Anytime Proximity Moving Horizon Estimation: Stability and Regret

In this article, we address the efficient implementation of moving horizon state estimation of constrained discrete-time linear systems. We propose a novel iteration scheme that employs a proximity-based formulation of the underlying optimization algorithm and reduces computational effort by performing only a limited number of optimization iterations each time a new measurement is received. We outline conditions under which global exponential stability of the underlying estimation errors is ensured. Performance guarantees of the iteration scheme in terms of regret upper bounds are also established. A combined result shows that both exponential stability and a sublinear regret, which can be rendered smaller by increasing the number of optimization iterations, can be guaranteed. The stability and regret results of the proposed estimator are showcased through numerical simulations.