Switching systems: active mode recognition, identification of the switching law

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Switching systems: active mode recognition, identification of the switching law

Elom Ayih Domlan, José Ragot, Didier Maquin

To cite this version:

Elom Ayih Domlan, José Ragot, Didier Maquin. Switching systems: active mode recognition, iden-

tification of the switching law. Journal of Control Science and Engineering, Hindawi Publishing

Corporation, 2007, 2007, pp.ID 50796. �10.1155/2007/50796�. �hal-00195472�

of the Swithing Law

Elom Ayih Domlan

∗1

, José Ragot

2

& DidierMaquin

1 1

Department of Chemial and Materials Engineering

University of Alberta

Edmonton, AB, T6G 2G6, Canada

elom.domlanualberta.a

2

Centre de Reherhe en Automatiquede Nany, UMR

7039

Nany-Université, CNRS

2, avenue de laForêt de Haye

54516

Vand÷uvre-les-Nany, Frane

{jose.ragot, didier.maquin}ensem.inpl-nany.fr

Abstrat

Theproblemoftheestimation ofthedisretestate ofaswithingsystem isstudied. The

knowledgeoftheswithinglawisessentialforthiskindofsystemasitsimpliestheirmanipu-

lationforontrolpurposes. Thispaperinvestigatestheuseofamodel-baseddiagnosismethod

forthedeterminationoftheativemodeateahtimepointbasedonthesysteminput/output

data. Theissueoftheparametriidentiationoftheswithinglawisalsoaddressed.

1 Introdution

The modelling of omplex systems often leads to omplex nonlinear models. To get rid of the

omplexity of the obtained model, one often resorts to a widely used modelling strategy whih

represents the system behaviour by using a set of models with a simple struture, eah model

desribing the behaviour of the system in a partiular operating zone. Within this modelling

framework,hybridmodels[9,19℄areverysuessfulin representingsuhproesses.

Hybridmodelsharaterizephysialproessesgovernedbyontinuousdierentialanddierene

equationsanddisretevariables. Theproessisdesribedbyseveraloperatingregimesalledmodes

and the transition from one mode to another is governed by the evolution of internal variables

(input, output, state) or externalvariables or events (ationof a humanoperator on the system

forinstane). Theglobal behaviourobtainedfor themodelled omplexsystemisstronglyrelated

to the nature of the proedure managing the transition from one mode to another. When this

transitionisabrupt,oneobtainsthelassofswithingmodels. Thislassofmodelsiswidelyused

beausethewellmastered toolsforanalysis andontroloflinearsystemsanbeextended,under

∗

Correspondingauthor.

bymodelsbelongingto thislass.

Researh onswithingsystemsismainly foused ontheelds ofidentiation[17, 22, 24,25℄,

ontrol [9,12℄, stabilityanalysis [12, 21℄ and stateestimation [1, 6℄. Theknowledge of themode

desribing theevolutionof the system at any moment, this mode being alled ative mode, is a

ruial piee ofinformation that simplies theappliation of thevarious resultsoming from the

elds of identiation, ontrol, stability analysis and state estimation. Akerson and Fu [1℄ were

thersttoonsiderthequestionofthedeterminationoftheativemodebystatingtheproblemin

theform ofastateestimationprobleminanoisyenvironment. Thesystemnoiseismodelledbya

setof Gaussiandistributions,withdierentmeansand varianesthat inuenethe systemoneat

atime, thetransition from onenoisesoure to anotherbeingdeterminedby aMarkovtransition

matrix. In [14, 15, 16, 24℄, the reognition of the ative mode is arried out by the means of

model-baseddiagnosis tehniques. A methodologyforthe designofdynami observersforhybrid

systemsisproposedin[6℄. Thesuggestedobserveronsistsoftwoparts: aloationobserverwhih

isdediatedtothereognitionoftheativemodeatanymomentandaontinuousobserverwhih

is devoted to the estimation of the ontinuous state, one the ative mode is reovered. Several

observability onepts were introdued in [3, 4, 26℄. Depending on the knowledge of the mode

sequeneandonthevariablestobereovered,severalobservabilityoneptsaregivenandtheyare

haraterizedthroughlinearalgebraitests.

Thereognitionoftheativemodeisloselyrelatedtotheproximityofthemodelsdesribingeah

mode. Itisobviouslyeasieriftheswithinglawisknown. Fromthere,oneanseetheimportane

oftheidentiation oftheswithinglaw.

This paperaddressestheissueof theativemodedetermination foraswithingsystem,using

onlythesysteminput/outputdata. Toperformthistask, amodel-baseddiagnosismethod [23℄is

extended to this lass of systems. We also put forward aproedure for the identiation of the

swithinglaw. Thepaperstartsin setion2.1withabriefreminderonthemodellingofswithing

systems. Thereognitionoftheativemodeisdevelopedin setion3. Theproposedmethodrests

on model-based diagnosis methods. Then, the onditions guaranteeing the disernability of the

variousmodesareformulated. An enhanementtothemethodis arriedoutin orderto takeinto

aountthepreseneofmeasurementnoise.Setion4isdevotedtotheidentiationoftheswithing

law. Theproposed methodoersanintervalapproahfortheestimation oftheparametersof the

swithinglaw. Anaademiexampleisshowninsetion5.

2 Problem statement

2.1 Modelling of swithing systems

Letusonsiderthesystemrepresentedbyequation(1):

x (k + 1) = A µ k x (k) + Bu (k) y (k) = Cx (k)

µ k ∈ {1, 2, . . . , s}, s ∈ N ^∗ \{1}

x ∈ R ⁿ , u ∈ R ^m , y ∈ R ^p

(1)

Equation(1)representsaswithingsystemwith

s

^{operatingregimesormodes. Thevariables}

u(·)

^,

y(·)

^and

x(·)

respetivelystandfortheinput,theoutputandthestateofthesystem. Theswithes areintroduedbymeansofthestatematrixwhihtakesitsvalueinaniteset

A = {A 1 , A 2 , . . . , A s }

systemandtheresultspresentedin thispaperanbeextended to theasewhere thematries

B

and

C

^{also takedierent values. The variable}

µ (·)

^{denotes the ative mode at any moment. For}

example,ifonehas

µ k = i

^,

i ∈ {1, 2, . . . , s}

^{,thesystemissaidtobeinthemode}

i

^attheinstant

k

^.

Theevolutionof themodeseletionvariable

µ (·)

^{anbedesribed in avarietyofways. Here,we}

assumethat

µ k

^isgivenby:

µ k = i if ξ k ∈ H i , i ∈ {1, 2, . . . , s} ,

⁽²⁾

wheretheswithinglawdependsonthevariable

ξ (·) ∈ R ^{n ξ}

. Eahregion

H i

^{isaonvexpolyhedron}

denedas:

H i = {ξ k ∈ R ^{n ξ} |H i ϕ ^T k ≤ 0}

⁽³⁾

with

ϕ k =

ξ k 1

,

H ∈ R ^{q×(n ξ +1)}

and the set

{H 1 , H 2 , . . . , H s }

^{is a omplete partition of}

H ⊂ R ^{n ξ}

,i.e.

S s i=1

H i = H

^and

H i ∩ H j = ∅, ∀i 6= j

^.

Inordertoletthepieewiseanemapdenedbyequation(3)bewellposed,weallowsomeofthe

≤

inequalitiesto bestrit,meaningtheyanbereplaedby

<

inequalities.

Thevariable

ξ (·)

^{an be externalto thesystem and, in that ase,themode sequeneis arbitrary}

andindependentofthesystemvariables(input,outputandstate). Theswithesfromonemodeto

anotheranalsobetriggeredbyinternalvariablesasthestate

x(·)

^{(pieewiseanesystems[8℄)or}

theinput

u(·)

^andtheoutput

y(·)

^(pieewiseautoregressiveexogenous systems). Weassumehere that

ξ k

^isdenedby:

ξ k =

Y k−1,k−n a U k−1,k−n a

,

⁽⁴⁾

where

Y k−1,k−n a = y k−1 y k−2 . . . y k−n a

and

U k−1,k−n a = u k−1 u k−2 . . . u k−n a

.

Itisworthnotingthatthedenitionof

ε k

^{inequation(4)donotlimitthesignianeoftheproposed}

ontributionin thispaper(espeially in setion4) astheproposedmethodremains appliableas

longas

ξ k

^{an beestimated(aseofpieewiseanesystems) ormeasured.}

Combining(1), (2)and(4),weretainedmodelofequation(5)asamodelforswithingsystemsin

theontinuationofthispaper:







x (k + 1) = A µ _k x (k) + Bu (k) y (k) = Cx (k)

µ k = i if ξ k ∈ H i , i ∈ {1, 2, . . . , s} , s ∈ N ^∗ \{1}

ξ k =

Y k−1,k−n a U k−1,k−n a

x ∈ R ⁿ , u ∈ R ^m , y ∈ R ^p

(5)

Themodel of equation(5) isintended in this paper to representaswithing system that do not

swithateverytimepointlikeitanbetheaseforstationverter. Hene,thesystemisassumed

tohaveaminimumdwelltimein amodeafter aswithinginstant.

Comingfrom(5),rst, wewishto reovertheativemodeatanymoment, usingonlythesystem

input/outputdataonaniteobservationwindow. Ifthesystem'smodesareassumedtorepresent

healthyoperatingmodesaswellasfaultyoperatingmodes,theativemodeestimationtaskanthe

beseenasafaultdetetiontask. Onetheativemodeisreovered,thegoalisto estimate,from

asuientlyrih sequene ofinput/output data, theparameters

H i , i = 1, . . . , s

^{of theswithing}

law,knowingitsstruture.

Weintroduethefollowingdenitions:

Denition1 Apath

µ

^{isanite sequene ofmodes:}

µ = (µ 1 · µ 2 · . . . · µ h )

^.

Thelengthof apath

µ

^isdenoted

|µ|

^and

Θ h

^{denotesthe setof allpaths of length}

|µ|

^.

µ _[i,j]

^{isthe inxof the path}

µ

^between

i

^and

j

^:

µ _[i,j] = (µ i · µ i+1 · . . . · µ j )

^.

Denition2 The observabilitymatrix

O µ,h

^{of apath}

µ ∈ Θ h

^{isdenedas :}

O µ,h =



 

 

 

C CA µ 1

.

.

.

C A µ _h A µ _h−1 · · · A µ ₁

| {z }

h



 

 

 

(6)

Denition3 On anite observation window

[k − h, k]

^{, the ative path}

µ ^∗

^{isthe one desribing}

the atualmode sequene onthe observationwindow.

From denitions 1 and 3, the estimation of the ativemode at any moment is equivalent to the

determination of thepath desribing thetrue mode sequene onanite observation window. In

ordertoahievethis, throughouttheremainderof thispaper,wewillfousonthereoveryofthe

ativepathonanobservationwindow.

3 Reognition of the Ative Mode

3.1 Detetion of the ative path

Theativepath determinationtask anbeformulatedasareursiveproblem applied toasliding

window. Onatimewindow

[k − h, k]

^{,equation(5)anbewrittenas:}

O µ,h x (k − h) =



 

y (k − h)

.

.

.

y (k)



  − T µ,h



 

u (k − h)

.

.

.

u (k)



 

⁽⁷⁾

where

T µ,h

^{isaToeplitzmatrixdenedby:}

T µ,h =



 

 

 

0 0 . . . 0 0

CB 0 0 0

.

.

.

.

.

. .

.

.

C A µ _k−1 . . . A µ _k−h+1

| {z }

h−1

B C A µ _k−1 . . . A µ _k−h+2

| {z }

h−2

B . . . CB 0



 

 

 

(8)

Equation(7)anbewrittenin amoreompatway:

Y k−h,k − T µ,h U k−h,k = O µ,h x(k − h)

⁽⁹⁾

Therelation (9) links on the time window the input and the outputof thesystem to the initial

state

x(k − h)

^ontheobservationwindow. Weintroduethefollowingproposition:

Assumption1 Theobservabilitymatries

O µ,h

^ofthepaths

µ

^{generatedonthe}observationwindow

[k − h, k]

^{are allof fullrank:}

rank (O µ,h ) = dim (x) = n, ∀h ≥ n

^.

Theexisteneof an integer

h

^{, suh that assumption 1holds, wasanalysed in [5℄ and islinked to}

pathwiseobservabilitythat havebeenfurthermoreshowntobedeidable.

Usingproposition1,oneandeneaprojetionmatrix

1

Ω µ,h

^{insuhawaythat}

Ω µ,h O µ,h = 0

^,

i.e.

Ω µ,h

^{isseletedasabasisfortheleft nullspaeof}

O µ,h

^.

Next,residuals

r µ,h (·)

^,independentoftheinitialstate

x(k − h)

^{,anbedened as:}

r µ,h (k) = Ω µ,h (Y k−h,k − T µ,h U k−h,k )

⁽¹⁰⁾

Theresiduals

r µ,h (·)

^{areusefulforthe}determinationoftheativepathontheobservationwindow andtheyonly depend onmeasurable variables, namelythe systeminputandoutput. Infat,for

theativepath

µ ^∗

^,theresidual

r µ ^∗ ,h (·)

^equalszero.

Theorem1 Theativepath

µ ^∗

^{desribingthe truemodesequeneonatimewindow}

[k − h, k]

^is

the onesatisfying:

r µ ^∗ ,h (k) = Ω µ ^∗ ,h (Y k−h,k − T µ ^∗ ,h U k−h,k ) = 0

⁽¹¹⁾

To reover the true mode sequene

µ ^∗

^{from the system} measurements, one an proeed in the followingway:

•

^{rst, all the possible paths of length}

h

^{are built on the time window}

[k − h, k]

^{. This is}

equivalenttondingallthematries

O µ,h

^.

•

^{knowingthematries}

O µ,h

^{,theprojetionmatries}

Ω µ,h

^{areeasilyalulated.}

•

^{from the matries}

O µ,h

^and

Ω µ,h

^{, oneanform the residuals}

r µ,h (·)

^{using thesystem mea-}

surements.

•

^{the ativepath is reovered from the system} measurements bytesting theresiduals

r µ,h (·)

anditorrespondstotheonewhihresidualequalszero.

Theorem1 impliitlysaysthat the observabilitymatries

O µ,h

^{do notshare thesamenullspae.}

Setion3.2willhighlighttheonditionsthat guaranteethisimpliitassumption.

3.2 On the number of paths

Itiseasytoseethattheenumerationofallpathsonatimewindow

[k − h, k]

^{introduesaproblem}

ofombinativeexplosionrelatedtothenumberofmodesandthelengthoftheobservationwindow.

Indeed,thenumberof residuals

r µ,h (·)

^,

µ ∈ Θ h

^{,to bealulatedis equalto}

s ^h

^{and quiklygrows}

withthelength

h + 1

^{of the}observationwindowandthenumber

s

^{of modes. Then, theuseofall}

pathsonatimewindowisawkwardandomputationallydemanding.

Inpratie,allpaths

µ ∈ Θ h

^{donothaveto beonsidered at everymoment. Whenatatime}

k 0

^,

theativepathonanobservationwindow

[k 0 − h, k 0 ]

^{isidentied,itisnotneessarytotestthe}

s ^h

residualsatthenextinstant

k 0 + 1

^{. Onlythepaths}

µ ∈ Θ h

^withinxes

µ [k 0 −h+1,k 0 −1]

^identialto

theinx

µ ^∗ _{[k 0 +1−h,k 0 −1]}

^ofthepath

µ ^∗

^{reoveredpreviouslyat}

k 0

^{areonsideredatthenextinstant}

1

Infat, the existeneofthe projetionmatrixisdiretlylinkedtothe observabilityofthe systemandto the

lengthoftheobservationwindow[18℄

k 0 + 1

^.

Moreover, assumingthat the minimumsojourntime in a mode is greater thanthe length of the

observation window, oneanlimitthenumber ofgenerated pathsbyonly onsidering pathsthat

desribethe mode sequene when the systemremains in the samemode alloverthe durationof

theobservationwindow,i.e.

µ = (i · i · . . . · i)

^,

i ∈ {1, 2, . . . , s}

^. Nevertheless,theredutionofthe numberofresidualsomes atthe expenseofadelay inthe estimationof theswithingtime from

onemodetoanother. Thereognitionoftheativepathannottakeplaeaslongastheswithing

instantisin theobservationwindow. Thus,when applyingthisredutionof thenumberofpaths,

amaximumdelayequalto thelengthoftheobservationwindowexists.

Priorknowledgeoftheproesssuh asprohibited swithing sequenesorminimaltime between

twoonseutiveswithes, analso help to limitthenumberof generated residualsorpathsto be

onsidered.

Inapratialimplementation,themethodologyshouldbetorstomputeareduedsetofresiduals

omposedoftheresidualslinkedtopathsthatdesribethemodesequenewhenthesystemremains

inthesamemodeonthetimewindow. Fromthisinitialsetofresiduals,areduedset ofresiduals

anbeonsideredateahtimeinstant,dependingonthepreviouslyreoveredpath. Thisoperation

onsiderablyreduestheomputingload.

3.3 Disernability of the modes

Inwhat follows, weare interestedin theonditionsguaranteeingthedisernabilityof thevarious

paths enumerated on anobservation window. These onditions ensure theuniqueness of the re-

overed ative path

µ ^∗

^{during the path reognition proess.} Disernability guarantees that two dierentmodesneverinduethesystemin thesamedynamisonanitetimewindow.

Denition4 Twopaths

µ ¹ ∈ Θ h

^and

µ ² ∈ Θ h

^{aredisernibleonan}observationwindow

[k − h, k]

iftheirrespetiveorrespondingresiduals

r µ ¹ ,h (·)

^and

r µ ² ,h (·)

^arenotsimultaneouslynullwhenone ofthe twopaths isativeonthe onsideredobservationwindow.

The study of paths disernability onditions have also been investigated by other authors like

BabaaliandEgerstedt[3℄,Hwangetal.[20℄,Vidalet al.[26℄. Thedierenehereisthatthestudy

ofthepathsdisernabilityonditionsisnotperformedindependentlyof theativemodeobserver

but alsotakesinto aounttheharateristisof themodeobserverthanksto theanalysis of the

residuals

r µ,h (·)

^.

Inordertoestablishthedisernabilityonditionsof twodierentpaths, letus onsidertwopaths

µ ¹ ∈ Θ h

^and

µ ² ∈ Θ h

^{on an} observation window

[k − h, k]

^{. We denote}

Y _k−h,k ^{µ 1}

(respetively

Y _k−h,k ^{µ 2}

^{) thesystem outputvetorwhen theativepath is}

µ ¹

(respetively

µ ²

^{). We suppose that}

ataninstant

k

^{,theativepathonthe}observationwindowisthepath

µ ¹

^{. This}informationbeing unknown,wehavetoanalysethepossibilitiesthatthepath

µ ¹

^orthepath

µ ²

^{areinadequaywith}

thesystemdata. From(10),theexpressionsoftheresiduals

r µ ¹ ,h (·)

^and

r µ ² ,h (·)

^aregivenby:

r µ ¹ ,h (k) = Ω µ ¹ ,h Y k−h,k − T µ ¹ ,h U k−h,k

r µ ² ,h (k) = Ω µ ² ,h Y k−h,k − T µ ² ,h U k−h,k

⁽¹²⁾

Sine

µ ¹

^{istheativemodeonthe}observationwindow,equation(12)anbewritten as:







r µ ¹ ,h (k) = Ω µ ¹ ,h

Y _k−h,k ^{µ 1} − T µ ¹ ,h U k−h,k

r µ ² ,h (k) = Ω µ ² ,h

Y _k−h,k ^{µ 1} − T µ ² ,h U k−h,k

⁽¹³⁾

and,bydenition,onealsohas

Ω µ ¹ ,h

Y _k−h,k ^{µ 1} − T µ ¹ ,h U k−h,k

= 0

^{. Fromwhere:}

( r µ ¹ ,h (k) = 0 r µ ² ,h (k) = Ω µ ² ,h

Y _k−h,k ^{µ 1} − T µ ² ,h U k−h,k

(14)

Addingandtakingaway

Y _k−h,k ^{µ 2}

^{fromtheexpressionof}

r µ ² ,h (·)

^,oneobtains:

( r µ ¹ ,h (k) = 0 r µ ² ,h (k) = Ω µ ² ,h

Y _k−h,k ^{µ 1} − Y _k−h,k ^{µ 2} + Y _k−h,k ^{µ 2} − T µ ² ,h U k−h,k

(15)

Asbydenition

Ω µ ² ,h

Y _k−h,k ^{µ 2} − T µ ² ,h U k−h,k

= 0

^,onehas:

( r µ ¹ ,h (k) = 0 r µ ² ,h (k) = Ω µ ² ,h

Y _k−h,k ^{µ 1} − Y _k−h,k ^{µ 2}

(16)

Equation (16) learly points out that the residual alulated for the path

µ ²

^{(non-ative path)}

diretly depends on the dierene between the systemoutputs when the mode sequene evolves

aordingto thetwopaths

µ ¹

^and

µ ²

^{,thesystembeingexited bythesameinputsinbothases.}

Fromequation(16),aneessaryandsuientonditionfor thedisernabilityofthepaths

µ ¹

^and

µ ²

^is:

Y _k−h,k ^{µ 1} − Y _k−h,k ^{µ 2} ∈ N / r (Ω µ ² ,h )

⁽¹⁷⁾

where

N r

^{standsfortheoperatorrightnullspae.}

Aordingto equation(9),onehas:

Y _k−h,k ^{µ 1} − Y _k−h,k ^{µ 2} = O µ ¹ ,h − O µ ² ,h

x(k − h) + T µ ¹ ,h − T µ ² ,h

U k−h,k

⁽¹⁸⁾

where

x(k − h)

^{isthevalueofthesystemstateat theinitial instantofthe}observationwindow.

Onededuesfrom(18)aftermultipliationontheleft by

Ω µ ² ,h

^:

Ω µ ² ,h (Y _k−h,k ^{µ 1} − Y _k−h,k ^{µ 2} ) = Ω µ ² ,h O µ ¹ ,h x(k − h) + Ω µ ² ,h T µ ¹ ,h − T µ ² ,h

U k−h,k

⁽¹⁹⁾

If

Y _k−h,k ^{µ 1} − Y _k−h,k ^{µ 2}

^{belongsto therightnullspaeof}

Ω µ ²

^,onehas:

Ω µ ² ,h O µ ¹ ,h x(k − h) + Ω µ ² ,h T µ ¹ ,h − T µ ² ,h

U k−h,k = 0

⁽²⁰⁾

Therelation(20)issatisedforalmosteveryinitialstate

2

x(k − h)

^{ifthefollowingneessaryand}

suientonditionissatised:

Ω µ ² ,h O µ ¹ ,h = 0 Ω µ ² ,h T µ ¹ ,h − T µ ² ,h

U k−h,k = 0

⁽²¹⁾

2

seeremark1fortheexplanationoftheexpressionforalmosteveryinitialstate

Therefore,thepaths

µ ¹

^and

µ ²

^{arenotdisernibleonatimewindow}

[k − h, k]

^{iftherelations(21)}

aresatised.

Theorem2 Twopaths

µ ¹

^and

µ ²

^{ofaswithingsystemaredisernibleonan}observationwindow

[k − h, k]

^{,foralmost everyinitial state}

x(k − h)

^,if:

Ω µ ⁱ ,h O _µ j ,h 6= 0, i, j ∈ {1, 2} , i 6= j

⁽²²⁾

or

Ω µ ⁱ ,h T µ ^j ,h − T µ ⁱ ,h

U k−h,k 6= 0 i, j ∈ {1, 2} , i 6= j

⁽²³⁾

Theproofofthis theoremdiretlyomesfromthepreeding remarks.

Whenthepaths

µ ¹

^and

µ ²

^areofthetype

(i · i · . . . · i)

^,

i ∈ {1, 2, . . . , s}

^{,theorem2isequivalent}

tothemodedisernabilityonditionsformulatedin [14℄.

Remark1 In theorem 2, the expression for almost every initial state holds owing to the fat

that the disernability of the paths annot be ensured for any initial state

x(k − h)

^{. In fat, for}

ertain partiular values of

x(k − h)

^{, the relation (20) is always satised} independently of the input sequene

U k−h,k

^{. For example, in the situation where}

O µ ¹ ,h

^{has full rank, for}

x(k − h) =

O µ ¹ ,h

†

Φ − T µ ¹ ,h − T µ ² ,h

U k−h,k

, equation (20) is satised for every input sequene

U k−h,k

^,

where

Φ

^{belongstothe right nullspaeof}

Ω µ ² ,h

^and

O µ ¹ ,h

†

isapseudo-inverse of

O µ ¹ ,h

^.

3.4 Determinationof the ative mode in a noisy environment

In setion 3.1, the determination of the ative mode at any moment was arried out within a

deterministi framework, i.e. there were no noise on thesystem measurement. Now, we assume

the presene of a bounded noise on the output of the system desribed by equation (5). The

onlyavailableinformationonthenoiseisitsmaximummagnitude. Noprobabilistiassumptionis

formulatedontheprobabilitydistributionofthemeasurementnoise:



 

 

x (k + 1) = A µ _k x (k) + Bu (k) y (k) = Cx (k) + n(k)

∀k, |n(k)| ≤ δ, δ > 0

(24)

where

δ

^{isthebound ofthe}measurementnoisemagnitude

n(·)

^.

Inthissituation,theresidual

r µ ^∗ ,h (·)

^{,denedby(11)andwhihorrespondstotheativepath}

µ ^∗

onthe time window

[k − h, k]

^{, is no longer equalto zero. Indeed, theexpression of the residual}

r µ ^∗ ,h (·)

^{,usingequation(10),beomes:}

r µ ^∗ ,h (k) = Ω µ ^∗ ,h (Y k−h,k − T µ ^∗ ,h U k−h,k + N k−h,k )

⁽²⁵⁾

wherethevaluestakenbythemeasurementnoiseontheobservationwindow

[k − h, k]

^arestaked

in

N k−h,k

^{. As}

Ω µ ^∗ ,h (Y k−h,k − T µ ^∗ ,h U k−h,k ) = 0

^,oneanwrite:

r µ ^∗ ,h (k) = Ω µ ^∗ ,h N k−h,k

⁽²⁶⁾

Usingtheboundofthemeasurementnoisemagnitude,weandeneanintervalresidual

[r µ ^∗ ,h (k)]

[2℄:

[r µ ^∗ ,h (k)] = [r _µ ∗ ,h , ¯ r µ ^∗ ,h ]

⁽²⁷⁾

where

r _µ ∗ ,h

^and

r ¯ µ ^∗ ,h

^{depends on the bound}

δ

^{of the} measurement noise and are given by :

r _µ ∗ ,h = − |Ω µ ^∗ ,h | U δ

^and

r ¯ µ ^∗ ,h = |Ω µ ^∗ ,h | U δ

^,

U

beingaolumnvetoroflengthequaltothenumber ofolumnsof

Ω µ ^∗ ,h

^{andalltheelementsof}

U

beingequalto

1

^.

Inanintervalframework,thedetermination of theativepath amountsto seeking thepath that

orresponds to an intervalresidual inluding the value zero. This test anbe performed by al-

ulating thesignof the produtof theupperandlowerbounds ofeah intervalresidual

[r µ,h (·)]

^.

Theintervalresidual

[r µ,h (·)]

^{assoiatedwiththeativepath}

µ ^∗

^{istheoneforwhihthesignofthe}

produtofitsupperandlowerbound isnegative.

Dependingontheevolutionofthevariousoperatingregimesdynamis,itanhappenthatmore

thanoneintervalresidualsontainsthevaluezero,thissituationbeinglinkedtothepathdisern-

abilityandtheboundofthemeasurementnoisemagnitude. Inthisase,onerefrainsfrom making

anydeisionontheativepath. Wehavetoonsiderthissituationfromalooserpointofviewand

we anonlyenumerate the set of allpossible ativepaths. However,onsidering suessivetime

instants

k + 1, k + 2, . . .

^{, thesituation maybelaried.}

Note that it is also oneivableto introduesomeprobabilisti modelling assumptions on the

outputnoiseandthenrefertoastatistialtestliketheCUSUM [7℄algorithmtoreovertheative

pathfromtheanalysisofthegenerated residuals.

4 Identiation of the swithing law

Onethereognitionoftheativemodeateverymomentisperformed,thenextstepistoproeed

totheidentiationoftheparametersoftheswithing lawdesribedby(5).

Theidentiation of theswithing lawaims at nding aomplete partition of theregressors set

into

s

^{polyhedral regionssuhthat}

µ k = i

^if

ξ k ∈ H i

^,

i ∈ {1, 2, . . . , s}

^{. Thisproblem amountsto}

separating

s

^{sets of pointsby means of linearlassiers} (hyperplanes). Depending onthe ative mode estimation proess, the resulting

s

^{sets of pointsmay be linearly separable ornot (due to}

noiseormislassiation). Intheliterature,RobustLinearProgramming(RLP)[10℄and Support

VetorMahines(SVM)[13℄methodsareemployed.

Weonsiderhereanotherwaytoproeedtothedeterminationoftheparametersofthe

s

^polyhedral

regions. An intervalapproahis adopted. The interval representationallowsto look for aset of

aeptablevaluesfortheswithinglaw,thissetbeingofasimplegeometrialform.Theomputation

ofthesetofallfeasibleseparatinghyperplanesisalsousefultotheaimofharaterisingthemodel

unertainties.

4.1 Determinationof the swithing law parameters in an interval form

Weassumethatfrom model(5),oneobtainsadataset

D =

ξ _k ^T , k = 1, . . . , N

^{. Afterproeeding}

totheativemodereognition,thedataset

D

^anbepartitionedinto

s

^lasses

C i , i = 1, . . . , s

^using

thefollowinglassiationrule:

ξ k ∈ C i

^if

µ k = i

⁽²⁸⁾

From the lasses

C i , i = 1 . . . , s

^{, the} determination of the parameters

H i

^,

i = 1 . . . , s

^amounts

to separating the

s

^{lasses using linear lassiers whih are, in this ase,} hyperplanes. This an bedone by eitheronsidering allthe

s

^{lassestogetherat thesametime} (one-against-all andall- together approah)or onsideringthem pairwise (one-against-one approah). Here,weadopt the

one-against-one approah. The one-against-one approah onsiders all possible ombinations of

pairoflasses. Letusonsidertwolasses

C i

^and

C j

^with

i 6= j

^.

Toseparate

C i

^and

C j

^{, weneed to omputeahyperplane}

H ij = {ξ k ∈ R ^{n a +n b}

h ij ϕ ^T _k = 0 , h ij ∈ R ^{n a +n b +1} }

^,with

ϕ k =

ξ k 1

,insuhawaythat:

h ij ϕ ^T _k > 0

^if

ξ k ∈ C i

h ij ϕ ^T _k < 0

^if

ξ k ∈ C j

(29)

where

h ij ∈ R ^{n a +n b +1}

.

Using thesystemdesription(5), oneanwrite therelation (30)foranydata

ξ k

^belongingto

C i

^or

C j

^:

ν k h ij ϕ ^T _k

> 0, k ∈ I = {k 1 , k 2 , . . . , k N _ij }

⁽³⁰⁾

where

ν k =

^sign

h ij ϕ ^T _k

oralternatively:

ν k =

1

^if

ξ k ∈ C i

−1

^if

ξ k ∈ C j

(31)

and

I

^{isasetontainingthetimeinstantsatwhihthemode}

i

^orthemode

j

^{weredetetedduring}

themode reognitionproess. The onstant

N ij

^{is thesumof theardinalof}

C i

^{and theardinal}

of

C j

^.

Considering(30),forallthe

N ij

^data

ξ k

^{,oneobtainsasetof}inequalitiesthatanbeexpressedin theform ofalinearmatrixinequality:

−



 

ν k ₁ ϕ k ₁

.

.

.

ν k _Nij ϕ k _Nij



  h ^T _ij <



  0

.

.

.

0



 

⁽³²⁾

TheresolutionoftheLMI(32)givesadomainofaeptablesolutionstowhihbelongtheparameters

h ij

^{. Generally, solving (32) leads to a omplex domain, i.e. a domain desribed with a huge}

numberofvertie. Toreduethisomplexity,oneanlookforasimplerpolytopiform desribing

aredueddomainofaeptablesolutions. Here,welookforazonotope.

For example, the rst graphof gure 1 representsthe projetion in

R ²

of the found domain for the datasetin table 1with

ϕ k = (y k−1 u k−1 1)

^,

h ij = (1 α β)

^,

α ∈ R

and

β ∈ R

. This domain is depited in the plan

{α, β}

^{on the graph onthe left of gure1 and orrespondsto the set of}

inequalities(33)obtainedfrom equation(32):



 



 

 β > 1 β < 2

−α + β > 0

−2α + β < 0

(33)

All the points belonging to this domain are partiular aeptable solutions. The symbol o

highlightsoneof those aeptable solutionsand maybe, for example,the oneresultingfrom the

implementation ofaninterior-pointalgorithm. Thegraphonthe rightof gure1presentsasub-

optimal solution (grey area) that simplies the desription of the found domain in the form of

independentinequalitiesinrespet to

α

^and

β

^:

1 < α < 1.5

1.5 < β < 2

⁽³⁴⁾

u k−1

^{0 0 -1 -2}

y k−1

^{-1 -2 0 0}

ν k

^{1 -1 1 -1}

0 1 2

−0.5 0 0.5 1 1.5 2 2.5

α −0.5 0 1 2

0 0.5 1 1.5 2 2.5 β

α β

Figure1: Aeptabledomainsfor

α

^and

β

Thedetermination ofazonotopeharaterizingthesetofaeptable solutionsisequivalentto

thedeterminationof theparameters

h ij

^{in anintervalform. Forthat, many}optimization riteria an be hosen. For example, one an fore the widths of the intervals to be determined to be

maximalwhilerespetingthesystemonstraints.

Theparameters

h ij

^{aredesribedinanintervalformby:}



 



 



h ij = h ij ₀ + λ h _ij ⊗ r h _ij

r h _ij > 0 λ h ij

∞ ≤ 1

(35)

where

h ij ₀

^{is thevetorontainingthe entres of the searhed intervals,}

r h ij

^{representsthe half-}

widthsof theintervalsand thevariables

λ (·)

^{are bounded normalizedvariablesthat allowto take}

intoaountallthevaluesinside aninterval. Theoperator

⊗

^{performs a}omponentwiseprodut oftwovetors. Wereall that foranyvetor

e ∈ R ⁿ

, one has

kek _∞ = max

1≤i≤n |e i |

^,

e i

^beingthe

i

^th

omponentofthevetor

e

^{. Theinequalityholdingonthevetor}

r h _ij

^isaomponentwiseinequality. Usingtheintervalform of

h ij

^{(35),equation(30)anberewrittenas:}



 

 

r h i,j > 0 λ h ij

∞ ≤ 1

ν k h ij ₀ + λ h _ij ⊗ r h _ij

ϕ k ≥ 0, k = k 1 , . . . , k N _ij

(36)



 

 

r h _i,j > 0 ν k h ij ₀ + r h ij

ϕ k ≥ 0, k = k 1 , . . . , k N ij

ν k h ij ₀ − r h ij

ϕ k ≥ 0, k = k 1 , . . . , k N ij

(37)

Finally,tond

h ij ₀

^et

r h ij

^{,wehavetolookforintervalswithmaximal}half-widthswhilerespeting theonstraints(37). A naturalhoie anbeto maximizethevolumeofthezonotope. Thisleads

totheonstrainedoptimizationproblem(38):



 

 

h _ij max ₀ ,r _hij

n a +n b +1

Y

m=1

r h ij ,m

s. t. (37)

(38)

where

r h ij ,m

^isthe

m

^{th omponentofthevetor}

r h ij

^.

Onean then use lassial algorithms in the eld of optimization [11℄ for the resolution of (38).

Theonstrainedoptimization problem (38)hasto besolvedfor allpairsof lasses

{C i , C j }

^,

i 6= j

and

i, j ∈ {1, 2, . . . , s}

^.

Remark2 Itis lear that the aim ofthe presentedmethodinthis setion is topropose asimpler

desription of a geometrial domain represented by a set of inequalities. Hene, the methodis in

somewayindependentofthe linearseparability ofthelasses anditwillwork,whetherthe lasses

are linearly separable or not,as longas the initial geometrial domain (32) exists. Moreover, the

innerzonotopi approximation introduessomeonservatismbutthis isnotahuge drawbak. Itis

the ostofthe obtainingof avery simplegeometrial desription ofthe initial domain(32).

5 Example

Wepresenthereanaademiexampleofaswithingsystem. Thesimulatedsystemisharaterized

bythree modesandthematriesof themodelsdesribingthedierentmodesare:

A 1 =

−0.211 0 0 0.521

, A 2 =

0.691 0 0 −0.310

, A 3 =

0.153 0 0 0.410

, B = 2 −1 T

, C = 1 2

(39)

Therefore, the modes

1

^,

2

^and

3

^{of the system are} represented by seond order models with

K 1 = −1.464

^,

K 2 = 2.002

^and

K 3 = −0.514

^{asrespetive gains andthe ouples}

(−0.211; 0.521)

^,

(0.691; −0.310)

^and

(0.153; 0.410)

^{asrespetivepairsof poles. Theswithing law is}haraterized by:



 

 

 

 

 

 

µ k = 1 if h 12

y k−1 u k−1 1 T

≥ 0 and h 13

y k−1 u k−1 1 T

≥ 0 µ k = 2 if

h 12

y k−1 u k−1 1 T

< 0 and h 23

y k−1 u k−1 1 T

< 0 µ k = 3 if

h 23

y k−1 u k−1 1 T

≥ 0 and h 13

y k−1 u k−1 1 T

< 0

(40)

Table2: Setofallpathsoflength

2

Path

µ ¹ µ ² µ ³ µ ⁴ µ ⁵ µ ⁶ µ ⁷ µ ⁸ µ ⁹

µ 1

^{1 1 1 2 2 2 3 3 3}

µ 2

^{1 2 3 1 2 3 1 2 3}

with

h 12 = 1 0.51 0

,

h 13 = 0 1 0

,

h 23 = 1 −0.29 0

.

Figure 2 shows the input

u(·)

^{, the output}

y(·)

^{, the state}

x(·)

^{and the mode sequene}

µ (·)

^{. The}

vertialdashedlinesonthethirdgraphofgure2markthetimeinstantsatwhihswithesour.

Thefourth graphplots themode sequene desribedby the modeseletion variable

µ (·)

^{. Forin-}

stane,onthetimewindows

[1, 8]

^and

[9, 17]

^,thesystemisrespetivelyin themodes

1

^and

2

^.

0 10 20 30 40 50 60 70 80 90 100

−2 0

2 u(k)

0 10 20 30 40 50 60 70 80 90 100

−10 0

10 y(k)

0 10 20 30 40 50 60 70 80 90 100

−10 0

10 x(k)

10 20 30 40 50 60 70 80 90 100

1 2

3 µ _k

Figure2: Input

u(·)

^,output

y(·)

^,state

x(·)

^{, modesequene}

µ (·)

As

Ω µ ⁱ ,h O µ ^j ,h 6= 0

^,

µ ⁱ , µ ^j ∈ Θ 2

^,

µ ⁱ 6= µ ^j

^,

Θ 2

^{beingthesetofallpathsoflength}

2

^,theondition

(22)oftheorem2isrespeted. Condition(23)istestedat everymoment. Ifitis notsatised,no

deisionistakenonerningthereognitionoftheativepath.

In order to perform the determination of the ative path at every moment from the system

inputandoutputsignals,weonsideranobservationwindowoflength

3

^{. Theset}

Θ 2

^ofallpaths

oflength

2

^ontheobservationwindoworrespondstothesetofthenine pathsin table2.

Figure3presentstheevolutionofthealulatedresiduals. Thedierentgraphsonthegureshow

theresiduals

r _{(i · j),h} (·)

^,

i, j ∈ {1, 2, 3}

orrespondingtothepathsoflength

2

^{intable2. Onlyoneof}

thenineresidualsequalszeroateahinstant,theindex

(i · j)

^{ofthisresidual}orrespondingtothe ativepath ontheonsidered timewindow. Forexample, fromtime

k = 1

^to

k = 6

^{,the residual}

r (1 · 1),h (·)

^{(rstrowand rstolumnofgure3) equalszero,meaningthat thepath}

(1 · 1)

^{is the}

ativeoneontheobservationwindow. Hene,theativemodeonthewindow

[1, 6]

^isthemode

1

^.

At

k = 7

^{,onlytheresidual}

r (1 · 2),h (·)

^{equalszero,meaningthatthepath}

(1 · 2)

^{beomestheative}

one. Fromthere,and takingintoaountthelengthof theobservationwindow,theourreneof

aswithat

k = 9

^ishighlighted.

0 50 100

−10

−5 0 5 10

r _{(1 ⋅ 1)}

0 50 100

−15

−10

−5 0 5

10 r _(1⋅2)

0 50 100

−5 0 5

10 r _(1⋅3)

0 50 100

−10 0 10

20 r _{(2 ⋅ 1)}

0 50 100

−10

−5 0 5 10

r _(2⋅2)

0 50 100

−10

−5 0 5

10 r _{(2 ⋅ 3)}

0 50 100

−5 0 5

10 r _{(3 ⋅ 1)}

0 50 100

−15

−10

−5 0 5 10

r _{(3 ⋅ 2)}

0 50 100

−5 0 5

10 r _(3⋅3)

Figure3: Residuals

r µ,h (·)

^,

µ ∈ Θ h

The mode sequene (rst graph of gure 4) and its estimation (seond graph of gure 4) while

analysingthe residualsare depited ongure 4. The gureshows that themodesequene is ex-

atlyreonstruted.

10 20 30 40 50 60 70 80 90 100

1 2

3 true µ _k

10 20 30 40 50 60 70 80 90 100

1 2

3 estimated µ _k

Figure 4: Modereognition

Asexplained in setion3.2, in order toredue the number ofresidualsto beanalysed during the

modereognitionproess,oneanonsideronlythepathsdesribingthemodesequenewhenthe

systemremains in the samemode all overthe durationof the observation window. Inthis ase,

onlythepaths

(1 · 1)

^,

(2 · 2)

^,and

(3 · 3)

^{havetobeonsidered.}

Forthesystemdesribedbymatries(39),weproeedto thereognitionoftheativepathin

anoisyenvironmentwhere the systemoutputis subjetto the eet ofabounded noise. Inthis

situation,tomaketheanalysissimpler,weonlyonsiderthethreepaths

(1 · 1)

^,

(2 · 2)

^and

(3 · 3)

^.

Ongure5,thethreeintervalresidualsareshownindashedlines. Oneannotiethatonlyoneof

thethree interval residualsinludes at anymoment thevaluezero, this residualbeingassoiated

withtheativepathontheobservationwindow.

0 10 20 30 40 50 60 70 80 90 100

−5 0 5 10

r (1 ⋅ 1)

0 10 20 30 40 50 60 70 80 90 100

−10

−5 0

r (2 ⋅ 2)

0 10 20 30 40 50 60 70 80 90 100

−5 0 5

10 r

(3 ⋅ 3)

Figure5: Evolutionoftheresiduals

Theseond and the third graphsof gure6 illustrate theresults of theativepath detetionby

analysingtheintervalresiduals. Theseondgraphshowsthemodesdetetedwhiletestingthemem-

bership of the valuezero to theintervalresiduals. Although the modes arerather welldeteted,

therearesituationswhereitwasimpossibletoprovideanestimateof

µ _(·)

^{beauseofthefat that}

morethan oneintervalresidualornoneof thethree intervalresidualsontainthe valuezero. On

theseond graphof gure6,thepoints, withY-oordinateequalto zero,emphasizesthiskindof

situation whih is due to thepresene of noiseand to the fat that all thepossiblepaths on the

observationwindowarenotonsidered intheanalysis. Thethird graphofgure6is obtainedby

testingtheoherenein thesuessionofthedetetedativepathsat onseutivemoments. This

isequivalenttothepathredutionmethodpresentedinsetion3.2 usingtheinxof thedeteted

ativepath. Oneannotie aperfetreonstrutionofthemodesequene.

One the proess of the ative mode reognition is performed, we an proeed now to the

identiationof theparametersof theswithing lawdened by equation(40). Thedatasetis as-

sumed to be representative enough of the system's various operating regimes. The onstrained

optimizationproblem(38)isresolvedbyusinganiterativealgorithm.Theresultsarepresentedin

table3.

Intable3,theentresoftheintervalsfoundareindiatedby

h (.) ₀

^,thehalf-widthsby

r h (·)

^and

10 20 30 40 50 60 70 80 90 100 0

1 2 3

true µ _k

10 20 30 40 50 60 70 80 90 100

0 1 2 3

estimated µ _k

10 20 30 40 50 60 70 80 90 100

0 1 2 3

estimated µ _k

Figure6: Ativepathreognition

Table3: Boundedparameters

h 12 h 12 0 r h 12

1 1.013 0.526 0.51 0.694 0.215 0 0.000 0.001

h 13 h 13 0 r h 13

0 0.013 0.180 1 1.052 0.381 0 0.011 0.021

h 23 h 23 0 r h 23

1 1.112 0.281

−0.29 −0.326 0.197

0 0.001 0.007

thereal valuesarerepresentedby

h (·)

^{. Fromtable3,theintervalsfoundare:}

[h 12 ] =

[0.604 , 1.656] [0.470 , 0.909] [−0.001 , 0.001]

[h 13 ] =

[−0.167 , 0.193] [0.671 , 1.433] [−0.010 , 0.032]

[h 23 ] =

[0.831 , 1.393] [−0.523 , −0.129] [−0.006 , 0.008]

(41)

Whileanalysing the estimated valuesin table 3,one ansee that the estimated intervals for the

swithing law parametersalways inlude the real values

h _(·)

^{. In fat, this situation depends on}

theloalization of thedata points in the regressor set. When there are many data points whih

are lose to the separatinghyperplanes in the estimation dataset, the estimated interval for the

swithinglawparametersarelikelyto ontaintherealparameters.

6 Conlusion

In this paper, weput forward amethod for the determination of the swithing instantsand the

ativemodeof aswithing system. Themethod rests ontheanalysis ofresidualsgenerated from

theparametersoftheswithing lawgoverningtheswithesfromonemodetoanotherbyusingan

approahthatomputesthebounds oftheparameterstobeidentied.

A point to be developed is the situation where all the modes of the system are not indexed

beforehand. Inthisase,onedoesnothaveompleteknowledgeofalltheoperatingregimesofthe

system. Therefore,whenanewmodeisdeteted,itisneessarytoproeedtotheidentiationof

thenon-indexed operatingmodesi.e. thestatematriesorrespondingtothesemodes.

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