## Competences required

MATLAB, linear algebra, signals and systems theory

## Project description

Signal processing and control methods are typically model-based. In a first `modeling' step, a mathematical model is built from the measured noisy data. In a second `design' step, the model is used to solve a specific application problem. In this project, we explore `data-driven methods', in which modeling and design are combined into a single task. By combining modelling and design into a single `direct’ problem, better solutions can be found than by solving the two problems separately.

The main tools that we use is structured low-rank matrix approximation and completion. Low-rank matrices have central role in data science: the data often has some underlying low-rank property, i.e., low complexity. The low-rank property can be used to remove noise and find unknown elements. This is used for example to design recommendation system [1].

In the context of linear dynamics, low-rank matrices come with a specific structure that captures time-invariance and linearity, namely the (block) Hankel matrix (having repeated elements along the anti-diagonals). Removing noise from signals is related to finding the nearest low-rank Hankel-structured matrix. Low-rank matrix completion appears in two ways: Firstly, there can be missing data because of sensor outages or problematic communication channels. Secondly, data completion can be used to find entire unknown signals, which allows us to come up with data-driven variants of classical problems such as system simulation (input u is given, output y is unknown), output tracking control (output y is given, input u is unknown), or Kalman filtering [2].

In this project you will learn more about the Hankel matrix view on linear systems and the data-driven approach. You will then zoom in on a specific data-driven signal processing task, and implement, explore and compare different methods for tackling the task at hand.

## References

- Netflix prize, see https://en.wikipedia.org/wiki/Netflix_Prize.
- I. Markovsky. A missing data approach to data-driven filtering and control. IEEE Trans. Automatic Control, 62:1972-1978, April 2017. http://homepages.vub.ac.be/~imarkovs/publications/ddsp.pdf
- I. Markovsky. Low-Rank Approximation: Algorithms, Implementation, Applications. Springer, 2019. http://homepages.vub.ac.be/~imarkovs/book/book2e-2x1.pdf

2020