SOCN course
“Behavioral approach to system theory”

The behavioral approach to systems theory, put forward 40 years ago by Jan C. Willems, remained till recently an esoteric niche of research. The renewed interest in the last years is because of its unique suitability for the newly emerged data-driven paradigm, specifically the representation-free perspective on dynamical systems as sets of trajectories. A result derived in the behavioral setting that became known as the fundamental lemma started a new class of subspace-type data-driven methods. The fundamental lemma gives conditions for a non-parametric representation of a linear time-invariant system by the image of a Hankel matrix constructed from raw time series data. This course reviews the fundamental lemma, its generalizations, and related data-driven analysis, signal processing, and control methods. A prototypical signal processing problem considered in the course is interpolation and approximation of trajectories. It includes simulation, state estimation, and output tracking control as special cases. The theory leads to computationally tractable methods. Regularized version of the methods is robust to noise and model assumptions. Participants will get hands-on experience with the methods via Matlab exercises and individual work on mini-projects.

The course consists of six 2.5-hour sessions. A session starts with an introduction, followed by questions-and-answers and exercises. After the third session, students will choose mini-projects to work on.

The course will not cover a fixed curriculum and will be adapted as much as possible to the need and interests of the participants. An indicative outline of topics that will be discussed in the course is:

• Dynamical systems as sets of trajectories
  • Complexity of a linear time-invariant system
  • Parametric representations of a bounded complexity linear time-invariant system
  • Controllability and control in the behavioral setting
• Data-driven non-parametric model representation
  • The fundamental lemma
  • Identifiability and the most powerful unfalsified model
  • Generalizations of the fundamental lemma
• Data-driven interpolation and approximation of trajectories
  • Existence and uniqueness of an exact interpolant
  • Mixed interpolation and approximation
  • Recursive computation
• Dealing with inexact/noisy data
  • Naive approach: using the pseudo-inverse
  • Structured low-rank pre-processing
  • Nuclear-norm and \(\ell_1\)-norm regularization
• Generalizations to nonlinear systems
  • Polynomial time-invariant systems
  • Hammerstein, finite-lag Volterra, and bilinear systems
  • Linear time-invariant embedding
• Applications
  • Input estimation
  • Internal navigation
  • Dynamic measurement

• Slides
• Exercises

• I. Markovsky, L. Huang, and F. Dörfler. Data-driven control based on behavioral approach: From theory to applications in power systems, Tutorial paper submitted to the IEEE Control Systems Magazine, 2022
• I. Markovsky and F. Dörfler. Behavioral systems theory in data-driven analysis, signal processing, and control. Annual Reviews in Control, 52:42–64, 2021.
• I. Markovsky. Low-Rank Approximation: Algorithms, Implementation, Applications. Springer, 2019.

Report and oral presentation on the mini-project.