Conservation Laws and
Lumped System Dynamics
A.J. van der Schaft, B.M. Maschke
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
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 , 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
Lab. d’Automatique etde GeniedesProc´ ed´ es, Universit´ e ClaudeBernard Lyon-1, F-69622 Villeur-
banne, Cedex, France, e-mail: firstname.lastname@example.org
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  we have laid down a framework for formulating con-
servation laws described by partial differential equations as infinite-dimensional
port-Hamiltonian systems. Furthermore, in  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
This paper is a follow-up of our previous paper . 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 , 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.
An oriented graph1G, see e.g. , 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.
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. .) 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,
while Kirchhoff’s voltage laws (KVL) take the form
V ∈ imBT.
A graphtheoreticinterpretationofKirchhoff’scurrentandvoltagelaws canbegiven
as follows . 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.
16 A.J. van der Schaft, B.M. Maschke
The components euare equal to the reciprocal of the temperature at each 2-face.
Since the temperatureis varyingover the faces, there is a thermodynamicdriving
force vector e ∈Λ1given as the vector of differences
e = ∂2eu
By Fourier’s law the heat flux is determined by the thermodynamic driving force
f = R(eu)e,
with R(eu) = RT(eu) ≥ 0 depending on the heat conduction coefficients. (Note the
sign-difference with (26).) The resulting system is a port-Hamiltonian system (of
relaxation type), with vector of state variables x given by the internal energy vector
u, and ’Hamiltonian’ s(u). By (28) the entropy s(u) satisfies
∂u(u) = fTR(eu) f ≥ 0
expressing the fact that the entropy is monotonously increasing. (Note again the
sign-differencewith thetreatmentabove,wheretheHamiltonianH was decreasing.)
The exchange of heat through the boundary of the system is incorporated as
above, cf. (29, 30), by splitting the edges (1-cells) into internal edges with the resis-
tive relation (31) and boundary edges. This leads to
with fb,ebdenoting the heat flux, respectively, thermodynamical driving force,
through the boundary edges.
A framework has been laid down for the formulation of open physical systems on
k-complexes, generalizing the graph-theoretic formulation of electrical circuit dy-
namics with terminals. It has been shown that Kirchhoff’s laws can be generalized
to openk-complexes,defininga Diracstructureinvolvingboundarycurrentsandpo-
tentials, thus generalizing the concept of ’terminal’ to the distributed case. This has
been illustrated on the example of heat transfer on a 2-complex. This simple exam-
conservationlaws, without the need to formulate the dynamicsas a set of pde’s (and
possibly to discretize the pde’s later on).
In future work we will apply and extend the framework to different classes of
port-Hamiltonian systems on k-complexes (corresponding to different physical set-
tings), and employ these models for boundary control.
Conservation Laws and Lumped System Dynamics17
Acknowledgements It is a great pleasure to contribute to this Liber Amicorum for Professor Okko
Bosgra at the occasion of his 65-th birthday. Okko has played a key role in the flourishing of the
Systems & Control community withinthe Netherlands, both by his scientific leadership at the Delft
University of Technology, as well as by his stimulating role in bringing control engineering and
mathematical systems and control theory together. Ihope thischapter pays tributetohisremarkable
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