In this course dynamical systems described by sets of coupled linear, differential or difference equations are considered. The goal is to approximate the behavior of such systems by that of simpler ones, the complexity being defined as the number of state variables needed to describe them. For such approximations to be useful in theory and in practice, the following properties need to be satisfied:

(1) Preservation of certain properties of the original system (for instance stability).

(2) Optimization or at least quantification, of appropriate measures of the error introduced by the approximation.

(3) Numerical reliability and efficiency.

There are many methods available for approximating linear dynamical systems. However, the majority fail to satisfy at least some of the above requirements.

In the first part of this course, we will present the theory of approximation in the so-called Hankel norm, which addresses the first two issues. The major difficulty involved in solving this problem is that it is non-convex. It can be solved nevertheless, using an ad-hoc tool, namely the singular value decomposition of a certain Hankel operator associated with the system. This generalizes the well known theory of optimal and sub-optimal approximation of constant matrices of finite size by matrices of lower rank.

In the second part, a different approximation method is introduced. This is the Lanczos-Arnoldi approach which is connected to Pade approximation. The first feature of this method is the construction of approximants by means of projections; the second is the explicit avoidance of matrix inversions.

The advantage of the first approach is that explicit formulae are provided for the approximants and in addition properties (1) and (2) are satisfied. The downside is that property (3) is not (consequently, this method is not applicable to systems with hundreds or thousands of states). In contrast, the second method addresses this latter concern, but fails to satisfy the first two requirements.

Both methods mentioned above will be analyzed and compared with one another. Examples and case studies from various engineering disciplines will complement the presentation; the algorithms used are available in Matlab.

Benchmark examples for model reduction:  href="http://www.icm.tu-bs.de/NICONET/benchmodred.html

Model reduction m-files: zip archive