR ⁿ

^,onehas

−x+y ∈ M ⁻¹ N K (x) ⇔ x =

^proj

_M (K; y) ⇔ x = argmin _z∈K 1

2 (z −y) ^T M (z −y)

(1)

where proj

M (K; y)

^{denotes theorthogonalprojetionof}

y

^on

K

^{in themetri}

dened by

M

^{. Foranyreals}

x

^and

y

^,onehas

x ∈

^sgn

(y) ⇔ y ∈ N [−1,1] (x).

⁽²⁾

For

x ∈ IR ^m

^{, Sgn}

(x) = (

^sgn

(x 1 ) ...

^sgn

(x m )) ^T

^,

||x|| ∞ = max(|x 1 |, ..., |x m |)

^,

||x|| 1 = P m

i=1 |x i |

^{. Foranymatrix}

M

^andvetor

x

^,thenorms

||M ||

^and

||x||

^are

supposed to be ompatible norms so that

||M x|| 6 ||M || ||x||

^{. Forafuntion}

f : IR → IR

^onehas

||f || ∞ =

^esssup

_t∈ IR |f (t)|

^.

I _n

^isthe

n × n

^{identitymatrix.}

2 A rst-order system

Weanalyzeinthissetionthesimplestasetoillustratehowthemethodworks.

Two ases are treated: without and with disturbane ompensation (in the

ontinuous-timesystem). Thebasiideasareillustratedonasimplerst-order

system.

2.1 The ase without disturbane ompensation

Letusstartbyonsideringthefollowingbasislidingmodesystem:

( x(t) = ˙ −aτ (t) + ϕ(t)

τ(t) ∈

^sgn

(x(t)),

⁽³⁾

where

ϕ(·)

^{is the}perturbation suh that

kϕk ∞ < ρ < a

^{. Theontrolinput is}

here

u(t) = τ(t)

^{. Itmaybeseen,inthelanguageofdierentialinlusionstheory,}

asaseletionoftheset-valuedright-hand-sideofthesystem. Choosingorretly

this seletionis the objetof the followingdisretization. Thesystem(3) has

x = 0

^{asits unique}equilibirum point, whih is globally asymptotially stable and is reahed in nite time (this may beshownwith theLyapunovfuntion

V (x) = x ²

^{). The}disrete-timesliding modesystemisimplementedasfollows:



 

 

˜

x k+1 = x k − ahτ k+1

τ k+1 ∈

^sgn

(˜ x k+1 )

x k+1 = x k − ahτ k+1 + hϕ k+1

(4)

Thersttwolinesof (4)maybe onsideredasthenominalunperturbedplant,

from whih oneomputestheinputattime

t k

^{. Thethirdline isthebakward}

Eulerapproximationoftheplant,onwhih thedisturbaneis ating. Onehas

u(t) = τ k+1

^onthetime-interval

[t k , t k+1 )

^.

Proposition1 Suppose that the initial state in (4 ) satises

|x 0 | > ah > 0

^.

Then after a nite number of steps

k 0

^{one obtains that}

x ˜ _k = 0

^and

x _k = hϕ _k

for all

k > k 0

^{. In other words, the disturbane is attenuated by a fator}

h

^.

Moreovertheapproximatedderivativeofthestatesatises

x k+1 −x _k

h = ϕ k+1 − ϕ _k

for all

k > k 0 + 1

^whereas

^{x ˜ k+1} _h ^{−˜ x k} = 0

^{for all}

k > k 0

^{. Theontrol inputtakes}

values inside the sign multifuntionmultivalued part on the sliding surfae for

all

k > k 0

^.

Proof:Thegeneralizedequation

x ˜ k+1 = x k − ahτ k+1

^and

τ k+1 ∈

^sgn

(˜ x k+1 )

is found to be equivalent, using (1) and (2), to the inlusion

τ k+1 − ^x _ah ^k ∈

−N [−1,1] (τ k+1 )

^{whih is equivalent to}

τ k+1 =

^proj

([−1, 1]; _ah ^{x k} )

^{. Thus one ob-}

tains:

ˆ If

x k > ah

^then

x ˜ k+1 = x k − ah

^{and sgn}

(˜ x k+1 ) = 1

^,

ˆ If

x _k < − ah

^then

˜ x k+1 = x _k + ah

^andsgn

(˜ x k+1 ) = −1

^,

ˆ If

0 > x _k > − ah

^then

x ˜ k+1 ∈ (0, ah)

^,andsgn

(˜ x k+1 ) = −1

^,

ˆ If

0 < x _k < ah

^then

x ˜ k+1 ∈ (− ah, 0)

^,andsgn

(˜ x k+1 ) = 1

^.

Fromtheaboveweinferthat:

ˆ If

x k > ah

^then

x k+1 = x k +hϕ k+1 −ah = x k +h(ϕ k+1 −a) < x k +h(ρ−a)

^.

Sine

ρ − a < 0

^{thestateisstritlydereasedfromstep}

k

^tostep

k + 1

^.

ˆ If

x k < −ah

^then

x k+1 = x k + hϕ k+1 + ah = x k + h(ϕ k+1 + a) > x k + h(a − ρ)

^{. Sine}

a − ρ < 0

^{thestateisstritlyinreasedfromstep}

k

^tostep

k + 1

^.

One dedues that if the initial data satises

|x 0 | > ah

^{then after}

k 0 =

⌈ _h|a−ρ| ^{x 0} ⌉

^stepsonegets

x ˜ _k 0 = 0

^{. Indeedat}

k 0

^thestate

x _k

^{reahestheinterval}

(−ah, ah)

^{andthentheuniquesolutionfor}

x ˜ k

^{iszero. From}

x ˜ k 0 = 0

^onededues

that

|x k 0 | < ah

^{. Toomputethenextvalueof}

x ˜ k

^{onehastosolvethe}generalized equation

( x ˜ k 0 +1 = x k 0 − ahτ k 0 +1

τ _{k 0} +1 ∈

^sgn

(˜ x _{k 0} +1 ),

⁽⁵⁾

whoseuniquesolutionis foundbyinspetiontobe

x ˜ _{k 0} +1 = 0

^{1. Thereasoning}

anberepeatedto onludethat

x ˜ k = 0

^forall

k > k 0

^{. Therefore}

˜ x k+1 −˜ x k

h = 0

forall

k > k 0

^{. Nowletusassumethatfor}

k > k 0

^wehave

˜

x k+1 = x k − ahτ k+1 = 0, k > k 0 ,

⁽⁶⁾

that is

τ k+1 = x k

ha .

⁽⁷⁾

Inthisase,thestate

x k+1

^isgivenby

x k+1 = hϕ k+1 ,

⁽⁸⁾

andtherefore

x _k = hϕ _k , τ k+1 = ϕ _k

a

^forall

k > k 0 + 1,

⁽⁹⁾

sothat

x k+1 −x _k

h = ϕ k+1 − ϕ _k

^forall

k > k 0 + 1

^.

Notie that the bakward (or impliit) Euler disretization of the unper-

turbedplantoinidesfor(3) withthezero-orderholder(ZOH)disretization.

Consideringtheperturbedplant,theonlydierenebetween(4)andtheZOH

disretizationisthat

hϕ k+1

^beomes

R

[t k ,t k+1 ) ϕ(t)dt

^{. And}

hϕ k+1

^in(8)and(9)

hastobereplaedby

R

[t k ,t k+1 ) ϕ(t)dt < ah

^{. The}attenuationofthedisturbane

still holds with the ZOH method. In other words the state

x(·)

^{of the plant}

satises

x(t) = R

[t _k ,t) ϕ(t)dt 6 hρ

^{. Inamoregeneralsetting, the}disretization of theontrollerandthe disretizationof theplanthaveto bethesame(both

impliitEuler, orbothZOH)in orderforthe disturbaneattenuationtohold.

Notie that the above shows that

V k = |˜ x k |

^{is a Lyapunov funtion for the}

nominalsystem.

1

Theunderlyingruialpropertythatmakesthisholdisthemaximalmonotoniityofthe

signmultifuntion.

2.2 The ase with disturbane ompensation

Let usonsider thease withdisturbane ompensation. Letus dene

x(t) = ˙ˆ

−aτ 1 (t)

^,

τ 1 (t) ∈

^sgn

(x(t))

^,

e = x − ˆ x

^{, and the ontroller}

u = −a

^sgn

(x(t)) − α

^sgn

(e(t))

^,

a > 0

^,

α > 0

^and

a < α

^{. Thusthelosed-loopsystemisgivenby:}



 

 

 

 

˙

x(t) = −aτ 1 (t) − ατ 2 (t) + ϕ(t)

˙

e(t) = −ατ 2 (t) + ϕ(t) τ 1 (t) ∈

^sgn

(x(t)) τ 2 (t) ∈

^sgn

(e(t))

(10)

where

ϕ(·)

^{is adisturbanesuh that}

k ϕ k ∞ < ρ < min(a, α)

^{. Thexed point}

(x, e) = (0, 0)

^{ofthesystemmaybeshowninaratherstandardway[27℄tobe}

globallystronglyasymptotiallystablewiththenonsmoothLyapunovfuntion

V (x, e) = |x| + |e|

^{. Moreover, the system attains in anite time the sliding}

surfae

e = 0

^{whereitevolvesaordingtotheslidingdynamis}

x(t) = ˙ −aτ 1 (t)+

ϕ(t)

^{. Theondition}

a < α

^{impliesthat theoriginisnotattaineddiretly,but}

rstthesystemslidesonthesurfae

e = 0

^{. Onthissurfaeitisapparentfrom}

(10)thatthedynamisin

x

^evolvesasadisturbane-freesystem. Thedisrete sliding modesystemisimplementedasfollows:



 

 

 

 

˜

x k+1 = x k − ahτ 1,k+1 − αhτ 2,k+1

˜

e k+1 = e _k − αhτ 2,k+1

τ 1,k+1 ∈

^sgn

(˜ x k+1 ) τ 2,k+1 ∈

^sgn

(˜ e k+1 ),

(11)

andtheupdateproedure representingtheplantdynamisisgivenby:

( x k+1 = x k − ahτ 1,k+1 − αhτ 2,k+1 + hϕ k+1

e k+1 = e _k − αhτ 2,k+1 + hϕ k+1 .

⁽¹²⁾

Proposition2 Assumethat

|e 0 | > αh > 0

^{. Thenafteranitenumberofsteps}

k 0

^{one obtains}

˜ e k = 0

^and

e k = hϕ k+1

^{for all}

k > k 0

^{. Let}

|x k 0 − hϕ k 0 +1 | >

ah > 0

^{. Then there exists}

k 1 < +∞

^suhthat

x ˜ k = 0

^{for all}

k > k 0 + k 1

^and

x k = hϕ k

^{for all}

k > k 0 + k 1

^.

Proof: From(11)wehave

( e ˜ k+1 = e k − αhτ 2,k+1

τ 2,k+1 ∈

^sgn

(˜ e k+1 ),

⁽¹³⁾

whih is exatly the rst two lines in (4). Therefore the onlusions drawn

for (4) apply, just replaing

a

^by

α

^{. Thus the}

e k −

^{dynamis is}

e k+1 = e k − αh

^proj

([−1, 1]; _αh ^{e k} ) + hϕ k+1

^{. After}

k 0

^{the disrete trajetory evolves on the}

slidingsurfae

˜ e k = 0

^while

τ 2,k+1 = _αh ^{e k}

^and

e k = hϕ k+1

^{,andoneobtainsusing}

(1):



 

 

˜

x k+1 = x _k − hϕ k+1 − ahτ 1,k+1

τ 1,k+1 ∈

^sgn

(˜ x k+1 ) x k+1 = x _k − ahτ 1,k+1

⇔ x k+1 = x _k − ah

^proj

([−1, 1]; ^{x k −hϕ} _ah ^k+1 )

(14)

ThenweanredoagainthesamealulationsasintheproofofProposition

1(byreplaing

x k

^by

x k − hϕ k+1

^{intherstlineof(4),and}

x k + hϕ k+1

^by

x k

in thethird line), toinfer thatafter anite numberof stepsonegets

x ˜ k = 0

^,

τ 1,k+1 = ^{x k −hϕ} _ah ^k+1

^,and

x k+1 = x _k − e _k − (x k − hϕ k+1 ) + hϕ k+1 = hϕ k+1

⁽¹⁵⁾

Indeedletusnowassumethat:

( x ˜ k+1 = x k − ahτ 1,k+1 − αhτ 2,k+1 = 0

˜

e k+1 = e _k − αhτ 2,k+1 = 0,

⁽¹⁶⁾

that is







τ 1,k+1 = x k − e k

ah τ 2,k+1 = e k

αh .

(17)

Aftertheupdateproedure (12),weget

( x k+1 = hϕ k+1

e k+1 = hϕ k+1 .

(18)

Weanonludethatonetheslidingmodein

x ˜

^and

˜ e

^{isreahedwehave,}

( x k+1 = hϕ k+1

e k+1 = hϕ k+1

,

^forall

k > k 0

⁽¹⁹⁾

and

( τ 1,k+1 = 0 τ 2,k+1 = ϕ _k

α

,

^forall

k > k ₀ + 1.

⁽²⁰⁾

Consequentlythedisrete-timeontrollerguaranteestheonvergeneofthe

stateofthenominal systemin nitetimeto theorigin,whiletheplant'sstate

is equaltothedisturbaneattenuatedbyafator

h

^{. Tosummarize, from(11)}

and(12)thedisrete-timelosed-loopsystemistherefore:



 



 



x k+1 = x k − ahτ 1,k+1 − αhτ 2,k+1 + hϕ k+1

e k+1 = e k − αhτ 2,k+1 + hϕ k+1

τ 1,k+1 =

^proj

([−1, 1]; ^{x k −αhτ} _ah ^2,k+1 ) τ 2,k+1 =

^proj

([−1, 1]; _αh ^{e k} )

(21)

Oneseesthat thisisveryeasilyimplementable withnestedprojetions.

2.3 Extension to higher-order systems

Inordertoshowthattheforegoingmethodextendsto

n−

^{thordersystemswith}

the equivalent-ontrol-based sliding-mode-ontroller (ECB-SMC [28, Chapter

2℄) andalsotobetterxtheideasonthestrutureoftheproposedontrollers,

letusonsiderthelineartime-invariantsystemwithdisturbane

x(t) = ˙ Ax(t) +

Bu(t)+Dϕ(t)

^with

||ϕ(t)|| 1 6 pϕ max

^forall

t

^,

ϕ max > |ϕ i | ∞

^forall

1 6 i 6 p

^and

D ∈ IR ^n×p

^{. Letushooseaslidingsurfae}

Σ = {x ∈ IR ⁿ | Cx = 0, C ∈ IR ^m×n }

^,

where

m

^{isthedimensionoftheinputvetor}

u(t)

^{. TheECB-SMCtakestheform}

u ∈ −(CB) ⁻¹ CAx − α(CB) ⁻¹

^Sgn

(Cx)

^{, provided}

CB

^{isfull-rank. Let}

z ^∆ = Cx

^.

Thereduedlosed-loopdynamisis

z(t) = ˙ −ατ + CDϕ(t)

^,

τ ∈

^Sgn

(z)

^,whih

is globally asymptotiallystable and

Σ

^{is reahed in nite time provided}

α >

p||CD|| ϕ max

^{(thisanbeshownwiththeLyapunovfuntion}

V (z) = ¹ ₂ z ^T z

^that

satises along the losed-loop trajetories

V ˙ (t) 6 || z || 1 (− α + p || CD || ϕ _max )

^).

Thesystemisdisretizedas

x k+1 = x k + hAx k + hBu k+1 + hDϕ k+1 ,

⁽²²⁾

andthenominalsystemissimplygivenby

x ˜ k+1 = (I + hA)x k + hBu k+1

^{. The}

impliitEulerontrollerisdenedas







u k+1 = −(CB) ⁻¹ CAx _k − α(CB) ⁻¹ τ k+1

τ k+1 ∈

^Sgn

(C x ˜ k+1 ).

(23)

Therefore

τ k+1

^{isgivenby(see(1)and(2)):}

τ k+1 ∈

^Sgn

(Cx k − αhτ k+1 ) ⇔ τ k+1 =

^proj

([−1, 1] ^m ; 1

αh Cx k ),

⁽²⁴⁾

where

[−1, 1] ^m = [−1, 1] × ... × [−1, 1] m−

^{times. Thus the ontroller to be}

applied attime

t k

^is

u k+1 = −(CB) ⁻¹ CAx _k − α(CB) ⁻¹

^proj

([−1, 1] ^m ; 1

αh Cx _k ).

⁽²⁵⁾

Wethereforeobtain,with

z _k = Cx _k

^and

z ˜ _k = C˜ x _k

^:







˜

z k+1 = z k − αhτ k+1

τ k+1 ∈

^Sgn

(˜ z k+1 )

z k+1 = z k − αhτ k+1 + hCDϕ k+1 ,

(26)

that is similar to (4). Thus the sameonlusions asin Proposition 1maybe

drawnforthisdisrete-timesystemprovidedthat

α > p ||CD||ϕ max

^{: thesliding}

surfae

C˜ x k = 0

^{isattainedaftera}nite-numberofstepswhateverthebounded initialstate,andthedisrete-timesystemevolvessmoothlyonthissurfaewhile

thedisturbaneeetsonthevariable

Cx k

^{areattenuatedbyafator}

h

^.

Remark 1 The disrete-time input obtained from [28, Equ.(9.36)℄ (see also

[4 , 17 ℄and [9 ℄for the original ontribution) whenapplied to(22 ) isalulated

tobe:

u k+1 = −(hCB) ⁻¹ C(I + hA)x k = −(CB) ^{−1 Cx} _h ^k − (CB) ⁻¹ CAx _k

^,whih

is linear. The disrepany with (25 ) is the projetion on the set

[−1, 1] ^m

^that

is intrinsially present in the impliit Euler input (that is nonlinear Lipshitz

ontinuous),andisnotaonsequeneofadding saturations beauseofatuator

limitations. Alsotheontrollerin(25)remainsboundedwhen

h → 0

^,aproperty

shared by all the ontrollers onsidered in this paper. One may say that both

ontroller designs share the same philosophy sine they are both alulated

in order to fore the disrete sliding surfae to be zero, with a suitable input.

However they are not atall equivalent. In pratie the ontrollers proposed in

thispapermaybealulatedusingasuitableomplementarityproblemsolver[2 ℄.

As alluded to in setion 2.1, the plant and the ontroller have to be dis-

retized with the same method (bakward Euler or ZOH) in order to assure

the disturbane attenuation. Let us investigate the zero-order-holdermethod

(ZOH)onthisexample. Theinputisassumed tobeonstanton

[t k , t k+1 ]

^and

is omputedat

t = t k

^{. TheZOH}disretization oftheECB-SMC ontrolleron

[t k , t k+1 ]

^{takestheform[32℄:}

x k+1 = A ^∗ (h)x k − αB ^∗ (h)τ k+1 + ϕ ^∗ (h),

⁽²⁷⁾

with

A ^∗ (h) = e ^Ah − R h

0 e ^At dt B(CB) ⁻¹ CA

^,

B ^∗ (h) = R h

0 e ^At dt B(CB) ⁻¹

^,

ϕ ^∗ _k (h) = R h

0 e ^At Dϕ((k+1)h−τ)dτ

^{. Notiethatas}

h → 0

^then

A ^∗ (h) ≈ I n +Ah−

hB(CB) ⁻¹ CA + O(h ² )

^,

B ^∗ (h) ≈ hB(CB) ⁻¹ +O(h ² )

^,onsequentlytheimpliit EulerandZOHmethodsyieldthesamedisrete-timesystemwhenthesampling

period issmall. Alsoonemayomputethat

||ϕ ^∗ _k (h)|| 1 6 hp ||D||ϕ max + O(h ² )

^.

Thisyieldsthegeneralizedequation:

(a)







˜

x k+1 = A ^∗ (h)x k − αB ^∗ (h)τ k+1

τ k+1 ∈

^Sgn

(C x ˜ k+1 )

x k+1 = A ^∗ (h)x k − αB ^∗ (h)τ k+1 + Cϕ ^∗ _k (h)

⇒ (b)



 



 



C˜ x k+1 = CA ^∗ (h)x k − αCB ^∗ (h)τ k+1

τ k+1 ∈

^Sgn

(C˜ x k+1 )

Cx k+1 = CA ^∗ (h)x k − αCB ^∗ (h)τ k+1

+Cϕ ^∗ _k (h)

(28)

Suppose that thematrix

CB ^∗ (h)

^{is symmetri positivedenite (sine}

CB ^∗ = hI m + O(h ² )

^{itfollowsthat for}

h

^smallenough

CB ^∗ > 0

^{isguaranteedif}

CB

^is

invertible). Then from(1)and (2)thersttwolines of(28)(b)areequivalent

to:

CA ^∗ (h)x k − αCB ^∗ (h)τ k+1 ∈ N [−1,1] ^m (τ k+1 ) ⇔ τ k+1 =

^proj

_CB ∗ (h) ([−1, 1] ^m ; _α ¹ (CB ^∗ (h)) ⁻¹ CA ^∗ (h)x k )

⇔ τ k+1 = argmin _z∈[−1,1] m 1

2 (z − _α ¹ (CB ^∗ (h)) ⁻¹ CA ^∗ (h)x k ) ^T CB ^∗ (h)(z − _α ¹ (CB ^∗ (h)) ⁻¹ CA ^∗ (h)x k ),

(29)

whereproj

CB ^∗ (h)

^{istheprojetioninthemetridenedby}

CB ^∗ (h)

^{. There-}

foreateahsteptheontrollerisalulatedasthesolutionofaquadratipro-

grammeand is unique. Notiethat when

h

^{issmall then}

CB ^∗ (h) ≈ hI _m

^and

CA ^∗ (h) ≈ C

^sothat

τ k+1 = argmin _z∈[−1,1] m

1 2 (z− 1

hα Cx k ) ^T (z− 1

hα Cx k ) ⇔ τ k+1 =

^proj

([−1, 1] ^m ; 1 hα Cx k ).

The inputremains bounded when the sampling time dereases. The next

resultisobviousfrom(28)(b):

Lemma1 Let

C˜ x k+1 = 0

^{for some}

k > 0

^{. Then}

||Cx k+1 || 1 = ||Cϕ ^∗ _k (h)|| = 6 hp ||C|| ||D||ϕ max

^.

Thusthedisturbaneattenuationonthenominaldisrete-timesystemslid-

ing surfae holds with the ZOH method. If thehigher order terms in

h ²

^are

negleted,oneseesthat(28)(b)isthesameas(26)whereonlythedisturbane

termismodied,sothatoneagaintheonlusionsofProposition1apply: the

disrete-timesystemreahesthenominalsystemslidingsurfaeinanitenum-

berofsteps. Theanalysisforany

h > 0

^{ismoreinvolvedbeausethetermsin}

h ²

^{introdueaouplingbetween(28)(b)and(a). Howeversinewearefousing}

ontheslidingmodesandnite-timeonvergeneto theslidingsurfaeonly,we

mayassumethatthesolution

x(·)

^{ofthelosed-loopsystemisboundedforany}

bounded initial data, andthat thesolution

x k

^{ofits ZOHounterpartin (28)}

(a) is bounded as well, i.e.

||x k || 6 M

^{for all}

k > 0

^{and some}

M

^{. Then the}

followingholds:

Proposition3 Let

h > 0

^{begiven. Supposethat thesolutionof (28 )(a)satis-}

es

|| x _k || 6 M

^forall

k > 0

^andsome

M < +∞

^,andthat

CB ^∗ (h)

^issymmetri

positivedenite,with

CB ^∗ (h) > γI _m > 0

^forsomeknown

γ

^{. Thenthereexistsa}

onstant

δ(h ² , M )

^suhthatif

α > ^m _γ ||C|| hρ ||D|| ϕ max + δ(h ² , M )

^,

C˜ x k+1 = 0

for some

k > 0

^implies

C x ˜ k+n = 0

^{for all}

n > 2

^.

Proof: fromLemma1therstline of(28)(b)rewritesatstep

k + 2

^as:

C x ˜ k+2 = (C + O(h ² ))x k+1 − αCB ^∗ (h)τ k+2

= Cϕ ^∗ _k (h) + O(h ² )x k+1 − αCB ^∗ (h)τ k+2

(30)

Thus(30)and

τ k+2 ∈

^Sgn

(C x ˜ k+2 )

^formageneralizedequationwhihpossesses auniquesolutionbeause

αCB ^∗ (h)

^{ispositivedenite. Wemayrewriteitas:}

0 ∈ 1

α (CB ^∗ (h)) ⁻¹ C˜ x k+2 − 1

α (CB ^∗ (h)) ⁻¹ (Cϕ ^∗ _k (h) + O(h ² )x k+1 ) +

^Sgn

(C x ˜ k+2 )

(31)

Thereforeif

1

α (CB ^∗ (h)) ⁻¹ (Cϕ ^∗ _k (h)+ O(h ² )x k+1 ) ∈ [−1, 1] ^m

^then

C˜ x k+2 = 0

is the unique solution of (31). From the proposition's assumptions one has

||(CB ^∗ (h)) ⁻¹ (Cϕ ^∗ _k (h) + O(h ² )x k+1 )|| 1 6 ^m

γ ||C|| hp ||D||ϕ max +δ(h ² , M )

^,where

δ(h ² , M )

^{isanupperboundfor}

O(h ² )x k+1

^{. Thisupperbounddependsonlyon}

M

^{, the system's matries, and}

h

^{. It is therefore uniform with respet to the}

stepnumber

k

^.

ThenLemma1maybeappliedto showthedisturbaneattenuationonthe

nominalsystemdisrete-timeslidingsurfae.

2.4 Numerial simulations

The numerialsimulationsareobtainedwith thesionossoftware pakageof

the INRIA 2

that is dediated to non-smooth dynamial systems. In order to

reproduetheontinuous-timenature oftheplant,the plantdynamis isinte-

gratedin allthesimulationswiththemahinepreision,whereastheontroller

sampling time is muh larger:

h = 10 ⁻¹

^{s. This is equivalentto} implementing aZOHmethod. Thedisturbaneistaken as

ϕ(t) = φ sin(ωt)

^{andwesimulate}

thesystemin (10).

Theabovedevelopmentsareillustratedongure2with

a = 1

^,

α = 2

^,

ω = 5

and

φ = 0.1

^. Illustrations are given on Figures 3 and 4 with

a = 1

^,

α = 2

^,

ω = 100

^and

φ = 0.1

^{. Thedisturbane}attenuationislearlyshown.

3 Seond order systems

Let us now fous on a more general lass of systems and perform the same

stepsasfortherstorderase(ashort realloftheontinuous-timease,and

2

http://sionos.gforge.inria.fr/

-0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10

x1

t

PSfragreplaements

x k e k

(a)state

x k

^anderror

e k

^{vs. time}

-0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10

PSfragreplaements

τ 1,k τ 2,k ϕ

k

α

(b)Multiplier

τ 1,k+1

^,

τ 2,k+1

^andperturbation

ϕ k /α

^vs.time

Figure2: Simulationofthesystem(10),

φ = 0.1

^,

ω = 5

^.

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10

PSfragreplaements

x k e k

(a)Statevs.time(nesampling)

-0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002

4 5 6 7 8 9 10

PSfragreplaements

x k e k

(b)Statevs. time(nesampling,zoom)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

0 2 4 6 8 10

u

PSfragreplaements

()Controlinput

u(t)

Figure 3: Simulationofthesystem(10)with

ϕ(t) = φ sin ωt

^,

φ = 0.1

^,

ω = 100

^.

-0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10

PSfragreplaements

τ

1,k

τ

2,k

(a)Multiplier

τ

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10

PSfragreplaements

se

k

su

k

(b)Slidingvariable

s(t)

Figure4: Simulationofthesystem(10)with

ϕ(t) = φ sin ωt

^,

ω = 100

^.

thenthetime-disretization). Thesimulationswillbegivenafterthetheoretial

presentations.

3.1 First-ordersliding-modestabilizationwithdisturbane

ompensation

3.1.1 The ontinuous-timesystem

Theplantdynamisisgivenby

¨

x(t) = u(t) + ϕ(x(t), t),

⁽³²⁾

where

x(t) ∈ IR

^{isthestatevetor,}

u(t) ∈ IR

^{is theontrolinput. The distur-}

bane

ϕ(x, t) ∈ IR

^{represents thesystem unertainty and its inuene on the}

ontrol proess should be rejeted. It is assumed that

ϕ(x, t)

^{is an unknown}

funtion withana prioriknownupperestimate

ϕ max > 0

^suhthat

|ϕ(x, t)| < ϕ max

⁽³³⁾

for almost all

x, t ∈ IR.

^{Themodel repeats the struture of the plant and is}

givenby:

¨ ˆ

x(t) = u(t) + v(t),

⁽³⁴⁾

where

v(t) ∈ IR

^{is the model input. The error dynamis is then written as}

follows:

¨

e(t) = − v(t) + ϕ(x(t), t),

⁽³⁵⁾

where

e = x − x ˆ

^{is thedeviationof themodel statefromthe plantstate. The}

errordynamis,drivenbythesliding-modeinput,isgivenby:

v(t) ∈ k e e(t) + ˙ k s s e (t) + M v

^sgn

(s e (t)),

⁽³⁶⁾

anditis globallyasymptotiallystabilizedprovidedthat

M _v > ϕ max

^and

s _e =

˙

e + k e e

^where

k e

^and

k s

^{arepositiveonstants. Toreproduethis onlusionit}

suestorewritethestateequationfor

s e

^{,thus arrivingattheequation}

˙

s _e (t) ∈ −k s s _e (t) − M _v

^sgn

(s e (t)) + ϕ(x(t), t),

⁽³⁷⁾

whihhas

s ^∗ _e = 0

^{asits uniquexed point,whihis globallynite-timestable.}

Thus,bytheequivalentontrolmethodonehasthat:

v _eq (t) = ϕ(x(t), t)

⁽³⁸⁾

onthesurfae

s e = 0

^{anditisexpetedthattheontrollaw}

u ∈ −v − M x

^sgn

(s x ) − k x x, ˙

⁽³⁹⁾

with

s _x = ˙ x + k _x x

^, asymptotially ompensates for the disturbane

ϕ(x, t)

^.

Indeed,onethesliding modeoursonthesurfae

s _e = 0

^{,theplantequation}

takesthedisturbane-freeform







˙

s _x (t) ∈ − M _x

^sgn

(s x (t)) − k _e e(t) ˙

˙

e(t) = −k e e(t).

(40)

beause on this sliding surfaeone has

M v

^sgn

(s e (t)) = ϕ(x(t), t)

^{. Sine the}

dynamis (40) has

s ^∗ _x = 0

^{as aglobally} asymptotiallystable xed point, the desireddisturbaneompensationisthusprovided. Summarizing,thefollowing

result, guaranteeingthe global asymptoti stability of thelosed-loopsystem,

is obtained. Letusdenoteby

z

^{thestatevetor}

z = [e s e x s x ] ^T

^{. Theoupled}

plant/errordynamisinthelosed-loopsystemisgivenby:



 

 

 

 

 

 

 

 

 

 

 

 

˙ z(t) =











−k e 1 0 0

0 −k s 0 0

0 0 −k x 1

−k e −k s 0 0









 z(t) −











0 0

M _v 0

0 0

M _v M _x









 τ(t) +









 0 ϕ(x(t), t)

0 ϕ(x(t), t)











τ(t) ∈

^Sgn

"

0 1 0 0 0 0 0 1

# z(t)

! ,

(41)

It is noteworthy that the

(e s e )

^{subdynamis is deoupled from the}

(x s x )

subdynamis.

Proposition4 Considerthelosed-loopsystem(41)withpositivegains

k e , k s , M x , M v

and an externaldisturbane

ϕ(x, t)

^{suhthat (33 )holds for almost all}

x ∈ IR

^,

t ∈ IR

^and

M _v > ϕ max

^{. Then after a nite time, this system evolves in the}

sliding mode along the surfaes

s _e = 0

^and

s _x = 0

^{, and along these surfaes,}

the system dynamis isgoverned by the asymptotially stable, disturbane-free

equations (40 ).

TheproofofProposition4isratherstandard[27℄anditisthereforeomitted.

Theparametersubordination

k _v >> k _x

^{ensuresafasteronvergeneoftheerror}

dynamis ompared to the state variables of the plant whereas the ontroller

magnitude

M x

^{is requiredto bepositive only. Asamatter offat, thehigher}

M x

^{thehighertheplantonvergenerate.}

3.1.2 The bakward Eulertime-disretization

Letusproeedwiththesamedisretizationasintheaboverst-orderexamples.

For this let us onsider the rst error dynamis in (37), and disretize it on

[t k , t k+1 )

^as:



 



 



˜

s e,k+1 = s e,k − hk s s e,k − hM v τ 1,k+1

τ 1,k+1 ∈

^sgn

(˜ s e,k+1 )

s e,k+1 = s e,k − hk s s e,k − hM v τ 1,k+1 + hϕ(x k+1 , t k+1 ) e k+1 = e k + h e ˙ k+1

(42)

forall

k > 0

^{. Thersttwolinesarea}generalizedequationwithunknown

˜ s e,k+1

^,

whih we may rewriteas

0 ∈ F(˜ s e,k+1 )

^forsome multifuntion

F : IR → 2 IR

.

Ithasauniquesolutionsinethesignmultifuntionismaximalmonotoneand

F (·)

^{is2-monotoneasthesumofamonotoneanda2-monotone}multifuntions (seeDenition2.3.1andTheorem2.3.3in[10℄,andExerise12.4in[26℄). Notie

that if

s ˜ e,k+1 = 0

^then

(1 − hk s )s e,k = hM v τ 1,k+1

^and

s e,k+1 = hϕ(x k+1 , t k+1 )

^.

Also

τ 1,k+1

^isafuntionof

s e,k

^{only,thatisof}

e ˙ k = ^{e k −e} _h ^k−1

^and

e k

^{. Sothereis}

not anexatompensation asin the ontinuous-timease, but adisturbane-

attenuation by a fator

h

^{. Notie that (42) is exatly (4), replaing}

x k

^by

(1 − hk s )s e,k

^,

−a

^by

−M v

^{. Hene the onlusions of} Proposition 1 hold for (42). Weinferthatafter anitenumberof steps

k 0

^{, oneobtains}

s ˜ e,k = 0

^and

s e,k = hϕ(x k+1 , t k+1 )

^sothat

|s e,k | < hϕ max

^forall

k > k 0

^forsomenite

k 0

^.

Thenextresultharaterizestheevolutionof

e k

^{ontheslidingsurfae}

s ˜ e,k = 0

^.

Lemma2 Suppose that the sliding surfae

Σ ˜ e = {˜ s e,k ∈ IR | s ˜ e,k = 0}

^is

attained at

k = k 0

^{andthat the systemstayson it. Take for simpliity}

k 0 = 0

^.

Then:

e k+1 = (1 + hk e ) ^−k−1 e 0 + h ² (1 + hk e ) ⁻¹

k

X

i=0

(1 + hk e ) ^i−k ϕ(x i , t i ).

⁽⁴³⁾

Proof: Onehas

e k = e k−1 + h e ˙ k

^and

s e,k = ˙ e k + k e e k

^{. Weinferthat}

e _k = (1 + hk _e ) ⁻¹ e k−1 + (1 + hk _e ) ⁻¹ h ² ϕ k+1

⁽⁴⁴⁾

fromwhih(43)follows.

Notie that if we implement

e k+1 = e k + h e ˙ k

^{then we obtain}

e k = (1 − hk _e )e k−1 + h ² ϕ _k

^{and similaralulationsmay bedone, usingthefat that for}

h > 0

^{small enough}

0 < 1 − hk _e < 1

^{. Therefore on the sliding surfae the}

disrete-timeerroristhesumofanasymptotiallyvanishingterm,plusaterm

thatdependsonthedisturbane,attenuatedbyafator

h ²

^{. Theseondpartof}

theerrordynamisin(40)isnowdisretizedasfollows:



 



 



˜

s x,k+1 = s _x,k − hk _e e ˙ _k − hk _s s _e,k − hM _v τ 1,k+1 − hM _x τ 2,k+1

τ 2,k+1 ∈

^sgn

(˜ s x,k+1 )

s x,k+1 = s _x,k − hk _e e ˙ _k − hk _s s _e,k − hM _v τ 1,k+1 − hM _x τ 2,k+1 + hϕ(x k+1 , t k+1 ) x k+1 = x _k + h x ˙ k+1

(45)

Notiethatif

k > k 0

^then

(1 −hk s )s e,k = hM _v τ 1,k+1

^and

s _e,k = hϕ(x k+1 , t k+1 )

^.

For

k > k 0

^{thesystemevolvesonthesliding surfae}

˜ s e,k = 0

^andweobtain:







˜

s x,k+1 = s x,k − hϕ(x k+1 , t k+1 ) − hM x τ 2,k+1

τ 2,k+1 ∈

^sgn

(˜ s x,k+1 )

s x,k+1 = s _x,k − hk _e e ˙ _k − hM _x τ 2,k+1

(46)

From(43)weinferthat

hk e e ˙ k = ǫ k + h ² α k

^where

|α k | 6 ϕ 0 P k

i=0 (1 + hk e ) ^i−k−1

and

ǫ _k

^is exponentially dereasingsine

1 + hk _e > 1

^{. It followsalsothat}

| α _k |

is upperbounded byaonstant notdepending on

k

^{and wemaywrite}

| α _k | 6 αϕ max

^{forsomeonstant}

α

^{. Wethereforerewrite(46)as}







˜

s x,k+1 = s _x,k − hϕ(x k+1 , t k+1 ) − hM _x τ 2,k+1

τ 2,k+1 ∈

^sgn

(˜ s x,k+1 )

s x,k+1 = s _x,k − ǫ _k − h ² α _k − hM _x τ 2,k+1 .

(47)

It is noteworthy that (47) is similar to (14) and to (4) exept for the expo-

nentially deaying term

ǫ k

^{. Thus the following holds, whih shows that the}

disturbaneeetsarestillattenuatedbyafator

h

^:

Proposition5 Considerthe disrete-timesystem(47)that representsthe sys-

tem's dynamis on the sliding surfae

Σ ˜ e = {˜ s e,k ∈ IR | ˜ s e,k = 0}

^{, .e. for}

k > k 0

^{. Supposethat}

M x > ϕ max

^{. There exists}

k 1 < +∞

^,

k 1 > k 0

^{,suh that}

for all

k > k 1

^{one has}

˜ s x,k = 0

^{. Then}

|s x,k+1 | 6 hϕ max + |ǫ k | + h ² αϕ max

^.

Proof:Therstpartoftheprooffollowsthesamelinesastheaboveproofs

ofnite-timeonvergenesandisomitted. Theseond partfollowseasilyfrom

(47)byimposing

s ˜ x,k+1 = 0

^{andinsertingthevalueof}

hM _x τ 2,k+1

^intothethird

lineof(47).

Thenextresultharaterizesthedynamisof

x _k

^{ontheslidingsurfae}

x ˜ _k = 0

^{. Forsimpliitywetake}

k 1 = 0

ⁱⁿProposition 5.

Lemma3 Suppose that for

k > 0

^{the system evolves on the sliding surfae}

˜

s x,k = 0

^{,sothat (negletingtermsin}

h ²

⁾

|s x,k+1 | 6 hϕ max + |ǫ k |

^{. Then}

x k = (1 + hk x ) ⁻¹ x 0 − h(1 + hk x ) ⁻¹

k−1

X

i=0

(1 + hk x ) ⁻ⁱ (ǫ k−1−i + h ² α k−1−i + hϕ k−i ).

(48)

Proof: From

s x,k+1 = 0

^{oneeasilyderives:}

x k+1 = (1 + hk _x ) ⁻¹ x _k − h(1 + hk _x ) ⁻¹ (ǫ k + h ² α _k + hϕ k+1 )

⁽⁴⁹⁾

fromwhih(48)isdedued.

Thedisturbaneisthereforeattenuatedbyafator

h ²

^{onthestateposition}

x _k

^{. Similarlyto (21), using (1) we may rewrite the} disrete-time losed-loop systemas:



 

 

 



 

 

 



s x,k+1 = s x,k − hk e e ˙ k − hk s s e,k − hM v τ 1,k+1 − hM x τ 2,k+1 + hϕ(x k+1 , t k+1 ) s e,k+1 = s e,k − hk s s e,k − hM v τ 1,k+1 + hϕ(x k+1 , t k+1 )

x k+1 = x k + h x ˙ k+1

e k+1 = e k + h e ˙ k+1

τ 1,k+1 =

^proj

([−1, 1]; ^{s e,k −hk} _hM ^{s s e,k}

v )

τ 2,k+1 =

^proj

([−1, 1]; ^{s x,k −hk e e ˙ k −hk} _hM ^{s s e,k −hM x τ 1,k+1}

x ).

(50)

Onehasalso

s e,k = ˙ e k +k e e k

^,

s x,k = ˙ x k +k x x k

^,sothat

x k+1 = (1+hk x ) ⁻¹ (x k + hs x,k+1 )

^and

e k+1 = (1 + hk e ) ⁻¹ (e k + hs e,k+1 )

^{. The ontroller has anested-}

projetionstrutureandiseasilyimplementableattime

t = t k

^{withtheknowl-}

edgeof

x k

^,

x k−1

^and

e k

^,

e k−1

^.

3.2 Position feedbak stabilization of a double integrator

Letusnowpasstoother typesofsliding-modedisontinuousontrollerswhih

havebeenproposedin theliterature,knownasthetwistingandsuper-twisting

algorithms[11℄[28,Ÿ3.6.2,3.6.3℄. Theypossessadvantages(nite-timestability

of the origin, better disturbaneattenuation), howevertheirstability analysis

ismoreintriate.

3.2.1 Finite-timestabilizingstate feedbak synthesis

Tobegin with, wepresent astati feedbak ontroller that globally stabilizes

thedoubleintegrator:

˙

x(t) = y(t), y(t) = ˙ u(t).

⁽⁵¹⁾

Afeedbaklaw

u(x, y)

^{isfurtherreferredtoasnite-time}stabilizingifitrenders the origin of the losed-loop system (51) a nite-time stable equilibrium as

dened in[22℄. Thefollowingstatefeedbak

u = −µ

^sgn

(y) − ν

^sgn

(x)

⁽⁵²⁾

withparameters

ν > µ > 0

^{isproposedtogloballystabilizethedoubleintegrator}

(51).

Theorem1 Considerthedynamisofthelosed-loopsystemin(51)(52 ). This

dynamis has a unique xed point

(x, y) = (0, 0)

^{whih is globally nite-time}

stable,providedthat the ontrollerparametersaresuhthat

ν > µ > 0

^.

Theproofmaybefoundin theExample3.2 andsetion4.6of [22℄. Letus

nowonsiderthedisturbane-orruptedversion:

˙

x(t) = y(t), y(t) = ˙ u(t) + ϕ(x(t), y(t), t)

⁽⁵³⁾

and investigate the robustnesspropertiesof thelosed-loopsystem (52), (53)

againstexternaldisturbanes

ϕ(x, y, t)

^{,beingaloallyintegrablefuntiononall}

potentialtrajetories

x(t), y(t)

^{. Aordingto [22, Theorem 4.2℄,thedisturbed}

systemin(52)(53)rendersthesystemnite-timestable,regardlessofwhihever

disturbane

ϕ(·)

^{withauniformupperbound}

ess

sup

t>0

|ϕ(x(t), y(t), t)| 6 ϕ max

⁽⁵⁴⁾

onitsmagnitudesuhthat

0 < ϕ max < µ < ν − ϕ max

⁽⁵⁵⁾

aets the system. This robustnessproperty is ahieveddue to the high fre-

quenyontrollerswithingintheslidingmodeoftheseondorderthat ours

in theorigin.

Theorem2 [22 ,Setion4.6℄Given

µ

^and

ν > 0

^{,thelosed-loopsystemin(52)}

(53 ) has aunique xed point

(x ^∗ , y ^∗ ) = (0, 0)

^{whih is globally} asymptotially nite-time stable, regardless of whihever disturbane

ϕ(·)

^{, satisfying (54 )and}

(55 ), aetsthe system.

LetusproposethefollowingimpliitEulertime-disretization,where

ϕ k+1

= ∆

ϕ(x k+1 , y k+1 , t k+1 )

^:



 

 

 



 



˜

x _k+1 = x _k + h˜ y _k+1

˜

y _k+1 = y _k − hντ _1,k+1 − hµτ _2,k+1 τ _1,k+1 ∈

^sgn

(˜ x _k+1 )

τ 2,k+1 ∈

^sgn

(˜ y k+1 ) x = x + hy

(56)

from whih it follows applying (1) to the seond and the fourth lines of (56)

that

τ 2,k+1 =

^proj

([−1, 1]; y k − hντ 1,k+1

hµ ).

⁽⁵⁷⁾

The disrete-time system in (56) is still onstruted along the same lines as

the ones in the foregoing setions: one omputes the input from a nominal

unperturbedsystem(therstfour linesof(56))andthenoneinjetstheom-

puted inputinto theplantdynamis (thelast twolinesof (56)). Howeverthis

time there is no deoupling betweenthe

x ˜ k −

^{dynamis andthe}

y ˜ k −

^dynamis.

Let us now alulate the ontrol input. One has

y ˜ k+1 = y _k − hντ 1,k+1 − hµ

^proj

([−1, 1]; ^{y k −hντ} _hµ ^1,k+1 )

^{. Therefore}

y ˜ k+1 = y _k −hντ 1,k+1 −

^proj

([−hµ, hµ]; y _k − hντ 1,k+1 )

^{. Thusthreemodes arepossible:}

ˆ (i)if

y k − hντ 1,k+1 > hµ

^onegets

y ˜ k+1 = y k − hντ 1,k+1 − hµ

^,

y k+1 = y k − hντ 1,k+1 −hµ+hϕ k+1

^and

x ˜ k+1 = x k +h(y k −hντ 1,k+1 )−h ² µ

^,

x k+1 = x k + h(y k − hντ 1,k+1 ) − h ² µ + h ² ϕ k+1

^{. Also}

τ 1,k+1 = _h ¹ 2 ν

^proj

([−h ² ν, h ² ν]; x k + hy k − h ² µ)

^,

τ 2,k+1 = 1

^.

Therearethreesub-modes:

(i-1) let

x _k + hy _k − h ² µ > h ² ν

^{: then}

τ 1,k+1 = 1

^,

y _k > h(ν + µ)

^,

x k+1 = x _k + hy _k − h ² (ν + µ) + h ² ϕ k+1

^,

y k+1 = y _k − h(ν + µ + ϕ k+1 )

^.

(i-2)let

x k + hy k − h ² µ < −h ² ν

^{: then}

τ 1,k+1 = −1

^,

y k > h(µ − ν)

^,

x k+1 = x k + hy k + h ² (ν − µ) + h ² ϕ k+1

^,

y k+1 = y k + h(ν − µ + ϕ k+1 )

^.

(i-3)let

|x k + hy k − h ² µ| 6 h ² ν

^{: then}

x k < 0

^,

τ 1,k+1 = ^{x k +hy} _h 2 ^k ν ^{−h 2 µ}

^,

x k+1 = h ² ϕ k+1

^,

y k+1 = − ^x _h ^k + hϕ k+1

^,

x ˜ k+1 = 0

^,

y ˜ k+1 = − ^x _h ^k

^.

ˆ (ii)if

y k −hντ 1,k+1 < −hµ

^onegets

y ˜ k+1 = y k −hντ 1,k+1 +hµ

^,

y k+1 = y k − hντ 1,k+1 +hµ+hϕ k+1

^and

x ˜ k+1 = x k +h(y k −hντ 1,k+1 )−h ² µ

^,

x k+1 = x k + h(y k − hντ 1,k+1 ) + h ² µ + h ² ϕ k+1

^{. Also}

τ 1,k+1 = _h ¹ 2 ν

^proj

([−h ² ν, h ² ν]; x k + hy k + h ² µ)

^,

τ 2,k+1 = −1

^.

Therearethreesub-modes:

(ii-1) let

x _k + hy _k + h ² µ > h ² ν

^{: then}

τ 1,k+1 = 1

^,

y _k < h(ν − µ)

^,

x k+1 = x _k + hy _k − h ² (ν − µ − ϕ k+1

^,

y k+1 = y _k + h(ν − µ − ϕ k+1 )

^.

(ii-2)let

x k +hy k + h ² µ < −h ² ν

^{: then}

τ 1,k+1 = −1

^,

y k < −h(ν + µ)

^,

x k+1 = x k + hy k + h ² (ν + µ + ϕ k+1

^,

y k+1 = y k + h(ν + µ + ϕ k+1 )

^.

(ii-3)let

|x k + hy k + h ² µ| 6 h ² µ

^{: then}

τ 1,k+1 = ^{x k +hy} _h 2 ^k ν ^{+h 2 µ}

^,

x k+1 = h ² ϕ k+1

^,

x ˜ k+1 = 0

^,

y k+1 = − ^x _h ^k + hϕ k+1

^,

y ˜ k+1 = − ^x _h ^k

^.

ˆ (iii) if

|y k − hντ 1,k+1 | 6 hµ

^{one gets}

y ˜ k+1 = y k − hντ 1,k+1 − (y k − hντ 1,k+1 ) = 0

^,

y k+1 = hϕ k+1

^and

x k+1 = x k + h ² ϕ k+1

^{. Also}

τ 1,k+1 ∈

sgn

(x k )

^,

τ 2,k+1 = ^{y k −hντ} _hµ ^1,k+1

^.

Oneseesthattheontroller

(τ 1,k+1 , τ 2,k+1 ) ^T

^{isaausalinputattime}

t = t k

andthereisnosingularityin

τ 1,k+1

^as

h

^{tendstozero. In(i)and(ii)thevalue}

for

τ 1,k+1

^{is obtained from the} generalizedequation

τ 1,k+1 ∈

^sgn

(x k + hy k − h ² ντ 1,k+1 − h ² µ)

^{andusing(1). Inallases}

τ 2,k+1

^{isobtainedfrom(57).}

It is easily heked that

(x ^∗ , y ^∗ ) = (0, 0)

^{is the unique xed point of the}

unperturbedsystem(56)(take

ϕ _k+1 = 0

^{in(56)). Thenextresultholds.}