The theory presented in this book arose as a product of a continued
collaboration between the two authors during the years 2003-2021.
The basis for this collaboration was our common interest in passive
linear time-invariant input/state/output systems theory. At the time
this project started, O. Staffans was preparing a joint article with Prof. J. Ball that, in particular, explored the
connections between conservative input/state/output systems theory on the
one hand and some results in the behavioral theory introduced by
J. Willems in the late 1980s on the other hand. After extensive
discussions on this approach, comparing it to the theory of
passive electrical networks, we understood that this opens up a new
direction in the study of passive linear time-invariant systems. We
called the new class of systems that arose in this way passive
state/signal systems. From the outset, it was clear that the
notion of passivity with an arbitrary supply rate fits more naturally
into the state/signal setting than in the input/state/output setting,
and that the standard ``diagonal transformation'' of Liv\v{s}ic, the
Potapov-Ginzburg transformation, and the Redheffer and chain-scattering transformations have natural interpretations as
transformations between input/output resolvents of different
input/state/output representations of a passive state/signal system.
We also soon discovered that virtually all the standard control theory
notions such as controllability and observability, minimality,
stability, stabilizability, detectability, and well-posedness have
natural state/signal counterparts.
Our first article on the state/signal system was
completed and submitted for publication in the fall of 2003, and it
was followed by many others. Some of the
results presented in this book were obtained in collaboration with
Ph.D. Mikael Kulula. The bulk of the work was done during D. Arov's
regular visits to Åbo Akademi during August-October 2003-2010 and
to Aalto University during August-October 2011-2017, with an
average length of almost three months. These visits were financed by the
Academy of Finland, the Magnus Ehrnrooth Foundation, and the Finnish
Society of Sciences and Letters.
In the fall of 2009, it was decided that the theory was sufficiently
mature to be presented in terms of a book, and the writing of this
book began on August 30, 2009. By the end of November 2009, a
preliminary list of contents was ready. Two significant factors in
this decision were the research grant from the Academy of Finland
that relieved O. Staffans from teaching duties during the academic
year 2009-2010 and the leave of absence for D. Arov for extensive
periods of time from the South Ukrainian Pedagogical University based
on a joint exchange agreement with \AA bo Akademi.
The book we originally planned to write was supposed to be
devoted to linear time-invariant systems in discrete time. In 2011, we
realized that it would be more important to, instead, write a book on
linear time-invariant systems in continuous time, and in 2013 it was
clear that it was not feasible to write only one book on systems in
continuous time. The continuous time theory contains a number of
mathematical difficulties that must first be sorted out, and this is
done in the present volume. The application of this theory to passive
state/signal systems in continuous time remains to be written
down.
We thank the Academy of Finland, the Magnus Ehrnrooth Foundation, and the
Finnish Society of Sciences and Letters for their financial support,
without which this work could not have been carried out. We also
thank \AA bo Akademi and Aalto University for excellent working
facilities, and the South Ukrainian Pedagogical University for giving
D. Arov ample time to devote to research.
Above all, we are grateful to our wives Nataliya and Satu-Marjatta for
their constant support, understanding, and patience while this work
was carried out.