Hamiltonian control systems - TU Ilmenau

Hamiltonian control systems

From modeling to analysis and control

Arjan van der SchaftJohann Bernoulli Institute for Mathematics and Computer Science,

University of Groningen, the Netherlands

Dimitri JeltsemaDelft Institute of Applied Mathematics,

Delft University of Technology, the Netherlands

Elgersburg School, March, 2012

Arjan van der Schaft Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands DimitriHamiltonian control systemsElgersburg School, March, 2012 1 /

108

Plan for the week

Outline

1 Plan for the week

2 Basics of port-based modeling

3 Definition of port-Hamiltonian systems

4 Scattering: from power variables to wave variables

5 Port-Hamiltonian systems and passivity

6 Other properties of port-Hamiltonian systems

7 Distributed parameter port-Hamiltonian systems

8 Conclusions and Outlook


108

Plan for the week

Table of Contents for the Week

• Monday: definition of port-Hamiltonian systems and examples

• Tuesday: interconnection of port-Hamiltonian systems; properties ofport-Hamiltonian systems

• Wednesday: control by interconnection and passivity-based control

• Thursday: control case studies; relations with other modelingapproaches

• Friday: distributed-parameter port-Hamiltonian systems


108

Plan for the week

Today

• Basics of port-based modeling

• Dirac structures

• Definition of port-Hamiltonian systems

• Classes of port-Hamiltonian systems

Overall message:first principles physical modeling leads to a generalized Hamiltoniansystem description with lots of useful information for analysis andcontrol; for linear, nonlinear and infinite-dimensional systems.


108

Basics of port-based modeling

Outline

1 Plan for the week









108


The basic elements

Port-based modeling is based on viewing the physical system as theinterconnection of ideal energy processing elements, all expressed in(vector) pairs of flow variables f ∈ F , and effort variables e ∈ E , where Fand E are linear spaces of equal dimension.Furthermore, there is a pairing between F and E defining the power

< e | f >

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

• Energy-storing elements:

x = −f

e = ∂H∂x

(x)

• Energy-dissipating elements:

R(f , e) = 0, eT f ≤ 0


108


The basic elements

• Energy-routing elements: generalized transformers, gyrators:

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

They are power-conserving:

eT f = 0

• Ideal interconnection constraints

0-junction :e1 = e2 = · · · = ek , f1 + f2 + · · ·+ fk = 01-junction :f1 = f2 = · · · = fk , e1 + e2 + · · ·+ ek = 0Ideal flow or effort constraints :f = 0, or e = 0

Also power-conserving:

e1f1 + e2f2 + · · ·+ ek fk = 0


108


From power-conserving elements to Dirac structures

All power-conserving elements/interconnection constraints have thefollowing properties in common.They are described by linear equations:

Ff + Ee = 0, f , e ∈ Rℓ

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

while furthermorerank

[F E

]= ℓ

All power-conserving elements/interconnection constraints will be groupedinto one geometric object: the Dirac structure.


108


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 .

For any skew-symmetric map J : E → F its graph given as(f , e) ∈ F × E | f = Je is a Dirac structure !


108


Alternative definition of Dirac structure

Symmetrized form of power

< e | f >= eT f , (f , e) ∈ F × E .

is 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 .


108


Alternative definition of Dirac structure

Definition

A (constant) Dirac structure is a subspace

D ⊂ F × E

such thatD = D⊥⊥,

where ⊥⊥ denotes orthogonal complement with respect to the bilinear form≪,≫.


108


Towards port-Hamiltonian systems

Four-port model: Consider a Dirac structure D ⊂ F × E , where

f = (fS , fR , fI , fC ), e = (eS , eR , eI , eC )

S (storage)

R (resistive)

C (control)

I (interaction)

D


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Definition of port-Hamiltonian systems

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108


The port-Hamiltonian system is defined by closing the energy-storage andresistive ports by their constitutive relations

−x = fS ,∂H

∂x(x) = eS

respectivelyR(fR , eR) = 0

This leads to the DAEs

(−x(t), fR(t), fI (t), fC (t),∂H∂x

(x(t)), eR (t), eI (t), eC (t)) ∈ Dt ∈ R

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

Thus DAEs of a special form, determined by the Dirac structure andenergy-conserving and energy-dissipating constitutive relations.Definition can be also extended to cover infinite-dimensional physicalsystems.


108


Energy-balance

Power-conservation

eTS fS + eTR fR + eTI fI + eTC fC = 0

implies the energy-balance

dHdt(x(t)) = ∂H

∂x(x(t))x(t) =

eTR (t)fR(t) + eTI (t)fI (t) + eTC (t)fC (t) ≤

eTI (t)fI (t) + eTC (t)fC (t)

This implies that if the Hamiltonian H has a minimum at the equilibriumunder study, then it can be used as a Lyapunov function.


108


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


108


Example

interconnected by the Dirac structure

fs = −em = −y , fm = es − u

(power-conserving since fses + fmem + uy = 0) yields the port-Hamiltoniansystem

[qp

]

=

[0 1−1 0

][∂H∂q

(q, p)∂H∂p

(q, p)

]

+

[01

]

u

y =[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)

qpϕ

=

0 1 0−1 0 00 0 − 1

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

2k1(1− qk2)


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Network modeling is prevailing in modeling and simulation oflumped-parameter physical systems (multi-body systems, electricalcircuits, 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 ofDAEs, seemingly without any structure.

Port-based modeling and port-Hamiltonian system theory identifies theunderlying structure of network models of physical systems, to be used foranalysis, simulation and control.


108


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 vectorsubbundle D ⊂ TM ⊕ T ∗M such that for every x ∈ M the vector space

D(x) ⊂ TxM × T ∗xM

is a Dirac structure as before.


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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 A(q)λ the constraint forces.Dirac structure is defined by the symplectic form on T ∗Q together withconstraints AT (q)q = 0 and force matrix B(q).

Can be systematically extended to general multi-body systems.


108


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 rollingconstraints (rolling without slipping) are

x = θ cosϕ, y = θ sinϕ

The total energy is

H =1

2p2x +

1

2p2y +

1

2p2θ +

1

2p2ϕ

and the constraints can be rewritten as

px = pθ cosϕ, py = pθ sinϕ.


108


Input-state-output port-Hamiltonian systems

Consider a Dirac structure D given as the graph of the skew-symmetricmap

[fSfC

]

=

[−J(x) −g(x)gT (x) 0

] [eSeC

]

,

leading (fS = −x , eS = ∂H∂x

(x)) to a port-Hamiltonian system as before

x = J(x)∂H∂x

(x) + g(x)eC , x ∈ X , eC ∈ Rm

fC = gT (x)∂H∂x

(x), fC ∈ Rm


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Energy-dissipation is included by terminating some of the ports by linearresistive elements

fR = −R(eR), R = RT ≥ 0

Writing out

x = J(x)∂H

∂x(x) + gR(x)fR + g(x)eC , eR = gT

R (x)∂H

∂x(x)

this leads to an input-state-output port-Hamiltonian system given as

x = [J(x)− R(x)]∂H∂x

(x) + g(x)eC

fC = gT (x)∂H∂x

(x)

whereR(x) = gR(x)Rg

TR (x) ≥ 0


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Multi-modal physical systems

Example (Bouncing pogo-stick)

Consider a vertically bouncing pogo-stick consisting of a mass m and amassless foot, interconnected by a linear spring (stiffness k and rest-lengthx0) and a linear damper d .

m

kd

g

xy sum of forces

zero on foot

spring/damperin series

foot fixedto ground

spring/damperparallel

Figure: Model of a bouncing pogo-stick: definition of the variables (left), situationwithout ground contact (middle), and situation with ground contact (right).


108


Example

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.


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Example

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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Example

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

We obtain a switching port-Hamiltonian system, specified by a Diracstructure Ds depending on the switch position s ∈ 0, 1n (here n denotesthe number of independent switches).

Another interesting example is switching electrical circuits and powerconverters.


108


Geometric definition of a port-Hamiltonian system

Consider a Dirac structure D ⊂ F × E , where

f = (fS , fR , fI , fC ), e = (eS , eR , eI , eC )

S (storage)

R (resistive)

C (control)

I (interaction)

D

(−x(t), fR(t), fI (t), fC (t),∂H∂x

(x(t)), eR (t), eI (t), eC (t)) ∈ D,

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


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Equational representations

Dirac structures, and therefore port-Hamiltonian systems, admit differentequational representations, with different properties for analysis,simulation, and control.Let D ⊂ F × F∗, with dimF = n, be a Dirac structure.

1. Kernel and Image representation

D = (f , e) ∈ F × F∗ | Ff + Ee = 0for 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) ∈ F × F∗ | f = ETλ, e = FTλ, λ ∈ R

n.Arjan van der Schaft Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands DimitriHamiltonian control systems

Elgersburg School, March, 2012 30 /108


In case of the presence of an energy-conserving port S , anenergy-dissipating port R and an external port the Dirac structure D canthus be given in kernel representation as

D =

(fS , eS , fR , eR , f , e) ∈ X × X ∗ ×FR ×F∗R ×F × F∗ |

FS fS + ESeS + FR fR + EReR + Ff + Ee = 0

with

(i) ESFTS + FSE

TS + ERF

TR + FRE

TR + EFT + FET = 0

(ii) rank[FS | ES | FR | ER | F | E

]= dim(X × FR ×F)

Then the port-Hamiltonian system is given by the set of DAEs

FS x(t) = ES

∂H

∂x(x(t)) + FR fR(t) + EeR(t) + Ff (t) + Ee(t),

where the vectors fR , eR satisfy at all time-instants the energy-dissipatingconstitutive relations; e.g., fR(t) = ReR(t).


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2. Constrained input-output representationEvery Dirac structure D can be written as

D = (f , e) ∈ F × F∗ | f = Je + Gλ,GTe = 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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In the absence of energy-dissipating and external ports it follows that anyport-Hamiltonian system can be represented as

x = J(x)∂H∂x

(x) + G (x)λ, J(x) = −JT (x)

0 = GT (x)∂H∂x

(x)

A typical example are mechanical systems with kinematic constraints

q = ∂H∂p

(q, p)

p = −∂H∂q

(q, p) + A(q)λ

0 = AT (q)∂H∂p

(q, p) (= AT (q)q)

where A(q)λ are the constraint forces.


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3. Hybrid input-output representationLet 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.


108


From one representation to another

In principle we can freely move from one representation to another.For example, consider a linear port-Hamiltonian system, withoutenergy-dissipating and external ports. This can be always written as

x = JQx + Gλ, J = −JT ,Q = QT

0 = GTQx , H(x) = 12x

TQx

The constraint forces Gλ can be eliminated by pre-multiplying the firstequation by the annihilating matrix G⊥, leading to the DAE system (inkernel representation)

[G⊥

0

]

x =

[G⊥JGT

]

Qx

The corresponding pencil

s

[G⊥

0

]

−[G⊥JQGTQ

]

is non-singular if GTQG has full rank, and in fact, the system has indexone if and only if GTQG has full rank.


108


More explicitly, define the coordinate transformation (assuming w.l.o.g.that G has full rank)

z =

[G⊥

GT

]

x =: Vx

This leads to the transformed system

z = (VJV T )(V−TQV−1)(Vx) + VGλ = JQz +

[0

GTG

]

λ

0 = GTQx =[0 GTG

]Qz

Since GTG has full rank, this means that the constraint amounts to

(Qz)2 = 0, where Qz =

[(Qz)1(Qz)2

]

.

Hence the system reduces to the Hamiltonian differential equations

z1 = J11(Q11 − Q12Q−122 Q21)z1


108


In general, port-Hamiltonian systems will have generally index one.Consider the nonlinear port-Hamiltonian system

x = J(x)∂H∂x

(x) + G (x)λ, J(x) = −JT (x)

0 = GT (x)∂H∂x

(x)

Then differentiation of the algebraic constraints yields

0 =d

dtGT (x)

∂H

∂x(x) = · · ·+ GT (x)

∂2H

∂x2(x)G (x)λ,

which can be solved for λ as long as the rank of

GT (x)∂2H

∂x2(x)G (x)

is equal to the rank of G (x).

In particular if the Hessian ∂2H∂x2

(x) is invertible, then the system has indexone.


108


Canonical coordinates

Any constant Dirac structure D ⊂ F × F∗ can be represented as follows.There exist linear coordinates x = (q, p, r , s) for F (with dim q = dim p),such that in the corresponding bases for (fq, fp , fr , fs) for TxF and(eq , ep , er , es) for F∗, the Dirac structure is given as

fq = −epfp = eqfr = 0es = 0

In such canonical coordinates a port-Hamiltonian system on X = Fwithout energy-dissipating and external ports can be written as

q = ∂H∂p

(q, p, r , s)

p = −∂H∂q

(q, p, r , s)

r = 0

0 = ∂H∂s

(q, p, r , s)


108


Excursion to integrability

A Dirac structure on a manifold X is integrable if it is possible to findlocal coordinates such that, in these coordinates, the Dirac structurebecomes a constant Dirac structure, that is, it is not modulated anymoreby the state variables.Thus then there also exist canonical coordinates.

First case

Let the modulated Dirac structure D be given for every x ∈ X as thegraph of a skew-symmetric mapping J(x) from the co-tangent space T ∗

xXto the tangent space TxX . Integrability in this case means that J(x)satisfies the conditions

n∑

l=1

[

Jlj(x)∂Jik∂xl

(x) + Jli (x)∂Jkj∂xl

(x) + Jlk(x)∂Jji∂xl

(x)

]

= 0, i , j , k = 1, . . . , n


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In this case we may find, by Darboux’s theorem around any point x0 wherethe rank of the matrix J(x) is constant, local canonical coordinatesx = (q, p, r) in which the matrix J(x) becomes the constantskew-symmetric matrix

0 −Ik 0Ik 0 00 0 0

Then J(x) defines a Poisson bracket on X , given for every F ,G : X → R

as

F ,G =∂TF

∂xJ(x)

∂G

∂x

Indeed, by the integrability condition the Jacobi-identity holds:

F , G ,K + G , K ,F + K , F ,G = 0

for all functions F ,G ,K .


108


Second case

A similar story holds for a Dirac structure given as the graph of askew-symmetric mapping ω(x) from the tangent space TxX to theco-tangent space T ∗

x X . In this case the integrability conditions take theform

∂ωij

∂xk(x) +

∂ωki

∂xj(x) +

∂ωjk

∂xi(x) = 0, i , j , k = 1, . . . , n

The skew-symmetric matrix ω(x) can be regarded as the coordinaterepresentation of a differential two-form ω on X , that isω =

∑ni=1,j=1 dxi ∧ dxj , and the integrability condition corresponds to the

closedness of this two-form (dω = 0).


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The differential two-form ω is called a pre-symplectic structure, and asymplectic structure if the rank of ω(x) is equal to the dimension of X . Bya version of Darboux’s theorem we may find, around any point x0 wherethe rank of the matrix ω(x) is constant, local coordinates x = (q, p, s) inwhich the matrix ω(x) becomes the constant skew-symmetric matrix

0 Ik 0−Ik 0 00 0 0


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For general Dirac structures, integrability is defined as

Definition

A Dirac structure D on X is integrable if for arbitrary pairs of smoothvector fields and differential one-forms (X1, α1), (X2, α2),(X3, α3) ∈ Dthere holds

< LX1α2 | X3 > + < LX2

α3 | X1 > + < LX3α1 | X2 >= 0

with LXidenoting the Lie-derivative.


108


The Dirac structure corresponding to mechanical systems with kinematicconstraints

D = (fq , fp, eq , ep) | fq = −ep, fp = eq + A(q)λ,AT (q)ep = 0

is integrable if and only if the kinematic constraints

AT (q)q = 0

are holonomic, which means that it is possible to find configurationcoordinates q = (q1, . . . , qn) such that the constraints are equivalentlyexpressed as

qn−k+1 = qn−k+2 = · · · = qn = 0 ,

In this case one may eliminate the configuration variables qn−k+1, . . . , qn,since the kinematic constraints are equivalent to the geometric constraints

qn−k+1 = cn−k+1, . . . , qn = cn ,

for certain constants cn−k+1, . . . , cn determined by the initial conditions.Arjan van der Schaft Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands DimitriHamiltonian control systems


Scattering: from power variables to wave variables

Outline

1 Plan for the week









108


Scattering representations

Main idea: Use in the total space of port-variables F × F∗ a differentsplitting than the ’canonical’ duality splitting (in flows f ∈ F and effortse ∈ F∗).Consider the canonical bilinear form

≪ (f1, e1), (f2, e2) ≫=< e1 | f2 > + < e2 | f1 >, (fi , ei ) ∈ F×F∗, i = 1, 2,

on F × F∗. This is an indefinite bilinear form, which has n singular values+1 and n singular values −1 (n = dimF).


108


A pair of subspaces Σ+,Σ− ⊂ F × F∗ is a pair of scattering subspaces if

(i) Σ+ ⊕ Σ− = F × F∗

(ii) ≪ σ+1 , σ

+2 ≫> 0 for all σ+

1 , σ+2 ∈ Σ+ unequal to 0.

≪ σ−1 , σ

−2 ≫< 0 for all σ−

1 , σ−2 ∈ Σ− unequal to 0.

(iii) ≪ σ+, σ− ≫= 0 for all σ+ ∈ Σ+, σ− ∈ Σ−.

It is readily seen that any pair of scattering subspaces (Σ+,Σ−) satisfies

dimΣ+ = dimΣ− = dimF


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The collection of pairs of scattering subspaces can be characterized asfollows.

Lemma

Let (Σ+,Σ−) be a pair of scattering subspaces. Then there exists aninvertible linear map

R : F → F∗

with< (R + R∗)f |f > > 0, for all 0 6= f ∈ F ,

such thatΣ+ := (R−1e, e) ∈ F × F∗ | e ∈ F∗

Σ− := (−f ,R∗f ) ∈ F × F∗ | f ∈ FConversely, for any invertible linear map R : F → F∗ the pair (Σ+,Σ−) isa pair of scattering subspaces.


108


Let (Σ+,Σ−) be a pair of scattering subspaces. Then every pair of powervectors (f , e) ∈ F × F∗ can be represented as

(f , e) = σ+ + σ−

for a uniquely defined σ+ ∈ Σ+, σ− ∈ Σ−, called the wave vectors. Usingorthogonality of Σ+ w.r.t. Σ− for all (fi , ei ) = σ+

i + σ−i , i = 1, 2

≪ (f1, e1), (f2, e2) ≫=

< σ+1 , σ

+2 >Σ+ − < σ−

1 , σ−2 >Σ−

where <,>Σ+ is the restriction of ≪,≫ to Σ+, and <,>Σ− is minus therestriction of ≪,≫ to Σ−.


108


Taking f1 = f2 = f , e1 = e2 = e, and thusσ+1 = σ+

2 = σ+, σ−1 = σ−

2 = σ−, leads to

< e | f >= 12 ≪ (f , e), (f , e) ≫=

12 < σ+, σ+ >Σ+ −1

2 < σ−, σ− >Σ−

This yields the following interpretation of the wave vectors: The vector σ+

can be regarded as the incoming wave vector, with half times its normbeing the incoming power, and the vector σ− is the outgoing wave vector,with half times its norm being the outgoing power.


108


Let D ⊂ F × F∗ be a Dirac structure, that is, D = D⊥⊥ with respect to≪,≫. What is its representation in wave vectors ?Since ≪,≫ is zero restricted to D it follows that for every pair ofscattering subspaces (Σ+,Σ−)

D ∩ Σ+ = 0, D ∩ Σ− = 0,

and hence D can be represented as the graph of an invertible linear mapO : Σ+ → Σ−, that is,

D = (σ+, σ−) | σ− = Oσ+, σ+ ∈ Σ+


108


Furthermore< σ+

1 , σ+2 >Σ+=< Oσ+

1 ,Oσ+2 >

for every σ+1 , σ

+2 ∈ Σ+, and thus

O : (Σ+, <,>Σ+) → (Σ−, <,>Σ−)

is a unitary map (isometry).Conversely, every unitary map O as above defines a Dirac structure. Thusfor every pair of scattering subspaces (Σ+,Σ−) we have a one-to-onecorrespondence between unitary maps and Dirac structures D ⊂ F ×F∗.


108


Inner product scattering representations

A particular useful class of scattering subspaces (Σ+,Σ−) are thosedefined by an invertible map R : F → F∗ such that R = R∗. In this caseR is determined by the inner product on F defined as

< f1, f2 >R :=< Rf1 | f2 >=< Rf2 | f1 >

or equivalently by the inner product on F∗

< e1, e2 >R−1 :=< e2 | R−1f1 >=< e1 | R−1f2 >


108


Define for every (f , e) ∈ F × F∗ the pair (s+, s−) by

s+ := 1√2(e + Rf ) ∈ F∗

s− := 1√2(e − Rf ) ∈ F∗

Let (s+i , s−i ) correspond to (fi , ei ), i = 1, 2.

2 < s+1 , s+2 >R−1 = < e1, e2 >R−1 + < f1, f2 >R

+ ≪ (f1, e1), (f2, e2) ≫

2 < s−1 , s−2 >R−1 = < e1, e2 >R−1 + < f1, f2 >R

− ≪ (f1, e1), (f2, e2) ≫

Hence, if (fi , ei ) ∈ Σ+, or equivalently s−i = ei − Rfi = 0, then2 < s+1 , s+2 >R−1= 2 ≪ (f1, e1), (f2, e2) ≫, while if (fi , ei ) ∈ Σ−, orequivalently s+i = ei + Rfi = 0, then2 < s−1 , s−2 >R−1= −2 ≪ (f1, e1), (f2, e2) ≫ .


108


Thus the mappings

σ+ = (f , e) ∈ Σ+ 7−→ s+ = 1√2(e + Rf ) ∈ F∗

σ− = (f , e) ∈ Σ− 7−→ s− = 1√2(e − Rf ) ∈ F∗

are isometries (with respect to the inner products on Σ+ and Σ−, and theinner product on F∗). Hence we may identify the wave vectors (σ+, σ−)with (s+, s−).

Remark Note that the pair of scattering subspaces (Σ+,Σ−)corresponding to R may be characterized as

Σ+ = (f , e) ∈ F × F∗| ≪ (f , e), (f , e) ≫=< e, e >R−1 + < f , f >R

Σ− = (f , e) ∈ F × F∗| ≪ (f , e), (f , e) ≫= − < e, e >R−1 − < f , f >R


108


Representation of a Dirac structure in terms of the wave

vectors

. For every Dirac structure D ⊂ F × F∗ there exist linear mappingsF : F → V and E : F∗ → V satisfying

(i) EF ∗ + FE ∗ = 0,

(ii) rank (F + E ) = dimF ,

where V is a linear space with the same dimension as F . Thus for any(f , e) ∈ D the wave vectors (s+, s−) are given as

s+ = 1√2(F ∗λ+ RE ∗λ) = 1√

2(F ∗ + RE ∗)λ

s− = 1√2(F ∗λ− RE ∗λ) = 1√

2(F ∗ − RE ∗)λ, λ ∈ V∗


108


It follows thats− = (F ∗ − RE ∗)(F ∗ + RE ∗)−1s+

Hence the unitary map O : F∗ → F∗ is given as

O = (F ∗ − RE ∗)(F ∗ + RE ∗)−1 = (FR−1 − E )−1(FR−1 + E )

satisfyingO∗R−1O = R−1


108

Port-Hamiltonian systems and passivity

Outline

1 Plan for the week









108


Passivity

A square nonlinear system

Σ :x = f (x) + g(x)u, u ∈ R

m

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 withV (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 =.


108


If H is differentiable then being passive is equivalent to

d

dtV ≤ yTu

which reduces to (Willems, Hill-Moylan)

∂TV∂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 + Buy = 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


108


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

d

dtH = −eTR fR + eTC fC ≤ eTC fC

and thus H is a storage function. Furthermore, if there are nopower-dissipating elements R , then a port-Hamiltonian system with H ≥ 0is lossless.


108


From passive to port-Hamiltonian in the linear case

Fixing a particular storage matrix Q > 0 and defining J to be theskew-symmetric part of the matrix AQ−1 and −R its symmetric part, thesystem can be written as the port-Hamiltonian system

x = [J − R ]Qx + Bu

y = BTQx

whereJ = −JT , R = RT ≥ 0

Thus there are different port-Hamiltonian realizations of the same passivesystem.


108


Similarly, most nonlinear passive systems can be written as aport-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


108

Other properties of port-Hamiltonian systems

Outline

1 Plan for the week









108


Casimirs and Algebraic Constraints

Define the smooth distributions

G0 := X ∈ TX | (X , 0) ∈ D

G1 := X ∈ TX | ∃α ∈ T ∗X s.t. (X , α) ∈ Dand 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 = ker P1

P0 = annG1

If D is integrable, then the (co-)distributions G0,G1,P0,P1 are allinvolutive.


108


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.


108


Algebraic constraints

In general (−x , dH(x)) ∈ D(x) induces algebraic constraints on the statevariables, leading to the constrained state space

Xc = x ∈ X | ∃f s.t. (f , dH(x)) ∈ D(x)

Throughout we assume that this defines a smooth submanifold. Thisconstrained state space is determined by the Hamiltonian H and by theco-distribution P1 (or, equivalently, the distribution G0).


108


Symmetries and conserved quantities

Definition

Let D be a Dirac structure on X . A vector field f on X is an infinitesimalsymmetry of D (briefly, a symmetry of D) if

(Lf X , Lf α) ∈ D, for all (X , α) ∈ D

Analogously we say that a diffeomorphism ϕ : X → X is a symmetry of Dif

(ϕ−1∗ X , ϕ∗α

)∈ D, for all (X , α) ∈ D

Theorem

Let f be a symmetry of the Dirac structure D, with associateddistributions G0,G1 and co-distributions P0,P1. Then

Lf Gi ⊂ Gi , Lf Pi ⊂ Pi , i = 0, 1


108


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.


108


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 Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands DimitriHamiltonian control 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.


108


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).


108


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


108


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.


108


Thus

DA ‖ DB := (f1, e1, f3, e3) ∈ F1 ×F∗1 ×F3 ×F∗

3 | ∃(f2, e2) ∈ F2 ×F∗2

such that

(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 bilinear

forms). Then DA ‖ DB is a Dirac structure with respect to the bilinearform on F1 ×F∗

1 ×F3 ×F∗3 .


108


Proof

Consider DA, DB defined in kernel representation by

DA =(f1, e1, fA, eA) ∈ F1 ×F∗

1 ×F2 ×F∗2 |F1f1 + E1e1 + F2AfA + E2AeA = 0

DB =(fB , eB , f3, e3) ∈ F2 ×F∗

2 ×F3 ×F∗3 |F2B fB + E2BeB + F3f3 + E3e3 = 0

Make use of the following basic fact from linear algebra:

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


108


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


108


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


108


Explicit expressions for the composition

The composition of DA with DB is given by

[F1 E1 F2A E2A 0 00 0 −F2B E2B F3 E3

]

f1e1f2e2f3e3

= 0,

Define

M =

[F2A E2A

−F2B E2B

]

and let LA, LB be matrices with

L = [LA|LB ] , ker L = imM

Premultiplication of the equations by the matrix L := [LA|LB ] results inLAF1f1 + LAE1e1 + LBF3f3 + LBE3e3 = 0


108


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 .


108

Distributed parameter port-Hamiltonian systems

Outline

1 Plan for the week









108


How to deal with infinite-dimensional components ?

Want to deal with multi-physics systems also incorporatingdistributed-parameter (infinite-dimensional) components:

1. Power-converter connected to an electrical machine via a transmissionline2. Electrical power-networks including generators and dynamic loads3. Hydraulic networks with pipes4. Multi-body systems with flexible componentsetc.


108



fa

ea

fb

eb

a b

Figure: Simplest example: Transmission line

Telegrapher’s equations define the 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)


108


Transmission line as port-Hamiltonian system

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 external (boundary) flows f = (fa, fb) and boundary effortse = (ea, eb). Define the infinite-dimensional subspace of(C∞[a, b])2 × (C∞[a, b])2 × R

2 × R2 by the equations

[fEfM

]

=

[0 ∂

∂z∂∂z

0

] [eEeM

]

[faea

]

=

[eE (a)eM(a)

]

,

[fbeb

]

=

[eE (b)eM(b)

]


108


This defines a Dirac structure:

Use integration by parts

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

aeE (z)fE (z) + eM(z)fM(z)− ebfb + eafa =

∫ b

aeE (z)

∂∂zeM(z) + eM(z) ∂

∂zeE (z)− ebfb + eafa =

∫ b

a[−eM(z) ∂

∂zeE (z) + ebfb − eafa] + eM(z) ∂

∂zeE (z)− ebfb + eafa = 0

Thus eT f = 0 for all (f , e) ∈ D. However, in general this implies that forall (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⊥⊥.Arjan van der Schaft Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands DimitriHamiltonian control systems



Still need to show that D⊥⊥ ⊂ D :

Let (fE , fM , eE , eM , fa, ea, fb, eb) ∈ D⊥⊥, that is

0 =∫ b

aeE fE + eE fE + eM fM + eM fM+

−ebfb − eb fb + eafa + ea fa

for all (fE , fM , eE , eM , fa, ea, fb, eb) ∈ D.Take first fa = ea = fb = eb = 0. Then

0 =

∫ b

a

eE∂

∂zeM + eE fE + eM

∂

∂zeE + eM fM

for all such (eE , eM). This implies

fE =∂

∂zeM , fM =

∂

∂zeE


108


Substitution yields

0 =∫ b

aeE

∂∂zeM + eE

∂∂zeM + eM

∂∂zeE + eM

∂∂zeE

−ebfb − eb fb + eafa + ea fa

which implies

eE (b)eM(b) + eM(b)eE (b)− eE (a)eM(a)− eM(a)eE (a)

−ebfb − eb fb + eafa + ea fa = 0

for all fa = eE (a), fb = eE (b), ea = eM(a), eb = eM(b). This yields

eb = eM(b), fb = eE (b), ea = eM(a), fa = eE (a)


108


Telegrapher’s equations as port-Hamiltonian system

Substituting (as in the lumped-parameter case)

fE = −∂Q∂t

fM = −∂ϕ∂t

fS = −x

eE = QC

= ∂H∂Q

eM = ϕL= ∂H

∂ϕ

eS =

∂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.


108


Energy-balance

ddt

∫ b

a12Q2(z)C

+ 12ϕ2(z)L

dz = −∫ b

aeE (z)fE (z) + eM(z)fM(z)dz =

−ebfb + eafa = −I (b, t)V (b, t) + I (a, t)V (b, t)

Remark

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

∂2Q∂t2

= − ∂∂z

∂I∂t

= − ∂∂z

∂∂t

φL=

− ∂∂z

1L∂φ∂t

= ∂∂z

1L

∂∂z

QC= 1

LC∂2Q∂z2

(provided L(z),C (z) do not depend on z), or similar expressions in φ, I orV .

The same equations hold for a vibrating string, or for a compressiblegas/fluid in a one-dimensional pipe.


108


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

[hv

]

+

[v hg v

]

∂z

[hv

]

= 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 totalenergy

H(h, v) =1

2

∫ b

a

[hv2 + gh2]dz


108


yielding the co-energy functions1

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 thetelegrapher’s equations, as

∂h∂t(z , t) = − ∂

∂z∂H∂v

∂v∂t(z , t) = − ∂

∂z∂H∂h

with boundary variables −hv|a,b and (12v2 + gh)|a,b.

1Daniel Bernoulli, born in 1700 in Groningen as son of Johann Bernoulli, professor inmathematics at the University of Groningen and pioneer of the Calculus of Variations(the Brachistochrone problem).


108


Paying tribute to history:

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

Figure: Daniel Bernoulli, born in Groningen in 1700.


108


We obtain the energy balance

d

dt

∫ b

a

[hv2 + gh2]dz = −(hv)(1

2v2 + gh)|ba

which can be rewritten as

−v(12gh2)|ba − v(12hv

2 + 12gh

2))|ba =

velocity × pressure + energy flux through the boundary


108


Conservation laws

All examples sofar 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 coupledconservation 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, β2voltage V and current I , respectively.)


108


For some purposes it is illuminating to rewrite the equations in terms ofthe 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 thecharacteristic velocities 1√

LCand − 1√

LCcorresponding to the

characteristic eigenvectors (and curves).


108


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 arenonlinear, the characteristic curves may intersect, corresponding to shockwaves.


108


Higher-dimensional spatial domain

Electromagnetic Field: Maxwell’s equations

∂D∂t

= curl H, E = ǫ−1D Faraday

∂B∂t

= − curl E , H = µ−1B Ampere


108


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]


108


General framework

Z is n-dimensional spatial domain with boundary ∂Z .The exterior derivative d : Ωk(Z ) → Ωk+1(Z ) incorporates all vectorcalculus operations ( grad, curl, div).Define a Dirac structure on the space of flows and efforts:

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

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

by setting

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

fb(t) = eE (t, ∂Z ), eb(t) = ±eM(t, ∂Z )

(Transmission line: n = p = q = 1Maxwell’s equations: n = 3, p = q = 2 )


108


Regulation of the shallow water equations

Consider again∂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

(12v2 + gh)|a,b

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


108


Suppose we want to control the water level h to a desired height h∗.We cannot use the total energy as a Lyapunov function.An obvious ’physical’ controller is to add to one side of the canal, say theright-end b, an infinite water reservoir of height h∗, corresponding to theport-Hamiltonian ’source’ system

ξ = ucyc = ∂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


108


By mass balance,∫ b

a

h(z , t)dz + ξ + c

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

V (h, v , ξ) := 12

∫ b

a[hv2 + gh2]dz + gh∗ξ − gh∗[

∫ b

ah(z , t)dz + ξ]

+12g(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, sincethe Hamiltonian Hc(ξ) = gh∗ξ is not bounded from below.


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An alternative, passive, choice of the Hamiltonian controller system is totake

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 beobtained 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 refinedresults.)


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Conclusions and Outlook

Outline

1 Plan for the week









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• Port-Hamiltonian systems theory as a modular theory for complexmulti-physics systems

• Inclusion of infinite-dimensional component systems:differential-algebraic partial differential equations

• Leads to new control paradigms and strategies

• Not covered here (see literature): structure-preserving modelreduction, spatial discretization, network dynamics (port-Hamiltoniansystems on graphs), chemical and thermodynamic systems.


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


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Some key references

• A.J. van der Schaft, L2-Gain and Passivity Techniques in NonlinearControl, Lect. Notes in Control and Information Sciences, Vol. 218,Springer-Verlag, Berlin, 1996, p. 168, 2nd revised and enlargededition, Springer-Verlag, London, 2000, p. xvi+249.

• A.J. van der Schaft, B.M. Maschke, The Hamiltonian formulation ofenergy conserving physical systems with external ports, Archiv furElektronik und Ubertragungstechnik, 49, pp. 362–371, 1995.

• M. Dalsmo, A.J. van der Schaft, On representations and integrabilityof mathematical structures in energy-conserving physical systems,SIAM J. Control and Optimization, vol.37, pp. 54–91, 1999.

• R. Ortega, A.J. van der Schaft, I. Mareels, & B.M. Maschke, Puttingenergy back in control, Control Systems Magazine, vol. 21, pp.18–33, 2001.

• A.J. van der Schaft, B.M. Maschke, Hamiltonian formulation ofdistributed-parameter systems with boundary energy flow, Journal ofGeometry and Physics, vol. 42, pp.166–194, 2002.


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• G. Golo, V. Talasila, A.J. van der Schaft, B.M. Maschke, Hamiltoniandiscretization of boundary control systems, Automatica, vol. 40/5,pp. 757–761, 2004.

• J. Cervera, A.J. van der Schaft, A. Banos, Interconnection ofport-Hamiltonian systems and composition of Dirac structuresAutomatica, vol. 43, pp. 212–225, 2007.

• R. Pasumarthy, A.J. van der Schaft, Achievable Casimirs and itsImplications on Control of port-Hamiltonian systems, Int. J. Control,vol. 80 Issue 9, pp. 1421–1438, 2007.

• D. Eberard, B. Maschke, A.J. van der Schaft, An extension ofpseudo-Hamiltonian systems to the thermodynamic space: towards ageometry of non-equilibrium thermodynamics, Reports onMathematical Physics, Vol. 60, No. 2, pp. 175–198, 2007.

• Geoplex Consortium, Modeling and Control of Complex PhysicalSystems; the Port- Hamiltonian Approach, Springer, 2009.

• A.J. van der Schaft, B. Maschke, Discrete conservation laws andport-Hamiltonian systems on graphs and complexes, submitted forpublication, 2011.


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Hamiltonian control systems - TU Ilmenau