port controlled hamiltonian systems and related nonlinear...

Introduction PCH systems Implicit PCH systems Control Conclusion

Port Controlled Hamiltonian systems andrelated nonlinear control approaches

Laurent Lefèvre

Laboratoire de Conception et d’Intégration des Systèmes (LCIS)Grenoble Institute of Technology (Grenoble INP - Esisar)

Ecole d’automatique francophone - Mai 2012 - Université "Politehnica" Bucarest

1 / 58


Acknowledgments and contributions ...

Some of the material presented here is the result of discussions andcollaborations with distinguished colleagues, including:

E. Mendes (LCIS, Grenoble INP)

B. Maschke, B. Hamroun and F. Couenne (LAGEP, UCB Lyon 1)

R. Nouaillétas and S. Brémond (IRFM, CEA Cadarache)

Y. Le Gorrec (FEMTO, ENS2M)

Many contributions came from MSc and PhD students, including:

A. Baaiu, R. Moulla, H. Hoang, H. Peng (LAGEP, UCB Lyon 1)

A. Dimofte, B. Hamroun, T. Vu, A. Nguyen (LCIS, Grenoble INP)

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Sketch of the talk

1 Introduction to PHS

2 Port-Controlled Hamiltonian (PCH) systems

3 Implicit Port Hamiltonian systems

4 Physical approaches to the nonlinear control problem

5 Conclusions, prospects and opportunities

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An introduction to distributed parameterssystems

4 / 58


Complex water systems

Figure: Schematic view of the Bourne irrigation system near Valence(Vercors mountains)

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Tokamak fusion reactors

Figure: Schematic view of the coming ITER or existing Tore Supratokamak reactors at CEA - Cadarache (Saint-Paul lez Durance)

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High performance heating/refrigeration systems

Figure:

7 / 58


Other examples of distributed parameters systems

academic examples: wave, heat, beam, membrane and telegrapherequations

piezzoelectric actuators

fluid flow and fluid/structure control systems

magneto-hydrodynamic flows, plasma control

classical Maxwell field equations

quantum mechanics (Schrödinger and Klein-Gordon equations)

free surface problems (Burger, Korteweg de Vries, Boussinesq,Saint-Venant/SWE)

thermodynamics (chemical reactions, transport phenomena, phasesequilibrium, etc.)

biology (preys-predators, population dynamics, bio-reactors)

⇒ most not simplified real world applications lead to DPS models!

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Some characteristics of distributed parameters systems

variables are non uniform in space, state equations are PartialDifferential Equations (PDEs)

Boundary Control Systems or distributed control

extensions of classical control results exist for linear DPS usingsemigroup theory: transfer functions, I/O operators, controllability orobservability Grammians, state space realization, LQG or H∞ control, ...

results exist for the regional analysis and control of linear (or bilinear)DPS (regional controllability or observability, spreadability, viability, etc.)

practical solutions for control laws or observers design remain hard toachieve (solution of operators equations, infinite dimensional control)

they are no general results for nonlinear DPS

⇒ Particular cases: systems of conservation laws with I/O or port variables

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Different approaches for the control of DPS

finite dimensional approximation (total or partial discretization)[Quarteroni et Valli 1994] Numerical Approximation of Partial Differential Equations, Springer-Verlag

[Zuazua 2002] Controllability of partial differential equations and its semi- discrete approximations, Disc

Cont Dyn Syst 8 (2) 469-513

discrete modelling approaches (cellular automata, Lattice Boltzmannmodels)[Chopard et Droz 1998] Cellular Automata Modeling of Physical Systems, Cambridge University Press

semigroup of linear operators (extensions to the semilinear case)[Curtain et Zwart 1995] An introduction to infinite-dimensional linear systems theory, Springer-Verlag

PDE / functional analysis approach[Lions 1988] Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev.30 pp.

1-68

regional analysis (mainly using PDE/semigroup approaches)[El Jai et al. 2009] Systèmes dynamiques : Analyse régionale des systèmes linéaires distribués, PUP

port-Hamiltonian approach[Duindam et al. 2009] Modeling and control of complex physical systems: the Port-Hamiltonian approach,

Springer-Verlag 10 / 58


The port-Hamiltonian approach: motivation

Use physical insight explicitly

modular modelling

decomposition of simulation codes: reusability, parallel computation

specific integration schemes conserving physical invariants

physically-based control design (Lyapunov functions, passivity basedapproach, energy shaping)

simultaneous design of process and control

Inspired from thermodynamics and mechanics

Irreversible thermodynamics: systems of balance equations andphenomenological laws

Analytical mechanics: Hamiltonian variational formulations in (q, p)coordinates

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Finite dimensional Port-Controlled Hamiltonian(PCH) systems

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Hamiltonian mechanics with port variables

Hamilton equations(∂q∂t∂p∂t

)=

(∂Hδp (q, p)

∂Hδq (q, p) + τ

)with q the generalized coordinates, p the corresponding momenta, τ thegeneralized forces, M(q) the inertia matrix, P(q) the potential energy andH(q, p) the total energy (Hamiltonian)

Balance equation and passivity

dHdt

=∂T H∂q

q +∂T H∂p

p =∂T H∂p

τ = qT τ = yT (t)u(t)

with y := q and u := τ . Therefore if P(q) is bounded from below, the systemis passive w.r.t. the pair of power-conjugated variables (u, y) and the storagefunction H(q, p)

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Conservative Port-Controlled Hamiltonian (PCH) systems

(∂q∂t∂p∂t

)=

(∂Hδp (q, p)

∂Hδq (q, p) + τ

)

Generalization in local state-space coordinates

x = J(x)∂H∂x

+ g(x)u

y = gT (x)∂H∂x

where u, y ∈ Rm are power conjugated input-output variables andx = (x1, . . . , xn) are local coordinates for the n-dimensional state spacemanifold X

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Poisson structure

The interconnection structure J(x) usually satisfies

1 skew-symmetry J(x) = −JT (x) which implies conservativeness andpassivity w.r.t. (u, y) and the storage function H:

dHdt

(x(t)) =∂T H∂x

x =∂T H∂x

(J(x)

∂H∂x

+ g(x)u)

= yT (t)u(t)

2 Jacobi identitiesn∑

l=1

[Jlj∂Jik

∂xl+ Jli

∂Jkj

∂xl+ Jlk

∂Jji

∂xl

]= 0 ∀i, j, k

which guarantee integrability and existence of canonical localcoordinates x = (q, p, s) s.t. J(x) reduces to (symplectic structure)

J =

0 I 0−I 0 00 0 0

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PCH example: LCTG circuits

Figure: open (controlled) LC circuit

Qϕ1

ϕ2

=

0 1 −1−1 0 01 0 0

︸︷︷︸

J skew-symmetric and constant

∂H∂Q∂H∂ϕ1∂H∂ϕ2

+

010

u(t)

where

H (ϕ1, ϕ2,Q) =ϕ2

1

2L1+

ϕ22

2L2+

Q2

2C; u(t) := V (t) ; y(t) :=

∂H∂ϕ1

= Iϕ1

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PCH systems with dissipation

Terminating some ports with resistive elements

g(x)u =[

g(x) gR(x)] [ u

uR

][

yyR

]=

[gT (x)

gTR (x)

]∂H∂x

(x)

uR = −S(x)yR ; S ≥ 0

leads to the input-state-output port-Hamiltonian formulation:

x = (J(x)− R(x))∂H∂x

+ g(x)u

y = gT (x)∂H∂x

where R(x) := gR(x)S(x)gTR (x) ≥ 0 and

dHdt

(x(t)) = yT (t)u(t)− ∂T H∂x

R(x)∂H∂x≤ yT (t)u(t)

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PCH example with dissipation: capacitor microphone

Figure: The capacitance C(q) is varying with the displacement q ofthe right plate with mass m. This plate is attached to a linear springwith constant k and a linear damper with constant c. It is actuated bya mechanical force F (air pressure arising from sound). E isconsidered as a voltage source

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Capacitor microphone example ... continued

The dynamical equations of motion can be written as the port-Hamiltoniansystem with dissipation

qpQ

=

0 1 0−1 0 00 0 0

︸︷︷︸

J

−

0 0 00 c 00 0 1/R

︸︷︷︸

R

∂H

∂q∂H∂p∂H∂Q

+

0 01 00 1/R

︸︷︷︸

g

[FE

]

where

H (ϕ1, ϕ2,Q) =ϕ2

1

2L1+

ϕ22

2L2+

Q2

2C; u(t) := V (t) ; y(t) :=

∂H∂ϕ1

= Iϕ1

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Implicit Port Hamiltonian systems

[Courant 1990] Dirac manifolds, Trans. American Math. Soc. 319:631–661

[Dorfman 1993] Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley

[Dalsmo & van der Schaft 1999] On representations and integrability of mathematical structures inenergy-conserving physical systems, SIAM J. of Contr. and Opt., 37(1):54–91, 1999

[Cervera & al. 2007] Interconnection of port-Hamiltonian systems and composition of Dirac structures,Automatica, 43:212-225

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The bond space of power conjugated effort and flow variables

Let F and E be two real vector spaces and assume that they are endowedwith a non degenerated bilinear form, called pairing denoted by:

〈.| .〉 : F × E → R(f , e) 7→ 〈e| f 〉 (1)

On the product space, called bond space:

B = F × E

the bilinear product leads to the definition of a symmetric bilinear form, calledplus pairing as follows:

� ·, · �: B × B → R((f1, e1) , (f ,2 e2)) 7→ � (f1, e1), (f2, e2)� := 〈e1| f2〉+ 〈e2| f1〉

(2)

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Dirac structure on vector spaces

Definition

A Dirac structure is a linear subspace D ⊂ B such that D = D⊥, with ⊥denoting the orthogonal complement with respect to the bilinear form�,�.

If a linear subspace D ⊂ B satisfies only the isotropy condition D ⊂ D⊥, whatmeans that it is not maximal, one say that it is a Tellegen structure.

This name arises from the fact that the condition D ⊂ D⊥ is equivalent to:〈e| f 〉 = 0, ∀ (f , e) ∈ D .This condition is known as Tellegen’s theorem for the admissible voltagese ∈ Rn and currents f ∈ Rn of an electrical network , endowed with theEuclidean product.

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Representations of finite-dimensional Dirac structures

Dirac structures admit algebraic definitions, based on some skew-symmetriclinear maps called representation of a Dirac structure.

[T.J. Courant. Dirac manifolds. Trans. American Math. Soc. 319, pages 631–661, 1990]

[M. Dalsmo and A.J. van der Schaft. On representations and integrability of mathematical structures in

energy-conserving physical systems. SIAM Journal of Control and Optimization, 37(1):54–91, 1999.]

Assume that the flow and effort spaces are finite-dimensional and choose

F = E = Rn

with the power pairing being the canonical Euclidean product in Rn

composed with a signature matrix σ:

〈e| f 〉 = eTσ f where f ∈ F = Rn, e ∈ E = Rn (3)

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Representations of a Dirac structure ... continued

Theorem

A Dirac structure D ⊂ B = Rn × Rn admits two n × n real matrices, denotedhere E and F, and satisfying

EσF T + FσET = 0

rank [E : F ] = n

D admits the image representation:

D = {(f , e) ∈ F × E|f = ETλ, e = F Tλ, λ ∈ Rn} (4)

and D also admits the kernel representation:

D = {(f , e) ∈ F × E| (F σ) f + (E σ) e = 0} (5)

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Representations of a Dirac structure ... continued

Theorem

A Dirac structure D ⊂ B = Rn × Rn admits a decomposition of the flow andeffort spaces:

F = F1 ⊕F2 3(

f10

)+

(0f2

)and E = E1 ⊕ E2 3

(e1

0

)+

(0e2

)and a skew-symmetric n × n real matrix denoted J which define theinput-output representation:

D = {((

f1f2

)+

(e1

e2

)) ∈ F × E|

(f1e2

)= J

(e1

f2

)} (6)

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Hamiltonian system defined w.r.t. a Dirac structure

We consider the case where the state space is a real vector space F ofdimension n. Then the flow space is again F and the effort space will bechosen as its dual space F∗. Then the bond space is simply F × F∗.The power pairing is defined using the duality product:

P(t) = 〈f |e 〉 := 〈e, f 〉 ; ∀(f , e) ∈ F × F∗

Definition

Consider a Dirac structure D ⊂ F × F∗ . A Hamiltonian system with respectto the Dirac structure D generated by the Hamiltonian functionH ∈ C∞ (F , R), is defined by the implicit differential equation:

( dxdt ,

∂H∂x

)∈ D.

The isotropy of the Dirac structure implies the conservation of theHamiltonian:

dHdt

=

⟨dxdt| ∂H∂x

⟩= 0 (7)

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Example: closed LCTG circuits

Figure: A simple LC circuit composed of two capacitors and an inductor

Kirchhoff’s laws define the kernel representation of a Dirac structure: 1 0 10 1 00 0 0

︸︷︷︸

F

iC1

vL

iC2

+

0 −1 01 0 01 0 −1

︸︷︷︸

E

vC1

iLvC2

= 0 (8)

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Example: LC circuit ... continued

The chosen state variables are:

x =

q1

φq2

charge of capacitor 1flux in inductorcharge of capacitor 2

thendxdt

=

iC1

vL

vC2

∈ FThe conjugated variables are the derivatives of the total electro-magneticenergy : H (x) w.r.t. the state variables, i.e.:

∂H∂x

(x) =

vC1

iL,iC2

∈ R3 = E

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Example: LC circuit ... continued

The dynamics of the LC circuit is an implicit Hamiltonian system.

(dxdt,∂H∂x

)∈ D ⇔

1 0 10 1 00 0 0

︸︷︷︸

F

ddt

q1

φq2

+

0 −1 01 0 01 0 −1

︸︷︷︸

E

∂H∂x

(x) = 0

(9)The rank degeneracy of F corresponds to the constraint:

∂H∂q1

(x)− ∂H∂q2

(x) = 0

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Extension to Port Hamiltonian Systems

Tangent and co-tangent spaces of the state space are augmented with a setof port variables used to represent the interaction of the system with itsenvironment

We augment the bond space according to

B :=

{((f i

f e

),

(ei

ee

))∈ F × E =

(F i ×Fe

)×(F i∗ × Ee

)}with the help of two other finite-dimensional port spaces Fe and Ee endowedwith the non degenerated bilinear form 〈.| .〉e.The augmented bond space B is then endowed with the bilinear form:⟨(

f i

f e

)|(

ei

ee

)⟩=⟨

ei | f i⟩

i+⟨ee| f e⟩

e

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Port Hamiltonian Systems

Definition

A port Hamiltonian system w.r.t. the Dirac structure D ⊂ B and generatedby the Hamiltonian function, H ∈ C∞

(F i , R

)is defined by the implicit

differential equation: (( dxdtf e

),

(∂H∂xee

))∈ D

Port Hamiltonian systems do not satisfy the Cauchy conditions as longas the system is not completed with some relations on the external portvariables (f e, ee) .Complex systems composed of a set of interacting subsystems may bedescribed using the Composition of Dirac structures rather thaninteraction Hamiltonian functionsThe isotropy property of the Dirac structure translates now into abalance equation for the Hamiltonian:

0 =

⟨∂H∂x| dx

dt

⟩+⟨ee| f e⟩

e =dHdt

+⟨ee| f e⟩

e (10)

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Example: open LCTG circuit

Figure: An open LC circuit with two external ports

Kirchhoff’s laws define the kernel representation of an extended Diracstructure

1 0 1 00 0 0 10 −1 0 00 0 0 0

︸︷︷︸

F

iCvL

i1i2

+

0 −1 0 00 1 0 0−1 0 0 1

1 0 −1 0

︸︷︷︸

E

vC

iLv1

v2

= 0

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Open LC circuit ... continued


The chosen state variables are:

x =

(qφ

)charge of capacitorflux in inductor

⇒ dxdt

=

(iC1

vL

)∈ F i

The conjugated variables are the derivatives of the total electro-magneticenergy H (x) w.r.t. the state variables, i.e.:

∂H∂x

(x) =

(vC

iL,

)∈ R2 = F i∗

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Open LC circuit ... continued


The two open ports define the vector spaces of external currentsFe = R2 3 (i1, i2) and voltages Ee = R2 3 (v1, v2). Then the Dirac structuredefined with the usual euclidian product is exactly

1 0 1 00 0 0 10 −1 0 00 0 0 0

︸︷︷︸

F

dqdtdφdti1i2

+

0 −1 0 00 1 0 0−1 0 0 1

1 0 −1 0

︸︷︷︸

E

∂H∂q∂H∂φ

v1

v2

= 0

(11) 34 / 58


Partial conclusion on implicit PCH systems

Port Hamiltonian systems are an extension of Hamiltonian systems which:

are defined by extending the internal spaces of tangent and cotangentspaces by a pair of conjugated port vector spaces

are using a Dirac structure on these extended effort and flow spaces

are similar to (constrained) Poisson Hamiltonian systems

allows to write balance equations including energy flows from theenvironment (hence to prove passivity properties)

allows interconnection (including feedback control!) throughcomposition of Dirac structures

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Nonlinear control based on PCH systemsproperties

[Ortega et al. 2001] Putting energy back in control, IEEE Control Systems Magazine, 21(2):18-32

[Maschke et al. 2002] Interconnection and damping assignment: passivity-based control of port-controlledHamiltonian systems, Automatica, 38:585?596

[Ortega et al. 2008] Control by interconnection and standard passivity-based control of port-hamiltoniansystems, IEEE Transactions on Automatic Control, 53(11):2527-2542

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PCH system model (reminder)

∑:

x = [J(x)− R(x)]

∂H∂x

(x) + g (x) u

y = gT (x)∂H∂x

(x)

where

x ∈ Rn are the energy state variables,

H is the Hamiltonian function

u, y are the input/output port variables

J = −JT a n × n is the structure skew matrix matrix

g is a n ×m input matrix

R = RT > 0 is the symmetric positive definite dissipation matrix.

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Passivity and energy Balance

Energy balance∫ t

0uT (τ)y(τ)dτ︸︷︷︸

suplied

= H [x (t)]− H [x (0)]︸︷︷︸stored

+

∫ t

0

[∂H∂x

]T

R[∂H∂x

]dτ︸︷︷︸

dissipated energy

Passive system

dHdt≤ uT y

Let x? = arg minH (x), set u = 0, then H (x) decreases and the systemreaches x?. The system is locally stable.

A Damping injection u = −Ky ,(K = K T > 0

)increases the

convergence rate to x?.

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Standard PBC

Given a desired equilibrium xd , find a control in form u = β (x) + v such thatthe closed loop dynamics satisfies:

Hd (x (t))− Hd (x (0)) =

∫ t

0vT (τ) z (τ) dτ − d (t)

Hd is the desired closed loop energy function

(v , z) is the new input-output passive pair of conjugated variables

d (t) is a dissipation term used to increase the convergence rate

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First solution : the energy shaping method

General principle

If there exists β (x) such that

−∫ t

0βT (x (τ)) y (τ) dτ = Ha (x) + κ

where κ is a constant determined by the initial condition,then the closed loop system with u = β (x) + v has the total shaped energy

Hd (x) = H (x) + Ha (x)

If xd = arg minHd (x), the system is stable at the desired equilibrium xd

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Example (Series RLC circuit)

∑:

x1 =

1L

x2

x2 = −1C

x1 −RL

x2 + u

y =1L

x2

If u = 0, the equilibrium isx? = (0, 0)

If u = Vd , the equilibrium isxd = (x1d , 0) , x1d = CVd

Shaped energy

Choosing

Hd (x) =12

(1C

+1

Ca

)(x1 − x1d )2 +

12L

x22 + κ

with xd = argmin (Hd (x)), one gets:

Ha (x1) =1

2Cax2

1 −(

1C

+1

Ca

)x1x1d

and the control law

u = − x1

Ca+

(1C

+1

Ca

)x1d

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Drawbacks for the energy shaping method

derivation of the control law from the shaped energy may be intricatedand constrained→ make use of the PCH model structure ("Control by interconnection")

the condition for the existence of a power shaping control law is

R (x)∂Ha

∂x(x) = 0

→ dissipation obstacle: Ha(x) should not depend on the coordinateswhere there is natural damping

The dissipation in energy balancing PBC is admissible only on thecoordinates which does not require "shaping of energy"

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Second solution : control by Interconnection (CbI)

˙ζ = [JC(ζ)− RC(ζ)]

∂HC

∂ζ(ζ) + gC (ζ) uC

yC = gTC (ζ)

∂HC

∂ζ(ζ){[

uuC

]=

[0 −ImIm 0

][yyC

]+

[v0

]Figure: Diagram of the CbI scheme

Closed loop system

x

ζ

=

J (x)− R (x) −g (x) gTC (ζ)

gC (ζ) gT (x) JC (ζ)− RC (ζ)

∂Hcl

∂x(x , ζ)

∂Hcl

∂ζ(x , ζ)

Hcl (x , ζ) = H (x) + HC (ζ) and∂Hcl

∂t≤ vT y

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Casimir function and static closed loop feedback

The closed loop Hamiltonian system may have Casimir’s functions C (x , ζ)such that

ddt

C (x(t), ζ(t)) = 0

along the trajectories. It is then always possible to write Casimir’s functions inthe form

C (x , ζ) = ζ − F (x) = 0

Dissipation obstacle ... again

Existence of a complete set of Casimir’s functions is related to integrabilityconditions. It requires

R(x)∂HC (F (x))

∂x= 0

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Parallel RLC circuit example

Example

A parallel RLC circuitH (x) =

12C

x21 +

12L

x22

∑:

x1 = − 1

RCx1 +

1L

x2

x2 = − 1C

x1 + u

y =1L

x2

The dissipation structure is

R(x) =

( 1R 00 0

)⇒ Dissipation obstacleThey are PBC methods which allow to overcome the dissipation obstacle:

CbI with a modified output to relax the DO condition [Ortega et al. 08]

IDA-PBC control

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Interconnection & Damping Assignment (IDA - PBC)

The Interconnection and Damping Assignment Passivity Based Control

uses a static feedback control law u = β(x)

uses the PCH structure of the model

to get a closed loop Hamiltonian system in the form:

x = [Jd (x)−Rd (x)]∂Hd

∂x(x)

with

Jd (x) = J (x) + Ja (x), Jd = −J Td , the desired interconnection

Rd (x) = R (x) +Ra (x) , Rd = RTd > 0, the desired damping

Hd (x) = H (x) + Ha (x), the closed loop Hamiltonian with desiredequilibrium properties

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IDA - PBC ... continued

To achieve these desired structure, damping and Hamiltonian, we need a

feedback law u = β(x) and a vector function K (x) =∂Ha

∂xwhich solve

the matching equation:

[Jd (x)−Rd (x)] K (x) = − [Ja (x)−Ra (x)]∂H∂x

(x) + g (x)β (x)

the integrability equation:

∂x K (x) = [∂x K (x)]T

the equilibrium assignment equation:

K (xd ) = −∂x H (xd )

the Lyapunov stability inequality

∂x K (xd ) > −∂2xx H (xd )

47 / 58


IDA - PBC ... continued

The closed loop system with u = β (x) is (locally) stable at the desiredequilibrium xd = argmin (Hd (x))

Ja (x) and Ra (x) are free provided that the constraints Jd = −JTd and

Rd = RTd > 0 are satisfied

Controller design and control law

When g⊥ (x) s.t. g⊥ (x) g (x) = 0 exists, the matching equations reduces to

g⊥ (x) [Jd (x)− Rd (x)] K (x) = −g⊥ (x) [Ja (x)− Ra (x)]∂H∂x

(x)

and the control law may be explicitely derived

β (x) =[gT (x) g (x)

]−1gT (x)

{[Jd (x)− Rd (x)] K (x) + [Ja (x)− Ra (x)]

∂H∂x

(x)

}

48 / 58


Parallel RLC circuit example

Model

H (x) =1

2Cx2

1 +1

2Lx2

2

∑:

x1 = − 1

RCx1 +

1L

x2

x2 = − 1C

x1 + u

y =1L

x2

Control design

natural damping (Ra = 0) andinterconnection structure (Ja = 0)

the matching equation reduces to

[J (x)− R (x)] ∂x Ha = g (x)β (x)

solutions are of the form

Ha (x) = Φ (Rx1 + x2)

where φ is any arbitrary integrationfunction

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Parallel RLC circuit example ... continued

Hamiltonian design

A choice

Ha (x) =Kp

2[(Rx1 + x2)− (Rx1d + x2d )]2 − RVd (Rx1 + x2)

with Kp >−1

(L + CR2)leads to a closed loop Hamiltonian:

Hd (x) = (x − xd )T

1C

+ R2Kp RKp

RKp1L

+ Kp

(x − xd ) + κ

with a global minimum xd . The control law is then

u (x) = −KpR [R (x1 − x1d ) + (x2 − x2d )] + R2Vd

50 / 58


Partial conclusion on PBC

passivity based control applies for all nonlinear physical(thermodynamical) systems

dissipation obstacle may be overcome (CbI, IDA-PBC)

structural approaches allow a deductive design of control laws and"Lyapunov" functions

robustness to parameters uncertainties (only the qualitative energeticbehaviour matters)

generalization are currently developped for distributed parameterssystems

no parametrization of the solutions for the matching equation

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Extension and prospects in the field of nonlinearcontrol for distributed parameters systems

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An hyperbolic example: the shallow water equations (SWE)[Lefèvre 2009] Méthodes géométriques pour la modélisation, la réduction et la commande des Systèmes à paramètres

distribués, Habilitation thesis

State (energy) variables are chosen as differential forms:

q(x , t) = ρS(x , t)dx mass density

p(x , t) = ρv(x , t)dx momentum density

The energy density may be written in this SWE example

H(x , t) = ρ

(g(

hA(x , h)−∫ h

0A(x , ξ)dξ

)+

S(x , t)v2(x , t)2

)dx

= H(q, p)

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The shallow water equations ... continued

Conservation equations

The simplified Saint-Venant equations is a system of 2 conservation laws

∂q∂t

= − ∂

∂z(S(x , t)v(x , t)) dx mass

∂p∂t

= − ∂

∂z

(ρ

(gh(x , t) +

v2(x , t)2

))dx momentum

Flow and effort variables

Flow variables could be defined as the differential forms (q, p).Effort variables are variational derivatives of H w.r.t. q(x , t) and p(x , t):

ep = δpH = S(x , t)v(x , t) water flow

eq = δqH = ρ

(gh(x , t) +

v2(x , t)2

)hydrodynamic pressure

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Hamiltonian formulation of SW conservation laws

The system of two conservation laws is defined using a very simple(canonical) 1D spatial interconnection structure with the help of these powerconjugated flow (fq , fp) := −(q, p) and effort (eq , ep) = (δqH, δpH) variables:

− ∂

∂t

[q(x , t)p(x , t)

]=

[0 ∂

∂x∂∂x 0

] [δqH(q, p)δpH(q, p)

]

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Power balance equation

dHdt

=ddt

∫ZH(q, p)

=

∫ZδqH ∧ q + δpH ∧ p

=

∫Z

eq ∧ fq + ep ∧ fp

= −∫

Zeq ∧ d (ep) + ep ∧ d (eq)

= −∫

Zd (eq ∧ ep)

= −∫∂Z

e∂ ∧ f∂

= pd (0, t)Q(0, t)− pd (L, t)Q(L, t)

where e∂ := eq |∂Z and f∂ := ep|∂Z56 / 58


The matrix differential operator

J =

(0 ∂

∂z∂∂z 0

)is skew-symmetric.Indeed, consider two vectors of smooth functions e = (e1, e2) ande′ = (e′1, e

′2) satisfying the homogeneous boundary conditions

e(a) = e(b) = e′(a) = e′(b) = 0, then:∫ ba

(et J e′ + e′t J e

)dz =

∫ ba

[e1(∂∂z e′2

)+ e2

(∂∂z e′1

)+e1

(∂∂z e′2

)+ e2

(∂∂z e′1

)]dz

= [e1 e′2 + e2 e′1]ba = 0

- Hamiltonian density function

- Canonical skew-symmetric interconnection structure

- Power balance equation and boundary port variables

⇒ PCH formalism for nonlinear distributed parameters systems!

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Ongoing research ...

Port-Hamiltonian formulation

moving boundary problems

plasma dynamics in tokamak

sediments and pollutants transport phenomena

Discretization/reduction

geometric reduction schemes

I/O properties analysis using Grammians

effective control computation

Control

spatial detection/localization problems (e.g. leaks in pipes)

control of non linear parabolic systems

control by interconnection of hyperbolic systems

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port controlled hamiltonian systems and related nonlinear...

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