Home page of H. L. Trentelman
Prof. dr. Harry L. Trentelman
University of Groningen
Johann Bernoulli Institute for Mathematics and Computer Science
P.O. Box 407
9700 AK Groningen
Visiting and delivery addres: Bernoulliborg, room 392
Nijenborgh 9, 9747 AG, Groningen
Phone : +31 (0)50 3633998
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Phone : +31 (0)50 3633939/50 (secretary)
Email : firstname.lastname@example.org
Harry Trentelman is a full professor in Systems and Control at the Johann Bernoulli Institute for Mathematics and Computer Science of the University of Groningen. From 1985 to 1991 he served as an assistent professor, and as an associate professor at the Mathematics Department of the University of Technology at Eindhoven, the Netherlands. He obtained his PhD degree in Mathematics from the University of Groningen in 1985. He serves as a senior editor of the IEEE Transactions on Automatic Control and as an associate editor of Automatica. He is past associate editor of the SIAM Journal on Control and Optimization and Systems and Control Letters. Dr. Trentelman is a Fellow of the IEEE.
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.