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Conservation Laws and

Lumped System Dynamics

A.J. van der Schaft, B.M. Maschke

1 Introduction

Physical systems modeling, aimed at network modeling of complex multi-physics

systems, has especially flourished in the fifties and sixties of the 20-th century, see

e.g. [11, 4] and referencesprovidedtherein. With the reinforcementof the ’systems’

legacy in Systems & Control, the growing recognition that ’control’ is not confined

to developing algorithms for processing the measurements of the system into con-

trol signals (but instead is concerned with the design of the total controlled system),

and facing the complexity of modern technological and natural systems, system-

atic methods for physical systems modeling of large-scale lumped- and distributed-

parameter systems capturing their basic physical characteristics are needed more

than ever.

In this paper we are concerned with the development of a systematic framework

for modeling multi-physics systems which is directly based on conservation laws.

Modeling based on conservation laws is prevalent in a distributed-parameter con-

text in areas such as fluid dynamics and hydraulic systems, chemical and thermody-

namical systems [2], as well as electromagnetism, but is also underlying the basic

structure of lumped-parameter systems such as electrical circuits. While the natu-

ral framework for formulating Kirchhoff’s laws for electrical circuits is the circuit

graph we will show in this paper how distributed-parameter conservation laws can

be discretized by using the proper generalization of the notion of graph to ’higher-

dimensional networks’, called k-complexes in algebraic topology. Furthermore, we

show how these discretized conservation laws define a power-conserving intercon-

A.J. van der Schaft

Institute of Mathematics and Computing Science, University of Groningen, PO Box 407, 9700

AK, the Netherlands, e-mail: A.J.van.der.Schaft@rug.nl

B.M. Maschke

Lab. d’Automatique etde GeniedesProc´ ed´ es, Universit´ e ClaudeBernard Lyon-1, F-69622 Villeur-

banne, Cedex, France, e-mail: maschke@lagep.univ-lyon1.fr

1

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2 A.J. van der Schaft, B.M. Maschke

nection structure, called a Dirac structure, which, when combined with the (dis-

cretized) constitutive relations, defines a finite-dimensional port-Hamiltonian sys-

tem [14, 13, 5].

In previous work [15] we have laid down a framework for formulating con-

servation laws described by partial differential equations as infinite-dimensional

port-Hamiltonian systems. Furthermore, in [8] we have shown how such infinite-

dimensional port-Hamiltonian systems can be spatially discretized to finite-dimen-

sional port-Hamiltonian systems by making use of mixed finite-element methods.

In this paper we show how alternatively we can directly spatially ’lump’the dynam-

ics described by conservation laws in a structure-preserving manner, again obtain-

ing a finite-dimensional port-Hamiltonian system description. This approach also

elucidates the concept of the spatial system boundary, and leads to the notion of

distributed terminals.

This paper is a follow-up of our previous paper [16]. Older references in this

spirit include [10, 12].

2 Kirchhoff’s laws on graphs and circuit dynamics

In this section we recall the abstract formulation of Kirchhoff’s laws on graphs,

dating back to the historical work of Kirchhoff [9], as can be found e.g. in [1, 3].

In order to deal with open electrical circuits we define open graphs, and we show

how Kirchhoff’s laws on open graphs define a power-conserving interconnection

structure, called a Dirac structure, between the currents through and the voltages

over the edges of the graph, and the boundary currents and potentials. This enables

us to describe the circuit dynamics as a port-Hamiltonian system.

2.1 Graphs

An oriented graph1G, see e.g. [3], consists of a finite set V of vertices and a finite

set E of directed edges, together with a mapping from E to the set of ordered pairs

of V . Thus to any branch e ∈ E there corresponds an ordered pair (v,w) ∈ V2

representing the initial vertex v and the final vertex w of this edge. An oriented

graph is completely specified by its incidence matrix B, which is an ¯ v× ¯ e matrix, ¯ v

being the numberof vertices and ¯ e being the number of edges, with (i, j)-th element

bijequal to 1 if the j-th edge is an edge towards vertex i, equal to −1 if the j-th

edge is an edge originating from vertex i, and 0 otherwise.

Given an oriented graph we define its vertex space Λ0as the real vector space of

all functions from V to R. Clearly Λ0can be identified with R¯ v. Furthermore, we

1In fact, we will be considering multi-graphs since we allow for the existence of multiplebranches

between the same pair of vertices.

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Conservation Laws and Lumped System Dynamics3

define its edge space Λ1as the vector space of all functions from E to R. Again,Λ1

can be identified with R¯ e.

In the context of an electrical circuit Λ1will be the vector space of currents

through the edges in the circuit. The dual space of Λ1will be denoted by Λ1, and

defines the vector space of voltages across the edges. (We have highlighted the

words ’through’ and ’across’ to refer to the classical use of ’through’ and ’across’

variables, see e.g. [11].) Furthermore,the duality product <V|I >=VTI of a vector

ofcurrentsI ∈Λ1with a vectorofvoltagesV ∈Λ1is the totalpoweroverthecircuit.

Similarly, the dual space of Λ0is denoted by Λ0and defines the vector space of

potentials at the vertices.

Remark 1. Since Λ0and Λ1have a canonical basis corresponding to the individual

vertices, respectively edges, there is a standard Euclidean inner product on both

spaces, and thus both Λ0and Λ1can be identified with Λ0, respectively Λ1, such

that the duality product becomes this standard inner product. In situations to be

treated later on this will not necessarily be the case.

The incidence matrix B can be also regarded as the matrix representation of a linear

map (denoted by the same symbol)

B : Λ1→Λ0

called the incidence operator or (boundary operator). Its adjoint map is denoted in

matrix representation as

BT: Λ0→ Λ1,

and is called the co-incidence (or co-boundary) operator.

2.2 Kirchhoff’s laws for graphs

ConsideranorientedgraphG specifiedbyits incidenceoperatorB.Kirchhoff’slaws

associated with the graph are expressed as follows. Kirchhoff’s current laws (KCL)

are given as

I ∈ kerB,

(1)

while Kirchhoff’s voltage laws (KVL) take the form

V ∈ imBT.

(2)

A graphtheoreticinterpretationofKirchhoff’scurrentandvoltagelaws canbegiven

as follows [3]. The kernel of the incidence operator B is the cycle space Z ⊂ Λ1

of the graph, while the image U ⊂ Λ1of the co-incidence operator BTis its cut

space (or, co-cycle space). Since kerB = (imBT)⊥(with⊥denoting the orthogonal

complementwith respect to the duality product between the dual spacesΛ1andΛ1)

the cycle space is the orthogonal complement of the cut space.

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4 A.J. van der Schaft, B.M. Maschke

This leads to the equivalent way of formulating Kirchhoff’s current laws as the

fact that the total currentI alongany cut is equalto zero,since I ∈kerB is equivalent

to I being orthogonal to the cut space U. The simplest elements of the cut space

U (which in fact are spanning the linear space U) are the cuts given by all edges

starting from or terminating on a single vertex v. Kirchhoff’s current laws for these

cut sets mean nothing else than the expression that the currents entering or leaving

any vertex v sum up to zero. Indeed, if v is numberedas the i-th vertex then the i-the

equation in the linear set of equations BI = 0 is precisely this.

On the other hand, since V ∈ imBTis equivalent to V being orthogonal to the

cycle space Z, Kirchhoff’s voltage laws can be equivalently described as the fact

that the total voltage over every cycle is zero.

The difference between Kirchhoff’s current and voltage laws is also reflected by

writing Kirchhoff’s voltage laws as

V = BTψ

(3)

for some vector ψ ∈Λ0, which has the physical interpretationof being the vector of

potentials at every vertex. Hence Kirchhoff’s voltage laws express that the voltage

distribution V over the edges of the graph corresponds to a potential distribution

over the vertices.

Of course, Tellegen’s theorem automatically follows from Kirchhoff’s laws. In-

deed, take any current distribution I satisfying Kirchhoff’s current laws BI = 0, and

any voltage distributionV satisfying Kirchhoff’s voltage laws V = BTψ. Then,

VTI = ψTBI = 0(4)

In particular, Tellegen’s theorem implies that for any actual current and voltage

distribution over the circuit the total powerVTI is equal to zero.

We summarize the Kirchhoff behavior BK(G) of a graph G with incidence ma-

trix B as

BK(G) := {(I,V) ∈ Λ1×Λ1| I ∈ kerB,V ∈ imBT}

(5)

It immediately follows that the Kirchhoff behavior defines a Dirac structure. Recall

[6, 14, 13] that a subspace D ⊂ V ×V∗for some vector space V defines a Dirac

structure if D = Dorthwhereorthdenotes the orthogonal complement with respect to

the indefinite inner product <<,>> onV ×V∗defined as

<< (v1,v∗

1),(v2,v∗

2) >>:=< v∗

1|v2> + < v∗

2|v1>,

with v1,v2∈V,v∗

V∗.

1,v∗

2∈ V∗, where <|> denotes the duality product between V and

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Conservation Laws and Lumped System Dynamics5

2.3 Kirchhoff’s laws for open graphs

Although in Kirchhoff’s original treatment of circuits and graphs external currents

entering the vertices of the graph were an indispensable notion, this has not been

articulated very well in the subsequent formalization of circuits and graphs2. Hence

a reinforcement of this systems point of view is definitely in order.

We will do so by extending the notion of graph to open graph. An open graph

G is obtained from an ordinary graph with set of vertices V by identifying a subset

Vb⊂ V of boundary vertices. The interpretation of Vbis that these are the vertices

thatareopentointerconnection(e.g.,withothergraphs).TheremainingsubsetVi:=

V −Vbare the internal vertices of the open graph.

Remark 2. Another way of defining open graphs is by identifying some of the edges

to be the boundary edges (open to interconnection).Such a definition is straightfor-

ward, and we will not elaborate on this. The distinction between the definitions of

an open graph using boundary vertices or boundary edges is analogous to the dif-

ference between boundary control of distributed-parameter systems and distributed

control; see also Section 3.

Kirchhoff’s current laws apply to an open graph G in a different manner than to

an ordinary graph, since the ordinary Kirchhoff’s current laws would imply that

the sum of the currents over all edges incident on a boundary vertex is zero, which

is not what we want for interconnection. Furthermore, by Tellegen’s theorem, the

ordinary KCL would imply that the total power in the circuit is equal to zero, thus

implying that there cannot be any ingoing or outgoing power flow. Hence we have

to modify Kirchhoff’s current laws by requiring that the incidence operator B maps

the vector of currents I to a vector that has zero components corresponding to the

internal vertices, while for the boundary vertices the image is equal to (minus) the

boundary current Ib. Decomposing the incidence operator B as

part of the incidence operator correspondingto the internal vertices, and Bbthe part

corresponding to the boundary vertices, we thus arrive at

?Bi

Bb

?

with Bithe

BiI = 0,

BbI = −Ib,

KCL (6)

HerethevectorIbis belongingtothevectorspaceΛboffunctionsfromtheboundary

vertices Vbto R (which is identified with R¯ vb, with ¯ vbthe number of boundary

vertices)3.

Kirchhoff’s voltage laws (KVL) remain unchanged, and will be written as

2Unfortunately, this holds for many formalizations of physical theories over the last century. A

proper theory of mechanics should include external forces from the very start, instead of restrict-

ing itself to closed mechanical systems. Thermodynamics cannot be properly formalized without

taking interaction with other systems into account.

3Alternatively, open graphs can be defined by attaching ’one-sided open edges’ (properly called

leaves) to every boundary vertex in Vb. Then the elements of the vector Ibare the currents through

these leaves, see also [17].