Geometric network modeling and control of physical
systems
Arjan van der Schaft
Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of Groningen, the Netherlands
La Cristalera, October 4-6, 2010
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 1 /
91
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 2 /
91
Overall contents of the course
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 3 /
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Overall contents of the course
Overall contents of the course
In collaboration with Bernhard Maschke, Romeo Ortega, StefanoStramigioli, Alessandro Macchelli, Peter Breedveld, Dimitri Jeltsema,Jacquelien Scherpen, Morten Dalsmo, Guido Blankenstein, DamienEberard, Goran Golo, Ram Pasumarthy, Javier Villegas, Gerardo Escobar,Guido Blankenstein, Aneesh Venkatraman, ..
• Part I: From network modeling to port-Hamiltonian systems
• Part IIa: Control of port-Hamiltonian systems.Part IIb: Distributed parameter port-Hamiltonian systems
• (Under construction !)Part IIIa: Port-Hamiltonian systems on k-complexesPart IIIb: Thermodynamical systems
Talks, together with references, will be made available at my homepage:www.math.rug.nl/˜arjan
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 4 /
91
Review on classical Hamiltonian systems
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 5 /
91
Review on classical Hamiltonian systems
Review on classical Hamiltonian systems
Common view on Hamiltonian systems
Classical Hamiltonian equations of motion
qi = ∂H∂pi
(q, p)
i = 1, · · · , n
pi = − ∂H∂qi
(q, p)
where
• q = (q1, · · · , qn) are the configuration coordinates,
• p = (p1, · · · , pn) are the generalized momenta,
• H(q, p) is the total energy of the system.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 6 /
91
Review on classical Hamiltonian systems
Review on classical Hamiltonian systems
Geometrically (coordinate-free) this is usually described by the triple
(T ∗Q, ω,H)
where
• Q is the configuration manifold(with local coordinates q = (q1, · · · , qn))
• ω is canonical symplectic form on the cotangent bundle T ∗Q(in local coordinates given by ω =
∑ni=1 dpi ∧ dqi)
• H : T ∗Q → R.
The Hamiltonian dynamics is defined by the vector field XH satisfying
ω(XH ,−) = −dH
Further generalizations: 1) Replace T ∗Q by a general symplectic manifold(M, ω), 2) Infinite-dimensional case.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 7 /
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Review on classical Hamiltonian systems
Review on classical Hamiltonian systems
Equivalently, let {, } denote the canonical Poisson bracket on T ∗Q,in canonical coordinates for T ∗Q given by
{F ,G} =
n∑
i=1
(∂F
∂qi
∂G
∂pi−
∂F
∂pi
∂G
∂qi),
then XH is determined by the requirement
XH(F ) = {F ,H}
for all F : T ∗Q → R.In an arbitrary set of local coordinates x the dynamics takes the form
x = J(x)∂H
∂x(x)
where J(x) = −JT (x) is the structure matrix of the Poisson bracket withelements
Jij = {xi , xj}, i , j = 1, · · · , n
Note that J(x) has full rank.Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems
La Cristalera, October 4-6, 2010 8 /91
Review on classical Hamiltonian systems
Non-symplectic Hamiltonian systems
On the other hand, it is well-known that many dynamical equations ofphysical interest are not precisely of this form.Typical example are the Euler equations for rigid body motion
pxpypz
=
0 −pz pypz 0 −px−py px 0
∂H∂px∂H∂py∂H∂pz
with p = (px , py , pz) the body angular momentum vector along the three
principal axes, and H(p) = 12
(p2xIx
+p2yIy
+ p2zIz
)
the kinetic energy (Ix , Iy , Iz
principal moments of inertia.)In general, many systems are of the Hamiltonian form
x = J(x)∂H
∂x(x)
with J(x) = −JT (x), but not of full rank.Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems
La Cristalera, October 4-6, 2010 9 /91
Review on classical Hamiltonian systems
Hamiltonian systems obtained by symmetry reduction
The Euler equations can be regarded as the reduction of classicalHamiltonian equations on a cotangent bundle.Reduced space is the orbit space of the action of a Lie group that leavesthe Hamiltonian invariant.In fact, Q = SO(3) and the cotangent bundle T ∗SO(3) can be reduced bythe action of SO(3) on T ∗SO(3) into so(3)∗,while the Hamiltonian is invariant under this action.
This holds in many situations, both in the finite- and infinite-dimensionalcase (“Marsden-Weinstein reduction by symmetry program”).
Thus the Poisson structures are still derivable from a standard symplecticstructure on a cotangent bundle.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 10 /
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From networks to geometric structure
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 11 /
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From networks to geometric structure
A different starting point
Network modeling of physical systems
Prevailing trend in modeling and simulation of lumped-parameter systems(multi-body systems, electrical circuits, electro-mechanical systems,robotic systems, cell-biological systems, etc.).
Advantages of network modeling:
• Systematic modeling procedure, which offers structural insight.
• Flexibility. Re-usability of components. Suited to design/control.
• Multi-physics approach.
• “Modularity can beat complexity.”
Originates from engineering, and calls for mathematical theory of
networks and dynamical systems.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 12 /
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From networks to geometric structure
A different starting point
Possible disadvantage of network modeling: it generally leads to a largeset of differential and algebraic equations (DAEs), seemingly without
any structure.This is a serious obstacle for analysis and control; especially for nonlinearmodels.
Aim: to identify the underlying Hamiltonian structure of network modelsof physical systems, and to use it for analysis, simulation and control.
What is the underlying geometric structure ? Not a cotangent bundle or areduced cotangent bundle !
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 13 /
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From networks to geometric structure
Port-based network modeling
Interaction between ideal system components is modeled by power-ports
modeling the energy exchange between the components.
Associated to every power-port there are conjugate pairs of variables(called flows f and efforts e), whose product eT f equals power.
For example,
• voltages and currents
• generalized forces and velocities
• pressure and volume change
This leads to a (generalized) Hamiltonian description of multi-physicssystems.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 14 /
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From networks to geometric structure
Example (LC-circuit)
Two inductors with magnetic energies H1(ϕ1),H2(ϕ2) (ϕ1 and ϕ2
magnetic flux linkages), and capacitor with electric energy H3(Q) (Qcharge).
ϕ1 ϕ2
L1 L2
C
Q
Question: How to write this LC-circuit as a Hamiltonian system in amodular way?
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 15 /
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From networks to geometric structure
Example (LC-circuit continued)
Storage equations for the components of the LC-circuit:
Inductor 1 ϕ1 = f1 (voltage)
(current) e1 = ∂H1∂ϕ1
Inductor 2 ϕ2 = f2 (voltage)
(current) e2 = ∂H2∂ϕ2
Capacitor Q = f3 (current)
(voltage) e3 = ∂H3∂Q
If the energy functions Hi are quadratic, e.g., H3(Q) = 12CQ
2, then the
elements are linear, e.g., voltage over capacitor = ∂H3∂Q
= QC, and similarly
for the inductors.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 16 /
91
From networks to geometric structure
Example (LC-circuit continued)
Kirchhoff’s voltage and current laws are
−f1−f2−f3
=
0 0 10 0 −1−1 1 0
e1e2e3
Substitution of eqns. of components yields Hamiltonian system
ϕ1
ϕ2
Q
=
0 0 −10 0 11 −1 0
∂H∂ϕ1
∂H∂ϕ2
∂H∂Q
with H(ϕ1, ϕ2,Q) := H1(ϕ1) + H2(ϕ2) + H3(Q) total energy.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 17 /
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From networks to geometric structure
Preliminary conclusions
• The structure matrix J is completely determined by theinterconnection structure of the system (in this case, Kirchhoff’scurrent and voltage laws).
• Skew-symmetry of J corresponds to the interconnection beingpower-conserving. (Tellegen’s theorem for Kirchhoff’s laws.)
• There is no clear underlying cotangent bundle or symplectic manifold !
• Building blocks of our theory should be open dynamical systems,instead of closed dynamical systems(the systems point of view).
• Complex Hamiltonian systems are obtained by interconnecting openHamiltonian systems.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 18 /
91
Port-Hamiltonian systems
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 19 /
91
Port-Hamiltonian systems
Port-Hamiltonian systems
The Hamiltonian equations for a closed dynamical system
x = J(x)∂H
∂x(x), J(x) = −JT (x), x ∈ X
are extended to open dynamical systems:
x = J(x)∂H∂x
(x) + g(x)f , f ∈ Rm
x ∈ X state space
e = gT (x)∂H∂x
(x), e ∈ Rm
where the external ports defined by the matrix g(x), and f ∈ Rm, e ∈ R
m
are the power-variables at the external ports (open to interconnection toother systems).By skew-symmetry of J(x) we obtain for any g(x) the energy-balance
dH
dt(x(t)) = eT (t)f (t) = power supplied to the system
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 20 /
91
Port-Hamiltonian systems
Feedback interconnection of two open Hamiltonian systems
xi = Ji (xi )∂Hi
∂xi(xi) + gi (xi )fi
ei = gTi (x)∂Hi
∂xi(xi )
xi ∈ Xi , i = 1, 2
via the feedback interconnection(power-conserving since f1e1 + f2e2 = 0 !)
f1 = −e2, f2 = e1
yields the Hamiltonian system
(x1x2
)
=
(
J1(x1) −g1(x1)gT2 (x2)
g2(x2)gT1 (x1) J2(x2)
)
︸ ︷︷ ︸
Jint(x1,x2)
(∂H1∂x1
(x1)
∂H2∂x2
(x2)
)
with state space X1 × X2, and total Hamiltonian H1(x1) + H2(x2).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 21 /
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Port-Hamiltonian systems
However, this class of Hamiltonian open systems isnot closed under arbitrary interconnection:
C1
Q1
φ
L
Q2
C2
Figure: Capacitors and inductors swapped.
Composition leads to algebraic constraints between the state variables; inthis case Q1 and Q2.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 22 /
91
Port-Hamiltonian systems
What is the appropriate generalization of the Poisson structure J ?
Answer: Dirac structures
(’From skew-symmetric mappings to skew-symmetric relations’)
Power is defined by
P = e(f ) =:< e | f >, (f , e) ∈ V × V∗.
where the linear space V is called the space of flows f (e.g. currents), andV∗ the space of efforts e (e.g. voltages).Symmetrized form of power is the indefinite bilinear form ≪,≫ on V ×V∗:
≪(f a, ea), (f b, eb) ≫ := < ea | f b > + < eb | f a >,
(f a, ea), (f b, eb) ∈ V × V∗.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 23 /
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Port-Hamiltonian systems
Definition (Weinstein, Courant, Dorfman)
A (constant) Dirac structure is a subspace
D ⊂ V × V∗
such thatD = D⊥,
where ⊥ denotes orthogonal complement with respect to the bilinear form≪,≫.
For a finite-dimensional linear space V this is equivalent to
(i) < e | f >= 0 for all (f , e) ∈ D,
(ii) dimD = dimV.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 24 /
91
Port-Hamiltonian systems
Examples
Mathematical
(a) Let J : V∗ → V be a skew-symmetric mapping. Then its graph{(f , e) ∈ V × V∗ | f = Je} is a Dirac structure.
(b) Let ω : V → V∗ be a skew-symmetric mapping.Then graph ω ⊂ V × V∗ is a Dirac structure.
(c) Let W ⊂ V be a subspace, and let annW be its annihilating subspaceof V∗. Then W × annW ⊂ V × V∗ is a Dirac structure.
Physical
(a) Kirchhoff’s laws
(b) Transformers and gyrators
(c) Kinematic pairs
(d) Ideal (workless) constraints
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 25 /
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Port-Hamiltonian systems
For many systems, especially those with 3-D mechanical components, theinterconnection structure will be modulated by the energy or geometricvariables.This leads to the notion of non-constant Dirac structures on manifolds.
Definition
Consider a smooth manifold M. A Dirac structure on M is a vectorsub-bundle D ⊂ TM ⊕ T ∗M such that for every x ∈ M the vector space
D(x) ⊂ TxM × T ∗
xM
is a Dirac structure as before.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 26 /
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Port-Hamiltonian systems
Geometric definition of a port-Hamiltonian system
x
∂H∂x(x)
fx
ex
f ∈ F
e ∈ F∗
D(x)H
Figure: Port-Hamiltonian system
The dynamical system defined by the relations
(−x(t),∂H
∂x(x(t)), f (t), e(t)) ∈ D(x(t)), t ∈ R
is called a port-Hamiltonian system.So we have generalized from (M, ω,H) to (X ,D,F ,H).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 27 /
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Port-Hamiltonian systems
Particular case is a Dirac structure D(x) ⊂ TxX × T ∗xX × F × F∗ given
as the graph of the skew-symmetric map
[fxe
]
=
[−J(x) −g(x)gT (x) 0
] [exf
]
,
leading (fx = −x , ex = ∂H∂x
(x)) to a Hamiltonian open system as before
x = J(x)∂H∂x
(x) + g(x)f , x ∈ X , f ∈ Rm
e = gT (x)∂H∂x
(x), e ∈ Rm
However, in general, the equations of a port-Hamiltonian system areDAEs.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 28 /
91
Port-Hamiltonian systems
Example: Mechanical systems with kinematic constraints
Constraints on the generalized velocities q:
AT (q)q = 0.
This leads to constrained Hamiltonian equations
q = ∂H∂p
(q, p)
p = −∂H∂q
(q, p) + A(q)λ+ B(q)f
0 = AT (q)∂H∂p
(q, p)
e = BT (q)∂H∂p
(q, p)
with H(q, p) total energy, and λ the constraint forces.Dirac structure is defined by the Poisson structure on T ∗Q together withconstraints AT (q)q = 0 and external force matrix B(q) (see later on fordetails).
Can be extended to general multi-body systems.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 29 /
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Port-Hamiltonian systems
Example (Rolling coin)
Let x , y be the Cartesian coordinates of the point of contact of the coinwith the plane. Furthermore, ϕ denotes the heading angle, and θ the angleof Queen Beatrix’ head (on the Dutch version of the euro).The rolling constraints are
x = θ cosϕ, y = θ sinϕ
(rolling without slipping). The total energy isH = 1
2p2x + 1
2p2y + 1
2p2θ + 1
2p2ϕ, and the constraints can be rewritten as
px = pθ cosϕ, py = pθ sinϕ.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 30 /
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Port-Hamiltonian systems
Jacobi identity and holonomic constraints
There is an important notion of integrability of a Dirac structure on amanifold.
Definition (Dorfman, Courant)
A Dirac structure D on a manifold M is called integrable (or, closed) if
< LX1α2 | X3 > + < LX2
α3 | X1 > + < LX3α1 | X2 >= 0
for all (X1, α1), (X2, α2), (X3, α3) ∈ D.
For constant Dirac structures the integrability condition is automaticallysatisfied.The Dirac structure D defined by the canonical symplectic structure andkinematic constraints AT (q)q = 0 satisfies the integrability condition ifand only if the constraints are holonomic; that is, can be integrated togeometric constraints φ(q) = 0.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 31 /
91
Port-Hamiltonian systems
Special cases; see Dalsmo & vdS for more info.
(a) Let J be a Poisson structure on M, defining a skew-symmetricmapping J : T ∗M → TM. Thengraph J ⊂ T ∗M ⊕ TM is a Dirac structure.Integrability is equivalent to the Jacobi-identity for the Poissonstructure.
(b) Let ω be a (pre-)symplectic structure on M, defining askew-symmetric mapping ω : TM → T ∗M. Thengraph ω ⊂ TM ⊕ T ∗M is a Dirac structure.Integrability is equivalent to the closedness of the symplecticstructure.
(c) Let K be a constant-dimensional distribution on M, and let annK beits annihilating co-distribution. Then K × annK ⊂ TM ⊕ T ∗M is aDirac structure.Integrability is equivalent to the involutivity of distribution K .
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 32 /
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Port-Hamiltonian systems
Integrability of the Dirac structure is equivalent to the existence ofcanonical coordinates:
If the Dirac structure D on X is integrable then there exist coordinates(q, p, r , s) for X such that
D(x) = {(fq , fp, fr , fs , eq , ep , er , es) ∈ TxX × T ∗
x X}
fq = −ep, fp = eq
fr = 0, 0 = es
Hence the Hamiltonian system corresponding to D and H : X → R is
q = ∂H∂p
(q, p, r , s)
p = −∂H∂q
(q, p, r , s)
r = 0
0 = ∂H∂s
(q, p, r , s)
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 33 /
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Port-Hamiltonian systems
REPRESENTATIONS AND TRANSFORMATIONS
Dirac structures, and therefore port-Hamiltonian systems, admit differentrepresentations, with different properties for simulation and control.Let D ⊂ V × V∗, with dimV = n, be a Dirac structure.
1. Kernel and Image representation
D = {(f , e) ∈ V × V∗ | Ff + Ee = 0}, for n × n matrices F and E(possibly depending on x) satisfying
(i) EFT + FET = 0,
(ii) rank[F...E ] = n.
It follows that D can be also written in image representation asD = {(f , e) ∈ V × V∗ | f = ETλ, e = FTλ, λ ∈ R
n}.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 34 /
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Port-Hamiltonian systems
2. Constrained input-output representation
Every Dirac structure D can be written as
D = {(f , e) ∈ V × V∗ | f = Je + Gλ,GT e = 0}
for a skew-symmetric matrix J and a matrix G such that
im G = {f | (f , 0) ∈ D}.
Furthermore, ker J = {e | (0, e) ∈ D}.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 35 /
91
Port-Hamiltonian systems
3. Hybrid input-output representation
Let D be given by square matrices E and F as in 1. Suppose rankF = m(≤ n). Select m independent columns of F , and group them into a
matrix F1. Write (possibly after permutations) F = [F1...F2], and
correspondingly E = [E1...E2], f =
[f1f2
]
, e =
[e1e2
]
.
Then the matrix [F1...E2] is invertible, and
D =
{[f1f2
]
,
[e1e2
] ∣∣∣∣
[f1e2
]
= J
[e1f2
]}
with J := −[F1...E2]
−1[F2...E1] skew-symmetric.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 36 /
91
Port-Hamiltonian systems
4. Canonical coordinates
For simplicity take F × F∗ to be void (no external ports).If the Dirac structure on X is integrable then there exist coordinates(q, p, r , s) for X such that
D(x) = {(fq , fp , fr , fs , eq , ep , er , es ) ∈ TxX × T ∗
xX |
fq = −ep , fp = eq
fr = 0, 0 = es
}
Hence the port-Hamiltonian system on X takes the form
q = ∂H∂p
(q, p, r , s)
p = −∂H∂q
(q, p, r , s)
r = 0
0 = ∂H∂s
(q, p, r , s)
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 37 /
91
Port-Hamiltonian systems
DAE representation of port-Hamiltonian systems
Represent the Dirac structure D in kernel representation as
D = {(fx , ex , f , e) | Fx(x)fx + Ex(x)ex + F (x)f + E (x)e = 0},
with(i) ExF
Tx + FxE
Tx + EFT + FET = 0,
(ii) rank [Fx...Ex
...F...E ] = dim(X × F).
Since the flows fx and efforts ex corresponding to the energy-storingelements are given respectively as fx = −x and ex = ∂H
∂x(x), it follows that
the system is described by the set of differential-algebraic equations(DAEs)
Fx(x(t))x(t) = Ex(x(t))∂H
∂x(x(t)) + F (x(t))f (t) + E (x(t))e(t)
with f , e the external port variables.Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems
La Cristalera, October 4-6, 2010 38 /91
Port-Hamiltonian systems
Mixture of constrained and hybrid input-output representation
By a hybrid input-output partition of the vector of port flows(f , e) ∈ F ×F∗ as (u, y) we can represent any port-Hamiltonian system inconstrained form as
x = J(x)∂H∂x
(x) + G (x)λ+ g(x)u, x ∈ X , u ∈ Rm
0 = GT (x)∂H∂x
(x) + D(x)u,
y = gT (x)∂H∂x
(x), y ∈ Rm
whereJ(x) = −JT (x), D(x) = −DT (x)
This is the form as encountered before in the case of kinematic constraints.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 39 /
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Port-Hamiltonian systems
Interconnection port-Hamiltonian systems, and
composition of Dirac structures
The composition of two Dirac structures with partially shared variables isagain a Dirac structure:
D12 ⊂ V1 × V∗
1 × V2 × V∗
2
D23 ⊂ V2 × V∗
2 × V3 × V∗
3
F1
F∗1
F2
F∗2
F3
F∗3
DA DB
︸ ︷︷ ︸
DA||DBFigure: Composed Dirac structure
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 40 /
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Port-Hamiltonian systems
Af2
Ae2
DA
DB
1f 3
f
1e 3
e
Bf2
Be2
Figure: Standard interconnection
f A2 = −f B2 ∈ F2
eA2 = eB2 ∈ F∗
2
The gyrating (or feedback) interconnection
f A2 = −eB2eB2 = f B2
can be easily transformed to this case.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 41 /
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Port-Hamiltonian systems
Thus
DA ‖ DB := {(f1, e1, f3, e3) ∈ F1 ×F∗
1 ×F3 ×F∗
3 | ∃(f2, e2) ∈ F2 ×F∗
2 s.t.
(f1, e1, f2, e2) ∈ DA and (−f2, e2, f3, e3) ∈ DB}
Theorem
Let DA, DB be Dirac structures (defined with respect toF1 ×F∗
1 ×F2 ×F∗
2 , respectively F2 ×F∗
2 ×F3 ×F∗
3 and their bilinearforms). Then DA ‖ DB is a Dirac structure with respect to the bilinearform on F1 ×F∗
1 ×F3 ×F∗
3 .
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 42 /
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Port-Hamiltonian systems
Proof
Consider DA, DB defined in matrix kernel representation by
DA = {(f1, e1, fA, eA) ∈ F1 ×F∗
1 ×F2 ×F∗
2 |F1f1 + E1e1 + F2AfA + E2AeA =
DB = {(fB , eB , f3, e3) ∈ F2 ×F∗
2 ×F3 ×F∗
3 |F2B fB + E2BeB + F3f3 + E3e3 =
Make use of the following basic fact from linear algebra:
(∃λ s.t. Aλ = b) ⇔ [∀α s.t. αTA = 0 ⇒ αTb = 0]
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 43 /
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Port-Hamiltonian systems
Note that DA, DB are alternatively given in matrix image representation as
DA = im
ET1
FT1
ET2A
FT2A
00
DB = im
00
ET2B
FT2B
ET3
FT3
Hence, (f1, e1, f3, e3) ∈ DA ‖ DB ⇔ ∃λA, λB such that
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 44 /
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Port-Hamiltonian systems
f1e100f3e3
=
ET1 0
FT1 0
ET2A ET
2B
FT2A −FT
2B
0 FT3
0 ET3
[λA
λB
]
⇔
⇔ ∀(β1, α1, β2, α2, β3, α3) s.t.
(βT1 αT
1 βT2 αT
2 βT3 αT
3 )
ET1 0
FT1 0
ET2A ET
2B
FT2A −FT
2B
0 FT3
0 ET3
= 0,
βT1 f1 + αT
1 e1 + βT3 f3 + αT
3 e3 = 0
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 45 /
91
Port-Hamiltonian systems
Explicit expressions for the composition of two Dirac structures
Consider Dirac structures DA ⊂ F1 ×F∗
1 ×F2 ×F∗
2 ,DB ⊂ F2 ×F∗
2 ×F3 ×F∗
3 , given by matrix kernel/image representations(FA,EA) = ([F1|F2A], [E1|E2A]), respectively(FB ,EB) = ([F2B |F3], [E2B |E3]). Define
M =
[F2A E2A
−F2B E2B
]
and let LA, LB be matrices with
L = [LA|LB ] , ker L = imM
F = [LAF1|LBF3]E = [LAE1|LBE3]
is a matrix kernel/image representation of DA ‖ DB .
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 46 /
91
Port-Hamiltonian systems
This relaxed kernel/image representation can be readily understood bypremultiplying the equations characterizing the composition of DA with DB
[F1 E1 F2A E2A 0 00 0 −F2B E2B F3 E3
]
f1e1f2e2f3e3
= 0,
by the matrix L := [LA|LB ]. Since LM = 0 this results indeed in therelaxed kernel representation
LAF1f1 + LAE1e1 + LBF3f3 + LBE3e3 = 0
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 47 /
91
Port-Hamiltonian systems
Consequence
The interconnection of a number of port-Hamiltonian systems(Xi ,Di ,Hi ), i = 1, · · · , k , through an interconnection Dirac structure DI isa port-Hamiltonian system (X ,D,H), with
H = H1 + · · · + Hk ,
X = X1 × · · · × Xk
and D the composition of D1, · · · ,Dk ,DI .
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 48 /
91
Port-Hamiltonian systems
Mechanical systems with kinematic constraints
The constrained Hamiltonian equations define a port-Hamiltonian system,with respect to the Dirac structure D (in constrained input-outputrepresentation)
D = {(fS , eS , fC , eC ) | 0 =[0 AT (q)
]eS , eC =
[0 BT (q)
]eS ,
−fS =
[0 In
−In 0
]
eS +
[0
A(q)
]
λ+
[0
B(q)
]
fC , λ ∈ Rk}
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 49 /
91
Port-Hamiltonian systems
Example: Mechanical systems with kinematic constraints
Constraints on the generalized velocities q:
AT (q)q = 0.
This leads to constrained Hamiltonian equations
q = ∂H∂p
(q, p)
p = −∂H∂q
(q, p) + A(q)λ+ B(q)f
0 = AT (q)∂H∂p
(q, p)
e = BT (q)∂H∂p
(q, p)
with H(q, p) total energy, and λ the constraint forces.Dirac structure is defined by the Poisson structure on T ∗Q together withconstraints AT (q)q = 0 and external force matrix B(q) (see later on fordetails).
Can be extended to general multi-body systems.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 50 /
91
Port-Hamiltonian systems
Mechanical systems with kinematic constraints
The algebraic constraints on the state variables (q, p) are
0 = AT (q)∂H
∂p(q, p)
and the constrained state space is Xc = {(q, p) | AT (q)∂H∂p
(q, p) = 0}.We may solve for the algebraic constraints and at the same time eliminatethe constraint forces A(q)λ in the following way. Since rank A(q) = k ,there exists locally an n × (n − k) matrix S(q) of rank n − k such that
AT (q)S(q) = 0
Now define p = (p1, p2) = (p1, . . . , pn−k , pn−k+1, . . . , pn) as
p1 := ST (q)p, p1 ∈ Rn−k
p2 := AT (q)p, p2 ∈ Rk
The map (q, p) 7→ (q, p1, p2) is a coordinate transformation. Indeed, therows of ST (q) are orthogonal to the rows of AT (q).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 51 /
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Port-Hamiltonian systems
Mechanical systems with kinematic constraints
In the new coordinates the constrained Hamiltonian system becomes
q˙p1
˙p2
=
0n S(q) ∗−ST (q)
(−pT [Si ,Sj ](q)
)
i ,j∗
∗ ∗ ∗
∂H∂q∂H∂p1
∂H∂p2
+
00
AT (q)A(q)
λ+
0Bc(q)
B(q)
u
AT (q)∂H∂p
= AT (q)A(q) ∂H∂p2
= 0
with H(q, p) the Hamiltonian H expressed in the new coordinates q, p.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 52 /
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Port-Hamiltonian systems
Mechanical systems with kinematic constraints
Here Si denotes the i -th column of S(q), i = 1, . . . , n − k , and [Si ,Sj ] isthe Lie bracket of Si and Sj , in local coordinates given as:
[Si ,Sj ](q) =∂Sj∂q
(q)Si(q)−∂Si∂q
Sj(q)
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 53 /
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Port-Hamiltonian systems
Mechanical systems with kinematic constraints
Since λ only influences the p2-dynamics, and the constraints
AT (q)∂H∂p
(q, p) = 0 are equivalently given by ∂H∂p2
(q, p) = 0, the
constrained dynamics is determined by the dynamics of q and p1
(coordinates for the constrained state space Xc)
[q˙p1
]
= Jc(q, p1)
[∂Hc
∂q(q,p1)∂Hc
∂p1(q,p1)
]
+
[0
Bc(q)
]
u,
where Hc (q, p1) equals H(q, p) with p2 satisfying ∂H
∂p2= 0, and where the
skew-symmetric matrix Jc(q, p1) is given as the left-upper part of the
structure matrix, that is
Jc(q, p1) =
[On S(q)
−ST (q)(−pT [Si ,Sj ](q)
)
i ,j
]
,
where p is expressed as function of q, p, with p2 eliminated from ∂H∂p2
= 0.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 54 /
91
Port-Hamiltonian systems
Mechanical systems with kinematic constraints
Furthermore, in the coordinates q, p, the output map is given in the form
y =[
BTc (q) B
T(q)][
∂H∂p1
∂H∂p2
]
which reduces on the constrained state space Xc to
y = BTc (q)
∂H
∂p1(q, p1)
These equations define a port-Hamiltonian system on Xc , withHamiltonian Hc given by the constrained total energy, and with structurematrix Jc .
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 55 /
91
Port-Hamiltonian systems
Example (Rolling coin)
Let x , y be the Cartesian coordinates of the point of contact of the coinwith the plane. Furthermore, ϕ denotes the heading angle, and θ the angleof Queen Beatrix’ head (on the Dutch version of the euro).The rolling constraints are
x = θ cosϕ, y = θ sinϕ
(rolling without slipping). The total energy isH = 1
2p2x + 1
2p2y + 1
2p2θ + 1
2p2ϕ, and the constraints can be rewritten as
px = pθ cosϕ, py = pθ sinϕ.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 56 /
91
Port-Hamiltonian systems
Example (The rolling coin continued)
p1 = pϕp2 = pθ + px cosϕ+ py sinϕp3 = px − pθ cosϕp4 = py − pθ sinϕ
The constrained state space Xc is given by p3 = p4 = 0, and the dynamicson Xc is computed as
xy
θϕp1p2
=
0 cosϕ0 sinϕ
O4 0 11 0
0 0 0 −1 0 0− cosϕ − sinϕ −1 0 0 0
∂Hc
∂x∂Hc
∂y∂Hc
∂θ∂Hc
∂ϕ∂Hc
∂p1∂Hc
∂p2
+
0 00 00 00 00 11 0
[u1u2
]
[y1y2
]
=
[12p2p1
]
where Hc (x , y , θ, ϕ, p1, p2) =12p
21 +
14p
22 .Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems
La Cristalera, October 4-6, 2010 57 /91
Port-Hamiltonian systems with dissipation
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 58 /
91
Port-Hamiltonian systems with dissipation
Inclusion of energy-dissipation
Energy-dissipation is included by terminating some of the ports byresistive elements
fR = −F (eR),
where the mapping F is such that
eTR fR = −eTR F (eR) ≤ 0, for all eR
Then the energy balance ddtH = eT f is replaced by
d
dtH ≤ eT f
More general and symmetric definition:
R(fR , eR) = 0 , eTR fR ≤ 0,
for all fR , eR satisfying the relation.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 59 /
91
Port-Hamiltonian systems with dissipation
Example (Electro-mechanical system: magnetically levitated ball)
qpϕ
=
0 1 0−1 0 0
0 0 − 1R
∂H∂q∂H∂p∂H∂ϕ
+
001
V , I =[0 0 1
]
∂H∂q∂H∂p∂H∂ϕ
Coupling of electrical and mechanical domain via the Hamiltonian !
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 60 /
91
Port-Hamiltonian systems with dissipation
Pole- and zero-dynamics of port-Hamiltonian systems
Start with a general port-Hamiltonian system in kernel representation
Fx x = Ex∂H
∂x(x)− FRF (eR) + EReR + FP fP + EPeP
Various pole/zero-dynamics, which inherit the port-Hamiltonian structure,can be defined. Simplest two possibilities:
fP = 0, or eP = 0
For eP = 0 (while leaving fP free) we obtain the port-Hamiltonian system
LFx x = LEx∂H
∂x(x)− LFRF (eR) + LEReR
where L is any matrix of maximal rank satisfying LFP = 0.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 61 /
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Port-Hamiltonian systems with dissipation
Indeed, the equations LFx fx + LExex + LFR fR + LEReR = 0 define thereduced Dirac structure
Dred ⊂ Fx × Ex ×FR × ER ,
which results from interconnection of the original Dirac structure D withthe Dirac structure on the space of external port variables FP × EPdefined by eP = 0.The choice fP = 0 is similar, the difference being that L should now satisfyLEP = 0.For a hybrid partitioning of the port-variables fP , eP , we may define forevery subset K ⊂ {1, · · · ,m} the reduced Dirac structure corresponding tosetting the variables ePi , i ∈ K , fPi , i /∈ K , equal to zero (while leaving thecomplementary part free).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 62 /
91
Multi-modal physical systems
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 63 /
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Multi-modal physical systems
Multi-modal physical systems
Physical systems with switching constraints and/or switching networktopology: locomotion behavior of robots and animals, power converterswith switches and diodes, systems with inequality constraints.
Many multi-modal physical systems can be formulated asport-Hamiltonian systems with switching Dirac structure.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 64 /
91
Multi-modal physical systems
Example (Boost converter)
The circuit consists of an inductor L with magnetic flux linkage φL, acapacitor C with electric charge qC and a resistance load R , together witha diode and an ideal switch S , with switch positions s = 1 (switch closed)and s = 0 (switch open).The diode is modeled as an ideal diode:
vD iD = 0, vD ≤ 0, iD ≥ 0.
Port-Hamiltonian model (with H = 12C q
2C + 1
2Lφ2L):
[qCφL
]
=
[− 1
R1− s
s − 1 0
][ ∂H∂qC
= qCC
∂H∂φL
= φL
L
]
+
[01
]
E +
[siD
(s − 1)vD
]
I = φL
L
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 65 /
91
Multi-modal physical systems
Example (Bouncing pogo-stick)
Consider a vertically bouncing pogo-stick consisting of a mass m and amassless foot, interconnected by a linear spring (stiffness k and rest-lengthx0) and a linear damper d .
m
kd
g
xy sum of forces
zero on foot
spring/damperin series
foot fixedto ground
spring/damperparallel
Figure: Model of a bouncing pogo-stick: definition of the variables (left), situationwithout ground contact (middle), and situation with ground contact (right).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 66 /
91
Multi-modal physical systems
Example (Bouncing pogo-stick continued)
The mass can move vertically under the influence of gravity g until thefoot touches the ground. The states of the system are x (length of thespring), y (height of the bottom of the mass), and p (momentum of themass, defined as p := my). Furthermore, the contact situation is describedby a variable s with values s = 0 (no contact) and s = 1 (contact). TheHamiltonian of the system equals
H(x , y , p) =1
2k(x − x0)
2 +mg(y + y0) +1
2mp2
where y0 is the distance from the bottom of the mass to its center of mass.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 67 /
91
Multi-modal physical systems
Example (Bouncing pogo-stick continued)
When the foot is not in contact with the ground total force on the foot iszero (since it is massless), which implies that the spring and damper forcemust be equal but opposite. When the foot is in contact with the ground,the variables x and y remain equal, and hence also x = y .For s = 0 (no contact) the system is described by the port-Hamiltoniansystem
ddt
[yp
]
=
[0 1−1 0
] [mgpm
]
−dx = k(x − x0)
while for s = 1 the port-Hamiltonian description is
d
dt
xyp
=
0 0 10 0 1−1 −1 −d
k(x − x0)mgpm
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 68 /
91
Multi-modal physical systems
Example (Bouncing pogo-stick continued)
The two situations can be taken together into one port-Hamiltoniansystem with variable Dirac structure:
d
dt
xyp
=
s−1d
0 s0 0 1−s −1 −sd
k(x − x0)mgpm
The conditions for switching of the contact are functions of the states,namely as follows: contact is switched from off to on when y − x crosseszero in the negative direction, and contact is switched from on to off whenthe velocity y − x of the foot is positive in the no-contact situation, i.e.when p
m+ k
d(x − x0) > 0.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 69 /
91
Multi-modal physical systems
In both examples above we obtain a switching port-Hamiltonian system,specified by a
• Dirac structure Ds depending on the switch position s ∈ {0, 1}n (heren denotes the number of independent switches)
• a common Hamiltonian H : X → R
• common resistive structure R.
Every switching may be internally induced (like in the case of a diode in anelectrical circuit or an impact in a mechanical system) or externallytriggered (like an active switch in a circuit or mechanical system).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 70 /
91
Multi-modal physical systems
Start with a Dirac structure D relating all flow and effort variables:
D ⊂ Fx × Ex ×FR × ER ×FP × EP ×FS × ES
where
• Fx × Ex is the space of flow and effort variables corresponding to theenergy-storing elements.
• FR × ER denotes the space of flow and effort variables of the resistiveelements.
• FP × EP is the space of flow and effort variables corresponding to theexternal ports (or sources).
• FS , respectively ES , denote the flow and effort spaces of the idealswitches.
Let s be the number of switches, then every subset π ⊂ {1, 2, . . . , s}defines a switch configuration, according to
e iS = 0, i ∈ π, f jS = 0, j 6∈ π
We will say that in switch configuration π, for all i ∈ π the i -th switch isclosed, while for j /∈ π the j-th switch is open.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 71 /
91
Multi-modal physical systems
Switching dynamics
For each fixed switch configuration π this leads to the following Diracstructure Dπ
Dπ = {(fx , ex , fR , eR , fP , eP ) | ∃fS ∈ FS , eS ∈ ES
such that e iS = 0, i ∈ π, f jS = 0, j 6∈ π, and
((fx , ex , fR , eR , fP , eP , fS , eS ) ∈ D}
Indeed, every switch configuration π defines a Dirac structure on the spaceof flow and effort variables fS , eS of the switches, and Dπ equals thecomposition of this Dirac structure with the overall Dirac structure D.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 72 /
91
Multi-modal physical systems
Switching dynamics
Let the Hamiltonian H : X → R denote the total energy at theenergy-storage elements with state variables x = (x1, · · · , xn); i.e., thetotal energy is given as H(x). The constitutive relations between the statevariables x , and the flow and effort vector of the energy-storing elementsare given as
x = −fx , ex =∂H
∂x(x)
This immediately implies the energy balance
d
dtH =
∂TH
∂x(x)x = −eTx fx ,
The constitutive relations for the linear resistive elements are given as
fR = −ReR , R = RT > 0,
implying the power-dissipating property
eTR fR = −eTR ReR < 0, for all eR ∈ ER , eR 6= 0
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 73 /
91
Multi-modal physical systems
Switching dynamics
Definition
The dynamics of the switching port-Hamiltonian system is given as
(−x(t),∂H
∂x(x(t)),−ReR(t), eR(t), fP(t), eP (t)) ∈ Dπ
at all time instants t during which the system is in switch configuration π.
It follows from the power-conservation property of Dirac structures thatduring the time-interval in which the system is in a fixed switchconfiguration
d
dtH = −eTR ReR + eTP fP ≤ eTP fP ,
thus showing passivity for each fixed switch configuration if theHamiltonian H is bounded from below.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 74 /
91
Multi-modal physical systems
The conditions for a particular switch configuration π may entail algebraicconstraints on the state variables x . These are characterized by theconstraint subspace
Cπ := {ex ∈ Ex | ∃fx , fR , eR , fP , eP , such that
(fx , ex , fR , eR , fP , eP) ∈ Dπ, fR = −ReR}
The subspace Cπ determines, together with H, the algebraic constraints ineach switch configuration π:
∂H
∂x(x(t)) ∈ Cπ
Hence if Cπ 6= Ex then in general this imposes algebraic constraints on thestate vector x(t).
What is happening if the switch configuration changes and the state
is not satisfying the constraints of the new switch configuration ?
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 75 /
91
Multi-modal physical systems
Jump rules
Define for each π the jump space
Jπ := {fx | (fx , 0, 0, 0, 0, 0) ∈ Dπ}
The following crucial relation between the jump space Jπ and theconstraint subspace Cπ holds true. Recall that Jπ ⊂ Fx while Cπ ⊂ Ex ,where Ex = (Fx )
∗.
Theorem
Jπ = C⊥
π
where ⊥ denotes the orthogonal complement with respect to the dualityproduct between the dual spaces Fx and Ex .
Note: the proof crucially depends on the linearity of the resistiverelations, as well as on the positivity of them.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 76 /
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Multi-modal physical systems
Definition (State transfer principle)
Consider the state x− of a switching port-Hamiltonian system at aswitching time where the switch configuration of the system changes intoπ. Suppose x− is not satisfying the algebraic constraints corresponding toπ, that is
∂H
∂x(x−) 6∈ Cπ
Then the new state x+ just after the switching time satisfies
x+ − x− ∈ Jπ,∂H
∂x(x+) ∈ Cπ
This means that at this switching time an instantaneous jump from x− tox+ with xtransfer := x+ − x− ∈ Jπ will take place, in such a manner that∂H∂x
(x+) ∈ Cπ.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 77 /
91
Multi-modal physical systems
The jump space Jπ is the space of flows in the state space X = Fx that iscompatible with zero effort ex at the energy-storing elements and zeroflows fR , fP and efforts eR , eP at the resistive elements and external ports.
In fact, the jump space consists of all flow vectors fx that may be added tothe present flow vector corresponding to a certain effort vector at theenergy storage and certain flow and effort vectors at the resistive elementsand external ports, while remaining in the Dirac structure Dπ, withoutchanging these other effort and flow vectors.
Since Dπ captures the full power-conserving interconnection structure ofthe system while in switch configuration π, reflecting the underlyingconservation laws of the system, the jump space Jπ thus corresponds to aparticular subset of conservation laws.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 78 /
91
Multi-modal physical systems
The state transfer principle thus proclaims that the discontinuous changein the state vector is an impulsive motion satisfying this particular set ofconservation laws.The state transfer principle formalizes and extends the classical charge andflux conservation principle in electrical circuit theory:
The discontinuous change in the charges whenever switches are closedcorresponds to an impulsive current satisfying Kirchhoff’s current laws forthe circuit (under the new switch configuration) where the inductors,resistors and external ports have been open-circuited. Dually, thediscontinuous change in the fluxes resulting from opening switchescorresponds to an instantaneous voltage drop satisfying Kirchhoff’s voltagelaws for the circuit where the capacitors, resistors and external ports havebeen short-circuited.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 79 /
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Multi-modal physical systems
Theorem
Let H be a convex function. Then for any x− and x+ satisfying the statetransfer principle
H(x+) ≤ H(x−)
Makes use of the following fact: convexity of f : Rn → R is equivalent to
f (y) ≥ f (x)+ <∂f
∂x(x) | y − x >
for all x , y .The state transfer principle in the linear case also allows for a variational
characterization.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 80 /
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Symmetries and conserved quantities
Outline of Part I
1 Overall contents of the course
2 Review on classical Hamiltonian systems
3 From networks to geometric structure
4 Port-Hamiltonian systems
5 Port-Hamiltonian systems with dissipation
6 Multi-modal physical systems
7 Symmetries and conserved quantities
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 81 /
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Symmetries and conserved quantities
Integrability
We restrict ourselves to systems without external and resistive ports.
Let D be a Dirac structure on a smooth manifold X . We will say that D isa integrable Dirac structure (or, true Dirac structure, in the terminologyof Courant, Weinstein, Dorfman) if it satisfies the closedness (orintegrability) condition
< LX1α2 | X3 > + < LX2
α3 | X1 > + < LX3α1 | X2 >= 0
for all (X1, α1), (X2, α2), (X3, α3) ∈ D.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 82 /
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Symmetries and conserved quantities
Define the smooth distributions
G0 := {X ∈ TX | (X , 0) ∈ D}
G1 := {X ∈ TX | ∃α ∈ T ∗X s.t. (X , α) ∈ D}
and the smooth co-distributions
P0 := {α ∈ T ∗X | (0, α) ∈ D}
P1 := {α ∈ T ∗X | ∃X ∈ TX s.t. (X , α) ∈ D}
Clearly, G0 ⊂ G1, P0 ⊂ P1, while by D = D⊥ one obtains
G0 = kerP1
P0 = annG1
If D is integrable, then the (co-)distributions G0,G1,P0,P1 are allinvolutive.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 83 /
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Symmetries and conserved quantities
Casimirs and Algebraic Constraints
Definition
Let X be a manifold with Dirac structure D, and let H : X → R be asmooth function (the Hamiltonian). The Hamiltonian systemcorresponding to (X ,D,H) is given as
(x , dH(x)) ∈ D(x), x ∈ X
The Casimirs are defined as all functions C : X → R such that dC ∈ P0.Indeed, this means that
< dC | f >= 0
for all f ∈ G1; i.e., along al possible evolutions of the system.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 84 /
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Symmetries and conserved quantities
Algebraic constraints
In general (x , dH(x)) ∈ D(x) induces algebraic constraints on the statevariables, leading to the constraint set
Xc = {x ∈ X | ∃f s.t. (f , dH(x)) ∈ D(x)}
Throughout we assume that this defines a smooth submanifold. Thisconstraint submanifold is determined by the Hamiltonian H and by theco-distribution P1 (or, equivalently, the distribution G0).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 85 /
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Symmetries and conserved quantities
Symmetries and conserved quantities
Definition
Let D be a Dirac structure on X . A vector field f on X is an infinitesimalsymmetry of D (briefly, a symmetry of D) if
(Lf X , Lf α) ∈ D, for all (X , α) ∈ D
Analogously we say that a diffeomorphism ϕ : X → X is a symmetry of Dif
(ϕ−1∗ X , ϕ∗α
)∈ D, for all (X , α) ∈ D
Theorem
Let f be a symmetry of the Dirac structure D, with associateddistributions G0,G1 and co-distributions P0,P1. Then
Lf Gi ⊂ Gi , Lf Pi ⊂ Pi , i = 0, 1
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 86 /
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Symmetries and conserved quantities
We have the following analog of the statement that any Hamiltonianvector field is a symmetry for the symplectic form:
Theorem
Let D be a closed Dirac structure on X . Let f be a vector field on X forwhich there exists a smooth function F : X → R such that (f , dF ) ∈ D.Then f is a symmetry of D.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 87 /
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Symmetries and conserved quantities
Noether type of result on the existence of conserved quantities:
Theorem
Let (X ,D,H) be a Hamiltonian system. Let f be a vector field on X forwhich there exists a smooth function F such that
(f (x), dF (x)) ∈ D(x), x ∈ Xc
Furthermore, let f be a symmetry for H on Xc , that is
Lf H(x) = 0, x ∈ Xc
Then LXHc(F ) = 0 on Xc , where Hc is the Hamiltonian H restricted to Xc .
Thus F is a conserved quantity for XHcon Xc .
Proof Because of D = D⊥ we have
< dH(x) | f (x) > + < dF (x) | XHc(x) >= 0, x ∈ Xc ,
since (f (x), dF (x)) ∈ D(x), x ∈ Xc , and (XHc(x), dH(x)) ∈ D(x), x ∈ Xc ,
by construction.Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems
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Symmetries and conserved quantities
Consider a symmetry Lie group G of the Dirac structure D; that is, the Liegroup G acts on X by diffeomorphisms Φg : X → X , g ∈ G , and Φg is asymmetry of D for every g ∈ G . Equivalently, for every ξ ∈ g (the Liealgebra of G ) the infinitesimal generator Xξ of the group action is an(infinitesimal) symmetry of D. Throughout assume that the quotientspace X := X/G of G -orbits on X is a manifold with smooth projectionmap ρ : X → X . Then the Dirac structure D reduces to X as follows.
Theorem
Let G be a symmetry Lie group of the generalized Dirac structure D onX , with quotient manifold X and smooth projection ρ : X → X . Thenthere exists a reduced Dirac structure D on X , defined as
(X , α) ∈ D if ∃X with ρ∗X = X s.t. (X , α) ∈ D, where α = ρ∗α
Furthermore, if D is integrable, then so is D.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 89 /
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Symmetries and conserved quantities
Theorem
Let (X ,D,H) be a Hamiltonian system. Let G be a symmetry Lie groupof the Dirac structure D on X , with quotient manifold X , smoothprojection ρ : X → X , and reduced Dirac structure D on X . Furthermore,suppose the action of G on X leaves H invariant, leading to a reducedHamiltonian H : X → R such that H = H ◦ ρ. Then the Hamiltoniansystem (X ,D,H) projects to the Hamiltonian system (X , D, H).
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 90 /
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Symmetries and conserved quantities
Summary Part I
• Network structure determines the geometric structure ofport-Hamiltonian systems.
• Geometric structure is formalized as a Dirac structure; generalizingsymplectic and Poisson structures.
• Dirac structures are closed under power-conserving interconnection.
• Provides systematic framework for modeling of multi-physics systems.
Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 91 /
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