Data-Driven Inference on Optimal Input-Output Properties of Polynomial Systems With Focus on Nonlinearity Measures

• Published in: IEEE transactions on automatic control Vol. 68; no. 5; pp. 2832 - 2847
• Main Authors: Martin, Tim, Allgower, Frank
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Data-driven system analysis; Dynamical systems; Estimation; identification for control; Linear approximation; Linear systems; Noise measurement; Nonlinear systems; Nonlinearity; Optimization; polynomial dynamical systems; Polynomials; Representations; Semidefinite programming; Time-domain analysis; Trajectory; Upper bounds
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3226652

In the context of dynamical systems, nonlinearity measures quantify the strength of nonlinearity by means of the distance of their input-output behavior to a set of linear input-output mappings. In this article, we establish a framework to determine nonlinearity measures and other optimal input-output properties for nonlinear polynomial systems without explicitly identifying a model but from a finite number of input-state measurements, which are subject to noise. To this end, we deduce from data for the unidentified ground-truth system three possible set-membership representations, compare their accuracy, and prove that they are asymptotically consistent with respect to the amount of samples. Moreover, we leverage these representations to compute guaranteed upper bounds on nonlinearity measures and the corresponding optimal linear approximation model via semidefinite programming. Furthermore, we extend the established framework to determine optimal input-output properties described by time domain hard integral quadratic constraints.