arjan/DownloadLectures/cristalera1.pdf · Overallcontentsof thecourse Overall contents of the course In collaboration with Bernhard Maschke, Romeo Ortega, Stefano Stramigioli, Alessandro

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


91

Overall contents of the course

Outline of Part I









91



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


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Review on classical Hamiltonian systems

Outline of Part I









91



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.


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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.


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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


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



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.


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From networks to geometric structure

Outline of Part I









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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.


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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 !


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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.


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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?


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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.


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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.


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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.


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Port-Hamiltonian systems

Outline of Part I









91



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


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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).


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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.


91


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∗.


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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.


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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


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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.


91


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).


91


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.


91


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.


91


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.


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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.


91


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 .


91


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)


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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}.


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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}.


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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.


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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)


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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



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.


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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


91


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.


91


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 .


91


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]


91


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


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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


91


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 .


91


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


91


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 .


91


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}


91


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.


91



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).


91



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.


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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)


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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.


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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 .


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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.


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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


Port-Hamiltonian systems with dissipation

Outline of Part I









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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.


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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 !


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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.


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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).


91

Multi-modal physical systems

Outline of Part I









91



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.


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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


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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).


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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.


91



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


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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.


91


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).


91


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.


91


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.


91


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


91


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.


91


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 ?


91


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.


91


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.


91


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.


91


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.


91


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.


91

Symmetries and conserved quantities

Outline of Part I









91


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.


91


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.


91


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.


91


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).


91



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


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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.


91


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



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.


91


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).


91


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.


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arjan/DownloadLectures/cristalera1.pdf · Overallcontentsof thecourse Overall contents of the course In collaboration with Bernhard Maschke, Romeo Ortega, Stefano Stramigioli, Alessandro