A Discontinuous Unscented Kalman Filter for Non-Smooth Dynamic Problems

For a number of applications, including real/time damage diagnostics as well as control, online methods, i.e., methods which may be implemented on-the-fly, are necessary. Within a system identification context, this implies adoption of filtering algorithms, typically of the Kalman or Bayesian class. For engineered structures damage or deterioration may often manifest in relation to phenomena such as fracture, plasticity, impact or friction. Despite the different nature of the previous phenomena, they are described by a common denominator: switching behavior upon occurrence of discrete events. Such events include for example, crack initiation, transitions between elastic and plastic response, or between stick and slide modes. Typically, the state-space equations of such models are non-differentiable at such events, rendering the corresponding systems non-smooth. Identification of non-smooth systems poses greater difficulties than smooth problems of similar computational complexity. Up to a certain extent, this may be attributed to the varying identifiability of such systems, which violates a basic requirement of online Bayesian Identification algorithms, thus affecting their convergence for non-smooth problems. Herein, a treatment to this problem is proposed by the authors, termed the Discontinuous D- modification, where unidentifiable parameters are acknowledged and temporarily excluded from the problem formulation. In this work the D- modification is illustrated for the case of the Unscented Kalman Filter UKF, resulting in a method termed DUKF, proving superior performance to the conventional, and widely adopted, alternative.