A Discontinuous Extended Kalman Filter for non-smooth dynamic problems

Abstract

Problems that result into locally non-differentiable and hence non-smooth state-space equations are often encountered in engineering. Examples include problems involving material laws pertaining to plasticity, impact and highly non-linear phenomena. Estimating the parameters of such systems poses a challenge, particularly since the majority of system identification algorithms 5 are formulated on the basis of smooth systems under the assumption of observability, identifiability and time invariance. For a smooth system, an observable state remains observable throughout the system evolution with the exception of few selected realizations of the state vector. However, for a non-smooth system the observable set of states and parameters may vary during the evolution of the 10 system throughout a dynamic analysis. This may cause standard identification (ID) methods, such as the Extended Kalman Filter, to temporarily diverge and ultimately fail in accurately identifying the parameters of the system. In this work, the influence of observability of non-smooth systems to the performance of the Extended and Unscented Kalman Filters is discussed and a novel algo-15 rithm particularly suited for this purpose, termed the Discontinuous Extended Kalman Filter (DEKF), is proposed.

A Discontinuous Unscented Kalman Filter for Non-Smooth Dynamic Problems

“…Hence, this investigation leads to a result similar to those of the DEKF in Chatzis et al (2017) for Elasto-plastic springs: it appears more favorable to use initial estimates of F y i smaller than the real value of those parameters, or at least smaller than the maximum force seen by an elastic spring for the displacements of the system. Of course while this is not possible a priori, as those values are not known one can be informed by the DUKF of the inability of the algorithm to identify the corresponding value of F y i , due to the lack of change of the parameters and the corresponding large covariance terms.…”

“…The observability of non-smooth systems and the points of differentiation to their smooth counterparts have been discussed in Chatzis et al (2014Chatzis et al ( , 2017. It should be noted that the notions of observability and identifiability used in these papers and in this work refer to the ability to distinguish the states and parameters from their neighbors at a specific time instance.…”

“…The dynamic states kxel i correspond to the product k i x el i and are used instead of the elastic displacements as it allows separating the states into observable and unidentifiable within all branches without having to use a non-linear transformation (Chatzis et al, 2017). In terms of the identifiability of the system it can easily be shown as in Chatzis et al (2017), that all the dynamic states together with c 1 and c 2 are always identifiable.…”

“…In terms of the identifiability of the system it can easily be shown as in Chatzis et al (2017), that all the dynamic states together with c 1 and c 2 are always identifiable. However, only one of [k i , F y i ] is identifiable depending on whether spring i lies in an elastic or plastic branch at that specific time instance.…”

“…In this work, and following original developments introduced in Chatzis et al (2017), a modified version of the standard UKF is proposed. The key to the formulation lies in adoption of a modular state-space formulation, evaluation of the observability within each time step of the analysis, and selection of an appropriate subspace of the full state vector to be used by the UKF.…”