Conlusion

0 2 4 6 8 10 12 14 16 18

−0.2

−0.1 0 0.1 0.2 0.3

Real and estimated model outputs - Fit = 62.95 % - Vaf = 86.29 %

System output Model output

0 2 4 6 8 10 12 14 16 18

−2

−1 0 1 2

Time (s)

Real and estimated model outputs - Fit = 94.89 % - Vaf = 99.74 %

System output Model output

Figure7.17: Global validationdata.

7.4.2.2 Blak-box ase

As a next step, let us present the validation results of the estimated blak-box

LPV models. The obtained averageloalBFTmeasurements aregiven inTable 7.3

while theglobalvalidationstepyieldsthe followingaveragetmeasurements

73.8%

and

97.2%

^for

φ ˙ ₁

^and

φ ˙ ₂

^, respetively. Notie that, in the blak-box framework, thereare moreparameterstotune duringtheoptimization. This isthe mainreason

why we ould obtain slightlybetter BFTs in this ase than in the gray-box one.

i 2 4 6 8 10

φ ˙ ₁ [BF T ]

^{84.64 82.34 82.33 85.03 84.80}

φ ˙ 2 [BF T ]

^{79.26 79.19 82.43 80.40 75.24}

Table7.3: Performane metris(BFT (%))forthe estimated frozen blak-box LPV

models on validationdata

φ ⁽ⁱ⁾ _2,0 ∈ [π/8 : π/16 : 6π/8]

^.

TheobtainedaverageBFTsequalto

[73.8% 97.2%]

^and

[59.9% 94.22%]

^inthe

blak-andgray-boxase,respetively. Notiethatwhentheverysimplemodel,depitedin

Figure7.15, isused, the obtained average BFTmeasurements are

[53.02% 84.31%]

respetivelyon

φ ˙ 1

^and

φ ˙ 2

^{whihislowerthanwhatisobtainedbyusingtheestimated}

LPV models. Therefore, the proposed modelreallyallows toapture the dynamis

of the system. All these results prove that, adding up prior information through

the knowledge of the LPV model struture, is an eient solution resulting in an

aurate LPV modelby applying loal experimental data and the

H ∞

-norm-based method developed in Chapter 5.

Chapter 8

Summary of the obtained results and

future researh objetives

8.1 Thesis Points

In this Setion, the developed new sienti results are summarized as thesis

points with the referenes to the orresponding Chapters in this manusript. The

rst thesis deals with the re-struturing of blak-box state-spae LTI models into

gray-box ones. The seond and third thesis form a separate thesis group beause

they takle the identiation problem of state-spae LPV models by involving a

lassial interpolation step. Even though, the fourth and fth thesis deal alsowith

the identiationof state-spae LPV models, they ompose adistint thesis group,

beause here, anew behavioral approahand the

H ^∞

^{-norm isused toestimatethe}

LPV model fromloalexperiments. In the following, the developed tehniques are

enumerated aording tothe abovepresented grouping.

Thesis 1 A new tehnique being able to restruture blak-box linear

time-invariant (LTI) state-spae models into gray-box ones, by traing bak the

identi-ation problem to a strutured

H ^∞

^{synthesis problem, has been developed. The}

blak- and gray-box LTI models are ompared in the frequeny domain. Then, by

minimizing the

H ^∞

-norm-based ost funtion dened by Eq. (4.16), the unknown parameters found inthe gray-box LTI modelare determined.

Thisthesispointisdeveloped andpresentedinChapter4. Asimulationexample

isusedtodemonstratethe eetivenessofthe proposed solutioninSetion6.2. The

obtained results have been published in[155, 153℄.

Thesis 2.1 Anew tehniqueperformingthe identiationof interpolatedlinear

parameter-varying (LPV) models from loally restrutured models by using

stru-tured

H ^∞

^{synthesishasbeendeveloped. Inthisase,theloallyidentiedblak-box}

LTImodels aretransformedintotheorrespondingloalfrozengray-boxLPV

mod-els, inevery working point,by usingloallythe

H ^∞

-norm-based LTI re-struturing tehnique developed during the rst thesis point. This step is then followed by a

lassialleast-squares-based interpolation inordertoderive the nalgray-box LPV

model.

is used to demonstrate the eetiveness of the proposed solution in Chapter 6.3.3.

The obtained resultshave been published in [154℄.

Thesis2.2 Anewtehniqueperformingthe identiationof interpolatedlinear

parameter-varyingmodels fromloallyrestrutured models by usingthe

null-spae-based tehnique has been developed. Here, the loally identied blak-box LTI

models are again transformed into the orresponding loally frozen gray-box LPV

models by using anull-spae-based tehnique, developed in[115℄. This step is then

followed by a lassial least-squares-based interpolation in order to derive the nal

gray-box LPV model.

ThisthesispointisdevelopedandpresentedinSetion5.4. Asimulationexample

is used to demonstrate the eetiveness of the proposed solution in Chapter 6.3.3.

The obtained resultshave been published in [157℄.

Thesis 3.1. A new tehnique being able to identify blak and gray-box linear

parameter-varyingmodelsfromloalexperiments,bytraingbaktheidentiation

problem to a strutured

H ^∞

^-norm optimization problem, has been developed.The loallyestimatedblak-boxLTIandthefrozengray-boxLPVmodelsareplaedinto

the

H ^∞

-norm-basedglobalostfuntion denedby Eq.(5.12). Then,the nalLPV model is estimated by optimizing one single ost funtion without the appliation

of the lassialinterpolation.

ThisthesispointisdevelopedandpresentedinSetion5.5. Asimulationexample

is used to demonstrate the eetiveness of the proposed solution in Chapter 6.3.3.

The obtained resultshave been published in [149, 151, 152, 156, 148, 150℄.

Thesis 3.2. A new

H ∞

-norm-based approah whih determines a set of loal models for linear parameter-varying model identiationhas been developed. The

developed algorithmis ableto determineiteratively areliableset of loaloperating

points. Then,the determined set of working pointsan beappliedduring anyloal

model-based LPV modelidentiationtehnique.

ThisthesispointisdevelopedandpresentedinSetion5.3. Asimulationexample

is used to demonstrate the eetiveness of the proposed solution in Chapter 6.3.2.

The obtained resultshave been published in [149, 148℄.

In document H ∞-norm for the (Pldal 128-132)