Back to the roots of data-driven dynamical system identification

ongoing

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). 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