port controlled hamiltonian systems and related nonlinear...
TRANSCRIPT
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
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
An introduction to distributed parameterssystems
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Introduction PCH systems Implicit PCH systems Control Conclusion
Complex water systems
Figure: Schematic view of the Bourne irrigation system near Valence(Vercors mountains)
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
High performance heating/refrigeration systems
Figure:
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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
Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
Open LC circuit ... continued
Figure: An open LC circuit with two external ports
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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Introduction PCH systems Implicit PCH systems Control Conclusion
Open LC circuit ... continued
Figure: An open LC circuit with two external ports
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
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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 )
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Introduction PCH systems Implicit PCH systems Control Conclusion
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)
}
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
Extension and prospects in the field of nonlinearcontrol for distributed parameters systems
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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
The shallow water equations ... continued
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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Introduction PCH systems Implicit PCH systems Control Conclusion
The shallow water equations ... continued
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
Introduction PCH systems Implicit PCH systems Control Conclusion
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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Introduction PCH systems Implicit PCH systems Control Conclusion
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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