### Lecturer

Ivan Markovsky

Vrije Universiteit Brussel

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Contact

### Schedule and place

This course will take place on March 12, 13, 14, 19, 20, 21, 2014 at K.U.Leuven (Room 00.62, ESAT, Kasteelpark Arenberg 10, B-3001 Heverlee-Leuven)

Schedule : 9h15 - 12h15

## Description

Established data modeling approaches are often derived in a stochastic setting. An alternative deterministic approximation approach, known in the systems and control literature as the behavioral approach, has been developed since the 80's by Jan C. Willems and co-workers. The behavioral approach differentiates between the abstract notion of a model and the concrete notion of a model representation. This distinction proves to be important for developing a coherent theory and effective algorithms for system identification, analysis, and control. The course presents a behavioral approach to system identification.

The highlight of the course is the low-rank approximation problem, which is a practical tool for modeling in the behavioral setting. A matrix constructed from the data being rank deficient implies that there is an exact low complexity linear model for that data. Moreover, the rank of the data matrix corresponds to the complexity of the model. In the generic case when an exact low-complexity model does not exist, the aim is to find a model that fits the data approximately. The corresponding computational problem is low-rank approximation. In the case of linear time-invariant dynamical models, the data matrix is, in addition, Hankel structured and the approximation should have the same structure.

Once the approximate system identification problem is formulated as a low-rank approximation problem, it is solved by generic methods. Except for a few special cases, however, low-rank approximation problems are nonconvex and a global solution is expensive to compute. In the course, we present methods based on local optimization, which lead to fast and effective algorithms. The cost function evaluation has the system theoretic interpretation of Kalman smoothing.

In addition to the theory and algorithms for exact and approximate system identification, the course presents examples from system theory (model reduction and distance to uncontrollability), computer algebra (approximate common divisor computation), and machine learning (recommender systems). Software implementation of the developed methods makes the theory applicable in practice. An essential part of the course are exercises, which give hands-on experience with the presented theory and methods.

## Course material

**Students are asked to take their own laptop for the exercise sessions**

The course is based on the following references :

- I. Markovsky,
*Low rank approximation: Algorithms, implementation, applications*, Springer, 2012 - I. Markovsky, et al.,
*Exact and approximate modeling of linear systems: A behavioral approach*, SIAM, 2006 - I. Markovsky and K. Usevich,
*Software for weighted structured low-rank approximation*, J. Comput. Appl. Math., 256:278-292, 2014

## Evaluation

To be discussed with the students at the beginning of the course.