  This paper proposes a primal-dual framework to learn a stable estimator for
linear constrained estimation problems leveraging the moving horizon approach.
To avoid the online computational burden in most existing methods, we learn a
parameterized function offline to approximate the primal estimate. Meanwhile, a
dual estimator is trained to check the suboptimality of the primal estimator
during execution time. Both the primal and dual estimators are learned from
data using supervised learning techniques, and the explicit sample size is
provided, which enables us to guarantee the quality of each learned estimator
in terms of feasibility and optimality. This in turn allows us to bound the
probability of the learned estimator being infeasible or suboptimal.
Furthermore, we analyze the stability of the resulting estimator with a bounded
error in the minimization of the cost function. Since our algorithm does not
require the solution of an optimization problem during runtime, state estimates
can be generated online almost instantly. Simulation results are presented to
show the accuracy and time efficiency of the proposed framework compared to
online optimization of moving horizon estimation and Kalman filter. To the best
of our knowledge, this is the first learning-based state estimator with
feasibility and near-optimality guarantees for linear constrained systems.
