Colloquium Mathematics, V.S. Gottimukala, RuG

Join us for coffee and tea at 15.45 p.m.

Date:                          Tuesday January 8, 2013

Speaker:                     V.S. Gottimukala, University of Groningen

Room:                         5161.0267 (Bernoulliborg)

Time:                           16.15

Title: Rational Representations and Behavioral Distance

Abstract:

In this thesis we treat a number of problems related to modeling, analysis and control of dynamical systems. In order to do so, we use an elegant mathematical approach called the “behavioral" approach. In the framework of the behavioral approach a dynamical system is determined uniquely by its behavior which is the set of trajectories that satisfy the physical laws that govern the system. We deal with a special class of dynamical systems called linear differential systems. The behavior of a linear differential system admits many different kinds of representations such as the kernel or image of a rational differential operator, or the external behavior of a state representation. In this thesis, we are primarily interested in the study of rational representations. As the first problem, we consider the equivalence of rational representations. We obtain necessary and sufficient conditions on the rational matrices appearing in the given kernel (image) representations under which they define one and the same behavior. In general, given a behavior, not every realization of the rational matrix appearing in a rational kernel (image) representation yields an output nulling (driving variable) representation of the behavior.

As a second problem, we obtain necessary and sufficient conditions on a realization to overcome this problem. We also address a similar question related to the minimality of state representations. Given a behavior and its proper real rational representation, we obtain necessary and sufficient conditions on the rational matrix under which a minimal realization of it yields a minimal output nulling (driving variable) representation of the behavior. The next important issue that we address in this thesis is the concept of distance between behaviors. This has applications in areas such as model reduction, and robust control and stabilization of dynamical systems. We discuss several ways of defining distance between behaviors using the concept of gap between closed subspaces of Hilbert spaces. Also, we compare the values and topologies associated with these notions of distance, and also express them in terms of rational representations as well as state representations. Finally, we treat the problem of robust stabilization in the behavioral framework. The problem of robust stabilization is to find a controller that regularly stabilizes all the plants within a neighborhood (of a given radius) defined around a given nominal plant. We call a controller that achieves this objective a robustly stabilizing controller. Using the notions of distance between behaviors introduced, we define neighborhoods around a nominal plant in a representation free way. For a given radius, we give a comparison of the various defined neighborhoods. Further, we solve the problem of robust stabilization for these neighborhoods.

Colloquium coordinators are Prof.dr. A.C.D. van Enter (e-mail : A.C.D.van.Enter@rug.nl) and

Dr. A.V. Kiselev (e-mail: a.v.kiselev rug.nl)

http://www.rug.nl/wiskunde/nieuws/actueel/wiskundecolloquia