Infinite dimensional port Hamiltonian systemscas.ensmp.fr/~chaplais/SeminarSlides/LE-GORREC_11_02_24.pdf · In nite dimensional port Hamiltonian systems 2D, 3D cases (cf A.v.d. Schaft

Introduction Infinite dimensional linear 1-D case Model reduction Conclusion

Infinite dimensional port Hamiltoniansystems

Y. Le Gorrec1

1FEMTO-ST, Automatic Control and Micro-Mechatronic Systems Department

CAS Seminar - Topic : PDE


1 IntroductionContextOutline

2 Infinite dimensional linear 1-D caseA simple exampleConsidered class of systemsDirac structure and PHS on Hilbert SpaceBoundary control systemsStability, control feedback

3 Model reductionMixed finite element methodSpectral properties

4 Conclusion


Context

Recent technological progresses and physical knowledge allow togo toward the use of complex systems :

Highly nonlinear.

Involving numerous physical domains and possibleheterogeneity.

With distributed parameters or organized in network.

New issue for system control theory

Modelling step is important → the physical properties can beadvantageously used for analysis, control or simulation purposes


Context

Recent technological progresses and physical knowledge allow togo toward the use of complex systems :

Highly nonlinear.

Involving numerous physical domains and possibleheterogeneity.

With distributed parameters or organized in network.

New issue for system control theory

Modelling step is important → the physical properties can beadvantageously used for analysis, control or simulation purposes


Context

Example 1 : Ionic Polymer Metal CompositeParcours

Activitésd’enseignement

Activités derecherche

Projetsd’intégration•ENSMM•FEMTO-ST/AS2M/SAMMI

Candidature au Poste de Professeur des Universités PU 61 - 0843, 2008 ENSMM p. 15/15

! Exemple de projet support: ProjetFranco-Japonais SAKURA

! Mise en oeuvre pratique - Pilote

Electromechanical system.

3 scales : Polymer-electrode interface, diffusion in thepolymer, beam bending.


Context

Example 2 : Nanotweezer for DNA manipulation

Multiphysic system.

Infinite dimensional system.


Context

Example 3 : Adsorption process

Parcours


Activités derecherche•Motivation•Développementsthéoriques

•Projets -Encadrements- Publications

Projetsd’intégration


Procédé d’adsorption par modulation de pression (A. Baaiu PHD)

Objectif : séparation par adsorption

(coll. IFP)

A

B

A B

B

! Système hétérogène

multi-échelles, régi par

thermodynamique irréversible

! EDP non linéaires

Microporous crystal

Bidispersepellet

rp z

rc

L!25 cm

R !1,24 mm

R !1 µmc

p

Extra granular phase

Macropore scale

Micropore scale

R !1 cmint

Multiscale heterogeneous system.

Dynamic behavior driven by irreversible thermodynamic laws


Context

Example 3 : Adsorption process

Parcours


Activités derecherche•Motivation•Développementsthéoriques

•Projets -Encadrements- Publications

Projetsd’intégration


Procédé d’adsorption par modulation de pression (A. Baaiu PHD)

Objectif : séparation par adsorption

(coll. IFP)

A

B

A B

B

! Système hétérogène

multi-échelles, régi par

thermodynamique irréversible

! EDP non linéaires

Microporous crystal

Bidispersepellet

rp z

rc

L!25 cm

R !1,24 mm

R !1 µmc

p

Extra granular phase

Macropore scale

Micropore scale

R !1 cmint

Multiscale heterogeneous system.

Considered phenomena :

Fluid scale : convection, dispersion.Pellet scale : diffusion (Stephan-Maxwell).Microscopic scale : Knudsen law.


Outline

A unified approach : the port Hamiltonian framework

Emphasis the geometric structure related to power exchanges powerful for analysis, model reduction and control.

Today : Infinite dimensional linear 1-D case and discretization


Outline




!

!"#$$

%&'$!()$

%&'$


Outline




Other works :

Infinite dimensional linear 2-D, 3-D cases (joint work with B.Maschke and H. Zwart).

Infinite dimensional port Hamiltonian systems 2D, 3D cases(cf A.v.d. Schaft and B. Maschke).

Infinite dimensional NL systems : Passivity and PHS vs.Stability and Riemann Invariants (Joint work : V. DosSantos, B. Maschke).

Non linear control : IDA-PBC, Entropy based control ofChemical reactors (Joint work : F. Couenne).


A simple example




4 Conclusion


A simple example

Vibrating string :!

!"#"

u(t,z)

The classical modelling is based on the wave equation : Newton’slaw + Hooke’s law (restoring force proportional to the deformation)

∂2u(z , t)

∂t2=

1

µ(z)

∂

∂z

(T (z)

∂u(z , t)

∂z

)

The structure of the model is not apparent. How to choose theboundary conditions ? ? ?


A simple example

Let choose as state variables the energy variables :

the strain ε = ∂u(z,t)∂z

the elastic momentum p = µ(z)v(z , t)

The total energy is given by : H(ε, p) = U(ε) + K (p)

U(ε) is the elastic potential energy :

U(ε) =

∫ b

a

1

2T (z)

(∂u(z , t)

∂z

)2

=

∫ b

a

1

2T ε(z , t)2

where T (z) denotes the elastic modulus.

K (v) is the kinetic co-energy :

K (p) =

∫ b

a

1

2µ(z)v (z , t)2 =

∫ b

a

1

2

1

µ(z)p2(z , t)

where µ(z) denotes the string mass.


A simple example

The vector of fluxes is given by :

β =

(v(t, z)σ

)where v(z , t) is the velocity and σ(z , t) = T (z)ε(z , t) the stress.The vector of fluxes β may be expressed in term of the generatingforces :

β =

(0 11 0

)︸︷︷︸

( δHδεδHδp

)︸︷︷︸

canonical generatinginerdomain coupling forces


A simple example

From the conservation laws :

∂

∂t

(εp

)+

∂

∂z

(vσ

)= 0

Consequently

∂

∂t

(εp

)= − ∂

∂z

(0 11 0

)( δHδεδHδp

)and

PDEs :

∂

∂t

(εp

)=

(0 − ∂

∂z

− ∂∂z 0

)( δHδεδHδp

)⇔ ∂2u(z , t)

∂t2= σ2∂

2u(z , t)

∂z2

+BC


A simple example

Underlying structure :

∂

∂t

(εp

)︸︷︷︸ =

(0 − ∂

∂z

− ∂∂z 0

)︸︷︷︸

(T (z) 0

0 1µ(z)

)(εp

)︸︷︷︸

f J = matrix e = drivingdifferential operator force

Hamiltonian operator J is skew-symmetric only for functionwith compact domain strictly in Z :∫ b

a

(e1 e2

)J(

e′1e′2

)+(

e′1 e′2)J(

e1

e2

)= − [e1e

′2 + e2e

′1]

ba


A simple example

Power balance equation :

ddt H(ε, p) =

∫ ba

(δHδε

∂ε∂t + δH

δp∂p∂t

)dz

= −∫ ba

(δHδε

∂∂z

δHδp + δH

δp∂∂z

δHδε

)dz

= −[δHδε

δHδp

]ba

If driving forces are zero at the boundary, the total energy isconserved, else there is a flow of power at the boundary. Definetwo port boundary variables as follows :(

f∂e∂

)=

( δHδεδHδp

)|a,b


A simple example

The linear space D 3 (f1, f2, e1, e2, f∂ , e∂)(f1e2

)=

(0 − ∂

∂z

− ∂∂z 0

)(e1

e2

)(

f∂e∂

)=

(e1

e2

)|a,b

defines a Dirac structure :D = D⊥ with respect to the symmetricpairing : ∫ b

ae1f1dz +

∫ b

ae2f2dz + [f∂e∂ ]ba

Port Hamiltonian system(∂

∂tα,δH

δα, f∂ , e∂

)∈ D


Considered class of systems




4 Conclusion


Considered class of systems

Considered class of systems :

∂x

∂t(t, z) = (J − GRSG∗R)L(z)x(t, z), x(0, z) = x0(z),

m(ffp

)= Je

(eep

)=

(J GR

−G∗R 0

)(eep

)with ep = Sfp where S is a coercive operator

„ffp

«∈ F ,

„eep

«∈ E and E = F = L2((a, b),Rn)× L2((a, b),Rn)

Covers models of : beams, wave, plates, (with or without damping)and also systems of diffusion/convection, chemical reactors ...


Dirac structure and PHS on Hilbert Space




4 Conclusion



The system is defined by :

f = J e

• Let the space of flow variables, F , and the space of effort

variables,E , be real Hilbert spaces.• Define the space of bond variables as B = F × E endowed by thenatural inner product⟨b1, b2

⟩=⟨f 1, f 2

⟩F+⟨e1, e2

⟩E , b1 =

(f 1, e1

), b2 =

(f 2, e2

)∈ B.

In order to define a Dirac structure, let us moreover endow thebond space B with a canonical symmetrical pairing, i.e., a bilinearform defined as follows :⟨b1, b2

⟩+

=⟨f 1, rE,Fe2

⟩F+⟨e1, rF,E f

2⟩E , b

1 =(f 1, e1

), b2 =

(f 2, e2

)∈ B.

(1)



Denote by D⊥ the orthogonal subspace to D with respect to thesymmetrical pairing :

D⊥ =

b ∈ B|⟨b, b′

⟩+

= 0 for all b′ ∈ D. (2)

Definition : A Dirac structure D on the bond space B = F×Eis a subspace of B which is maximally isotropic with respect tothe canonical symmetrical pairing, i.e.,

D⊥ = D. (3)

(fe

)∈ D ⇐⇒ Power conservation



PHS Definition based on Dirac structure and Hamiltonianfunction (total energy of the system).

Definition : Let B = E ×F be the bound space defined above andconsider the Dirac structure D and the Hamilonian function H(x)with x the energy variables. Define the flow variables, f ∈ F as thetime variation of the energy variables and the effort variables e ∈ Eas the variational derivative of H(x). The system

(f , e) =

(∂x

∂t,δHδx

)∈ D

is a Port Hamiltonian system with total energy H(x)



Parametrization :

J e =N∑

i=0

P(i)d ie

dz i(z) z ∈ [a, b] ,

where e ∈ HN((a, b); Rn) and P(i), i = 0, . . . ,N, is a n × n realmatrix with PN non singular and Pi = PT

i (−1)i+1. Let define

Q =

0BBB@P1 P2 · · · PN

−P2 −P3 · · · 0... · · ·

. . ....

(−1)N−1PN 0 · · · 0

1CCCABack to the Vibrating string

∂

∂t

„εp

«| z =

„0 −1−1 0

«| z

∂∂z

T (z) 0

0 1µ(z)

!„εp

«| z

f P1 e

,Q = P1



We define the symmetrical pairing (not depending on J ) of andthe port variables associated with J .Let F = E = L2((a, b); Rn)× RnN and define B = F × E with thefollowing canonical symmetrical pairing :⟨(

f 1, f 1∂ , e

1, e1∂

) (f 2, f 2

∂ , e2, e2

∂

)⟩+

= 〈e1, f 2〉L2 + 〈e2, f 1〉L2 − 〈e1∂ , f

2∂ 〉 − 〈e2

∂ , f1∂ 〉,

Definition : The port variables (e∂ , f∂) ∈ RnN associated with Jare defined by :

„f∂e∂

«= Rext

0BBBBBBBBBB@

e(b)...

dN−1edzN−1 (b)

e(a)...

dN−1edzN−1 (a)

1CCCCCCCCCCA, Rext =

1√

2

`Q −QI I

´



The subspace DJ of B defined as

DJ =

8>><>>:0BB@

ff∂ee∂

1CCA ˛ e ∈ HN((a, b); Rn),J e = f ,

„f∂e∂

«= Rext

0BB@e(b)

...

∂N−1z e(a)

1CCA9>>=>>;

is a Dirac structure, that means that D = D⊥.

Back to the Vibrating string

∂

∂t

(εp

)︸︷︷︸ =

(0 −1−1 0

)︸︷︷︸ ∂

∂z

(T (z)ε

1µ(z)p

)︸︷︷︸

f P1 e

,Q = P1

„f∂e∂

«=

1√

2

„P1 −P1

I I

«„e(b)e(a)

«=

1√

2

0BBB@T (a)ε(a)− T (b)ε(b)

p(a)µ(a)− p(b)µ(b)

T (a)ε(a) + T (b)ε(b)p(a)µ(a)

+ p(b)µ(b)

1CCCA


Boundary control systems




4 Conclusion



Boundary Control Systems :Theorem : Let W be a nN × 2nN full rank matrix. The system

x(t) = JLx(t)

u(t) = Bx(t) = W

(f∂(t)e∂(t)

)is a boundary control system, where AW = (JL)kerB is the

generator of a contraction semigroup on L2 ((a, b) ,Rn) if and onlyif

W ΣW T ≥ 0 where Σ =

(0 II 0

)



Let define the linear mapping C : HN((a, b),Rn)→ RnN as

Cx(t) := W

(f∂(t)e∂(t)

)and the output as y(t) = Cx(t), then for u ∈ C 2((0,∞); RnN) andx(0)− Bu(0) ∈ D(JW ) the following balance equation is satisfied :

1

2

d

dt‖x(t)‖2 =

(uT (t) yT (t)

)PW

(u(t)y(t)

).

where PW ,fW =

(W ΣW T W ΣW

W T ΣW T W ΣW T

)−1



Using W = S (I + V , I − V ) :

V = 0

x(t) = J x(t),u(t) = 1

2 (f∂(t) + e∂(t))y(t) = 1

2 (f∂(t)− e∂(t))=⇒

boundary control system, withthe associated semigroupa contraction12

ddt ‖x(t)‖2 = ‖u(t)‖2 − ‖y(t)‖2.

V = I

x(t) = J x(t)u(t) = f∂(t)y(t) = −e∂(t)

=⇒boundary control system, withthe associated semigroupunitary12

ddt ‖x(t)‖2 = u(t)T y(t)

Vibrating string :

V = 0 ⇒ u =1√2

(T (a)ε(a)

p(a)µ(a)

)and y =

1√2

(−T (b)ε(b)

− p(b)µ(b)

)


Stability, control feedback




4 Conclusion



Control using feedback at the boundary!

"#$%!&''#!()*+$,!

!!

!" #" $"

%""""""""&"

Open loop system Closed loop systemx = JLxu = Wimp

(f∂e∂

)y = Cimp

(f∂e∂

)

x = JLxr = (Wimp + αCimp)

(f∂e∂

)y = Cimp

(f∂e∂

)



The closed loop system described byx = JLxr = (Wimp + αCimp)

(f∂e∂

)y = Cimp

(f∂e∂

)is a boundary control system. Furthermore, the operatorAs = JL|D(As) generates a contraction semigroup onX = L2 ((a, b); Rn) where

D(As) =

x ∈ D(J )|

(f∂e∂

)∈ kerW

and W = (Wimp + αCimp) is a full rank nN × 2nN matrix



Theorem : (Assumption (λ− As)−1 : X → X is a compactoperator for λ > 0). Then the system described by (VV T = 0) :

x = J x

r = (Wimp + αCimp)

(f∂e∂

)y = Cimp

(f∂e∂

)with r = 0 and α > 0 is globally asymptotically stable. For anyx(0) ∈ X the unique (classical or weak) solution x(t) = T (t)x(0)of the closed loop system asymptotically approaches to zero, i.e.

lim∞‖x(t)‖X = 0






4 Conclusion



Let us restrict the previous class to :

∂x

∂t(t, z) = P1

∂z

∂t(Lx)(t, z) + (P0 − G0)Lx(t, z)

Let consider a BCS as defined previously, assuming thatu(t) = 0, ∀t ≥ 0. Then the system is exponentially stable if

either ‖ (Lx(b)) ‖2R ≤ k1 (〈αy , y〉R + 〈G0Lx(t),Lx(t)〉R)

or ‖ (Lx(a)) ‖2R ≤ k1 (〈αy , y〉R + 〈G0Lx(t),Lx(t)〉R)



As state variables we choose

x1 = ∂w∂z − φ : shear displacement,

x2 = ρ∂w∂t : transverse momentum distribution,

x3 = ∂φ∂z : angular displacement,

x4 = Iρ∂φ∂t : angular momentum distribution.

Then the model of the beam can be rewritten as

∂

∂t

0BB@x1

x2

x3

x4

1CCA =

26640 1 0 01 0 0 00 0 0 10 0 1 0

3775| z

P1

∂

∂z

0BBB@K x11ρ

x2

EI x31Iρ

x4

1CCCA+

26640 0 0 −10 0 0 00 0 0 01 0 0 0

3775| z

P0

0BBB@K x11ρ

x2

EI x31Iρ

x4

1CCCA| z

x

.



One can define the boundary port variables :

„f∂e∂

«=

1√

2

»P1 −P1

I I

–„(Lx)(b)(Lx)(a)

«=

1√

2

0BBBBBBBBBB@

(ρ−1x2)(b)− (ρ−1x2)(a)(Kx1)(b)− (Kx1)(a)

(I−1ρ x4)(b)− (I−1

ρ x4)(a)(EIx3)(b)− (EIx3)(a)(Kx1)(b) + (Kx1)(a)

(ρ−1x2)(b) + (ρ−1x2)(a)(EIx3)(b) + (EIx3)(a)

(I−1ρ x4)(b) + (I−1

ρ x4)(a)

1CCCCCCCCCCA.

(4)

Let us consider stabilization by applying velocity feedback i.e.following BC :

1ρ(a) x2(a) = 0, 1

Iρ(a)x4(a) = 0,

K (b)x1(b, t) = −α11

ρ(b)x2(b, t), EI (b)x3(b, t) = −α21

Iρ(b) x4(b)



Input mapping :

W =1√

2

2664−1 0 0 0 0 1 0 0

0 0 −1 0 0 0 0 1α1 1 0 0 1 α1 0 0

0 0 α2 1 0 0 1 α2

3775 then W ΣW T = 2

26640 0 0 00 0 0 00 0 α1 00 0 0 α2

3775As output we can choose

y =

0BBB@−K(a)x1(a)−(EI )(a)x3(a)

1ρ(b)

x2(b)1

Iρ(b)x4(b)

1CCCA , with fW =1√

2

26640 1 0 0 −1 0 0 00 0 0 1 0 0 −1 01 0 0 0 0 1 0 00 0 1 0 0 0 0 1

3775 .

Then

P−1W ,W =

[2α I

I 0

],PW ,W =

[0 II −2α

]



Energy balance :

d

dtE (t) =

d

dt‖x(t)‖2

L = 〈u(t), y(t)〉U − 〈αy(t), y(t)〉R

where

〈αy(t), y(t)〉R = α1|(ρ−1x2)(b, t)|2 + α2|(I−1x4)(b, t)|2

Then

‖ (Lx(b)) ‖2R = |(kx1)(b)|2 + |(ρ−1x2)(b)|2 + |(EIx3)(b)|2 + |(I−1

ρ x4)(b)|2= (α2

1 + 1)|(ρ−1x2)(b, t)|2 + (α22 + 1)|(I−1

ρ x4)(b)|2≤ κ〈αy(t), y(t)〉R

⇒ Stability


Mixed finite element method




4 Conclusion



Back to the example of the adsorption column :

Extragranular phase

Adsorption column

Pellet

Macropore

Phase

Flowing fluid

Classical formumlation.

∂q(z,t)∂t

+ ∂N∂z

+ αFads = 0∂qp(z,t)

∂t− Fads = 0

(5)

q mole density in the fluide phase, qp in the adsorbed phase, total fluxN = Nconv + Ndisp = vq − D ∂µ

∂z, Fads = k1a(µ− µp)

with Dankwert boundary conditions N|0 = vqin et ∂q∂z|L = 0





∂q(z,t)∂t

+ ∂N∂z


∂t− Fads = 0

(5)






∂q(z,t)∂t

+ ∂N∂z


∂t− Fads = 0

(5)


Port Hamiltonian formulation2664fqFdisp

αfqp

Fads

3775 =

26640 d 0 −1d 0 0 00 0 0 11 0 −1 0

37752664µNµp

Nads

3775 (6)

with constitutive relations

N = −D ∗ Fdisp + v ∗ q = −D ∗ Fdisp + vbµ

Nads = α a k1 Fads

µ = bq(7)





∂q(z,t)∂t

+ ∂N∂z


∂t− Fads = 0

(5)


Dirac structure : interconnection structure2664fqFdisp

αfqp

Fads

3775 =

26640 d 0 −1d 0 0 00 0 0 11 0 −1 0

37752664µNµp

Nads

3775 (6)

with the boundary port variables

„f∂e∂

«=

0BB@N(0)N(L)µ(0)−µ(L)

1CCA



DA R10

adsF

dN

t

q

pellet

accumulationinterconnection

dissipation

d

N

0

L

0N

LNConstitutive

thermodynamical law

Constitutive dispersion

law

)(qf )(dfNdispN

0

d

dconvN



DA R10

adsF

dN

t

q

pellet

accumulationinterconnection

dissipation

d

N

0

L

0N

LNConstitutive

thermodynamical law

Constitutive dispersion

law

)(qf )(dfNdispN

0

d

dconvN

Goal of the discretization scheme : preserve the structure



Remark 3.1 The Hodge star operator ! is a linear operator mapping p forms on V to (n" p)forms i.e. :

! : !p(V ) # !n!p(V )

In cartesian coordinates, consider the functions g(z) and the 1-form f(z) = g(z)dz then !f(z) =g(z). qL

s and qL being 1-forms, !qLs and !qL are 0-forms.

(3) Closure equation associated with the di!usion - We use Knudsen law [19] to presentthe di"usion in the adsorbed phase (microporous scale). That is to say we only consider thefriction exerted by the solid on the adsorbed species. So the constitutive relation representingthe di"usion is:

fmic2 = "Dmic ! qL

RT! emic

2 (15)

where Dmic is the di"usion constant.

4 Model reduction based on geometrical properties

4.1 General concept

As previously mentioned, the port based modelling of the adsorption column is based on basicelements having well defined energetic behavior as already depicted in Figure 3. The dis-tributed aspect of this port based model is essentially supported by the Dirac structure whichlinks power exchanges within the spatial domain and through the boundaries. The proposeddiscretization method consists in splitting the initial structured infinite dimensional model intoN finite dimensional sub-models (finite elements) with the same energetic behavior (cf Figure 4).Furthermore, the support functions used for the interpolation of both e"ort and flow variablesare di"erent to have enough degrees of freedom to guarantee the conservation of the structuralproperties.

e1

ab

f1

ab

e2

ab

f2

ab-f

1

ab

e1

ab -e2

ab

f2

ab

e1

a

e1

b

f2

a

f2

b

DabC Rab

a b

0

Rmic

a b

Power flow exchanged with

the following volume

Power flow exchanged with the previous volume

0 1

Figure 4: Principle of the spatial discretization

The interconnection between these sub-models is done using the power conjugate boundaryport variables. They correspond to the energy flowing across the boundary of one submodel tothe boundary of the next sub-model. To each submodel is associated the same generic structure(and consequently parts of the global mass and energy balances) as the global structure, thedi"erence lying in the fact that submodels are finite dimensional i.e. the Dirac structure Dab

on Figure 4 is finite dimensional and the reduced e"ort and flow variables (eab, fab) are no morespatially distributed.

For simplicity reasons superscript mic will be omitted in the remaining of the section. Letus now explain the reduction scheme.

7



The variables dN = Nd and dµ = µd are one-forms and are approximated onΩab by :

Nd(t, z) = Nabd (t) wNd

ab (z)

e2(t, z) = µabd (t) wµd

ab (z)(7)

where the support one-forms wNdab and wµd

ab satisfy :R

ΩabwNd

ab =R

Ωabwµd

ab = 1.The variables µ and N are zero-forms and are approximated on Ωab by :

µ(t, z) = µ(a)wµa (z) + µ(b)wµ

b (z),

N(t, z) = N(a)wNa (z) + N(b)wN

b (z)(8)

where the support zero-forms wµa ,wµ

b ,wNa et wN

b satisfy :

wµa (a) = 1, wµ

a (b) = 0, wµb (a) = 0, wµ

b (b) = 1wN

a (a) = 1, wNa (b) = 0, wN

b (a) = 0, wNb (b) = 1,



The approximated variables must verify the relation induced by theinterconnection structure :

Nd = dNµd = dµ

=⇒

8>><>>:Nab

d = N(b)− N(a) = Nb∂(t) − Na

∂(t)

µabd = µ(b)− µ(a) = − µb

∂(t) − µa∂(t)

Write the net power using the previous interpolation formulae :

Pnetab = µab Nab

d + µabd Nab +

hµb∂N

b∂ − µa

∂Na∂

iand identify with the real one, we obtain for µab and Nab :

µab = αabµa∂(t)− (1− αab)µb

∂(t)

Nab = (1− αab)Na∂(t) + αabN

b∂(t)

where the parameter αab =R

Ωabwµ

a wNdab ∈ [0, 1].

Relations between fab =ˆNab

d Nab Na∂ Nb

∂

˜T, eab = [µab µab

d µa∂ µ

b∂ ]T define

again a finite dimensional Dirac structure.



Objective : compute Nabdisp from µab

d such that the power behavior of the

dissipation element(dispersion law) is preserved : Nabdispµ

abd =

R b

aµd ∧ Ndisp as

well as the constitutive law of the dissipative element. Let us write theapproximate instantaneous power

PabR = −KabD(µab

d (t))2 (9)

with Kab depending on forms w only.

Nabdisp =

∂PabR

∂µabd

= −2KabDµabd (t) (10)

Objective : compute µab with respect to qab.

q(t, z), q and Nd lies in the same space =⇒ q(t, z) = qab(t)wNdab (z)

The energy of the element and its approximation on [a, b] are given by :

Gab =R t

0

`Rab

q(t, z)µ(t, z)´dt, g ab =

Z t

0

qab[bqab(t)| z µab

]wNdab (z)dt =

Z t

0

qabµabdt

dG ab

dt= qabµ

ab ⇒ dG ab

dqab= µab (11)

So the thermodynamic relations linking the gibbs density, concentration andchemical potentials are preserved by discretization.



Global schemePrincipe de la discrétisation structurée (Figure reprise deBaaiu2007)

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Simulations

With 10 meshes

0 20 40 60 80 1000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Time: sec

Con

cent

ratio

n of

com

pone

nt Q

: mol

/Vol

ume

Finite difference with N=10

0 20 40 60 80 1000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Time: sec

Con

cent

ratio

n of

com

pone

nt Q

: mol

/Vol

ume

Structural method N=10


Spectral properties

Analysis on simple cases :

Hyperbolic system (like undamped wave equation) :(f1f2

)=

(0 − ∂

∂z

− ∂∂z 0

)(e1

e2

)Parabolic system (like diffusion) :(

f1e2

)=

(0 − ∂

∂z

− ∂∂z 0

)(e1

f2

)

Question

What about spectral properties with respect to closure equations

Case 1 : f = ∂x∂t , e = δxH,H =

∫ ba

(αx2

1 + βx22

)dz

Case2 : f1 = ∂x1∂t , e1 = bx1, f2 = De2


Spectral properties

Case 1 : The associated PDE (without closure equation) is givenby : (

x1

x1

)=

(0 − ∂

∂z

− ∂∂z 0

)(x1

x2

)+ BC

The spectrum associated to this operator is defined by :(ϕ1

1 ϕ12

ϕ21 ϕ2

2

)(λ1 00 λ2

)=

(0 − ∂

∂z

− ∂∂z 0

)(ϕ1

1 ϕ12

ϕ21 ϕ2

2

)+BC

Now let us consider the closure equation

(e1

e2

)=

(δx1Hδx1H

)with : H(x1, x2) =

∫ ba

(αx2

1 + βx22

)dz , then(

ψ11 ψ1

2

ψ21 ψ2

2

)(µ1 00 µ2

)=

(0 − ∂

∂z

− ∂∂z 0

)(2β 00 2α

)(ψ1

1 ψ12

ψ21 ψ3

2

)


Spectral properties

Finally we obtain by identification :

=

0@ ϕ11 ϕ1

2

ϕ21 ϕ2

2

1A︷︸︸︷( √2β 0

0√

2α

)(ψ1

1 ψ12

ψ21 ψ2

2

)=

0@ λ1 00 λ2

1A︷︸︸︷(µ1

2√αβ

0

0 µ2

2√αβ

)

=

(0 − ∂

∂z

− ∂∂z 0

)=

0@ ϕ11 ϕ1

2

ϕ21 ϕ2

2

1A︷︸︸︷( √2β 0

0√

2α

)(ψ1

1 ψ12

ψ21 ψ2

2

)

Geometrical transformation :

Homothecy of the spectrum with factor 2√αβ


Spectral properties

0.2 0.1 0 0.1 0.21

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Real part

Imaginaty part

Spectrum of the canonical system

0.2 0.1 0 0.1 0.21

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Real part

Imaginary part

Specrum of the hyperbolic system

Homothety


Spectral properties

Case 2 : In a same way(f1e2

)=

(0 − ∂

∂z

− ∂∂z 0

)(e1

f2

)(12)

with the constraintf2 = De2 (13)

With the hypothesis of separation of variables (12) can berewritten with x = ψ0 expµt, e1 = bx :(

µ 00 b

)(ψ0∂ψ0∂z

)=

(0 − ∂

∂z

− ∂∂z 0

)(b 00 Db

)(ψ0∂ψ0∂z

)+ BC

(14)The eigenvalues and eigenfunctions satisfy :(

µ 00 1

)(ψ0

e2

)=

(0 ∂

∂z∂∂z 0

)(b 00 D

)(ψ0

e2

)


Spectral properties

which leads finally to( √b 0

0√

Dµ

)(ψ0ee2

µ

)µ1/2√

bD=

(0 − ∂

∂z

− ∂∂z 0

)( √b 0

0√

Dµ

)(ψ0ee2

µ

)

which has to be identified to :(ϕ1

ϕ2

)λ =

(0 − ∂

∂z

− ∂∂z 0

)(ϕ1

ϕ2

)

Geometrical transformation :

Dilation and rotation of the spectrum

λk = |λk |e±i π2 −→ µk = Db|λk |2e±iπ


Spectral properties

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Real part

Imag

inar

y pa

rt

Canonical system associated toStokes−Dirac structure Parabolic system

Dilatation and rotation


Spectral propertiesPrincipe de la discrétisation structurée (Figure reprise deBaaiu2007)

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(%345

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$*3%5

(*3%5$ % ( %

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9:

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PENG HUI (LAGEP-UCBL) Fevrier 20 / 48


Spectral properties

Let first consider one element of discretization.

Input : input mass flow

Output : output effort

This leads to :(f ab1

eab2

)=

(0 1

α− 1α 0

)(eab

1

f ab2

)+

(0 − 1

α− 1α 0

)(e∂b

f∂a

)(

f∂b

e∂a

)=

(0 1

α1α 0

)(eab

2

f ab2

)+

(0 α−1

α1−αα 0

)(e∂b

f∂a

)This element is interconnected with the next one using continuityof flow and equality of efforts : e∂

0b = e∂

1a and f∂

0b = −f∂

1a


Spectral properties

In the case α = 1f 11

e12...

f N1

eN2

=

(0 M−MT 0

)

e11

f 12...

eN1

f N2

+ g

(e∂

1N

f∂00

)

(f∂

1N

e∂00

)= gT

e1

1

f 12...

eN1

f N2

with M =

1 −1 0 0 · · ·0 1 −1 0 · · ·0

. . .. . .

. . .. . .

0 · · · 0 1 −1


Spectral properties

The spectrum associated with the Dirac structure :[0 dd 0

](eq

ep

)= λ

(eq

ep

)is computed using :

deq

dz2 = λ2eq, eq = Aeλz + Be−λz and

ep = Aeλz − Be−λz . With homogeneous boundary conditionsep(0) = eq(L) = 0 :[

1 −1eλL e−λL

](AB

)= 0

And

λk =2k + 1

2Lπi


Spectral properties

The interconnection of n elements gives rise to the following statematrix :

A =

[0 M−MT 0

](15)

with

M =

1 −1 0 0 · · ·0 1 −1 0 · · ·0

. . .. . .

. . .. . .

0 · · · 0 1 −1

(16)

The eigenvalues of the finite dimensional system are given by :

λk =n

L

√2 cos

(2k + 1

2n + 1π

)− 2

if k n then :

λk '2k + 1

2Lπi

= first k eigenvalues of the infinite dimensional system





4 Conclusion


Conclusion :

For 1D case :

A powerful tool to highlight physical properties

Parametrization of all admissible BC for a large class ofsystems

Simple tools (matrix conditions) to check stability

Model reduction

Preserve energetic properties and the geometric structure

Link the solutions of very different systems (parabolic andhyperbolic)

Numerically efficient


Future work :

Many extensions :

Generalization to 2D-3D linear systems.

Stabilization of a class of non linear systems.

Extensions to irreversible thermodynamic systems.

Control using infinite dimensional Port Hamiltonian Systems(Imersion/reduction + Casimir functions).

Model reduction of complex interconnected systems.


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