An introduction to port-Hamiltonian modeling of multi-physics … · 2019-02-13 · An introduction...

An introduction to port-Hamiltonian modeling of

multi-physics systems

Arjan van der Schaft

in collaboration with Bernhard Maschke, ..

Bernoulli Institute for Mathematics, CS and AIJan C. Willems Center for Systems and Control

University of Groningen, the Netherlands

Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian modelingin collaboration with Bernhard

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• Port-Hamiltonian modeling as a systematic framework for modeling ofinterconnected multi-physics systems

• Is based on viewing energy and power as ’lingua franca’ betweendifferent physical domains

• Combines Hamiltonian dynamics with network structure

• Extends Hamiltonian dynamics and geometric mechanics by includingenergy-dissipation and interaction with environment (or controller)

• Allows for port-Hamiltonian DAE systems


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Outline

1 Basics of port-based modeling and Dirac structures

2 Definition of port-Hamiltonian systems

3 Distributed-parameter port-Hamiltonian systems: Stokes-Dirac structures

4 Towards thermodynamics


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The basic picture

Dstorage dissipation

eS

fS

eR

fR

eP fP

Figure: Port-Hamiltonian system.


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The basic elements

Port-based modeling is based on viewing physical system asinterconnection of ideal energy processing elements, all linked by (vector)pairs of flow variables f ∈ F , and effort variables e ∈ E .

’Modeling for simulation and control’

F and E are linear spaces of equal dimension, with pairing < ·, · >defining power

< e | f >

Canonical choice: E = F∗ with < e | f >= eT f .

• Energy-storing elements:

x = −f

e = ∂H∂x

(x), H energy function

and hence ddtH = eT f .

• Energy-dissipating elements:

R(f , e) = 0, eT f ≤ 0


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• Energy-routing elements: generalized transformers, gyrators, · · · ,

f1 = Mf2, e2 = −MT e1, f = Je, J = −JT

Power-conserving:eT f = 0

• Ideal interconnection constraints, e.g.,

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

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

Ideal flow or effort constraints :f = 0, or e = 0

Also power-conserving:

eT f = e1f1 + e2f2 + · · · + ek fk = 0


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From energy-routing elements to Dirac structures

All energy-routing elements and interconnection constraints have followingproperties in common.Described by linear equations:

Ff + Ee = 0, f , e ∈ Rk

satisfyingeT f = e1f1 + e2f2 + · · · + ek fk = 0,

while furthermorerank

[F E

]= k

All energy-routing elements and interconnection constraints will begrouped into one geometric object: the Dirac structure.


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Definition of Dirac structures

Definition

A (constant) Dirac structure is a subspace

D ⊂ F × E

such that

(i) eT f = 0 for all (f , e) ∈ D,

(ii) dimD = dimF .

Example; for any skew-symmetric map J : E → F its graph{(f , e) ∈ F × E | f = Je} is Dirac structure.


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Alternative definition of Dirac structure; e.g., for

infinite-dimensional case

Symmetrized form of power

< e | f >= eT f , (f , e) ∈ F × E .

Symmetrization leads to the indefinite bilinear form ≪,≫ on F × E :

≪(f a, ea), (f b, eb) ≫ := < ea | f b > + < eb | f a >,

(f a, ea), (f b, eb) ∈ F × E .


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Alternative definition of Dirac structure

Definition

A (constant) Dirac structure is subspace

D ⊂ F × E

such thatD = D⊥⊥,

where ⊥⊥ denotes orthogonal companion with respect to ≪,≫.

All can be generalized to Dirac structures on manifolds X :

D(x) ⊂ TxX × T ∗x X

is a Dirac structure as before for any x ∈ X .


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Outline






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The basic picture

Dstorage dissipation

eS

fS

eR

fR

eP fP

Figure: Port-Hamiltonian system.


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Closethe energy-storing and energy-dissipating ports of the Dirac structureD by their constitutive relations:

−x = fS ,∂H

∂x(x) = eS

respectivelyR(fR , eR) = 0

This leads to the port-Hamiltonian DAEs

(−x(t), fR(t), fP(t), ∂H∂x

(x(t)), eR (t), eP(t)) ∈ Dt ∈ R

R(fR(t), eR(t)) = 0

Thus DAEs of a special form.


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Example (The ubiquitous mass-spring 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


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Example

Dirac structure linking fs , es , fm, em,F , v as

fs = −em = −v , fm = es − F

Power-conserving since fses + fmem + vF = 0. Yields port-Hamiltoniansystem [

q

p

]=

[0 1

−1 0

][∂H∂q

(q, p)

∂H∂p

(q, p)

]+

[0

1

]F

v =[0 1

][∂H∂q

(q, p)

∂H∂p

(q, p)

]

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


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Example (Electro-mechanical systems)

q

p

ϕ

=

0 1 0−1 0 00 0 −R

∂H∂q

(q, p, φ)∂H∂p

(q, p, φ)∂H∂ϕ

(q, p, φ)

+

001

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

2L(q)


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DC motor

_

V

I

J

b

R L

K

ω

τ

+

Figure: DC motor.

6 interconnected subsystems:◦ 2 energy-storing elements: inductor L with state ϕ (flux), and rotationalinertia J with state p (angular momentum);◦ 2 energy-dissipating elements: resistor R and friction b;◦ gyrator K ;◦ voltage source V .


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The energy-storing elements (here assumed to be linear) are given by

Inductor:

ϕ = −VL

I =d

dϕ

(1

2Lϕ2

)=

ϕ

L,

Inertia:

p = −τJ

ω =d

dp

(1

2Jp2)

=p

J

Hence corresponding total Hamiltonian H(p, φ) = 12Lφ

2 + 12J p

2.Energy-dissipating relations (also assumed linear) are

VR = −RI , τb = −bω,

with R , b > 0, where τb damping torque.Energy-routing gyrator (converting magnetic power into mechanical, andconversely) are

VK = −Kω, τK = KI


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The subsystems are interconnected by

VL + VR + VK + V = 0, τJ + τb + τK + τ = 0.

Dirac structure is defined by these interconnection equations, togetherwith equations for gyrator.

Results in port-Hamiltonian model

[ϕ

p

]=

[−R −K

K −b

]

ϕ

Lp

J

+

[1 00 1

] [V

τ

],

[I

ω

]=

[1 00 1

]

ϕ

Lp

J


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Triggered by Mark Peletier’s talk

Special type of port-Hamiltonian systems

x = −∂R

∂v

(∂H

∂x(x)

)

For example, a nonlinear mass-damper system (with x the momenta ofmasses, v velocities):

H(x) =1

2xTM−1x , v = M−1x , e.g. R(v) =

1

4v4

Similar to Brayton-Moser formulation of electrical networks; e.g., nonlinearRL-networks.


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Two main properties of port-Hamiltonian systems

Power-conservation of Dirac structure

eTS fS + eTR fR + eTP fP = 0

implies energy-balance

dHdt

(x(t)) = ∂H∂xT

(x(t))x(t) =

eTR (t)fR(t) + eTP (t)fP(t) ≤

eTP (t)fPt)

Implies passivity if H is bounded from below.


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Port-Hamiltonian systems are compositional

The interconnection of port-Hamiltonian systems through anyinterconnection Dirac structure is again port-Hamiltonian:

- Total Hamiltonian H is sum of Hamiltonians of subsystems:

H = H1 + · · · + HN

- Total Dirac structure is composition of Dirac structures of subsystems,together with interconnection Dirac structure.- Total energy-dissipating (resistive) part is direct product ofenergy-dissipating parts of subsystems.


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DAE analysis

Algebraic constraints are specified by Dirac structure D and H.

Define projection π∗(D) ⊂ E = X ∗.

Algebraic constraints on x are

∂H

∂x(x) ∈ π∗(D)

’Dual’ to conserved quantities F determined as

∂F

∂x(x) ∈ (π(D))⊥


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Interesting generalization (see also Volker Mehrmann, Hans Zwart, ..):

Replace in the definition of a port-Hamiltonian system the graph specifiedby the energy-storage

(x ,∂H

∂x(x)) ∈ F = X × E = X ∗

by a general Lagrangian submanifold L ⊂ T ∗X .

Leads to the definition of generalized port-Hamiltonian system (D, L).

Algebraic constraints are now induced by D as well as by L:

x ∈ π(L)

In some sense ’easier’ form of algebraic constraints.


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Any Dirac structure D ⊂ X × X ∗ can be embedded into the graph ofskew-symmetric map:

Let π∗(D) ⊂ X ∗ be (n − k)-dimensional. Define Λ := Rk .

Then there exists full-rank n × k matrix G and skew-symmetric n × n

matrix J such that D is given as set of all points (f , e) ∈ X × X ∗

satisfying for some λ∗ ∈ Λ∗

−f = Je + Gλ∗

0 = GT e

Conversely, any such equations for skew-symmetric map J : X ∗ → Xdefine Dirac structure.Hence Dirac structures correspond to extended skew-symmetric maps

J =

[−J −G

GT 0

]: X ∗ × Λ∗ → X × Λ


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Now for any H : X → R define the extended Lagrangian submanifoldL ⊂ T ∗X × T ∗Λ as the set of all points

{(x ,∂H

∂x(x), 0, λ∗)

Defines a generalized port-Hamiltonian system, with algebraic constraintsdetermined as

λ = 0

Conversely, Lagrangian algebraic constraints can be transformed into Diracalgebraic constraints.


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Outline






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Motivation

In many applications the system, or some of its subsystems, isdistributed-parameter.

Examples:

1. Power-converter connected to an electrical machine via a transmissionline,2. Hydraulic networks with fluid pipes,3. Multi-body systems with flexible components,etc.


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Distributed-parameter port-Hamiltonian systems

Simplest example: transmission line

fa

ea

fb

eb

a b

Telegrapher’s equations define boundary control system

∂Q∂t

(z , t) = − ∂∂zI (z , t) = − ∂

∂zφ(z ,t)L(z)

∂φ∂t

(z , t) = − ∂∂zV (z , t) = − ∂

∂zQ(z ,t)C(z)

fa(t) = V (a, t), ea(t) = I (a, t)fb(t) = V (b, t), eb(t) = I (b, t)


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Stokes-Dirac structure

Define internal flows fS = (fE , fM) and efforts eS = (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 boundary flows f = (fa, fb) and efforts e = (ea, eb).

Define infinite-dimensional subspace D of(C∞[a, b])2 × (C∞[a, b])2 × R

2 × R2 by equations

[fEfM

]=

[0 ∂

∂z∂∂z

0

] [eEeM

]

[faea

]=

[eE (a)eM(a)

],

[fbeb

]=

[eE (b)eM(b)

]


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D is Dirac structure: D = D⊥⊥

Use integration by parts

For any (fE , fM , eE , eM , fa, fb, ea, eb) ∈ D

∫ b

a[eE (z)fE (z) + eM(z)fM(z)]dz − ebfb + eafa =

∫ b

a[eE (z) ∂

∂zeM(z) + eM(z) ∂

∂zeE (z)]dz − ebfb + eafa =

∫ b

a[−eM(z) ∂

∂zeE (z)dz + eM(z) ∂

∂zeE (z)]dz(+ebfb − eafa) − ebfb + eafa = 0

Thus eT f = 0 for all (f , e) ∈ D. This implies for all (f1, e1), (f2, e2) ∈ D

0 = (e1 + e2)T (f1 + f2) = eT1 f1 + eT2 f2 + eT1 f2 + eT2 f1 =

eT1 f2 + eT2 f1 =≪ (f1, e1), (f2, e2) ≫

Hence D ⊂ D⊥⊥. Similar argument for D⊥⊥ ⊂ D.Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian modeling

in collaboration with Bernhard/ 37

Telegrapher’s equations as port-Hamiltonian system

Substituting (as in the finite-dimensional case)

fE = −∂Q∂t

fM = −∂ϕ∂t

fS = −x

eE = QC

= ∂H∂Q

(Q, ϕ)

eM = ϕL

= ∂H∂ϕ

(Q, ϕ)

eS =

∂H

∂x(x)

with energy density

H(Q, ϕ) =1

2

Q2

C+

1

2

ϕ2

L

we recover the telegrapher’s equations.


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Outline






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Consider two heat compartments, interacting by heat flow throughconducting wall. The two systems, indexed by 1 and 2, exchange a heatflow q given by Fourier’s law

q = λ(T1 − T2),

with temperatures

Ti =∂Ui

∂Si(Si), i = 1, 2,

with U1(S1),U2(S2) the internal energies of the two compartments.

Leads to pseudo port-Hamiltonian system

[S1

S2

]=

[− q

T1

qT2

]=

[−λT1−T2

T1

λT1−T2T2

]=

[0 λ( 1

T1− 1

T2)

−λ( 1T1

− 1T2

) 0

][ ∂U∂S1

∂U∂S2

]

with total energy U(S1,S2) := U1(S1) + U1(S2).


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Pseudo port-Hamiltonian, since the skew-symmetric map

[0 λ( 1

T1− 1

T2)

−λ( 1T1

− 1T2

) 0

]

does not depend on state variables S1,S2, but on co-energy variablesTi = ∂Ui

∂Si(Si).

Therefore does not define Dirac structure on state space R2 with

coordinates S1,S2.

Instead, example of the type (with J(e) = −JT (e) )

x = J(e)e, e =∂H

∂x(x), J(e)e nonlinear in e

As a consequence

S1 + S2 =(T1 − T2)2

T1T2≥ 0

Total entropy non-decreasing; i.e., irreversibility.

Port-Hamiltonian framework is not general enough !Arjan van der Schaft (Univ. of Groningen) Port-Hamiltonian modeling

in collaboration with Bernhard/ 37

Conclusions

• Port-based modeling: energy-storage, energy-dissipation,energy-routing

• Underlying geometry determined by Dirac structure

• Transformation of Dirac algebraic constraints into constraintsdetermined by Lagrangian submanifold

• Infinite-dimensional: Stokes-Dirac structure or extensions

• Extension to thermodynamics; see talk by Siep Weiland

• Structure-preserving model reduction of port-Hamiltonian systems


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Some key references

• vdS, D. Jeltsema, Port-Hamiltonian Systems Theory: An Introductory

Overview, now publishers, 2014; see my website for pdf.

• vdS, L2-Gain and Passivity Techniques in Nonlinear Control, 3rdedition, 2017.

• vdS, B. Maschke, ’Hamiltonian formulation of distributed-parametersystems with boundary energy flow’, Journal of Geometry and

Physics, 2002.

• vdS, B. Maschke, ’Port-Hamiltonian systems on graphs’, SIAM J.

Control Optim., 2013.

• vdS, B. Maschke, ’Generalized port-Hamiltonian DAE systems’,Systems & Control Letters, 2018.

• vdS, B. Maschke. ’Geometry of thermodynamic processes’, Entropy,2018.


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An introduction to port-Hamiltonian modeling of multi-physics … · 2019-02-13 · An introduction...

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