Online Bayesian identification of non-smooth systems

The robustness of online Bayesian Identification algorithms has been illustrated for a wide range of physical problems. The successful convergence of such algorithms for problems of highly nonlinear nature is tied to the precision of the approximation of the observed system via the employed state-space model. More sophisticated approximations, result in an increase of both the convergence rate and the associated computational cost. Nonetheless, the latter is a price worth paying for ensuring the former in the case of highly nonlinear problems. The assumption placed by most Bayesian filtering algorithms is that the parameters to be estimated are identifiable at each updating step. This however is a property that does not necessarily hold for systems involving nonsmooth nonlinearities, i.e., systems whose state-space or measurement equations are not differentiable. Such systems are linked to the modelling of damage-related phenomena such as plasticity, impact and sliding amongst other. Hence, a separate approach is proposed herein, namely the modification of algorithms to account for the lack of identifiability encountered for parameters of a non-smooth system at a specific step. This modification is termed by the authors as, the Discontinuous, D- modification and relies on the idea that unidentifiable parameters should remain invariant in the corresponding updating steps. This work will illustrate the benefits of the D- modification on the convergence of the Unscented Kalman Filter for non-smooth problems. An example from the dynamics of rocking bodies will be used to demonstrate the advantages of the method