An H ∞ -norm-based approah

4.4 From blak-box state-spae models to gray-box ones

4.4.1 An H ∞ -norm-based approah

As explained in Setion 4.1, instead of resortingto the standard

L ²

^-norm, ^it ^is

suggested herein using the

H ^∞

^-norm ⁱⁿ ^order ^to ^estimate ^the ^unknown ^parameter

vetor

θ

^aurately. ^More ^preisely^, ^by ^assuming ^that, ^from ^the ^available

data-set

{ u(k), y(k) } ^N−1 k=0

^a ^reliable ^blak-box ontinuous-time representation

( A , B , C )

has been estimated beforehand, i.e., a blak-box model able to apture aurately

the frequeny response of the real system is available, we aim at minimizing the

H ^∞

^-norm ^of ^the ^dierene ^given ^by ^Eq. ^(4.9), ^i.e.,

J(θ) = k G(θ, ω) − G (ω) k ² ∞ = k F(θ, ω) ^H F(θ, ω) k ^∞ ,

^(4.16)

with

F(θ, ω) = G(θ, ω) − G (ω).

^(4.17)

Astandardapproahforthiskindof

H ^∞

-norm-basedoptimizationproblemonsists in using the Kalman-Yakubovith-Popov lemma whih would herein lead to

bilin-ear matrix inequalities [20℄. In this Chapter, we have hosen to minimize

J(θ)

^by

using the proximity ontrol algorithm. As shown in [5, 6℄, the applied tehnique

in this hapter avoids the use of Lyapunov funtions, feature whih is beneial

beause these variables are often a soure of numerial troubleand inreased

om-putational load. The prie to pay for avoiding the previously mentioned numerial

troubles is the introdution of a semi-innite and non-smooth program. Indeed,

the optimization problem (4.16) is non-smooth and non-onvex with two soures

of non-smoothness, the innite max operator and the maximum eigenvalue

fun-tion, respetively [5℄. The use of this spei ost funtion dened by Eq. (4.16),

more preisely the

H ^∞

^-norm, ^an ^be ^motivated ^by ^the availability of quite new and eient optimization algorithmsdediated to the minimizationof non-smooth

and non-onvex funtions with additionalstruture [4, 5, 6, 53, 8℄. The use of the

H ^∞

^-norm ^an ^also ^be ^justied ^by ^notiing^that, ^by ^denition, ^the

H ^∞

^-norm ^is ^the

maximal singular value of the omplex gain matrix over all the frequenies. Thus,

thanks tothenormproperty,if theost funtion

J (θ)

ⁱⁿ^Eq.^(4.16)^is^small^enough

(i.e.,alotsmallerthan1),thenthedistanebetweentheatualsystemandthe

iden-tied one issmallaswell. Notie thatin [14,15℄asimilar ost funtionisproposed

to estimate the unknown parameter vetor of an identiable gray-box struture by

employing the hinfstrut Matlab funtion. Notie also that during the

devel-opments proposed in this thesis, the proximity ontrol algorithm[6℄ is used whih

an be seen as a novel version of the algorithm realized by the hinfstrut. This

hoie an mainly be explained by the fat that the hinfstrut funtion annot

dealwithunstable modelsduringitsoperation. Thisispartiularlyimportantwhen

J (θ)

^is initialized, before the optimization, by uniform random numbers resulting sometimes in unstable models.

Before desribing into detail the

H ∞

-norm-based method, it is important to point out that the availability of an aurate blak-box form

( A , B , C )

^is ^not ^a

strong assumption. Indeed, this estimation step an be performed eiently by

using one of the many identiationtoolboxesavailablein the literature(the

Mat-lab System Identiation Toolbox 12

, the CONTSID Toolbox 13

or the LTI

Sys-tem Identiation Toolbox 14

to name few). In this thesis, more preisely in

Se-tion 6.2, a spei attention is paid to the Multivariable OutputError State-sPae

(MOESP) algorithms[144, 93℄ aswell as the SimpliedRened Instrumental

Vari-able method for Continuous-time systems (SRIVC) [46, 47℄ (see Setions 9.2-9.3 in

the Appendixforabriefoverviewofthe identiationmethods). Thishoieanbe

justied asfollows. First, theMOESP lass of subspae-basedalgorithms, available

through the LTI System Identiation Toolbox, is indeed full of tehniques able to

deliverreliable(suboptimal)blak-boxstate-spaerepresentationsfromthedata-set

{ u(k), y(k) } ^N−1 k=0

^under^many^dierent^disturbaneassumptions(whiteoutputnoise, olored output noise, white state noise, error in variables) [144℄. Notie also that

MOESP-like algorithmsyielding diretly ontinuous-time 15

blak-box models have

beendeveloped duringthelastdeade[98,13℄. Ontopofthat,theappliationofthe

subspae-based algorithmto get this initialstate-spae representation an be

justi-ed by two otherreasons. First,this step is sharedby the standard

L 2

-norm-based tehniques, at least in the blak-box model framework [78℄. Thus, this estimation

step of

( A , B , C )

^, ^whih ^is ^ompulsory ^for ^future developments, does not inrease the omputational load of this novel

H ∞

^-based ^tehnique ⁱⁿ ^omparison ^with ^the

aforementioned

L ²

-norm-basedtehnique. Seond, the subspae-based methodsare numerially robust and are one-shot estimation proedures whih do not require,

apart from an upper bound of the system order, any prior knowledge about the

system toidentify. Thus, the state-spae form

( A , B , C )

^an ^be^estimated ^without

resortingtoany iterativealgorithm. Seond, theSRIVCmethod,whihan be

on-sideredasone ofthemosteientidentiationtehniquesforthediretestimation

of ontinuous-time models, is known to yield onsistent estimates when the model

belongs to the system lass [46℄. This tehnique is, more preisely, available in the

CONTSID ToolboxforidentifyingMISOmodels,aonstraintsatisedbythe

simu-12. http://www.mathworks.fr/fr/produts/sysid/.

13. http://www.iris.ran.uhp-nany.fr/ontsid/.

14. http://www.ds.tudelft.nl/~datadriven/lti/ltitoolbox_produt_page.html.

15. TheuseranfurthermoreombineaMOESPalgorithmyieldingadisrete-timestate-spae

latedsystemstudiedinSetion6.2. Notienallythattheimplementationavailable

inthe CONTSIDToolboxhas theadvantageofnot requiringany designparameters

tobespeied. Ofourse,users familiarwith othertehniquesan usetheirfavorite

algorithms to estimate the initial state-spae representations

( A , B , C )

^as ^long ^as

these blak-box models mimi the system behavior aurately.

Thetransformationoftheinformationontainedintheinput-output(I/O)

data-set into a blak-box model an be interesting for three main reasons. First, the

(potentiallyhuge) availabledata-setof size

N (n u + n y )

^is ^summed^up ^into ^a^model

ontaining

n ² _x + n x (n y + n u ) + n u n y

parameters. Thus, the dimension of the in-formation used by the iterative algorithm required for the

H ^∞

^-norm optimization, i.e, the seond step of the identiation proedure developed in this Chapter, is a

lot smaller. The omputational load of the proximity ontrolalgorithm used

here-after should then be redued. Seond, by having aess to blak-box models quite

easily from the available data-sets, it is straightforward to hek if the available

I/Oinformationisgoodenough by usingstandard toolsfor modelvalidation[78℄.

Third, the noise treatment, i.e., the attenuation of the noise eet for the

onsis-teny ofthe estimated parameters,is performedduringthis initialstep by hoosing

eientlythe tehniqueyieldingaonsistentform

( A , B , C )

^from^the^(noisy)

data-set

{ u(k), y(k) } ^N−1 k=0

^. ^This ^problem ^is ^quite ^standard ⁱⁿ ^LTI ^system identiation and, as said previously, an be solved eiently and in one shot by using spei

subspae-based algorithms(see, e.g., [144℄,when the MOESP lass of tehniques is

handled). Of ourse, this rst step is not omputationally ost free. For instane,

the omputationalost of the dierent steps omposing the PO-MOESPalgorithm

have been evaluated and gathered in Table 4.2. These values show that this step

anbequitenumeriallydemanding 16

when,e.g.,MOESP algorithmsareusedbut,

fortunately, should be performedonly one.

Step Multipliations

RQfatorization

2(α ² + β ² )(N − (α + β)/3)

Singularvalue deomposition

αn ² _y f ²

Extration of

A

and

C n y f n ² _x

Estimationof

B n ³ _u (n x + n y ) ³

Table 4.2: Computation load of the PO-MOESP algorithm [144℄.

f

^and

p

^are

the user-dened future and past window hyper-parameters hosen greaterthan

n _x

α = (n u + n y )p

^and

β = (n u + n y )f

^. ^See ^[144, ^Paragraph ^9.6.1℄ ^for ^a ^summary ^of

the main steps omposing the PO-MOESPalgorithm.

As mentioned in the previous Chapter, the proximity ontrol algorithm is

de-signed to minimizesuh ost-funtions as the one dened by Eq. (4.16). Here, the

followingspeiset-upisintroduedbeausethe proximityontrolmethodisused

tominimizethefrequeny-domaindistane"betweenblakandgray-boxmodelsby

tuningtheparametersgathered inthevetor

θ

^. ^As ^a^onsequene,^the Hamiltonian matrix dened by Eq. (3.13) and used during the proximity ontrol algorithm has

16. Ifneessary,thePO-MOESPalgorithmanbereplaedbyareursivesubspae-based

algo-the following inner matries, e.g.,

E _A = A(θ) − A , E _B = B(θ) − B , E _C = C(θ) − C .

^(4.18)

In the next Subsetion, the onvergene properties of the proximity ontrol

algo-rithm is desribed by onsidering the spei ase when

F ( θ , ω)

^has ^a ^state-spae

representation.

In document H ∞-norm for the (Pldal 53-56)

4.4 From blak-box state-spae models to gray-box ones

4.4.1 An H ∞ -norm-based approah

L 2

H ∞

θ

{ u(k), y(k) } N−1 k=0

( A , B , C )

H ∞

J(θ) = k G(θ, ω) − G (ω) k 2 ∞ = k F(θ, ω) H F(θ, ω) k ∞ ,

F(θ, ω) = G(θ, ω) − G (ω).

H ∞

J(θ)

H ∞

H ∞

H ∞

J (θ)

J (θ)

H ∞

( A , B , C )

{ u(k), y(k) } N−1 k=0

L 2

( A , B , C )

H ∞

L 2

( A , B , C )

( A , B , C )

N (n u + n y )

n 2 x + n x (n y + n u ) + n u n y

H ∞

( A , B , C )

{ u(k), y(k) } N−1 k=0

2(α 2 + β 2 )(N − (α + β)/3)

αn 2 y f 2

A

C n y f n 2 x

B n 3 u (n x + n y ) 3

f

p

n x

α = (n u + n y )p

β = (n u + n y )f

θ

E A = A(θ) − A , E B = B(θ) − B , E C = C(θ) − C .

F ( θ , ω)

L ²

H ^∞

{ u(k), y(k) } ^N−1 k=0

H ^∞

J(θ) = k G(θ, ω) − G (ω) k ² ∞ = k F(θ, ω) ^H F(θ, ω) k ^∞ ,

H ^∞

H ^∞

H ^∞

H ^∞

{ u(k), y(k) } ^N−1 k=0

L ²

n ² _x + n x (n y + n u ) + n u n y

H ^∞

{ u(k), y(k) } ^N−1 k=0

2(α ² + β ² )(N − (α + β)/3)

αn ² _y f ²

C n y f n ² _x

B n ³ _u (n x + n y ) ³

n _x

E _A = A(θ) − A , E _B = B(θ) − B , E _C = C(θ) − C .