Possible researh diretions

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℄.

8.2.1 Improvements of the null-spae-based tehnique

Let us rst mention the extension of the null-spae-based approah proposed

in [115℄. It is, more preisely, developed until now only for the identiation of

ane LPV models. This method should be modied to be able to takle with

frational representations ahieving a more general solution of the stated problem.

Said dierently, the goal is to provide a solution to the following set of bilinear

equations

A T = TA(θ ^g ), B , = TB(θ ^g ), C T = C(θ ^g ),

^(8.1)

where the matries

( A , B , C , A(θ ^g ), B(θ ^g ), C(θ ^g ),

^with

θ ^g ∈ R ⁿ ^θg )

^are, ^more

pre-isely,the innermatries of the LFR struture dened by Eq. (2.28) and wherethe

similaritytransformation

T ∈ S _∆

^(see^Eq.^(2.31))^is^blok^diagonal^aording ^to^the

denitions introduedin Setion2.2.

8.2.2 Strutural identiability and well-posedness of LFRs

By going further on the pathway introdued above, beause the identiability

property is stated as a very important key assumption of the tehnique developed

in [115℄, the strutural identiabilityof gray-box LFRs (both light- and

dark-gray-box as well) should also be investigated (see Subsetion 2.4.1 for the denition

of the strutural identiability). On top of that, when the identiation of LPV

models using frationalrepresentationsis takled,from apratialpoint of view, it

wouldalsobeinterestingtoexaminewhetherthevariationoftheshedulingvariable

improves the strutural identiability of the onsidered gray-box LPV model. In

other words, under whih onditions onthe modelstruture or onthe evolution of

the shedulingvariables an be ensured that, even if a loal frozen gray-box

state-spaeLPVmodelisnotidentiable,byinvolvingtwoormoreotheroperatingpoints,

the gray-box model struture nally beomes identiable. Asking dierently, is it

possible to derive some onditions, by onsidering several working points, in order

toestimate uniquely the physial parameters found on gray-box LPV modelwhih

is,basially, not identiable when onlyone singlefrozen loalmodelis onsidered?

Then, asmentionedinSetion 2.1, the investigationofthe well-posedness

prop-ertyof the linearfrationalLPVmodels isalsoaninterestingand hallenging

prob-lem. Sine, in the LPV framework, the matrix

∆(d, Υ)

^dened ^by ^Eq. ^(2.6)

ex-pliitlydepends ontheshedulingvariable

p(t)

^,^the well-posedness ondition ofthe problemshouldberelatedtotheevolutionrangeof

p(t)

^dened ^on^the^ompat ^set

P

(see Eq. (2.14)). Aording to the Author's knowledge, no neessary and su-ientonditionsare availableforthis probleminthe literature. Nevertheless, inthe

gray-box framework, when the model struture is obtained from the physial laws

governingthebehaviorofthe system,the strutureof

∆(d, Υ)

^an^be^xed ^a^priori

and suientLMI onditions ouldbeobtained toensure thewell-posedness of the

LFR by applying a S-proedure tool [21℄. So, this study, more preisely the link

between, e.g., the ondition numberof

I _n − A ∆(d, Υ)

^and ^the

p(t)

^-tra^jetory^,^an

beseenasaninterestingstartingpointforananalysisoftheoptimal

p (t)

^-trajetory

design ensuring aonsistent identiation. At the same time, suh aninvestigation

8.2.3 Handling the possible blak-box model estimation

er-rors

When the identiationof gray-box LTI and LPV models are performed based

on blak-box LTI models estimated beforehand, it is assumed that the initial

fully-parametrized models applied in the methods presented in Chapters 4-5 denoted

G (s)

^are ^perfetly ^identied. ^However, ⁱⁿ ^pratie, ^this ^assumption ^is ^hardly

fullled. Thus, the eet ofthe possible estimationerrors shouldalsobestudied by

alulating the sensitivity of the resulting ost-funtion to this error term denoted

∆ G

. Similarly,the modelingerrorsof the gray-box modelstrutures analsobe onsidered as

∆ G ( θ ^g )

^. ^Notie ^that, ^deteting ^the ^order ^or ^the ^magnitude ^of ^suh

modelingorestimationunertaintiesouldbeusefulforfurtherrobustLTIandLPV

ontrollersynthesis. A possible solutiontotake intoaountthe estimation erroris

the introdutionof an

l 2

-norm-basedterm in the following ost-funtion, e.g.,

min θ ^g

k ( G (s) + ∆ G (s)) − G(s, θ ^g ) k ^∞ − µ

k Y(s) k ²

k U(s) k ² − k G (s) k ^∞

,

^(8.2)

where

kY(s)k 2

kU(s)k 2 − k G (s) k ^∞ = k ∆ G (s) k ^∞

^with

θ ^g ∈ R ⁿ ^θg

ontains the parameters to identify and,

s

^stands^for ^the^Laplae ^transform^variable. ^In ^the^novel^ost ^funtion

dened by Eq. (8.2), the slak variable

µ

^is, ^more ^preisely, ^used ^to ^ut ^down ^or

trade o the eet of the estimation errors.

8.2.4 Improvements of the LPV model identiation

teh-niques based on loal experiments

As far as the identiation of LPV models applying the loal approah is

on-erned, several aspets have not been investigated in this thesis. Some of them are

listed below.

In this thesis, the sheduling variables are assumed to be measurable signals

of the system under study. This assumption should be relaxed by applying

signals estimated orlteredfromother measured signals(states and outputs)

of the onsidered proess, for instane, by involvingaKalman-lter [61℄.

It is alsoassumed that the shedulingvariablesare available real-timeand it

is measured without any noise. So, it would also be an interesting researh

diretionto examine the eets of time-delayed ornoisy shedulingvariables.

Inthisthesis, whenblak-boxLPVmodelsare identied,the dimensionofthe

∆(d, Υ)

^,^dened ^by^Eq.^(2.6),^blok^is^xed^a ^priori ^. ^In^order^to^yield^a^more

generi identiation framework, it is neessary to develop a omplementary

tehnique that is able to determine the optimaldimension of

∆(d, Υ)

^during

the estimation proedure.

After that, onerning the operating point seletion approah presented in

Subse-tion 5.3.2, the appliation and omparison of dierent dynami measures, suh as

e.g., the Binet-Cauhy kernel [147℄ and the Martin distane [91℄, are also possible.

Byfollowingthisline,abenhmarktothisoperatingpointseletionapproahanbe

developedbyinvolvingstatistialinvestigationandnoisyI/Odatasequenes.

More-over, it would also be welome to develop suh a seletion algorithm that is muh

more robust to the presene of measurement noises ating even on the sheduling

In document H ∞-norm for the (Pldal 132-135)

H ∞

H ∞

H ∞

A T = TA(θ g ), B , = TB(θ g ), C T = C(θ g ),

( A , B , C , A(θ g ), B(θ g ), C(θ g ),

θ g ∈ R n θg )

T ∈ S ∆

∆(d, Υ)

p(t)

p(t)

P

∆(d, Υ)

I n − A ∆(d, Υ)

p(t)

p (t)

G (s)

∆ G

∆ G ( θ g )

l 2

min θ g

k ( G (s) + ∆ G (s)) − G(s, θ g ) k ∞ − µ

k Y(s) k 2

k U(s) k 2 − k G (s) k ∞

,

kY(s)k 2

kU(s)k 2 − k G (s) k ∞ = k ∆ G (s) k ∞

θ g ∈ R n θg

s

µ

∆(d, Υ)

∆(d, Υ)

H ^∞

H ^∞

A T = TA(θ ^g ), B , = TB(θ ^g ), C T = C(θ ^g ),

( A , B , C , A(θ ^g ), B(θ ^g ), C(θ ^g ),

θ ^g ∈ R ⁿ ^θg )

T ∈ S _∆

I _n − A ∆(d, Υ)

∆ G ( θ ^g )

min θ ^g

k ( G (s) + ∆ G (s)) − G(s, θ ^g ) k ^∞ − µ

k Y(s) k ²

k U(s) k ² − k G (s) k ^∞

kU(s)k 2 − k G (s) k ^∞ = k ∆ G (s) k ^∞

θ ^g ∈ R ⁿ ^θg