A Computational Framework for Generalized Constrained Inverse Problems

This paper generalizes and extends the theory of approximating measurement data with constrained basis functions. The new method is particularly well suited for modelling sensor data in cyber-physical systems, where the physics of the system being monitored needs to be embedded. The extension includes both co-located and interstitial constraints. The complete derivation of all required equations is presented in a matrix algebraic framework. The new approach enables the reconstruction of curves with a lower statistical uncertainty from fewer measurement points. The method is demonstrated here with examples from structural monitoring. A Monte Carlo simulation is presented where the new approach is compared with unconstrained approximation. The new approach has a sig-nificantly narrow band of uncertainty around the reconstructed curve. Furthermore, approximations are presented for real data acquired in a laboratory bending beam setup. This data is used to validate the claim that, with this method, reliable reconstructions can be obtained from a low number of measurement points.