Introduction
To obtain data-driven dynamic models for simulation, prediction, monitoring, classification or control tasks, in applications e.g. in Industry 4.0 and eHealth, most identification methods solve an optimization problem, relying on some nonlinear iterative algorithm. Undeniably, too many heuristics prevail: What do we mean by ‘solved’? Where did the algorithm converge to? Is the model globally optimal, unique and reproducible?
To tackle these scientific deficiencies, we design a framework to deal with inexact data. We solve a longstanding open problem of least squares optimality in system identification: for polynomial dynamical models, the optimal model derives from an eigenvalue problem.
Hereto, we generalize notions from Algebraic Geometry (multivariate polynomials), Operator Theory (model spaces), System Theory (multidimensional realization) and Numerical Linear Algebra (matrix computations). These four mathematical discplines are central in the figure below, where we give an overview of the three objectives of the Back to the Roots project, each containing four challenges.
![](https://homes.esat.kuleuven.be/~sistawww/bdm/backtotheroots/wp-content/uploads/2023_08_17_diagram_objectives-1024x576.png)
The first objective is to develop a mathematically rigorous realization approach that maps data onto new mathematical structures (multi-shift invariant projective subspaces).
The second objective is to conceive a misfit-latency framework to optimally map inexact data to these mathematical structures. We prove this to be a multiparameter eigenvalue problem. We expect breakthroughs in system theoretic characterizations of optimality (covering all existing methods), in the generalization to multiple input-output and multidimensional models and in finding the global optimum in the linear dynamic H2 model reduction problem.
The third objective is to implement matrix computation algorithms for the results of the first two objectives, to root sets of multivariate polynomials, to solve multiparameter eigenvalue problems and to isolate only the minimizing roots. We focus on matrix aspects of large scale, sparsity and structure.
Deliverables will be publications, software, graduate course material and science outreach initiatives, in line with the PI’s excellent track record.
General information
- Advanced Grant from the European Research Council (ERC)
- January 2021 – December 2025
- Project objectives