Research projects of A.C. Antoulas

Approximation of large-scale dynamical systems

Principal Investigators: A.C. Antoulas and D.C. Sorensen

(A) Krylov subspace projection techniques for constructing reduced order models of dynamical systems

This project seeks to develop reduced models for LTI systems through low rank approximation of certain system Grammians. The approach has high potential for addressing two fundamental difficulties with existing dimension reduction techniques. These two topics are central to the potential development of robust and widely applicable software. This approach seeks to:

  1. Provide rigorous bounds on on the H-infinity norm of the difference between the response of the full order system and the reduced order system in terms of the Hankel singular values neglected by the reduced order system;
  2. Extend naturally from single input, single output systems (SISO) to multiple input, multiple output systems (MIMO).
Our goal is to investigate subspace projection methods of Krylov and non-Krylov type. Implicit restarting methods will be developed that iteratively produce approximations of specified rank k to controllablity-, observability-, and cross- Grammians that are best approximations of rank k to the full Grammians. We seek methods related to the implicitly restarted Arnoldi, and the Jacobi-Davidson methods for large eigenvalue problems. These methods are well suited to large matrices A such as those arising from discretization of the spatial differential operator in a time dependent PDE.

(B) On the approximation of linear operators with applications

The goal of this project is to investigate problems of approximation of linear operators. The approximation method considered is based on the singular value decomposition of the operator in question. More precisely, we propose to study the existence of a unifying framework for the optimal and sub-optimal approximation of (i) unstructured operators in finite dimensions, and (ii) structured (Hankel) operators in infinite dimensions. It is expected that this study will shed light on a large class of singular-value-decomposition based approximation problems.

In order to predict the behavior of dynamical systems such as, a compact disc (CD) player, a chemical reaction, or a multi-story building, and appropriately modify (control) this behavior according to given specifications, a mathematical model is needed. Often, (detailed) models for these dynamical systems can be obtained using so-called finite-element methods. The resulting complexity of such models is very high. However, for both simulation and control purposes low-complexity models are needed, an important requirement being that these simplified models retain key features of the original high-complexity systems. The purpose of this project is to study approximation methods which yield low-complexity models with (theoretically) guaranteed properties. This is a necessary step towards addressing, in a second phase, the problem of control of such dynamical systems. It is expected that this project, besides its significance for mathematics, will have an impact on application problems arising in several engineering disciplines, and on the theorerical foundations of high performance computing.

On the Role of Normal Forms In Control and Systems Theory

Principal Investigators: A. Astolfi, P. Colaneri, and A.C. Antoulas

Several linear and non-linear control problems, including identification and robust control problems, have been widely studied in the last decades and numerous (co-ordinates free) characterisations have been proposed. These provide a set of necessary and sufficient conditions which must be tested to verify the solvability of the problem and are exploited in the construction of a solution. These tests involve coupled Linear Matrix Inequalities (in the case of linear systems) or Partial Differential Equations (in the case of non-linear systems), hence are very difficult to assess. Nevertheless, it has been shown that several problems (for example the stabilisation problem) are easily solved if the underlying system is described in a set of co-ordinates adapted to the problem. Goal of this research project is to study the role of co-ordinates transformations in the solution of identification and control problems. More precisely, the problems of parameters identification, static output feedback stabilisation, mixed H2-Hinf control and model reduction for linear and non-linear systems will be revisited in co-ordinates ie the problem will be studied in the co-ordinates system which is better suited to expose it. As a result, simple to test necessary conditions (obstructions) and simple constructive algorithms for the above mentioned problems will be derived.

Back to A.C. Antoulas' Home Page

Last modified: May 1, 2002