A Kalman Filter Approach for Sparse Input and State Tracking

The application of interest in the paper is estimating the unknown dynamic input and state vector (displacements and velocities) by using partial noisy acceleration measurements for a structural system. A dual Kalman filter scheme is employed for state-parameter identification, but, in addition, we propose a sparse Bayesian learning framework to impose spatially-sparse input (e.g., impulse excitation), and also we would like our model to capture the evolution of the sparse input changes with shared "common sparseness", i.e., the input changes between two successive time instants are also sparse. To this end, we present a hierarchical Bayesian state-space model for computing the marginal posterior distributions of the state and input parameters, where the two sparseness constraints mentioned above are effectively incorporated for each time instant. The measurement and state prediction error parameters (noise parameters) are learned solely from the available data up to the current time, where Bayesian Ockham razor is automatically implemented. Finally, numerical investigation of the proposed algorithm is presented. It is shown that reasonable estimates of impulse and seismic input as well as structural state vector can be accomplished. It is also shown that the well-known drift problem in the estimated input commonly encountered by existing filter methods is effectively alleviated.