Estimation of the nal LPV model

presented in Table 6.4, one an onlude that when

∆β

^{exeeds a ertain value,}

here 0.5, theseletionalgorithmrequiresfewerloalmodelidentiationsteps than

with the onstant grid-based approah (whih is 17 by using a xed step equals to

0.05 in terms of the sheduling variable). Notie that this property depends also

highly on the onsidered noise harateristis; and thus, on the applied LTI model

identiationmethod.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

BestBFT

[%]

^{90.07 90.07 90.08 90.06 90.05 90.07}

Median BFT

[%]

^{90.02 90.03 90.03 90.03 90.03 90.02}

Worst BFT

[%]

^{63.65 55.61 89.91 54.98 59.84 64.97}

Table 6.5: Best, median and worst I/O BFT measurements of the estimated loal

blak-box LTI models.

6.3.3.1 By using loal re-struturing-based tehniques

Herein,thedemonstrationoftheLPVmodelidentiationtehniquesontaining

the lassial interpolation step is proposed. In order to reah this goal, the loal

models are needed to be transformed into a oherent basis beforehand. So, having

aess to

3 × N op

^{loal LTI models for eah value of}

β ∈ [0.3 : 0.3 : 0.9]

^{, we have}

nowto transformthese worst, best and median models toget a oherentbasis

representation for eah. Hereafter, inorder tobe able toompare the performane

ofthedevelopedmethodswithanexistingtehniquefromtheliterature,themethod

developed in [84℄ is seleted tobe applied rst. In this ase, the loalfrozen

blak-box models (Eq. (6.17)) are balaned before the interpolation step. Seond, the

performane of the loally applied null-spae-based tehnique is presented. Then,

the

H ^∞

-norm-basedloallyre-struturing tehnique is takled. These two methods aim at transforming the loal frozen blak-box models (Eq. (6.17)) into gray-box

ones (Eq. (6.16)). Using these tehniques for the urrent simulationexample leads

to

2 × 10

^{blak-boxLPVmodelsforeahvalueof}

∆β ∈ [0.3 : 0.3 : 0.9]

^{. Notethatthe}

worst sets ofloalmodelsarenot usedwhenthelassialinterpolation-basedloal

approahes aretakledbeausethey donotyieldaeptable,even stable,nal LPV

models. Notie also that, by using the lassial interpolation-based loalapproah,

only ane or polynomial LPV models an be estimated (Eq. (2.21)). However, as

is shown hereafter, by using the

H ^∞

-norm-based behavioral interpolation method introdued inSetion 5.5, amore generi LPV/LFR modelan be estimated.

Loal balaning-based tehnique

First,themethoddevelopedin[84℄isseletedtobeonsideredhereininorderto

ompare the performane of the developed tehniques with a previously developed

one from the literature. In [84℄, more speially, the estimated blak-box loal

LTI models are balaned before the interpolation step yielding a oherent basis

representationforeveryloalLTImodels. Afterthere-struturingstep,an

element-wiseinterpolationofthe system matriestakesplaeresultinginapolynomialLPV

model(see Eq.(2.21)). This interpolationproedureis traedbak toalinearleast

squares problem. More preisely,the lsqlin funtionof Matlabhas been applied

herein to alulate the global LPV system matries. Then, the question of LPV

modelvalidationarises. Herein, this validation step isperformedby omparing the

time evolution of the model outputwith the outputof the real system. In order to

persistentlyexitethesysteminthewholeoperatingdomain,theshedulingvariable

seleted to be asimple sine havingan amplitude in the range of

[0 0.8]

^{and having}

a frequeny equals to

5

^{rad/s. Seond, the shedulingvariableis onsideredto bea}

hirp signal with the same amplitude as before and a linearly inreasing frequeny

in the range of

[1 − 55]

^{rad/s. Notie that the evolution of the rst sheduling}

variable is muh slower than the seond one. The main reason why we use two

dierentkinds of shedulingvariables isto enhane the fatthat several previously

developed tehniques, yieldingablak-boxLPVmodel,havegoodperformaneonly

for slow variations of the sheduling variables ( see [131, 139, 54, 84, 105, 31, 99℄).

Thus, intherstase,i.e.,whentheshedulingvariableisasine,theobtainedBFTs

are gathered in Table 6.6while, in the other ase, Table 6.7 gathersthe alulated

BFT values. Notie that there is no noise during this LPV model validation. An

I/O t measurement, similar to the one dened in Eq. (6.14), is used to quantify

the apabilitiesof themodeltomimithesystem behavior. The obtainedBFTs are

gathered inTable 6.6-6.7. In this ase, the results orresponding tothe worst set

of estimated loalblak-box models are not shown inthe tables beause, those sets

ofloalmodels,annotyieldastableLPVmodel. Aordingtotheobtainedresults,

one an onlude that the estimated LPV models are highly sensitive to both, the

numberofthe involved operatingpointsand thevariationofthe shedulingvariable

aswell. Theestimated LPV modelshaveaeptable BFTmeasurementsonlywhen

∆β = 0.3

^{, and} onsequently when

N _op = 10

^{operating points have been involved}

into the identiation and a simple sine has been used to validate the global LPV

model. However, in the other ases, when only 2 and 3 loal operating points are

used, the BFTs are worse. On top of that, even worse BFTs an be observed in

Table6.7, i.e.,when the sheduling variable isa hirp, and hene muh faster than

the simplesine onsidered in Table 6.6.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

Best BFT

[%]

^{90.10 61.32 59.72 92.56 64.20 60.01}

MedianBFT

[%]

^{89.7 64.56 61.43 92.31 63.97 63.51}

Table 6.6: Best and median I/O BFT measurements of the estimated blak-box

aneLPVmodelsbyloallybalaningtheblak-boxmodelsbeforetheinterpolation

step. Slowly varying sheduling variable.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

Best BFT

[%]

^{35.21 26.64 26.61 40.43 29.39 22.52}

MedianBFT

[%]

^{35.02 30.21 25.42 27.03 0 24.34}

Table 6.7: Best and median I/O BFT measurements of the estimated blak-box

aneLPVmodelsbyloallybalaningtheblak-boxmodelsbeforetheinterpolation

step. The sheduling variable varies fast.

Null-Spae-based approah

Seondly, let us onsider the null-spae-based tehnique developed in [115℄, in

order to transformthe estimated blak-box loalLTI models into a oherent basis.

Here, more preisely, a null-spae-based formulation of Eq. (4.7) is introdued for

whihthesimilaritytransformationmatrixandthephysialparametersarefoundin

thesolutionvetorbelongingtothenull-spaeofamatrixontainingtheparameters

of the blak-boxmodel

( A , B , C )

^.

Γκ = 0,

^(6.23)

where

Γ

^{matrix ontains the} informationfrom the estimated blak-box state-spae model, while

κ

^{gathers the unknown strutured gray-box system matries and the}

similaritytransformation. Basedon the assumptionthat the dimension of the

null-spae is larger than 1, the novelty of the algorithm given in [115℄ resides in the

extration of the physial parameters by using a non-onvex optimization. On top

of that, by assuming that the initial blak-box realization of the system is

on-sistent and the nal gray-box state-spae form is identiable, it is proved in [115℄

that uniqueness of the solution an be ensured. Then, the interpolation is again

traed bak to alinear least squares problem. After that, the estimated LPV

mod-els are validated similarly to the previous ase by using both sheduling variables.

Contrarily to the previous ase, it an be seen that the estimated LPV models are

sensitive only to the number of the involved operating points. More preisely, in

Tables 6.8-6.9,when

∆β = 0.3, N _op = 10

^{,the LPV models apture suiently well}

the dynamis of the original system whatever the sheduling variable is. However,

inthe other ases,when only2 and 3loaloperatingpointsare used, theBFTsare

worse. Ontopof that,even worse BFTsanbeobservedinTable 6.9,i.e.,whenthe

shedulingvariableisahirp, andhenemuhfaster thenthe simplesineonsidered

inTable 6.8.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

BestBFT

[%]

^{95.11 64.01 64.35 96.89 61.05 62.34}

Median BFT

[%]

^{94.98 63.87 64.81 96.98 60.98 64.75}

Table6.8: BestandmedianI/OBFTmeasurementsoftheestimateddark-gray-box

ane LPV models by using the null-spae-based re-struturing tehnique. Slowly

varying sheduling variable.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

BestBFT

[%]

^{95.61 29.31 28.93 96.43 31.01 32.26}

Median BFT

[%]

^{95.01 28.83 27.30 96.98 29.67 31.92}

Table 6.9: Best and median I/O BFT measurements of the estimated

dark-gray-box ane LPV modelsby usingthe null-spae-based re-struturing tehnique. The

shedulingvariable varies fast.

H ∞

-norm-based re-struturing approah

Finally, the

H ^∞

-norm-based tehnique introdued in Chapter 4 is employed to derive the frozen LPV models (Eq. (6.16)) in eah operating points before the

element-wise interpolation step. As mentioned in Chapter 4 , the applied method,

namely,theproximityontrolalgorithm,isneededtobeproperlyinitialized. Herein,

totest the robustness of this tehnique w.r.t. the initialization,a Monte-Carlo

sim-ulationofdimension10isperformed. Notie that,asmentionedinSubsetion6.3.1,

it is assumed that

m 1

^and

m 2

^{are equal and known a priori. So, hereafter, the}

initialvalues ofthe unknown parameters are alulated,in eah operatingpoint,as

follows,

θ i init = θ i real (1 + 4(r _j ⁽ⁱ⁾ − 0.5)),

^(6.24)

where,

r ⁽ⁱ⁾ _j

^,

i ∈ { 1, · · · , 6 }

^,

j ∈ { 1, · · · , 10 }

^{,denotesauniformrandomnumberinthe}

range

[0, 1]

^while

θ i real

^,

i ∈ { 1, · · · , 6 }

^{,stands for the real values of the} parameters, respetively (see Subsetion 6.3.1 for the true values of the physial parameters

and for further details of the onsidered system). After the re-struturing step, the

resultinggray-boxloalsystemmatriesareinterpolatedyieldingapolynomialLPV

model(seeEq.(2.21))byusingagainthelsqlinfuntion. Eventually,theestimated

globalLPVmodelisvalidatedbyusingtheaboveintroduedtwodierentevolutions

of the sheduling variable. The obtained mean BFTs (based on 10 gray-box LPV

models)are gatheredinTables 6.10-6.11,respetively,when theshedulingvariable

is a simple sine and a hirp signal. From allthese gures, itan be onluded that

the estimated global LPV models are also sensitive to the number of the involved

operatingpoints. Similarlytothepreviousase,when

∆β = 0.3, N op = 10

^,theLPV

models apture well the dynamis of the original system whatever the sheduling

variable is (see Tables 6.10-6.11). On the ontrary, in the other ases, when only

2 and 3 loal operating points are used, the BFTs are muh worse again. The

appliation of the faster hirp sheduling variable yields again even worse BFT

measurements.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

Best BFT

[%]

^{95.91 63.53 64.21 95.89 59.97 61.27}

MedianBFT

[%]

^{95.43 64.40 60.61 94.98 60.34 63.21}

Table 6.10: Best and median I/O BFT measurements of the estimated

dark-gray-boxaneLPVmodelsbyusingthe

H ^∞

-norm-basedre-struturingtehniqueloally.

Slowly varying sheduling variable.

p min → p max p max → p min

∆β

^{0.3 0.6 0.9 0.3 0.6 0.9}

Best BFT

[%]

^{95.89 32.13 28.30 96.06 29.14 30.12}

MedianBFT

[%]

^{95.36 33.02 32.20 95.78 28.80 29.97}

Table 6.11: Best and median I/O BFT measurements of the estimated

dark-gray-boxaneLPVmodelsbyusingthe

H ^∞

-norm-basedre-struturingtehniqueloally.

The sheduling variable varies fast.

6.3.3.2 By using the

H ^∞

^{-BI tehnique}

Let us now present the performane of the

H ^∞

-norm-based behavioral interpo-lation tehnique. By having aess the previously estimated

3 × 10

^{blak-box LTI}

modelsforeahvalue of

∆β ∈ [0.3 : 0.3 : 0.9]

^{,the seondstepofthe proedureaims}

at estimating the nal LPV model dened by Eq. (6.15) written as an LFR (see

(6.19)-(6.18)) through the minimization of the ost-funtion dened by Eq. (5.14)

fromaset ofloalmodelsby applyingthetehniqueintroduedinSetion5.5. This

behavioral interpolation suggested in this part of the thesis involves also the

mini-mizationofan

H ^∞

-norm-basedost funtion. Beause ofthe non-onvexity of suh a ost funtion, initialization issues an our. Herein as well, in order to test the

robustness of our tehnique w.r.t. the initialization, a Monte-Carlo simulation of

dimension10isperformed. ForthisLPV modelidentiationstep,twoLPVmodel

strutures presented inSubsetion 6.3.1,ablak-boxand alight-gray-boxstruture

(see Eqs. (6.19)-(6.18)) are handled. In the blak-box framework, the 25 sought

parametersareinitializedrandomlybypikingupuniformlydistributedvalues(

r _j ⁽ⁱ⁾

^,

i ∈ { 1, · · · , 25 }

^,

j ∈ { 1, · · · , 10 }

^{)inthe range}

[0, 1]

^,while,inthelight-gray-boxone, the parameters are initializedby using the following expression

θ i init = θ i real (1 + 4(r ⁽ⁱ⁾ _j − 0.5)),

^(6.25)

where, again,

r ⁽ⁱ⁾ _j

^,

i ∈ { 1, · · · , 6 }

^,

j ∈ { 1, · · · , 10 }

^{, denotes a uniform random}

num-ber in the range

[0, 1]

^while

θ i real

^,

i ∈ { 1, · · · , 6 }

^{, stands for the real values of the}

parameters, respetively (see Subsetion 6.3.1 for the true values of the physial

parameters and for further details of the onsidered system). Notie again that, as

mentionedinSubsetion6.3.1,itis assumed thatweknowa priori that

m 1

^and

m 2

are equal.

Blak-box LPV model

In the blak-box framework, Table 6.12 ontains the mean (based on 10 LPV

blak-box models)of the obtained BFTmeasurements. Notie that, hereafter,only

the faster (hirp) shedulingvariableis employed during the validationstep.

p min → p max p max → p min

β

^{0.3 0.6 0.9 0.3 0.6 0.9}

Best BFT

[%]

^{96.6 91.5 91.2 96.8 90.7 90.8}

MedianBFT

[%]

^{96.3 90.2 90.3 96.2 90.1 90.01}

Worst BFT

[%]

^{89.2 53.2 48.9 61.4 56.2 49.2}

Table 6.12: I/O BFT measurements of the estimated blak-box LPV/LFR models

by applyingthe

H ^∞

^{-BI algorithm. Sheduling variable varies fast.}

Thegures gathered inTable 6.12 show that,exept whenthe worstloal

mod-els are used for the

H ^∞

-norm-based optimization, the proedure dediated to the LPV blak-box modelidentiationleads toLPVmodels able,inaverage,tomimi

the behavior of the real system eiently even when few loalmodels are involved.

Furthermore,theonsideredmethodisnotsensitivetotheinitializationwhilethe25

soughtparametersareinitializedintherange

[0, 1]

^{. Notiethat,asitispresented in}

Subsetion6.3.1,theonsideredmodeldependsrationallyontheshedulingvariable

(seeEq.(6.15))whileinthepreviousasesananeLPVmodelhas beenestimated.

Thus, one of the huge advantages of this tehnique omparing to the previous ones

thathereinanLPV/LFRmodelhas beenestimated. Asaonsequene,the rational

dependene isintegrated intothe modelstrutureto identify, even in the blak-box

ase. In the following, it is shown that this feature an further be exploited when

more prior informationabout the struture of the system under study is provided.

Light-gray-box LPV model

Inthe gray-box framework, our goalistofous onthe apabilities of the

identi-ationalgorithmtoestimatethephysialparametersofthe realsystemaurately.

Indeed, inthis ontext, the struture of the LPV modelis xed a priori. By using

the same loal models as the ones onsidered in the blak-box ase, light-gray-box

LPV models are estimated by resorting, again, to the ost funtion (5.14). T

a-ble 6.13 gathersthe mean of the estimated physial parameters (aswell asthe real

parametervalues)byonsideringthebest andmedian setsofloalmodelswhile,

in Table 6.14, I/Otmeasurements(see Eq (6.14))satised by the estimated LPV

models (byfollowingthe blak-boxLPV modelI/Ovalidationproedureonsidered

previously) an be seen. Notie that the worst set of loal models is disarded in

this study beause this set of loal models leads to very bad physial parameter

estimations (of magnitude, even sign, far from or inontradition with the physis

of the real system) whih an be disriminated easily thanks to the available prior

informationon the sought physial parameters.

Aording to the obtained results,it an be onludedthat the physial

param-eters of the system under study are well-estimated based on both the "best" and

byEq (6.14)are onerned,itan beseeninTable6.14thatthe estimatedgray-box

LPV models are able to apture the behavior of the system even when little loal

informationis involved.

Notie that inthe gray-box framework,loalminima have arisen resultingin 1,

maximum 2 (out of 10) erroneous sets of estimated parameters. Again, these loal

minimaproblemsanbedisardedthankstotheavailablepriorknowledgeaboutthe

magnitude and/orsign of the real parameters. The resulting estimated parameters

are indeed alot out of the range of the real parameter values.

p min → p max p max → p min

Parameter

m 1,2 k K f 2 f m 1,2 k K f 2 f

RealValue 0.2 10 4 0.04 0.1 0.2 10 4 0.04 0.1

β

^0.3

Best 0.19 9.96 4.00 0.038 0.1 0.19 10.02 4.01 0.04 0.098

Median 0.19 9.89 3.98 0.037 0.11 0.2 9.9 4.04 0.039 0.102

β

^0.6

Best 0.2 10.01 3.93 0.04 0.09 0.19 9.79 3.97 0.039 0.009

Median 0.19 9.98 3.98 0.039 0.09 0.19 9.88 3.99 0.04 0.101

β

^0.9

Best 0.21 9.75 4.09 0.039 0.098 0.2 9.83 4.02 0.038 0.098

Median 0.2 9.86 3.93 0.039 0.10 0.199 9.88 3.92 0.04 0.102

Table 6.13: Estimated physial parameters based on the best and median sets of

loalmodels.

p min → p max p max → p min

β

^{0.3 0.6 0.9 0.3 0.6 0.9}

Best BFT

[%]

^{97.4 96.8 96.1 97.2 96.9 96.2}

Median BFT

[%]

^{97.4 96.9 96.4 97.1 96.1 96.1}

Table6.14: I/OBFTmeasurementsofthe estimatedlight-gray-boxLPVmodelsby

applying the

H ^∞

^{-BI algorithm. The shedulingvariable varies fast.}

In document H ∞-norm for the (Pldal 100-107)