Mikael Kurula - Åbo Akademi

2 Chapter 1. Introduction
the state/signal setting, rather than a separate proof for every possible input/output
case.
We introduce the concepts of passivity and conservativity in Chapter 3. The intuitive
interpretation of a passive system is that it has no internal energy sources.
A conservative system has neither internal energy sources nor internal energy sinks.
Passivity brings very useful extra structure to state/signal systems and exploiting this
structure seems to be the most promising way to obtain a sensible state/signal theory.
The state/signal theory is conceptually similar to the so-called behavioural theory
for finite-dimensional systems that has been developed by Willems and his co-operators.
See [PW98] for a good introduction to the behavioural theory and the references in
[Wil07] for more recent work in the field. The behavioural framework incorporates
distributed-parameter systems by considering also the spatial variables of the system
as “time variables”. In this way one obtains what is called an n−D (n-dimensional)
system, which often has a finite-dimensional state space but multi-dimensional “time”.
This is one of the main differences from the state/signal approach, where one keeps
time one-dimensional and makes the state space infinite-dimensional.
Willems describes the so-called “tearing, zooming and linking” approach to modelling
complex systems in [Wil07]. The main idea is to model a complex system as the
interconnection of simpler standard modules, whose individual behaviours are wellknown.
Therefore, the approach of tearing, zooming and linking is a major motivation
for the interconnection theory, and by extension, for the theory presented in this thesis.
The interconnection theory itself is also interesting to develop further, because control
is performed by interconnection in the behavioural setting; see [MM05] or [BT02].
The idea of modular modelling is also prevalent in the theory of port-Hamiltonian
systems. This theory has its roots in energy-based modelling of mainly nonlinear
physical systems; see [vdS00, MvdS05]. A port-Hamiltonian system consists of two
parts: the Hamiltonian, which measures the total energy of the system when it is
in a given state, and the Dirac structure. The Dirac structure encodes the relations
between the different variables present in the system and it also describes how the
system acts under interconnection. We present Dirac structures which are defined on
Hilbert spaces in Chapter 4, and we also describe how interconnection of two port-
Hamiltonian systems corresponds to a so-called composition of their respective Dirac
structures. Dirac structures and state/signal nodes are connected via an example in
Chapter 5.
Systems theory in the setting of this thesis offers quite a few technical challenges.
Linear operator theory provides the main tools for our study and, unlike the corresponding
discrete-time analogue, many of the involved operators are unbounded. The
finite-dimensional linear network theory is classical by now, see [Bel68], and thus most
of the substance of the thesis lies in the technical details of the included articles.