MHE under parametric uncertainty -- Robust state estimation without informative data

In this paper, we study joint state and parameter estimation for general nonlinear systems with uncertain parameters and persistent process and measurement noise. In particular, we are interested in stability properties of the resulting state estimate in the absence of persistency of excitation (PE). With a simple academic example, we show that existing moving horizon estimation (MHE) approaches for joint state and parameter estimation as well as classical adaptive observers can result in diverging state estimates in the absence of PE, even if the noise is small. We propose an MHE formulation involving a regularization based on a constant prior estimate of the unknown system parameters. Only assuming the existence of a stable state estimator, we prove that the proposed MHE approach results in practically robustly stable state estimates irrespective of PE. We discuss the relation of the proposed MHE formulation to state-of-the-art results from MHE and adaptive estimation. The properties of the proposed MHE approach are illustrated with a numerical example of a car with unknown tire friction parameters.