Truncated Unscented Kalman Filter for Incorporating Constraints in Joint State-Parameter Estimation

In joint state-parameter estimation using the Unscented Kalman Filter (UKF), considerable tuning of the initial state vector and covariance matrices may be required to ensure proper convergence. Constraints on the parameters, e.g. bounds on stiffness values, which may be known a priori from physics or engineering judgement, can be incorporated to have improved convergence. In this work, the Truncated Unscented Kalman Filter (TUKF) is introduced to handle such constraints within the UKF framework. The constraints are enforced by appropriately truncating the posterior probability density functions of the system states. The method is illustrated using two different class of systems: 1) shear frames with hysteretic Bouc-Wen elements exhibiting degradation and pinching; and 2) train-bridge systems, which can behave as linear time-varying systems because of the moving heavy train. Numerical examples are used to illustrate the robustness of the TUKF against noisy measurements and high uncertainty in the initial state vector. For the hysteretic models, as complex models involving a large number of parameters are prone to numerical instabilities; for easier practical implementations, estimation is explored with simple and incomplete model classes as well. Furthermore, shake table tests on a reinforced concrete frame, progressive damaged using inputs of increasing amplitude, are used to illustrate the effectiveness of the method in assessing the level of damage. For the train-bridge system, mode shape-based constraints are also utilized to further improve the estimation of the train and bridge parameters. This method is illustrated using experimental data from a lab-scale two-span continuous bridge, excited using a moving train, where it is also seen that the identified parameters can be used for damage detection. Overall, the results show that the TUKF improves on the convergence of the UKF, while maintaining the same level of accuracy in the state and parameter estimates.