Operating point seletion for LPV model identiation

iden-tiation

As mentioned in the previous Setion, the information about the

parameter-a pratial onstraint makes the working point seletion step ruial in order to

ensurethatthenalLPVisabletoapturethedynamisofthenon-linearsystemto

identify eiently. A naiveapproahwouldbetoinrease, asmuhaspossible, the

numberofloalmodelsby onsidering averytiny step between two loaldata-sets.

In order to avoidthis kind of numerially and pratiallyumbersome approah,it

issuggested hereaftertomeasure the omplementaryinformationbroughtby anew

loaloperatingand deide, aording tothis measurement,

(i)

^if ^this ^loal^data-set

iskept for the identiationofthe LPV model, (

ii

⁾ ^if^the ^user-dened ^step ^between

theurrentmodelandthenextonemustbeinreasedordereased. Intheliterature

dediatedtoLPVmodelidentiationfromloalexperiments,mostofthe time(see

amongotherse.g.,[82,143,135, 23℄),the Authorshoose equidistantworkingpoints

byseletingauser-denedbutxedstep

∆p

^between^two^loalexperimentsinterms of the sheduling variables. More preisely, by applying suh a simple operating

pointseletionapproah, theuser doesnot haveany informationof howthe system

dynamis evolve between the seleted working points. As is shown in Figure 5.1a,

byusing the sameuniedstep interms ofthe shedulingvariable

∆p

^between ^eah

working point, the next loal model may be found too far or too lose in terms

of the system dynamis

∆β

^. ^In ^the ^sequel, ⁱⁿ ^order ^to ^address ^this ^problem, ^a

novelapproah isproposed wherethe seletion isperformedby xing the allowable

dynamidistane

∆β

^rather^than^a^xed^step^sizeⁱⁿ^terms^of^the^sheduling^variable

vetor

∆p

^. ^In ^other ^words, ^by ^xing

∆β

^between ^the ^loal^operating ^points,^it ^an

be ensured that the estimated loal models are evenly distributed in terms of the

system dynamis along the trajetory of the sheduling variable vetor. This

may result in a set where the xed sheduling variables

p _i , i ∈ { 1, · · · , N op }

^are

distributed unevenly. This novel approah isdepited inFigure5.1b. So, basially,

the proposedmethodappliesa xed stepsize grid samplingintermsof the system

dynamis

∆β

^and^seeks ^theorrespondingxedvaluesofshedulingvariablevetor

p _i , i ∈ { 1, · · · , N op }

^. ^Now, ^the ^next ^question^whih^arises ^is^the ^following: ^how ^an

wequantify thedynamis of thesystem? Inthisthesis, morepreisely,the nu-gap

metri[146℄has been seletedforthis purpose. Thanks tothis mathmeasurement,

the step between two loal experiments is automatially modied and tuned with

respet to the information brought by the next loal model. For this seletion

proedure, it is assumed that the maximal (

p max = p

⁾ ^and ^minimal ⁽

p min = p

⁾

possiblevaluesoftheshedulingvariablesareknown(seeEq.(2.14)forthedenition

p

^and

p

^, respetively). By further assuming that an initial operating point is available,the next one should be

1. far enough from the rst one to derease the omplexity of the LPV model

identiationstep;

2. lose enough tothe rst one to ensurethat the global behaviorof the system

is well-aptured.

Depending on the value of the seleted

∆β

^, ^this ^rule ^allows ^us, ^on ^the ^one ^hand,

to redue the number of loal models in the end (ompared with a standard

teh-nique involving a onstant step), and on the other hand, to aquire more reliable

informationabout the variationof the system dynamisalong the trajetory of the

shedulingvariables. Wemayneedmoreloalexperimentsthannallyseletedloal

modelsbutitistheprietopayiftheuserwantstoseletreliableinformationabout

the nonlinearsystemtoidentifywhenthe shedulingvariableannotbepersistently

(a)Thelassial approah. (b)Our approah.

Figure5.1: OperatingpointseletionapproahesforLPVmodelidentiationbased

onloalexperiments.

Remark 5.1. Hereafter in this Chapter, we fous on ontinuous-time

representa-tions for the sake of oniseness. Notie however that the following developments

anbeextended todisrete-timemodelsdiretly. Inthis ontext,

γ = _dt ^d

^and

η = s

^is

the Laplae variable where

η

^is ^the ^Laplae ^transform ^of

γ

^based ^on ^the ^denitions

given in Setion 2.3.

5.3.1 Nu-gap metri

In order to desribe the working point seletion tehnique, let us rst

intro-due the measure of model t, the present seletion method is based on. Being a

omplex-topologialtool, the nu-gap metri [146℄, as a dynami distane

measure-mentbetween two LTI models 2

,is alulated in [145℄ by

ν g = δ( G ₁ (s), G ₂ (s)) =

k (I + G ₂ (s) G ₂ (s)) ⁻ ¹ ² ( G ₂ (s) − G ₁ (s))(I + G ₁ (s) G ₁ (s)) ⁻ ¹ ² k ^∞ ,

^(5.3)

where

G _i (s)

i ∈ { 1, 2 }

^, ^stands ^for ^the ^I/O representations of two LTI model while

• i ∈ { 1, 2 }

^, ^and

k • k ∞

^stand ^for ^the ^omplex ^onjugate ^and ^the

H ∞

^-norm ^of

•

respetively. Thismeasure is bounded between 0 and1 [145℄. In addition,itan be

shown that if

ν g

^is^lose ^to^0, ^then ^the ^two ^onsidered^models ^have^similar ^dynami

behavior. Contrarily, when

ν _g

^is ^lose ^to ^1, ^then ^the ^two ^models ^behave

dier-ently [145, 146℄. By this way, the nu-gap metri an be onsidered as a normalized

H ^∞

^-norm ^of ^the ^dierene ^between ^two ^LTI ^models. ^Thanks ^to ^its advantageous properties, this tool an eiently be employed during the operating point

sele-tion proedure as adistane measurement,in terms of the dynamisof the system,

denoted by

∆β

ⁱⁿ^Figure ^5.1.

2. Here,thegeneral(alligraphi)notationisappliedbeausethismetrianbealulatedfor

5.3.2 Seletion algorithm

First, it is important to notie that the following seletion algorithm an be

used with any identiation method based on loal experiments. For this seletion

proedure, it is assumed that the maximal (

p max = p

⁾ ^and ^minimal ⁽

p min = p

⁾

possiblevaluesoftheshedulingvariablesareknown(seeEq.(2.14)forthedenition

p

^and

p

^, respetively). Byassumingthatthis (weak)assumption issatised, the natural questionswhiharise when dealingwith aloalapproah are the following:

Howmany working points doyou need toget a onsistentnal LPVmodel?

Howan you seletthe good working points forthe loalexperiments?

How do you measure the informationbrought by a new working point and a

new loalmodel?

These important questions an guide the user in developing an heuristi tehnique

for seleting the working points inan eient way. They more preisely show that

a distane between two onseutive loal models must be introdued to deide if a

loal model must be kept beause it gives aess to new and reliable information

abouttheglobalbehaviorofthesystemtoidentify. Inthisthesis,ithasbeenhosen

to use the nu-gap metri [146℄ to measure suh a distane beause of its ability to

quantify (between 0and 1)the behavioralsimilaritiesordierenes between toLTI

models. Notie, however, that any other similar metris, suh as e.g., the Martin

distane [91℄, may be applied instead. One this frequeny tting measurement is

introdued, aording to prior knowledge or spei pratial onstraints satised

bythe system toidentify,the user must hoose aspei rangeof admissiblevalues

for this mathing measurement

ν g

^(related ^to^a ^user-dened ^threshold ^alled

∆β

ⁱⁿ

the following,see Figure5.1), rangewhihmust piturethe ondene the user has

in the apability of the seleted loal models to apture the global behavior of the

system. In the sequel, depending onwhether the urrent value of

ν g

^is ⁱⁿ ^or ^out ^of

the user-dened range of admissible values, the distane between the loal models

must be updated(inreased or dereased).

This senario being set, it is now neessary to translate it into an iterative

al-gorithm. Hereafter, the atual operating point and the step size are denoted by

p _act

^and

p _step

^, respetively. The following proedure an be performed in order to determine thebest operatingpointsandtoidentifythe loalblak-boxmodelsfrom

the loalinformationavailableatthese points. Notiehere thatby best seletion,it

is meant thatthe nal set of working pointsisthe best interms ofthe user-dened

∆β

^whih ^is ^a ^maximal ^allowable ^distane, ⁱⁿ ^the ^nu-gap ^metri ^sense ^[145℄,

be-tween eahloalmodel. In the sequel, the algorithmispresented by usingonly one

shedulingvariable. Notiethatthe extensionofthealgorithmtoseveral sheduling

variablesan bearried out straightforwardly.

By onstrution, suh an algorithm is heuristi. A statistial investigation of

the eet of the parameters

∆β

^and ^the ^eet ^of ^noisy ^data ^during ^loal ^model

identiationistakled inSetion6.3, inPart II, by resortingtosimulations. Now,

in order to guide the user, some important hints to eiently selet the

hyper-parameters of the tehnique are given. First, the initial value of

p step

^should ^be

hosen quitesmallw.r.t.

p max

^, ^e.g.,

p step ∈ [0.01, 0.05]p max .

^(5.4)

This hoie an be justied by the fat that the step size should be initially small

will determine (inrease or derease) at eah iteration a new value of

p step

^if ^it ^is

neessary. Seond,asfaras

∆β

^is^onerned,^as^an^be^seen^from^the^results^obtained

in Setion 6.3, the admissiblehoie of this hyper-parameter depends on the noise

involved inthe loalI/Odata sequenes and on the onsidered framework 3

The operatingpoint determinationproedure an be startedfrom the maximal

value(

p max

⁾^or^from^any^otherintermediatevalueofthe shedulingvariablesaswell by(only) slightlymodifyingtheafore-desribedalgorithm. Whenthealgorithman

notndanydynamialhangesenrouteto

p max

^,^two^working^points^will^be^seleted

anyway, the ones orresponding to

p min

^and

p max

Notie that this seletion algorithm an be easily implemented by using the

gapmetri funtionof the RobustControlToolbox availablein Matlab. As

men-tioned above, the proposed algorithm an be applied with any identiation

teh-niquebased onloalexperiments(suhmethodsan be,amongothers,[97, 96,157,

156℄). In this thesis, the proedure is embedded into the

H ^∞

-norm-based identi-ation method developed rst in [156℄ and also utilized for [157℄. The obtained

simulationresults are presented inSetion 6.3.

Seletion Algorithm

1. Estimate a blak-box LTI state-spae model

G _i (s)

i = 1

^, ⁱⁿ ^the

ini-tialoperatingpoint(

p act = p min

⁾^by ^using^any ^dediatedidentiation method[78,46,45,93℄. Denetheinitial

p step

^as

p step ∈ [0.01, 0.05]p max

andsettheoperatingpointindexas

i = i + 1

^. ^Compute^the^position^of

thenext workingpoint(

p i = p act + p step

^).

2. Performthefollowingsteps untilitholdstrue that

p act ≤ p max

Estimateablak-boxLTIstate-spaemodel

( G _i (s))

ⁱⁿ^the^atual

op-eratingpoint(

p act = p i

⁾^by^using^any^dediatedidentiationmethod [78,46,45,93℄.

Computethenu-gapmetribyusingthedenitiongiveninEq.(5.3),

σ = δ( G _i−1 (s), G _i (s)),

^(5.5)

∆β < σ < ∆β

^, ^then^the ^atual^model⁽

G _i

) iskept,

i = i + 1

^and

p i = p act + p step

^, ^where ^the ^user-dened ^threshold

∆β

^and

∆β

^are

hosen tobeequalto

(∆β − 0.1)

^and

(∆β + 0.1)

^,respetively. Else if

σ > ∆β

^, ^then ^the ^atual^model ⁽

G _i (s)

⁾^is ^disarded,

p _step =

p step − ∆p step

^and

p i = p act − p step

Else if

σ < ∆β

^, ^then ^the ^atual^model ⁽

G _i (s)

⁾^is ^disarded,

p step = p step + ∆p step

^and

p i = p act + p step

Theresultingsetontainsthekeptmodels,thedimensionofwhihisdenoted

N op

ⁱⁿ^the^following.

In document H ∞-norm for the (Pldal 61-65)

Operating point seletion for LPV model identiation

(i)

ii

∆p

∆p

∆β

∆β

∆p

∆β

p i , i ∈ { 1, · · · , N op }

∆β

p i , i ∈ { 1, · · · , N op }

p max = p

p min = p

p

p

∆β

γ = dt d

η = s

η

γ

ν g = δ( G 1 (s), G 2 (s)) =

k (I + G 2 (s) G 2 (s)) − 1 2 ( G 2 (s) − G 1 (s))(I + G 1 (s) G 1 (s)) − 1 2 k ∞ ,

G i (s)

i ∈ { 1, 2 }

•

i ∈ { 1, 2 }

k • k ∞

H ∞

•

ν g

ν g

H ∞

∆β

p max = p

p min = p

p

p

ν g

∆β

ν g

p act

p step

∆β

∆β

p step

p max

p step ∈ [0.01, 0.05]p max .

p step

∆β

p max

p max

p min

p max

H ∞

G i (s)

i = 1

p act = p min

p step

p step ∈ [0.01, 0.05]p max

i = i + 1

p i = p act + p step

p act ≤ p max

( G i (s))

p act = p i

σ = δ( G i−1 (s), G i (s)),

∆β < σ < ∆β

G i

i = i + 1

p i = p act + p step

∆β

∆β

(∆β − 0.1)

(∆β + 0.1)

σ > ∆β

G i (s)

p step =

p step − ∆p step

p i = p act − p step

σ < ∆β

p _i , i ∈ { 1, · · · , N op }

p _i , i ∈ { 1, · · · , N op }

γ = _dt ^d

ν g = δ( G ₁ (s), G ₂ (s)) =

k (I + G ₂ (s) G ₂ (s)) ⁻ ¹ ² ( G ₂ (s) − G ₁ (s))(I + G ₁ (s) G ₁ (s)) ⁻ ¹ ² k ^∞ ,

G _i (s)

ν _g

H ^∞

p _act

p _step

H ^∞

G _i (s)

( G _i (s))

σ = δ( G _i−1 (s), G _i (s)),

G _i

G _i (s)

p _step =

G _i (s)