port-hamiltonian systems: from geometric network modeling

Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April 6-10, 2010 1

Port-Hamiltonian Systems: fromGeometric Network Modeling to Control

Arjan van der Schaft, University of Groningen

Dimitri Jeltsema, Delft University of Technology

In collaboration with Bernhard Maschke, Romeo Ortega,

Jacquelien Scherpen, Stefano Stramigioli, Alessandro Macchelli,

Peter Breedveld, Hans Zwart, Morten Dalsmo, Guido Blankenstein,

Damien Eberard, Goran Golo, Ram Pasumarthy, Javier Villegas,

Gerardo Escobar, Guido Blankenstein, Aneesh Venkatraman,

Rostyslav Polyuga ..


From Port-Based Network Modeling to

Port-Hamiltonian Systems

1. From junction structures to Dirac structures

2. Port-Hamiltonian systems

3. Examples

4. Input-state-output port-Hamiltonian systems

5. Multi-modal physical systems

6. Representations and transformations


Port-based network modeling of lumped-parameter

physical systems leads to a representation of a multi-physics

system as a graph, where each edge is decorated with a (vector)

pair of flow variables f ∈ Rm, and effort variables e ∈ R

m.

H1 fH1

eH1

0

R1

1

IC : f = 0

T H2

H3 0fR2

eR2

R2

Figure 1: Port-based network modeling; bond graphs of Paynter


Each vertex corresponds to one of the following ideal elements:

• Energy-storing elements H:

x = fH

eH = ∂H∂x

(x)

• Power-dissipating elements R:

R(fR, eR) = 0, eTRfR ≥ 0

• Power-conserving elements: transformers T, gyrators GY,

ideal constraints IC.

• 0- and 1-junctions 0, 1:

e1 = e2 = · · · = ek, f1 + f2 + · · · + fk = 0

f1 = f2 = · · · = fk, e1 + e2 + · · · + ek = 0


• Transformers, gyrators, etc., are energy-routing devices, and

may correspond to exchange between different types of energy.

• Ideal powerless constraints such as kinematic constraints.

• 0- and 1-junctions correspond to basic conservation laws such

as Kirchhoff’s laws.

All power-conserving elements have the following properties in

common. They are described by linear equations:

Ff +Ee = 0, f, e ∈ Rl

satisfying

eT f = e1f1 + e2f2 + · · · + elfl = 0,

rank[

F E]

= l


A geometric definition of port-based network models

Take all power-conserving elements

(T, G, IC, 0- and 1-junctions)

together in a single power-conserving interconnection structure:

H1fH1

eH1

0

R1

1

IC : f = 0

T H2

H3 0fR2

eR2

R2

D

Figure 2: Power-conserving interconnection structure


Geometric definition:

Definition 1 A (constant) Dirac structure on a finite-dimensional

space V is a subspace

D ⊂ V × V∗

such that

(i) eT f = 0 for all (f, e) ∈ D,

(ii) dimD = dimV.

The elements of V are denoted as flows f , and of V ∗ as efforts e.


Equivalent, more general, definition of Dirac structure

Power is defined by

P = e(f) =:< e | f >= eT f, (f, e) ∈ V × V∗.

where the linear space V is called the space of flows f (e.g.

currents), and V∗ the space of efforts e (e.g. voltages).

Symmetrized form of power is the indefinite bilinear form , on

V × V∗:

(fa, ea), (f b, eb) := < ea | f b > + < eb | fa >,

(fa, ea), (f b, eb) ∈ V × V∗.


Definition 2 A (constant) Dirac structure is a subspace

D ⊂ V × V∗

such that

D = D⊥,

where ⊥ denotes orthogonal complement with respect to the

bilinear form ,.

Key element in the definition of port-Hamiltonian systems


An k dimensional storage element is determined by a

k-dimensional state vector x = (x1, · · · , xk) and a Hamiltonian

H(x1, · · · , xk) (energy storage), defining the lossless system

xi = −fHi, i = 1, · · · , keHi = ∂H

∂xi(x1, · · · , xk)

ddtH = −

∑ki=1 fHieHi

Such a k- dimensional storage component is written in vector

notation as:

x = −fH

eH = ∂H∂x

(x)

The elements of x are called energy variables, those of ∂H∂x

(x)

co-energy variables.


Geometric definition of a port-Hamiltonian system

H(x) DfH

eH

fP

eP

fR eR

The dynamics is given by the DAEs

(−x(t) = fH(t),∂H

∂x(x(t)) = eH(t), fR(t), eR(t), fP (t), eP (t)) ∈ D(x(t)), t ∈ R


Basic property

dH

dt(x(t)) =

∂H

∂x(x(t))x(t) = −eT

H(t)fH(t) = eTR(t)fR(t)+eT

P (t)fP (t) ≤ eTP (t)fP (t)

Example: The ubiquitous mass-spring-damper system:

Two storage elements:

• Spring Hamiltonian Hs(q) = 12kq

2 (potential energy)

q = fs = velocity

es = dHs

dq(q) = kq = force

• Mass Hamiltonian Hm(p) = 12mp2 (kinetic energy)

p = fm = force

em = dHm

dp(p) = p

m= velocity


interconnected by the Dirac structure

fs = em = y, fm = −es + u

(power-conserving since fses + fmem = uy) yields the

port-Hamiltonian system

q

p

=

0 1

−1 0

∂H∂q

(q, p)

∂H∂p

(q, p)

+

0

1

u

y =[

0 1]

∂H∂q

(q, p)

∂H∂p

(q, p)

with

H(q, p) = Hs(q) +Hm(p)


Energy-dissipation is included by adding an extra port to the

Dirac structure, terminated by power-conserving relations:

R(fR, eR) = 0, eTRfR ≤ 0

Example: For the mass-spring system, the addition of the damper

ed = −dRdfd

= −cfd, R(fd) =1

2cf2

d (Rayleigh function)

via the extended interconnection (Dirac structure)

fs = em = −fd = y, fm = es − ed + u

leads to the mass-damper-spring system

q

p

= (

0 1

−1 0

−

0 0

0 c

)

∂H∂q

(q, p)

∂H∂p

(q, p)

+

0

1

u

y =[

0 1]

∂H∂q

(q, p)

∂H∂p

(q, p)


Example: Electro-mechanical systems

q

p

ϕ

=

0 1 0

−1 0 0

0 0 − 1R

∂H∂q

(q, p, φ)

∂H∂p

(q, p, φ)

∂H∂ϕ

(q, p, φ)

+

0

0

1

V, I =

∂H

∂ϕ(q, p, φ)


Coupling electrical/mechanical domain via Hamiltonian H(q, p, φ).

H(q, p, ϕ) = mgq +p2

2m+

ϕ2

2k1(1 − qk2

)


Example: LC circuits

Two inductors with magnetic energies H1(ϕ1), H2(ϕ2) (ϕ1 and ϕ2

magnetic flux linkages), and capacitor with electric energy H3(Q)

(Q charge).

V denotes the voltage of the source.

Q

C

ϕ1 ϕ2

V

L1 L2


Hamiltonian 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

All are port-Hamiltonian systems with J = 0 and g = 1.

If the elements are linear then the Hamiltonians are quadratic, e.g.

H1(ϕ1) = 12L1

ϕ21, and ∂H1

∂ϕ1

= ϕ1

L1

= current , etc.


Kirchhoff’s interconnection laws in f1, f2, f3, e1, e2, e3, f = V, e = I are

−f1−f2−f3e

=

0 0 1 −1

0 0 −1 0

−1 1 0 0

1 0 0 0

e1

e2

e3

f

Substitution of eqns. of components yields port-Hamiltonian

system

ϕ1

ϕ2

Q

=

0 0 −1

0 0 1

1 −1 0

∂H∂ϕ1

∂H∂ϕ2

∂H∂Q

+

1

0

0

f

e = ∂H∂ϕ1

with H(ϕ1, ϕ2, Q) := H1(ϕ1) +H2(ϕ2) +H3(Q) total energy.


However, this class of port-Hamiltonian systems is not closed

under interconnection:

Figure 3: Capacitors and inductors swapped

Interconnection leads to algebraic constraints between the state

variables Q1 and Q2.


Network modeling is prevailing in modeling and simulation of

lumped-parameter physical systems (multi-body systems, electrical

circuits, electro-mechanical systems, hydraulic systems, robotic

systems, etc.), with many advantages:

• Modularity and flexibility. Re-usability (‘libraries’).

• Multi-physics approach.

• Suited to design/control.

Disadvantage of network modeling: it generally leads to a large set

of DAEs, seemingly without any structure.

Port-based modeling and port-Hamiltonian system theory

identifies the underlying structure of network models of

physical systems, to be used for analysis, simulation and

control.


Example of ideal constraints: Mechanical systems with

kinematic constraints

Ideal constraints on the generalized velocities q:

AT (q)q = 0.

This can be modelled as the composition of a transformer T and

an ideal constraint IC

q

F = A(q)λ

T

v = AT (q)q

λ

IC v = 0

Figure 4: Kinematic constraints


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 Lagrange multipliers.

The Dirac structure is defined by the symplectic form on the phase

space T ∗Q together with constraints AT (q)q = 0 and force matrix

B(q).

This can be systematically extended to general multi-body

systems, where the kinematic constraints are more generally given

by so-called kinematic pairs.


Example Unicycle

The rolling-without-slipping kinematic constraints are

x = ψ cosϕ, y = ψ sinϕ


Example of a transformer

Figure 5: Conversion from translational into rotational kinetic en-

ergy; and vice versa.

Transformer from (F, v) to (τ, ω); transformer ratio 2πl.


Example of a gyrator

Figure 6: Port-based model of a DC motor


Figure 7: A gyrator

Gyrator from (V, I) to (τ, ω): τ = KI, V = −Kω


For many systems, especially those with 3-D mechanical

components, the interconnection structure will be modulated by

the energy or geometric variables.

This leads to the notion of non-constant Dirac structures on

manifolds.

Definition 3 Consider a smooth manifold M . A Dirac structure on

M is a vector subbundle D ⊂ TM ⊕ T ∗M such that for every x ∈M

the vector space

D(x) ⊂ TxM × T ∗xM

is a Dirac structure as before.


Mathematical intermezzo: Jacobi identity and

holonomic constraints

There is an important notion of integrability of a Dirac structure

on a manifold.

Definition 4 A Dirac structure D on a manifold M is called

integrable 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

automatically satisfied.

The Dirac structure D defined by the canonical symplectic

structure and kinematic constraints AT (q)q = 0 satisfies the

integrability condition if and only if the constraints are holonomic;

that is, can be integrated to geometric constraints φ(q) = 0.


Examples

(a) Let J be a (pseudo-)Poisson structure on M , defining a

skew-symmetric mapping J : T ∗M → TM . Then

graph J ⊂ T ∗M ⊕ TM is a Dirac structure.

Integrability is equivalent to the Jacobi-identity for the Poisson

structure.

(b) Let ω be a (pre-)symplectic structure on M , defining a

skew-symmetric mapping ω : TM → T ∗M . Then

graph ω ⊂ TM ⊕ T ∗M is a Dirac structure.

Integrability is equivalent to the closedness of the symplectic

structure.

(c) Let K be a constant-dimensional distribution on M , and let

annK be its annihilating co-distribution. Then

K × annK ⊂ TM ⊕ T ∗M is a Dirac structure.

Integrability is equivalent to the involutivity of distribution K.


Input-state-output 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

fx

eP

=

−J(x) −g(x)gT (x) 0

ex

fP

,

leading (fx = −x, ex = ∂H∂x

(x)) to a port-Hamiltonian system as

before

x = J(x)∂H∂x

(x) + g(x)fP , x ∈ X , fP ∈ Rm

eP = gT (x)∂H∂x

(x), eP ∈ Rm


Power-dissipation is included by terminating some of the ports by

static resistive elements

fR = −F (eR), where eTRF (eR) ≥ 0, for all eR.

d

dtH ≤ eT

P fP

This leads, e.g. for linear damping, to input-state-output

port-Hamiltonian systems in the form

x = [J(x) −R(x)]∂H∂x

(x) + g(x)fP

eP = gT (x)∂H∂x

(x)

where J(x) = −JT (x), R(x) = RT (x) ≥ 0 are the interconnection and

damping matrices, respectively.


Multi-modal physical systems

Physical systems with switching constraints and/or switching

network topology: locomotion behavior of robots and animals,

power converters with switches and diodes, systems with inequality

constraints.

Many multi-modal physical systems can be formulated as

port-Hamiltonian systems with switching Dirac structure.


Example 5 (Boost converter) The circuit consists of an

inductor L with magnetic flux linkage φL, a capacitor C with

electric charge qC and a resistance load R, together with a 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:

vDiD = 0, vD ≤ 0, iD ≥ 0. (1)

we Port-Hamiltonian model (with H = 12Cq2C + 1

2Lφ2

L):

qC

φL

=

− 1

R1 − s

s− 1 0

∂H∂qC

= qC

C

∂H∂φL

= φL

L

+

0

1

E +

siD

(s− 1)vD

I = φL

L

(2)


Example 6 (Bouncing pogo-stick) Consider a vertically

bouncing pogo-stick consisting of a mass m and a massless foot,

interconnected by a linear spring (stiffness k and rest-length x0)

and a linear damper d.

m

kd

g

x

y sum of forces

zero on foot

spring/damper

in series

foot fixed

to ground

spring/damper

parallel

Figure 8: Model of a bouncing pogo-stick: definition of the variables

(left), situation without ground contact (middle), and situation with

ground contact (right).


The mass can move vertically under the influence of gravity g until

the foot touches the ground. The states of the system are x

(length of the spring), y (height of the bottom of the mass), and p

(momentum of the mass, defined as p := my). Furthermore, the

contact situation is described by a variable s with values s = 0 (no

contact) and s = 1 (contact). The Hamiltonian of the system

equals

H(x, y, p) =1

2k(x− x0)

2 +mg(y + y0) +1

2mp2 (3)

where y0 is the distance from the bottom of the mass to its center

of mass.


When the foot is not in contact with the ground total force on the

foot is zero (since it is massless), which implies that the spring and

damper force must 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-Hamiltonian system

ddt

y

p

=

0 1

−1 0

mg

pm

−dx = k(x− x0)

(4)


while for s = 1 the port-Hamiltonian description is

d

dt

x

y

p

=

0 0 1

0 0 1

−1 −1 −d

k(x− x0)

mgpm

(5)


The two situations can be taken together into one

port-Hamiltonian system with variable Dirac structure:

d

dt

x

y

p

=

s−1d

0 s

0 0 1

−s −1 −sd

k(x− x0)

mgpm

(6)

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 crosses zero in the negative direction, and contact is switched

from on to off when the velocity y − x of the foot is positive in the

no-contact situation, i.e. when pm

+ kd(x− x0) > 0.


In both examples above we obtain a switching port-Hamiltonian

system, specified by a Dirac structure Ds depending on the switch

position s ∈ 0, 1n (here n denotes the number of independent

switches), a Hamiltonian H : X → R and a resistive structure R.

Furthermore, every switching may be internally induced (like in the

case of a diode in an electrical circuit or an impact in a mechanical

system) or externally triggered (like an active switch in a circuit or

mechanical system).

Problems

• Well-posedness questions: e.g., systems with reverse Coulomb

friction may have multiple solutions.

• Computation of the next mode may be difficult.

• Collision rules.

• Investigation of limit cycles/periodic orbits.


REPRESENTATIONS AND TRANSFORMATIONS

Dirac structures, and therefore port-Hamiltonian systems, admit

different representations, 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 as

D = (f, e) ∈ V × V∗ | f = ETλ, e = FTλ, λ ∈ Rn.


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, kerJ = e | (0, e) ∈ D.


3. Hybrid input-output representation

Let D be given by square matrices E and F as in 1. Suppose rank

F = 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 =

f1

f2

, e =

e1

e2

.

Then the matrix [F1

...E2] is invertible, and

D =

f1

f2

,

e1

e2

∣∣∣∣∣∣

f1

e2

= J

e1

f2

with J := −[F1

...E2]−1[F2

...E1] skew-symmetric.


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)


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) ExFTx + 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-storing elements 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.


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 in constrained 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

where

J(x) = −JT (x), D(x) = −DT (x)

This is the form as encountered before in the case of kinematic

constraints.


Intermezzo: Relation with classical Hamiltonian equations

x = J(x)∂H

∂x(x)

with constant or ’ integrable’J- matrix admits coordinates

x = (q, p, r) in which

J =

0 I 0

−I 0 0

0 0 0

,

q = ∂H∂p

(q, p, r)

p = −∂H∂q

(q, p, r)

r = 0

For constant or integrable Dirac structure one gets Hamiltonian

DAEs

q = ∂H∂p

(q, p, r, s)

p = −∂H∂q

(q, p, r, s)

r = 0

0 = ∂H∂s

(q, p, r, s)


Recall of Hamiltonian dynamical systems from analytical

mechanics

Historically, the Hamiltonian approach starts from the principle of

least action, via the Euler-Lagrange equations and the Legendre

transformation, towards the Hamiltonian equations of motion.

The standard Euler-Lagrange equations are given as

d

dt

(∂L

∂q(q, q)

)

− ∂L

∂q(q, q) = τ,

where q = (q1, . . . , qk)T are generalized configuration coordinates for

the system with k degrees of freedom, the Lagrangian L equals the

difference T − P between kinetic co-energy T and potential energy

P , and τ = (τ1, . . . , τk)T is the vector of generalized forces acting on

the system.


The vector of generalized momenta p = (p1, . . . , pk)T is defined as

p =∂L

∂q=∂T

∂q

By defining the state vector (q1, . . . , qk, p1, . . . , pk)T the k

second-order equations transform into 2k first-order equations

q = ∂H∂p

(q, p)

p = −∂H∂q

(q, p) + τ

where the Legendre transform

H(q, p) = K(q, p) + P (q)

is the total energy of the system.


In standard mechanical systems the kinetic co-energy T is of the

form

T (q, q) =1

2qTM(q)q

where the k× k inertia (generalized mass) matrix M(q) is symmetric

and positive definite for all q. Hence

p = M(q)q

and because of the fact that the kinetic co-energy T is a quadratic

function of the velocities q it equals the kinetic energy

K(q, p) = 12p

TM−1(q)p.


The above equations are called the Hamiltonian equations of

motion, and H is called the Hamiltonian. The state space with

local coordinates (q, p) is called the phase space.

The following energy balance immediately follows:

d

dtH =

∂TH

∂q(q, p)q +

∂TH

∂p(q, p)p =

∂TH

∂p(q, p)τ = qT τ,

expressing that the increase in energy of the system is equal to the

supplied work (conservation of energy).


A Hamiltonian system with collocated inputs and outputs is more

generally given in the following form

q = ∂H∂p

(q, p) , (q, p) = (q1, . . . , qk, p1, . . . , pk)

p = −∂H∂q

(q, p) +B(q)u, u ∈ Rm,

y = BT (q)∂H∂p

(q, p) (= BT (q)q), y ∈ Rm,

Here B(q) is the input force matrix. In case m < k we speak of an

underactuated system.

By definition of the output y = BT (q)q we again obtain

dH

dt(q(t), p(t)) = uT (t)y(t)


A major generalization of the class of Hamiltonian systems consists

in considering systems which are described in local coordinates as

x = J(x)∂H∂x

(x) + g(x)u, x ∈ X , u ∈ Rm

y = gT (x)∂H∂x

(x), y ∈ Rm

Here J(x) is an n× n matrix which is skew-symmetric:

J(x) = −JT (x),

and x = (x1, . . . , xn) are local coordinates for an n-dimensional state

space manifold X . We recover the energy-balance

dHdt

(x(t)) = uT (t)y(t). In the previous case we had J =

0 I

−I0

.


Example 7 Consider a rigid body spinning around its center of

mass in the absence of gravity. The energy variables are the three

components of the body angular momentum p along the three

principal axes: p = (px, py, pz), and the energy is the kinetic energy

H(p) =1

2

(

p2x

Ix+p2

y

Iy+p2

z

Iz

)

,

where Ix, Iy, Iz are the principal moments of inertia. Euler’s

equations are

px

py

pz

=

0 −pz py

pz 0 −px

−py px 0

︸︷︷︸

J(p)

∂H∂px

∂H∂py

∂H∂pz

+ g(p)u, y = gT (p)∂H

∂p


J(p) is the canonical Lie-Poisson structure matrix on the dual of

the Lie algebra so(3) corresponding to the configuration space

SO(3) of the rigid body.)

Equations arise from the standard (6-dimensional) Hamiltonian

equations by reduction (’symmetry’).


Large-scale port-Hamiltonian systems, and composition of

Dirac structures

The composition of two Dirac structures with partially shared

variables is again a Dirac structure:

D12 ⊂ V1 × V∗1 × V2 × V∗

2

D23 ⊂ V2 × V∗2 × V3 × V∗

3

V1

V∗1

V2

V∗2

V3

V∗3

D12 D23

︸︷︷︸D12||D23

Figure 9: Composed Dirac structure


Af2

Ae2

DA

DB

1f 3

f

1e 3

e

Bf2

Be2

Figure 10: Standard interconnection

fA = −fB ∈ F2

eA = eB ∈ F∗2

The gyrating (or feedback) interconnection

fA = −eB

eB = fB

can be easily transformed to this case.


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 8 Let DA, DB be Dirac structures (defined with respect

to F1 ×F∗1 ×F2 ×F∗

2 , respectively F2 ×F∗2 × F3 ×F∗

3 and their

bilinear forms). Then DA ‖ DB is a Dirac structure with respect to

the bilinear form on F1 ×F∗1 ×F3 ×F∗

3 .


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 = 0DB = (fB , eB, f3, e3) ∈ F2 × F∗

2 ×F3 × F∗3 | F2BfB +E2BeB + F3f3 + E3e3 = 0

Make use of the following basic fact from linear algebra:

(∃λ s.t. Aλ = b) ⇔ [∀α s.t. αTA = 0 ⇒ αT b = 0]


Note that DA, DB are alternatively given in matrix image

representation as

DA = im

ET1

FT1

ET2A

FT2A

0

0

DB = im

0

0

ET2B

FT2B

ET3

FT3

Hence, (f1, e1, f3, e3) ∈ DA ‖ DB ⇔ ∃λA, λB such that


f1

e1

0

0

f3

e3

=

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 α

T1 β

T2 α

T2 β

T3 α

T3 )

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 ⇔


⇔ ∀(α1, β1, α2, β2, α3, β3) s.t.

F1 E1 F2A E2A 0 0

0 0 −F2B E2B F3 E3

α1

β1

α2

β2

α3

β3

= 0,

βT1 f1 + αT

1 e1 + βT3 f3 + αT

3 e3 = 0 ⇔

⇔ ∀(α1, β1, α3, β3) ∈ DA ‖ DB , βT1 f1 + αT

1 e1 + βT3 f3 + αT

3 e3 = 0 ⇔

⇔ (f1, e1, f3, e3) ∈ (DA ‖ DB)⊥

Thus DA ‖ DB = (DA ‖ DB)⊥, and so it is a Dirac structure.


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

(7)

and let LA, LB be matrices with

L = [LA|LB ] , kerL = imM

F = [LAF1|LBF3]

E = [LAE1|LBE3]

is a relaxed matrix kernel/image representation of DA ‖ DB.


This relaxed kernel/image representation can be readily understood

by premultiplying the equations characterizing the composition of

DA with DB

F1 E1 F2A E2A 0 0

0 0 −F2B E2B F3 E3

f1

e1

f2

e2

f3

e3

= 0, (8)

by the matrix L := [LA|LB]. Since LM = 0 this results indeed in the

relaxed kernel representation

LAF1f1 + LAE1e1 + LBF3f3 + LBE3e3 = 0 (9)


Consequence

The interconnection of a number of port-Hamiltonian systems

(Xi,Di, Hi), i = 1, · · · , k through an interconnection Dirac structure

DI is a port-Hamiltonian system (X ,D, H), with

H = H1 + · · · +Hk,

X = X1 × · · · × Dk

and D the composition of D1, · · · ,Dk,DI.


Example 9 (1-D mechanical system) Consider a spring with

elongation q and energy function Hs(q) = 12kq

2. Let (vs, Fs)

represent the external port through which energy can be exchanged

with the spring, where vs is equal to the rate of elongation

(velocity) and Fs is equal to the elastic force. This

port-Hamiltonian system can be written in kernel representation as

1 1

0 0

−qvs

+

0 0

1 −1

kq

Fs

= 0 (10)


Similarly we model a moving mass m with scalar momentum p and

kinetic energy Hm(p) = 12p

2 as the port-Hamiltonian system

1 1

0 0

−pFm

+

0 0

1 −1

pm

vm

= 0 (11)

where (Fm, vm) are respectively the external force exerted on the

mass and the velocity of the mass.


The mass and the spring can be interconnected to each other

using the symplectic gyrator

vs

Fm

=

0 1

−1 0

Fs

vm

(12)

Collecting all equations we have obtained a port-Hamiltonian

system with energy variables x = (q, p), total energy

H(q, p) = Hs(q) +Hm(p) and with interconnection port variables

(vs, Fs, Fm, vm). After elimination of the interconnection variables

(vs, Fs, Fm, vm) one obtains the port-Hamiltonian system

1 0

0 1

−q−p

+

0 1

−1 0

kq

pm

= 0 (13)

which is the ubiquitous mass-spring system.


Example: Coupled masses Consider two point masses m1 and m2

which are rigidly linked to each other. When decoupled the masses

are described by the port-Hamiltonian systems

pi = Fi,

vi = pi

mi

i = 1, 2 (14)

with Fi the force exerted on mass mi. Rigid coupling amounts to

F1 = −F2, v1 = v2 (15)

This leads to the port-Hamiltonian system

p1

p2

=

1

−1

λ

0 =[

1 −1]

p1

m1

p2

m2

(16)

where λ = F1 = −F2 now denotes the internal constraint force.


The resulting interconnnected system does not have external ports

anymore. On the other hand, external ports for the interconnected

system can be included by either extending (14) to

pi = Fi + F exti

vi = pi

mi

vexti = pi

mi

i = 1, 2 (17)

with F exti and vext

i denoting the external forces and velocities, or by

modifying the interconnection constraints (15) to e.g.

F1 + F2 + F ext = 0, v1 = v2 = vext, (18)

with F ext and vext denoting the external force exerted on the

coupled masses, respectively the velocity of the coupled masses.


Consider the port-Hamiltonian system

Σ :

x = J(x)∂H∂x

(x) + g(x)u+ b(x)λ

y = gT (x)∂H∂x

(x)

0 = bT (x)∂H∂x

(x)

x ∈ X

The Lagrange multipliers λ can be eliminated by constructing a

matrix b⊥(x) of maximal rank such that

b⊥(x)b(x) = 0


By premultiplication with b⊥(x) one obtains

Σ :

b⊥(x)x = b⊥(x)J(x)∂H∂x

(x) + b⊥(x)g(x)u

y = gT (x)∂H∂x

(x)

0 = bT (x)∂H∂x

(x)

x ∈ X (19)

This is a kernel representation of the port-Hamiltonian system.


Example 10 (Coupled masses continued) Consider the system

of two coupled masses. Premultiplication of the dynamic equations

by the row vector[

1 1]

yields the equations

p1 + p2 = 0,p1

m1− p2

m2= 0, (20)

which constitutes a kernel representation of the port-Hamiltonian

system.


A more difficult question concerns the possibility to solve for the

algebraic constraints of a port-Hamiltonian system:

0 = bT (x)∂H

∂x(x) (21)

Under constant rank assumptions the set

Xc := x ∈ X | bT (x)∂H

∂x(x) = 0

defines a submanifold of the total state space X ; the constrained

state space. In order that this constrained state space qualifies as

the state space for a port-Hamiltonian system without further

algebraic constraints one needs to be able to restrict the dynamics

of the port-Hamiltonian system to the constrained state space.

This is always possible under the condition that the matrix

bT (x)∂2H

∂x2(x)b(x) (22)


has full rank, since in this case the differentiated constraint

equations

0 =d

dt(bT (x)

∂H

∂x(x)) = ∗ + bT (x)

∂2H

∂x2(x)b(x)λ (23)

can be always uniquely solved for λ.


Example 11 (Coupled masses continued) Differentiating the

constraint equation p1

m1

− p2

m2

= 0 and using p1 = λ and p2 = −λ one

obtains

(1

m1+

1

m2)λ = 0 (24)

which determines the constraint force λ (to be equal to 0).

Defining the total momentum p = p1 + p2 one obtains the reduced

system p = p1 + p2 = 0.


On the other hand, suppose that the mass m1 is connected to a

linear spring with spring constant k1 and elongation q1 and that the

mass m2 is connected to a linear spring with spring constant k2 and

elongation q2. Then the dynamical equations change into

p1 = −k1q1 + λ and p2 = −k2q2 − λ, and differentiation of the

constraint p1

m1

− p2

m2

= 0 leads to

− k1

m1q1 +

k2

m2q2 + (

1

m1+

1

m2)λ = 0 (25)

which determines the constraint force λ as λ = m1m2

m1+m2

( k1

m1

q1 − k2

m2

q2),

and results in the dynamical equation for the total momentum p

given by

p = −k1q1 − k2q2 (26)


Consider the equations of a general mechanical system subject to

kinematic constraints. The constrained Hamiltonian equations

define a port-Hamiltonian system, with respect to the Dirac

structure D (in constrained input-output representation)

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


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

eliminate the 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,

the rows of ST (q) are orthogonal to the rows of AT (q).


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

+

0

0

AT (q)A(q)

λ+

0

Bc(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.


Here Si denotes the i-th column of S(q), i = 1, . . . , n− k, and [Si, Sj ]

is the Lie bracket of Si and Sj, in local coordinates given as:

[Si, Sj ](q) =∂Sj

∂q(q)Si(q) −

∂Si

∂qSj(q)


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.


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, with

Hamiltonian Hc given by the constrained total energy, and with

structure matrix Jc.


Example 12 (Example ?? continued: The rolling euro) Define

the new p-coordinates

p1 = pϕ

p2 = pθ + px cosϕ+ py sinϕ

p3 = px − pθ cosϕ

p4 = py − pθ sinϕ

The constrained state space Xc is given by p3 = p4 = 0, and the


dynamics on Xc is computed as

x

y

θ

ϕ

p1

p2

=

0 cosϕ

0 sinϕ

O4 0 1

1 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 0

0 0

0 0

0 0

0 1

1 0

u1

u2

y1

y2

=

12p2

p1

where Hc(x, y, θ, ϕ, p1, p2) = 12p

21 + 1

4p22.


Analysis of port-Hamiltonian systems

Port-Hamiltonian systems and passivity

A square nonlinear system

Σ :

x = f(x) + g(x)u, u ∈ Rm

y = h(x), y ∈ Rm

where x ∈ Rn are coordinates for an n-dimensional state space X , is

passive if there exists a storage function V : X → R with

V (x) ≥ 0 for every x, such that

V (x(t2)) − V (x(t1)) ≤∫ t2

t1

yT (t)u(t)dt

for all solutions (u(·), x(·), y(·)) and times t1 ≤ t2.

The system is lossless if ≤ is replaced by =.


If H is differentiable then ’passive’ is equivalent to

d

dtV ≤ yTu

which reduces to (Willems, Hill-Moylan)

∂T V∂x

(x)f(x) ≤ 0

h(x) = gT (x)∂V∂x

(x)

while in the lossless case ≤ is replaced by =.

In the linear case

x = Ax+Bu

y = Cx

is passive if there exists a quadratic storage function V (x) = 12x

TQx,

with Q = QT ≥ 0 satisfying the LMIs

ATQ+QA ≤ 0, C = BTQ


Clearly, any port-Hamiltonian system with Hamiltonian H ≥ 0 is

passive, sinced

dtH = −eT

RfR + eTP fP ≤ eT

P fP

and thus H is a storage function. Furthermore, if there are no

power-dissipating elements R, then a port-Hamiltonian system with

H ≥ 0 is lossless.


Every linear passive system with storage function V (x) = 12x

TQx,

satisfying

kerQ ⊂ kerA

can be rewritten as a linear port-Hamiltonian system

x = (J −R)Qx+Bu, J = −JT , R = RT ≥ 0

y = BTQx,

in which case the storage function V (x) = 12x

TQx is called the

Hamiltonian H.

Passive linear systems are thus port-Hamiltonian with

non-negative Hamiltonian.


Mutatis mutandis ’most’ nonlinear lossless systems can be written

as a port-Hamiltonian system

x = J(x)∂H∂x

(x) + g(x)u

y = gT (x)∂H∂x

(x)

with J(x) = −JT (x) and ∂H∂x

(x) the column vector of partial

derivatives. Note that

x = J(x)∂H

∂x(x)

is the internal Hamiltonian dynamics known from physics, which in

classical mechanics can be written as

q = ∂H∂p

(q, p)

p = −∂H∂q

(q, p)

with the Hamiltonian H the total (kinetic + potential) energy.


Similarly, most nonlinear passive systems can be written as a

port-Hamiltonian system (with dissipation)

x = [J(x) −R(x)]∂H∂x

(x) + g(x)u

y = gT (x)∂H∂x

(x)

with R(x) = RT (x) ≥ 0 specifying the energy dissipation

d

dtH = −∂

TH

∂x(x)R(x)

∂H

∂x(x) + uT y ≤ uT y


Pole- and zero-dynamics of port-Hamiltonian

systems

Start with a general port-Hamiltonian system in kernel

representation

Fxx = Ex

∂H

∂x(x) − FRF (eR) +EReR + FP fP +EP eP

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

LFxx = LEx

∂H

∂x(x) − LFRF (eR) + LEReR (27)

where L is any matrix of maximal rank satisfying LFP = 0.


Indeed, the equations LFxfx + LExex + LFRfR + LEReR = 0 define

the reduced Dirac structure

Dred ⊂ Fx × Ex ×FR × ER,

which results from interconnection of the original Dirac structure Dwith the Dirac structure on the space of external port variables

FP × EP defined by eP = 0.

The choice fP = 0 is similar, the difference being that L should now

satisfy LEP = 0.

For a hybrid partitioning of the port-variables fP , eP , we may define

for every subset K ⊂ 1, · · · ,m the reduced Dirac structure

corresponding to setting the variables ePi, i ∈ K, fPi, i /∈ K, equal to

zero (while leaving the complementary part free).


Model reduction of port-Hamiltonian systems

• Network modeling of complex lumped-parameter systems

(circuits, multi-body systems) often leads to high-dimensional

models.

• Structure-preserving spatial discretization of

distributed-parameter port-Hamiltonian systems yields

high-dimensional port-Hamiltonian models.

• Lumped-parameter modeling of systems like MEMS gives

high-dimensional port-Hamiltonian systems.

• Controller systems may be in first instance

distributed-parameter, and need to be discretized to low-order

controllers.


In many cases we want the reduced-order system to be again

port-Hamiltonian:

• Port-Hamiltonian model reduction preserves passivity.

• Port-Hamiltonian model reduction may (approximately)

preserve other balance laws /conservation laws.

• Physical interpretation of reduced-order model.

• Reduced-order system can replace the high-order

port-Hamiltonian system in a larger context.

Thus there is a need for structure-preserving model reduction of

high-dimensional port-Hamiltonian systems.


General structure-preserving model reduction

Let us assume that we have been able to find a splitting of the

state space variables x = (x1, x2) having the property that the x2

coordinates hardly contribute to the external port behavior of the

system, and thus could be omitted from the state space

description.

In which way is it possible to retain the port-Hamiltonian structure

in model reduction ?


Recall that the vector of flow and effort variables is required to be

in the Dirac structure

(f1x , f

2x , e

1x, e

2x, fR, eR, fP , eP ) ∈ D,

while the flow and effort variables fx, ex are linked to the

constitutive relations of the energy-storage by

x1 = −f1x ,

∂H∂x1 (x1, x2) = e1x

x2 = −f2x ,

∂H∂x2 (x1, x2) = e2x,

The basic idea for structure-preserving model reduction is to ’cut’

the interconnection

x2 = −f2x ,

∂H

∂x2(x1, x2) = e2x

between the energy storage corresponding to x2 and the Dirac

structure, in such a way that no power is transferred.


This is done by making both power products ( ∂H∂x2 )T x2 and (e2x)T f2

x

equal to zero.

H DR

−x1

−x2

∂H∂x1

∂H∂x2

e1x

e2x

f1x

f2x

fR

fP

eR

eP

Figure 11: Model Reduction Scheme


The following main scenario’s arise:

1 Set∂H

∂x2(x1, x2) = 0, e2x = 0

The first equation imposes an algebraic constraint on the space

variables x = (x1, x2). Under general conditions this constraint

allows one to solve x2 as a function x2(x1) of x1, leading to the

reduced Hamiltonian

Hecred(x1) := H(x1, x2(x1))

Furthermore, the second equation defines the reduced Dirac

structure

Decred := (f1

x , e1x, fR, eR, fP , eP ) | ∃f2

x such that

(f1x , e

1x, f

2x , 0, fR, eR, fP , eP ) ∈ D


leading to the reduced port-Hamiltonian system

(−x1,∂Hec

red

∂x1(x1),−F (eR), eR, fP , eP ) ∈ Dec

red

This reduction method is the Effort-constraint reduction method.


2 (Flow-constraint reduction method) Set

x2 = 0, f2x = 0

The first equation imposes the constraint

x2 = c (constant)

and thus defines the reduced Hamiltonian

H fcred(x1) := H(x1, c),

while the second equation leads to the reduced Dirac structure

Dfcred := (f1

x , e1x, fR, eR, fP , eP ) | ∃e2x such that

(f1x , e

1x, 0, e

2x, fR, eR, fP , eP ) ∈ D

and the corresponding reduced port-Hamiltonian system

(−x1,∂H fc

red

∂x1(x1),−F (eR), eR, fP , eP ) ∈ Dfc

red


3 Set

x2 = 0, e2x = 0

This leads to the reduced-order port-Hamiltonian system with

reduced Hamiltonian H fcred(x1) and reduced Dirac structure Dec

red.

4 Set∂H

∂x2(x1, x2) = 0, f2

x = 0

This leads to the port-Hamiltonian system with reduced

Hamiltonian Hecred(x1) and reduced Dirac structure Dfc

red.


The above reduction schemes have different physical

interpretations and consequences.

Consider an electrical circuit where x2 corresponds to the charge Q

of a single (linear) capacitor.

Application of the Effort-constraint method corresponds to

removing the capacitor (and setting its charge equal to zero) and

short-circuiting the circuit at the location of the capacitor.

The Flow-constraint method corresponds to open-circuiting the

circuit at the location of the capacitor, and keeping the charge of

the capacitor constant.

Method 3 is in this case very similar to the Effort-constraint

method, and corresponds to short-circuiting, with the minor

difference of setting the charge of the capacitor equal to a

constant.

Method 4 corresponds to open-circuiting while setting the charge

equal to zero (similar to Flow-constraint method).


Explicit equational representations of the four methods

starting from the full-order model:

Fxx = Ex

∂H

∂x(x) − FRF (eR) +EReR + FP fP +EP eP

Corresponding to the splitting of the state vector x into x = (x1, x2)

and the splitting of the flow and effort vectors fx, ex into f1x , f

2x and

e1x, e2x we write

Fx =[

F 1x F 2

x

]

, Ex =[

E1x E2

x

]

The reduced Dirac structure Decred corresponding to the

effort-constraint e2x = 0 is given by the explicit equations

LecF 1xf

1x + LecE1

xe1x + LecFRfR + LecEReR + LecFP fP + LecEP eP = 0

where Lec is any matrix of maximal rank satisfying

LecF 2x = 0


Similarly, the reduced Dirac structure Dfcred corresponding to the

flow-constraint f2x = 0 is given by the equations

LfcF 1xf

1x + LfcE1

xe1x + LfcFRfR + LfcEReR + LfcFP fP + LfcEP eP = 0

where Lfc is any matrix of maximal rank satisfying

LfcE2x = 0


It follows that the reduced-order model resulting from applying the

Effort-constraint method is given by

LecF 1x x

1 = LecE1x

∂Hecred

∂x1(x1)−LecFRF (eR)+LecEReR+LecFP fP +LecEP eP ,

whereas the reduced-order model resulting from applying the

Flow-constraint method is given by

LfcF 1x x

1 = LfcE1x

∂H fcred

∂x1(x1)−LfcFRF (eR)+LfcEReR +LfcFP fP +LfcEP eP

Similar expressions follow for the reduced-order models arising from

applying Methods 3 and 4.


Control of port-Hamiltonian systems

Contents

• Use of passivity for control

• Control by interconnection: set-point stabilization

• The dissipation obstacle

• A state feedback perspective; shaping the Hamiltonian

• New control paradigms


Use of passivity for control and beyond

• The storage function can be used as Lyapunov function,

implying some sort of stability for the uncontrolled system.

• The standard feedback interconnection of two passive systems

is again passive, with storage function being the sum of the

individual storage functions.

• Passive systems can be asymptotically stabilized by adding

artificial damping. In fact,

d

dtH ≤ uT y

together with the additional damping u = −y yields

d

dtH ≤ − ‖ y ‖2

proving asymptotic stability provided an observability

condition is met.


Example The Euler equations for the motion of a rigid body

revolving about its center of gravity with one input are

I1ω1 = [I2 − I3]ω2ω3 + g1u

I2ω2 = [I3 − I1]ω1ω3 + g2u

I3ω3 = [I1 − I2]ω1ω2 + g3u,

Here ω := (ω1, ω2, ω3)T

are the angular velocities around the principal

axes of the rigid body, and I1, I2, I3 > 0 are the principal moments

of inertia. The system for u = 0 has the origin as an equilibrium

point. Linearization yields the linear system

A =

0 0 0

0 0 0

0 0 0

B =

I−11 g1

I−12 g2

I−13 g3

.

Hence the linearization does not say anything about stabilizability.


Stability and asymptotic stabilization by damping injection

Rewrite the system in port-Hamiltonian form by defining the

angular momenta

p1 = I1ω1, p2 = I2ω2, p3 = I3ω3

and defining the Hamiltonian H(p) as the total kinetic energy

H(p) =1

2(p21

I1+p22

I2+p23

I3)

Then the system can be rewritten as

p1

p2

p3

=

0 −p3 p2

p3 0 −p1

−p2 p1 0

∂H∂p1

∂H∂p2

∂H∂p3

+

g1

g2

g3

u, y =

[

g1 g2 g3

]

∂H∂p1

∂H∂p2

∂H∂p3


Since H = 0 and H has a minimum at p = 0 the origin is stable.

Damping injection amounts to the negative output feedback

u = −y = −g1p1

I1− g2

p2

I2− g3

p3

I3= −g1ω1 − g2ω2 − g3ω3,

yielding convergence to the largest invariant set contained in

S := p ∈ R3 | H(p) = 0 = p ∈ R

3 | g1p1

I1+ g2

p2

I2+ g3

p3

I3= 0

It can be shown that the largest invariant set contained in S is the

origin p = 0 if and only if

g1 6= 0, g2 6= 0, g3 6= 0,

in which case the origin is rendered asymptotically stable (even,

globally).


Beyond control via passivity: What can we do if the

desired set-point is not a minimum of the storage function ??

Recall the proof of stability of an equilibrium (ω∗1 , 0, 0) 6= (0, 0, 0) of

the Euler equations.

The total energy H = 2I1

p2

1

+ 2I2

p2

2

+ 2I3

p2

3

= 12I1ω

21 + 1

2I2ω22 + 1

2I3ω23 has a

minimum at (0, 0, 0). Stability of (ω∗1 , 0, 0) is shown by taking as

Lyapunov function a combination of the total energy K and

another conserved quantity, namely the total angular momentum

C = p21 + p2

2 + p23 = I2

1ω21 + I2

2ω22 + I2

3ω23

This follows from

[

p1 p2 p3

]

0 −p3 p2

p3 0 −p1

−p2 p1 0

= 0


In general, for any Hamiltonian dynamics

x = J(x)∂H

∂x(x)

one may search for conserved quantities C, called Casimirs, as

being solutions of∂TC

∂x(x)J(x) = 0

Then ddtC = 0 for every H, and thus also H + C is a candidate

Lyapunov function.

Note that the minimum of H + C may now be different from the

minimum of H.


Control by interconnection: set-point stabilization:

Consider first a lossless Hamiltonian plant system P

x = J(x)∂H∂x

(x) + g(x)u

y = gT (x)∂H∂x

(x)

where the desired set-point x∗ is not a minimum of the

Hamiltonian H, while the Hamiltonian dynamics x = J(x)∂H∂x

(x) does

not possess useful Casimirs.

How to (asymptotically) stabilize x∗ ?


Control by interconnection:

Consider a controller port-Hamiltonian system

C :

ξ = Jc(ξ)∂Hc

∂ξ(ξ) + gc(ξ)uc, ξ ∈ Xc

yc = gT (ξ)∂Hc

∂ξ(ξ)

via the standard feedback interconnection

u = −yc, uc = y

cc

P

C

u

u

y

y


Then the closed-loop system is the port-Hamiltonian system

x

ξ

=

J(x) −g(x)gT

c (ξ)

gc(ξ)gT (x) Jc(ξ)

∂H∂x

(x)

∂Hc

∂ξ(ξ)

with state space X × X c, and total Hamiltonian H(x) +Hc(ξ).

Main idea: design the controller system in such a manner

that the closed-loop system has useful Casimirs C(x, ξ) !

This may lead to a suitable candidate Lyapunov function

V (x, ξ) := H(x) +Hc(ξ) + C(x, ξ)

with Hc to-be-determined.


Thus we look for functions C(x, ξ) satisfying

[∂T C∂x

(x, ξ) ∂T C∂ξ

(x, ξ)]

J(x) −g(x)gT

c (ξ)

gc(ξ)gT (x) Jc(ξ)

= 0

such that the candidate Lyapunov function

V (x, ξ) := H(x) +Hc(ξ) + C(x, ξ)

has a minimum at (x∗, ξ∗) for some (or a set of) ξ∗ ⇒ stability.

Remark: The set of such achievable closed-loop Casimirs C(x, ξ)

can be fully characterized.

Subsequently, one may add extra damping (directly or in the

dynamics of the controller) to achieve asymptotic stability.


Example: the ubiquitous pendulum

Consider the mathematical pendulum with Hamiltonian

H(q, p) =1

2p2 + (1 − cos q)

actuated by a torque u, with output y = p (angular velocity).

Suppose we wish to stabilize the pendulum at a non-zero angle q∗

and p∗ = 0.

Apply the nonlinear integral control

ξ = uc = y

−u = yc = ∂Hc

∂ξ(ξ)

which is a port-Hamiltonian controller system with Jc = 0.


Casimirs C(q, p, ξ) are found by solving

[∂C∂q

∂C∂p

∂C∂ξ

]

0 1 0

−1 0 −1

0 1 0

= 0

leading to Casimirs C(q, p, ξ) = K(q − ξ), and candidate Lyapunov

functions

V (q, p, ξ) =1

2p2 + (1 − cos q) +Hc(ξ) +K(q − ξ)

with the functions Hc and K to be determined.


For a local minimum, determine K and Hc such that

Equilibrium assignment

sin q∗ + ∂K∂z

(q∗ − ξ∗) = 0

−∂K∂z

(q∗ − ξ∗) + ∂Hc

∂ξ(ξ∗) = 0

Minimum condition

cos q∗ + ∂2K∂z2 (q∗ − ξ∗) 0 −∂2K

∂z2 (q∗ − ξ∗)

0 1 0

−∂2K∂z2 (q∗ − ξ∗) 0 ∂2K

∂z2 (q∗ − ξ∗) + ∂2Hc

∂ξ2 (ξ∗)

> 0

Many possible solutions.


The dissipation obstacle

Surprisingly, the presence of dissipation R 6= 0 may pose a problem !

C(x) is a Casimir for the Hamiltonian dynamics with dissipation

x = [J(x) −R(x)]∂H

∂x(x), J = JT , R = RT ≥ 0

iff

∂TC

∂x[J −R] = 0 ⇒ ∂TC

∂x[J −R]

∂C

∂x= 0 ⇒ ∂TC

∂xR∂C

∂x= 0 ⇒ ∂TC

∂xR = 0

and thus C is a Casimir iff

∂TC

∂x(x)J(x) = 0,

∂TC

∂x(x)R(x) = 0

The physical reason for the dissipation obstacle is that by using a

passive controller only equilibria where no energy-dissipation takes

place may be stabilized.


Similarly, if C(x, ξ) is a Casimir for the closed-loop port-Hamiltonian

system then it must satisfy

[∂T C∂x

(x, ξ) ∂T C∂ξ

(x, ξ)]

R(x) 0

0 Rc(ξ)

= 0

implying by semi-positivity of R(x) and Rc(x)

∂T C∂x

(x, ξ)R(x) = 0

∂T C∂ξ

(x, ξ)Rc(ξ) = 0

This is the dissipation obstacle, which implies that one cannot

shape the Lyapunov function in the coordinates that are directly

affected by energy dissipation.

Remark: For shaping the potential energy in mechanical systems

this is not a problem since dissipation enters in the differential

equations for the momenta.


To overcome the dissipation obstacle

Suppose one can find a mapping C : X → Rm, with its (transposed)

Jacobian matrix KT (x) := ∂C∂x

(x) satisfying

[J(x) −R(x)]K(x) + g(x) = 0

Construct now the interconnection and dissipation matrix of an

augmented system as

Jaug :=

J JK

KTJ KTJK

, Raug :=

R RK

KTR KTRK

By construction

[KT (x) | −I]Jaug = [KT (x) | −I]Raug = 0

implying that the components of C are Casimirs for the

Hamiltonian dynamics


x

ξ

= [Jaug −Raug]

∂H∂x

(x)

∂Hc

∂ξ(ξ)

Furthermore, since [J(x) −R(x)]K(x) + g(x) = 0

Jaug −Raug =

J −R [J −R]K

KT [J −R] KTJK −KTRK

=

J −R −g

[g − 2RK]T KTJK −KTRK

Thus the augmented system is a closed-loop system for a different

output !


Port-Hamiltonian systems with feedthrough term take the form

x = [J(x) −R(x)]∂H∂x

(x) + g(x)u

y = (g(x) + 2P (x))T ∂H∂x

(x) + [M(x) + S(x)]u,

with M skew-symmetric and S symmetric, while

R(x) P (x)

P T (x) S(x)

≥ 0


The augmented system is thus the feedback interconnection of the

nonlinear integral controller

ξ = uc

yc = ∂Hc

∂ξ(ξ)

with the plant port-Hamiltonian system with modified output with

feedthrough term

x = [J(x) −R(x)]∂H∂x

(x) + g(x)u

ymod = [g(x) − 2R(x)K(x)]T ∂H∂x

(x) + [−KT (x)J(x)K(x) +KT (x)R(x)K(x)]u


Generalization to feedback interconnection with

state-modulation.

Recall that KT (x) := ∂C∂x

(x) is a solution to

[J(x) −R(x)]K(x) + g(x) = 0. This can be generalized to

[J(x) −R(x)]K(x) + g(x)β(x) = 0

with β(x) an m×m design matrix.

The same scheme as above works if we extend the standard

feedback interconnection u = −yc, uc = y to the state-modulated

feedback

u = −β(x)yc, uc = βT (x)y

Note that K(x) is a solution for some β(x) iff

g⊥(x)[J(x) −R(x)]K(x) = 0

(In fact, β(x) := −(gT (x)g(x))−1gT (x)[J(x) −R(x)]K(x) does the job.)


A state feedback perspective: shaping the

Hamiltonian

Restrict (without much loss of generality) to Casimirs of the form

C(x, ξ) = ξj −Gj(x)

It follows that for all time instants

ξj = Gj(x) + cj , cj ∈ R

Suppose that in this way all control state components ξi can be

expressed as function

ξ = G(x)

of the plant state x. Then the dynamic feedback reduces to a

state feedback, and the Lyapunov function H(x) +Hc(ξ) + C(x, ξ)

reduces to the shaped Hamiltonian

H(x) +Hc(G(x))


A direct state feedback perspective:

Interconnection-Damping Assignment (IDA)-PBC control

A direct way to generate candidate Lyapunov functions Hd is to

look for state feedbacks u = uIDA(x) such that

[J(x) −R(x)]∂H

∂x(x) + g(x)uIDA(x) = [Jd(x) −Rd(x)]

∂Hd

∂x(x)

where Jd and Rd are newly assigned interconnection and damping

structures.

Remark: For mechanical systems IDA-PBC control is equivalent to

the theory of Controlled Lagrangians (Bloch, Leonard, Marsden, .).


For Jd = J and Rd = R (Basic IDA-PBC) this reduces to

[J(x) −R(x)]∂(Hd −H)

∂x(x) = g(x)uBIDA(x)

and thus in this case, there exists an uBIDA(x) if and only if

g⊥(x)[J(x) −R(x)]∂(Hd −H)

∂x(x) = 0

which is the same equation as obtained for stabilization by Casimir

generation with a state-modulated nonlinear integral controller !

Conclusion: Basic IDA-PBC ⇔ State-modulated Control by

Interconnection.


Shifted passivity w.r.t. a controlled equilibrium

(see Jayawardhana, Ortega). Consider a port-Hamiltonian system

x = Fz + gu, z = ∂H∂x

(x)

y = gT z

where F = J −R, g are constant, and a controlled equilibrium x0:

Fz0 + gu0 = 0, z0 =∂H

∂x(x0)

Define the shifted storage function

V (x) := Hp(x) − (x− x0)T ∂Hp

∂x(x0) −Hp(x0)

Note that ∂V∂x

= z − z0. It follows that

ddtV = (z − z0)

T x = (z − z0)T (Fz + gu) =

(z − z0)TF (z − z0) + (z − z0)

T g(u− u0) + (z − z0)T (Fz0 + gu0) ≤ (y − y0)

T (u− u0)

implying passivity w.r.t. the shifted inputs u− u0 and outputs y− y0.


New control paradigms

Example: Energy transfer control

Consider two port-Hamiltonian systems Σi

xi = Ji(xi)∂Hi

∂xi(xi) + gi(xi)ui

yi = gTi (xi)

∂Hi

∂xi(xi), i = 1, 2

Suppose we want to transfer the energy from the port-Hamiltonian

system Σ1 to the port-Hamiltonian system Σ2, while keeping the

total energy H1 +H2 constant.


This can be done by using the output feedback

u1

u2

=

0 −y1yT

2

y2yT1 0

y1

y2

It follows that the closed-loop system is energy-preserving.

However, for the individual energies

d

dtH1 = −yT

1 y1yT2 y2 = −||y1||2||y2||2 ≤ 0

implying that H1 is decreasing as long as ||y1|| and ||y2|| are

different from 0. On the other hand,

d

dtH2 = yT

2 y2yT1 y1 = ||y2||2||y1||2 ≥ 0

implying that H2 is increasing at the same rate. Has been

successfully applied to energy-efficient path-following control of

mechanical systems (cf. Duindam & Stramigioli).


Impedance control

Consider a system with two (not necessarily distinct) ports

x = [J(x) −R(x)]∂H∂x

(x) + g(x)u+ k(x)f, x ∈ X , u ∈ Rm

y = gT (x)∂H∂x

(x) u, y ∈ Rm

e = kT (x)∂H∂x

(x) f, e ∈ Rm

(28)

The relation between the f and e variables is called the

’impedance’ of the (f, e)-port. In Impedance Control (Hogan) one

tries to shape this impedance by using the control port

corresponding to u, y.

Typical application: the (f, e)-port corresponds to the end-point of

a robotic manipulator, while the (u, y)-port corresponds to

actuation.

Basic question: what are achievable impedances of the

(f, e)-port ?


Conclusions

• Beyond passivity by port-Hamiltonian systems theory.

• Control by interconnection and Casimir generation, IDA-PBC

control.

• Allows for ’physical’ interpretation of control strategies.

Suggests new control paradigms for nonlinear systems.

Use of passivity generally yields good robustness, but

performance theory is yet lacking.

See www.math.rug.nl/˜arjan for further info.

See recent book: Modeling and Control of Complex Physical

Systems; the Port-Hamiltonian Approach, Geoplex consortium,

Springer, 2009.


Part Distributed-parameter port-Hamiltonian

systems

fa

ea

fb

eb

a b

Figure 12: Simplest example: Transmission line

Telegrapher’s equations define the boundary control system

∂Q∂t

(z, t) = − ∂∂zI(z, t) = − ∂

∂z

φ(z,t)L(z)

∂φ∂t

(z, t) = − ∂∂zV (z, t) = − ∂

∂z

Q(z,t)C(z)

fa(t) = V (a, t), e1(t) = I(a, t)

fb(t) = V (b, t), e2(t) = I(b, t)


Transmission line as port-Hamiltonian system

Define internal flows fx = (fE , fM ) and efforts ex = (eE , eM ):

electric flow fE : [a, b] → R

magnetic flow fM : [a, b] → R

electric effort eE : [a, b] → R

magnetic effort eM : [a, b] → R

together with external boundary flows f = (fa, fb) and boundary

efforts e = (ea, eb). Define the infinite-dimensional Dirac structure

fE

fM

=

0 ∂

∂z

∂∂z

0

eE

eM

fa,b

ea,b

=

eE|a,b

eM |a,b


This defines a Dirac structure on the space of internal flows and

efforts and boundary flows and efforts.

Substituting (as in the lumped-parameter case)

fE = −∂Q∂t

fM = −∂ϕ∂t

fx = −x

eE = QC

= ∂H∂Q

eM = ϕL

= ∂H∂ϕ

ex =

∂H

∂x

with, for example, quadratic energy density

H(Q,ϕ) =1

2

Q2

C+

1

2

ϕ2

L

we recover the telegrapher’s equations.


Of course, the telegrapher’s equations can be rewritten as the

linear wave equation

∂2Q∂t2

= − ∂∂z

∂I∂t

= − ∂∂z

∂∂t

φL

=

− ∂∂z

1L

∂φ∂t

= ∂∂z

1L

∂∂z

QC

= 1LC

∂2Q∂z2

(provided L(z), C(z) do not depend on z), or similar expressions in

φ, I or V .

The same equations hold for a vibrating string, or for a

compressible gas/fluid in a one-dimensional pipe.

Basic question:

Which of the boundary variables fa, fb, ea, eb can be considered to

be inputs, and which outputs ?


Example 2: Shallow water equations; distributed-parameter

port-Hamiltonian system with non-quadratic Hamiltonian

The dynamics of the water in an open-channel canal can be

described by

∂t

h

v

+

v h

g v

∂z

h

v

= 0

with h(z, t) the height of the water at position z, and v(z, t) the

velocity (and g gravitational constant).

This can be written as a port-Hamiltonian system by recognizing

the total energy

H(h, v) =1

2

∫ b

a

[hv2 + gh2]dz


yielding the co-energy functionsa

eh = ∂H∂h

= 12v

2 + gh Bernoulli function

ev = ∂H∂v

= hv mass flow

It follows that the shallow water equations can be written, similarly

to the telegraphers equations, as

∂h∂t

(z, t) = − ∂∂z

∂H∂v

∂v∂t

(z, t) = − ∂∂z

∂H∂h

with boundary variables −hv|a,b and ( 12v

2 + gh)|a,b.

aDaniel Bernoulli, born in 1700 in Groningen as son of Johann Bernoulli, profes-

sor in mathematics at the University of Groningen and forerunner of the Calculus

of Variations (the Brachistochrone problem).


Paying tribute to history:

Figure 13: Johann Bernoulli, professor in Groningen 1695-1705.

Figure 14: Daniel Bernoulli, born in Groningen in 1700.


We obtain the energy balance

d

dt

∫ b

a

[hv2 + gh2]dz = −(hv)(1

2v2 + gh)|ba

which can be rewritten as

−v( 12gh

2)|ba − v( 12hv

2 + 12gh

2))|ba =

velocity × pressure + energy flux through the boundary


Conservation laws

All examples have the same structure

∂α1

∂t(z, t) = − ∂

∂z∂H∂α2

= − ∂∂zβ2

∂α2

∂t(z, t) = − ∂

∂z∂H∂α1

= − ∂∂zβ1

with boundary variables β1|a,b, β2|a,b, corresponding to two

coupled conservation laws:

ddt

∫ b

aα1 = −

∫ b

a∂∂zβ2 = β2(a) − β2(b)

ddt

∫ b

aα2 = −

∫ b

a∂∂zβ1 = β1(a) − β1(b)

(In the transmission line, α1 and α2 is charge- and flux-density, and

β1, β2 voltage V and current I, respectively.)


For some purposes it is illuminating to rewrite the equations in

terms of the co-energy variables β1, β2:

∂β1

∂t

∂β2

∂t

=

∂2H∂α2

1

∂2H∂α1α2

∂2H∂α2α1

∂2H∂α2

2

∂α1

∂t

∂α2

∂t

= −

∂2H∂α2

1

∂2H∂α1α2

∂2H∂α2α1

∂2H∂α2

2

∂β2

∂z

∂β1

∂z

For the transmission line this yields

∂V∂t

∂I∂t

= −

0 1

C

1L

0

∂V∂z

∂I∂z

The matrix is called the characteristic matrix, whose eigenvalues

are the characteristic velocities 1√LC

and − 1√LC

corresponding to

the characteristic eigenvectors (and curves).


For the shallow water equations this yields

∂β1

∂t

∂β2

∂t

= −

v g

h v

∂β1

∂z

∂β2

∂z

with

β1 =1

2v2 + gh, β2 = hv

being the Bernoulli function and mass flow, respectively.

This corresponds to two characteristic velocities v ±√gh, which

are, like in the transmission line case, of opposite sign (subcritical

or fluvial flow) if

v2 ≤ gh

Because the Hamiltonian is non-quadratic, and thus the pde’s are

nonlinear, the characteristic curves may intersect, corresponding

to shock waves.


Higher-dimensional spatial domain

Electromagnetic Field: Maxwell’s equations

∂D∂t

= curl H, E = ε−1D Faraday

∂B∂t

= − curl E, H = µ−1B Ampere


Differential version of

∫

∂SE = − d

dt

∫

SB Faraday

∫

∂SH = d

dt

∫

SD Ampere

This means that D and B are differential two-forms,

and E and H are differential one-forms!

Similar phenomenon in the telegrapher’s equations:

Voltage / current: functions on [a, b]

Charge / flux density: one-forms Qdz, ϕdz on [a, b]


General framework: Z is n-dimensional spatial domain with

boundary ∂Z.

The exterior derivative d : Ωk(Z) → Ωk+1(Z) incorporates all vector

calculus operations (grad, curl, div).

Define a Dirac structure on the space of flows and efforts:

f = (fE , fM , f) ∈ Ωp(Z) × Ωq(Z) × Ωn−q(∂Z)

e = (eE , eM , e) ∈ Ωn−p(Z) × Ωn−q(Z) × Ωn−p(∂Z)

by setting

fE(t, z) = ±deM (t, z), fM (t, z) = deE(t, z),

f(t) = eE(t, ∂Z), e(t) = ±eM (t, ∂Z)

(Transmission line: n = p = q = 1

Maxwell’s equations: n = 3, p = q = 2 )


Mixed lumped- and distributed-parameter

port-Hamiltonian systems

Typical example: power-converter connected via a transmission

line to a resistive load or an induction motor:

• The power-converter is a switching port-Hamiltonian

system.

• Transmission line is distributed-parameter port-Hamiltonian

system.

• Induction motor is a port-Hamiltonian system, with

Hamiltonian being the electro-mechanical energy.


Power converter connected to the load via transmission line

Il

VC = VL

Vl

E

L

CS

D

Rline

Figure 15: The Boost converter with a transmission line


Control by interconnection of

distributed-parameter port-Hamiltonian systems

Example: stabilization of the shallow water equations

∂h∂t

(z, t) = − ∂∂z

∂H∂v

(h, v)

∂v∂t

(z, t) = − ∂∂z

∂H∂h

(h, v)

with the 4 boundary variables

hv|a,b

−( 12v

2 + gh)|a,b

(mass flow and Bernoulli function at the boundary points a, b).


Suppose we want to control the water level h to a desired

height h∗.

An obvious ’physical’ controller is to add to one side of the canal,

say the right-end b, an infinite water reservoir of height h∗,

corresponding to the port-Hamiltonian ’source’ system

ξ = uc

yc = ∂Hc

∂ξ( = gh∗)

with Hamiltonian Hc(ξ) = gh∗ξ, by the feedback interconnection

uc = y = h(b)v(b), yc = −u =1

2v2(b) + gh(b)

This yields a closed-loop port-Hamiltonian system with total

Hamiltonian∫ l

0

1

2[hv2 + gh2]dz + gh∗ξ


By mass balance,∫ b

a

h(z, t)dz + ξ + c

is a Casimir for the closed-loop system. Thus we may take as

Lyapunov function

V (h, v, ξ) := 12

∫ b

a[hv2 + gh2]dz + gh∗ξ − gh∗[

∫ b

ah(z, t)dz + ξ] + 1

2g(b− a)h∗2

= 12

∫ b

a[hv2 + g(h− h∗)2]dz

which has a minimum at the desired set-point (h∗, v∗ = 0, ξ∗)

(with ξ∗ arbitrary).

Remark Note that the source port-Hamiltonian system is not

passive, since the Hamiltonian Hc(ξ) = gh∗ξ is not bounded from

below.


An alternative, passive, choice of the Hamiltonian controller system

is to take e.g.

Hc(ξ) =1

2gh∗ξ2

leading to the Lyapunov function

V (h, v, ξ) =1

2

∫ b

a

[hv2 + g(h− h∗)2]dz +1

2gh∗(ξ − 1)2

Asymptotic stability of the equilibrium (h∗, v∗ = 0, ξ∗ = 1) can be

obtained by adding ’damping’, that is, replacing uc = y = h(b)v(b) by

uc := y − ∂V

∂ξ(ξ) = h(b)v(b) − gh∗(ξ − 1)

leading to (if there is no power flow through the left-end a)

d

dtV = −gh∗(ξ − 1)2

(See also the work of Bastin & co-workers for related and more

refined results.)


Lumped port-Hamiltonian modeling of distributed

phenomena

Main idea: Discretize the spatial domain, while still identifying the

’correct’ nature of the lumped variables. Thus provide the direct

discrete analogue of the 0-forms, 1-forms, 2-forms, etc. employed in

the description of distributed-parameter port-Hamiltonian systems.

Alternative route: Discretize the port-Hamiltonian pde model in a

structure-preserving manner (using mixed finite elements).


Conclusions

• Port-Hamiltonian systems provide a unified framework for

modeling, analysis, and simulation of complex

lumped-parameter multi-physics systems.

• Starting point for (nonlinear) control. Suggests new control

paradigms.

• Inclusion of distributed-parameter components.

• Lumping of distributed-parameter systems to

finite-dimensional PH systems.

Structure-preserving model reduction

Extensions to thermodynamic systems and chemical reaction

networks.

Further exploration of the network graph information.


THANK YOU !, MERCI !See www.math.rug.nl/˜arjan for further info.

See selected references on webpage.

port-hamiltonian systems: from geometric network modeling

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