Mathematical models must meet several requirements in a rapid equationbased development environment and in future high-tech initiatives in systems engineering, such as the digital twin paradigm. A bottom-up modeling approach requires coupling models across different scales and physical domains. At the same time, we wish to replace a model within a network of models with a surrogate model while maintaining important structural properties of the overall system, such as stability or passivity. Replacing a model becomes necessary whenever the original model is not feasible for simulation, optimization, or control or cannot reproduce the actual phenomena to a satisfactory level. Towards the latter aspect, we discuss in this talk a system identification framework to learn passive dynamical systems from measurements either in the time or in the frequency domain.
In the time domain, we leverage the framework of port-Hamiltonian (pH) systems [1], which constitutes an innovative energy-based model paradigm that offers a systematic approach for the interactions of (physical) systems with each other and the environment via interconnection structures. By informing dynamic mode decomposition with a given quadratic energy functional as physical prior, we propose a physics-informed approach that learns passive dynamical systems from data; see [2]. The resulting optimization problem can be efficiently solved with a fast-gradient method and a clever initialization strategy. In the frequency domain, we exploit the equivalence of passivity with the existence of a solution of the KYP inequality and directly infer the low-rank KYP matrix from measurements by solving a nonlinear least-squares problem. The methods are demonstrated with numerical examples.
[1] van der Schaft, A. and Jeltsema, D. 2014. Port-Hamiltonian systems theory: An introductory overview. Foundations and Trends® in Systems and Control, 1(2-3):173–378.
[2] Morandin, R., Nicodemus, J., and Unger, B. 2023. Port-Hamiltonian Dynamic Mode Decomposition. To appear in SIAM J. Sci. Comput.. Also available as ArXiV e-print 2204.13474.