**Research Program **
Our primary research interest is systems- and control theory. The
purpose of systems- and control theory is the study of dynamical
systems that interact with their environment. The traditional way of
modeling a dynamical system is by an input-output map or
relation. Many systems that one encounters in physics or engineering,
however, do not exhibit the information flow direction that is
pre-supposed by the input-output structure. This objection has led to
the development of a new approach to systems and control, the so
called behavioral approach. In this setting, all external (or: manifest)
system variables are a priori treated on an equal footing. The model
then specifies a subset of the set of all manifest variable
time-trajectories as being possible. This subset is called the
behavior of the system.

Thus, the central point of view in the
behavioral approach is that a system is no longer identified with one
of its representations, i.e., a set of algebraic and differential
equations used to model it, but with the set of solutions to these
equations. By taking this point of view, the mathematical results
that are obtained tend to be more general than in most classical
approaches. In fact, many classical results in systems - and control
can be obtained as special cases of the behavioral approach.

Within the context of this approach to systems- and control, our
scientific work is divided into the following four research areas.

**1. Control as interconnection.**

The first research project that we have been working on
in the past few years is to set up a behavioral approach to control theory.
Since in the behavioral context the paradigm of
input-output structure has been put aside, first a new notion of
control had to be introduced. This has led to the concept of 'control as interconnection',
with the usual notion of feedback control as an important special case.

Around 1997 we came to the insight that, in order to be able to
formulate and study optimal control problems in the behavioral
framework, one needs a general theory of quadratic expressions
involving the manifest variables of the system and its higher order
derivatives. This has led to the notion of quadratic differential form (QDF).
Between 1998 and 2001, we have applied the theory
of quadratic differential forms to formulate and
resolve the so called problem of synthesis of dissipative
systems, which has the classical feedback H-infinity control
problem for input/output systems as a special case.

A central result in the context of control as interconnection that we obtained
is the fact that a behavior is regularly implementable by a controller that acts only on the control variables if and only if it is wedged in between the hidden behavior and the
plant behavior and the sum of the behavior and the controllable
part of the plant behavior equals the plant behavior. We have applied this basic result to obtain necessary and sufficient conditions for the existence of stabilizing controllers. A
nice result that we have obtained here is that, for a given full plant
behavior, there exists a regularly implementable, stable behavior if
and only if in the full plant behavior the variable to be controlled
is detectable from the control variable and the manifest plant
behavior is stabilizable.

We have also extended the results on the synthesis of dissipative
systems to the synthesis of strictly dissipative systems. In addition, we have worked on finding conditions for optimal synthesis of
dissipative systems, and on developing algorithms to check solvability of
the synthesis problems and to compute the desired dissipative
sub-behaviors and controllers. More recently we have worked on extending the results o
obtained on the synthesis of dissipative systems to the case that the weighting
functional defining the control performance is allowed to contain
higher order derivatives of the plant variable.

In the past four years, we have also worked on algorithms
for controller synthesis in a behavioral framework. We have developed algorithms to compute
controllers for the problems of pole placement and stabilization, both for the full interconnection as well as for the partial interconnection case. These algorithms are mainly based on the so-called unimodular and stable embedding problems, which are the problems to extend a given polynomial matrix with addtional rows or columns in such a way that the resulting square matrix becomes unimodular or Hurwitz. We have developed a numerically stable algorithm to compute such embeddings. We have also established a behavioral analogue of the famous Youla parametrization of all stabilizing controllers.

A current direction of research is in the development of a behavioral theory of robust control.

**2. Algorithms in Systems and Control**

A project closely conneced to the previous one deals with algorithmic issues in systems-
and control. The combination of system models with performance
functionals lies at the basis of most of the synthesis algorithms in
control and signal processing. The emphasis in our work has been on
linear differential systems in combination with functionals that are
integrals of quadratic differential forms (QDF's). This combination
brings about an elegant interplay between one- and two-variable
polynomial matrices. The one-variable polynomial matrices model the
system dynamics, and the two-variable polynomial matrices model the
performance. A particular example where this framework is used in our
work is in the research that uses the theory of dissipative systems, a
central concept in system theory that has been introduced about 30
years ago.

The theory of dissipative systems can be used fruitfuly in the problem
of factoring a given polynomial matrix in a symmetric way, with one of
the factors expressing the signature of the polynomial matrix along
the imaginary axis in the complex plane. This algorithmic problem is
known as the polynomial J-spectral factorization problem. It plays
an essential role in filtering and estimation problems, and in problem
of optimal and robust control. By assigning to the one-variable
polynomial matrix to be factored a two-variable-polynomial matrix in a
particular way, and by subsequently interpreting this two-variable
polynomial matrix as a quadratic differential form, we have managed to
embed the problem of polynomial J-spectral factorization into the
time-domain. Doing this makes it possible to interpret the
factorization problem directly as a problem of finding dissipation
rates for dissipative systems. Finding such dissipation rates can be
done by computing storage functions. All this can be cast into a
framework of ordinary matrix computations, like solving linear matrix
inequalities and algebraic Riccati equations. These results have been
obtained in collaboration with P. Rapisarda between 1997 and 1999, and have
been applied to the ordinary spectral factorization problem and to the J-spectral
factorization problem.

By applying general results on quadratic differential forms and their
associated two-variable polynomial matrices to dynamical systems
represented in classical state space form, in 2001 (in collaboration
with P. Rapisarda) we have established a new classification of the real
symmetric solutions of the well-known algebraic Riccati equation. This
classification uses so called Pick matrices obtained from QDF's
associated with the Riccati equation.

More recently, we have extended the results on J-spectral factorization for one-variable
polynomial matrices described above, to multivariate polynomial matrices.
As in the one-variable case, the idea is to reduce this type of problems to
problems of finding storage functions for dissipative distributed (nD) systems,
and subsequently to ordinary matrix computation problems. It has turned out
that the spectral factorization problem for multivariate polynomial matrices is
closely connected to the so-called SOS (sum of squares) problem of writing
a nonnegative polynomial as the sum of squares of polynomials, and to D. Hilbert's
17th problem.

**3. Modeling and Control of Distributed Systems**

A third research area deals with the study of open, distributed,
dynamical (or: nD) systems, i.e., dynamical systems described by
linear, constant coefficient, partial differential equations, in
which some of the variables are free. Dynamical systems described by
such equations appear in many problems of mathematical physics, such
as continuum mechanics and fluid dynamics. The aim is to establish, for this class
of systems, a systems- and control theory, based on the
behavioral approach, as has been developed in the past
decade for finite-dimensional, linear, time-invariant systems. In
particular, the project aims at generalizing to the class of
distributed systems important basic systems theoretic notions as input
and output, interconnection, and implementability. Also, our aim is to
generalize the successful concept of quadratic differential form
to the nD context, and establish a theory of lossless and
dissipative nD systems. For finite-dimensional, linear,
time-invariant systems, these notions have shown to be instrumental in
problems of H-infinity control and filtering, and we expect to be
able to develop similar theories for nD systems. In particular, the
idea of H-infinity filtering for nD systems will be relevant in
signal and image processing.

At this moment several results have already been obtained. we have made (in
collaboration with P. Rapisarda) a study of autonomous linear 1D
differential systems, and have introduced for this class of systems
the notion of Hamiltonian system. Different from classical
approaches, the property of being a Hamiltonian system is not defined
in terms of one of the system representations, but in terms of the
interplay of the system behavior with a certain bilinear differential
form (BDF) on that behavior. We have managed to characterize the
property of being Hamiltonian in terms of properties of the
representation of the system. We have also given a formal proof of the
statement that a system behavior is Hamiltonian if and only if there
exists a Lagrangian functional of the system variables and its higher
order derivatives such that the system behavior consists of the
solutions of the (higher order) Euler-Lagrange equations associated
with this Lagrangian functional. More recently, we have worked on the problem of generalizing
these results to nD system behaviors. By using Grobner bases applied to the polynomial
ring of polynomials in more than one variable, we have succeeded in
generalizing some of the results in the 1D case.

In the recent past we have worked on
extending results on implementability and regular implementability for 1D
systems to the context of nD systems. In particular, we have tried to generalize
the result that a desired behavior is regularly implementable by partial interconnection
if and only if it is plain implementable by partial interconnection and regularly
implementable by full interconnection. Recently, for the nD case, we have
obtained counterexamples to this statement in both directions. We have also
succeeded in finding reasonable assumptions under which the equivalence does hold.

**4. Model Reduction and Approximation for Behavioral Systems**

A fourth research area is model reduction and approximation for system behaviors.
The research in this project deals with reduction and
approximation of mathematical models for linear dynamical
systems. Given a model of a linear dynamical system with a certain
complexity, the problem is investigated of approximating this
model by a lower order, less complex one in such a way that the lower
order model retains or closely approximates the behavior of the
original model. The distinctive feature of this research proposal is
that it intends to study the model reduction problem from a behavioral point of view.
The purpose is to establish a
representation-free approach to model reduction and
approximation, one which considers the system itself as the starting
point, instead of one of its particular representations. We also
avoid the use of input/output partitions of the system variables. The
advantages of this approach are the wider variety of model classes
that can be considered, and the flexibility with which the results can
be adapted to the particular system representation at hand. The
behavioral approach to systems and control provides conceptual and
mathematical tools to deal with systems in an intrinsic way.
Especially the recently introduced notion of quadratic
differential form is expected to be useful in the context of
model reduction and approximation. Important applications are
envisioned in the form of algorithms to be implemented in symbolic
manipulation packages such as Mathematica or Maple.

Within this project, currently we are working on the problem to approximate
dissipative system behaviors by reduced order system behaviors that
have retained the property of dissipativeness.
Recently, in work of Antoulas and Sorensen, a new technique and
efficient numerical algorithms have been presented in order to perform
model reduction with passivity- and stability
preservation. This novel approach is based on the idea of combining
Krylov projection methods with
positive-real interpolation techniques; the reduced-order model is
obtained by interpolating a subset of the spectral zeros of the
original system. Sorensen has shown that there is no need for explicit
interpolation in the implementation: rather, the reduced-order model
can be found by computing a basis for its invariant subspace
corresponding to the spectral zeroes in the open left half-plane.
This idea renders Antoulas' technique efficient, and thus applicable
also to high-order systems.

In our research we have taken a related point of view using ideas from
the theory of behavioral dissipative systems. We have shown
that the model reduction approach of Antoulas can be interpreted as
special case of a general method for model reduction applicable also
when the original system is not passive. In this approach, it is
required that the set of anti-stable stationary trajectories of the
reduced-order model is a subset of the set of anti-stable stationary
trajectories of the original system; we have called this general technique
model reduction by retention of stationary trajectories. In
this setting, one is given a supply rate induced by a quadratic
form and a controllable system of
McMillan degree n which is half-line dissipative with
respect to this supply rate. Moreover, k < n anti-stable stationary
trajectories of the system behavior are specified. One is required to find a system
with McMillan degree k which is again half-line dissipative with respect to the given supply rate,
and such that its space of anti-stable stationary trajectories with
respect to the supply rate consists precisely of
the k selected trajectories of the original system. It can
be shown that under the given assumptions, the reduced order model
exists and is also half-line dissipative.

**5. Systems and Control Theory for Linear Systems**

A long standing research interest has been the `classical' approach to
systems- and control theory for input/state/output systems, in
particular problems of disturbance attenuation, linear quadratic
optimal control problems, H-2 optimal control problems, the
H-infinity control problem, problems of robust stabilization, the
H-2 optimal control problem for sampled-data systems, etc. On this
classical approach to systems and control we have finished a textbook
on the level of advanced undergraduate, and graduate
courses. This book, `Control Theory for Linear Systems', co-authored
with M.L.J. Hautus and A.A. Stoorvogel, has been published with
Springer Verlag, London, in January 2000:

H.L. Trentelman, A.A. Stoorvogel and M.L.J. Hautus, * Control
Theory for Linear Systems*, Springer, London, 2001.