A Time Splitting Based Optimization Method for Nonlinear MHE

Moving Horizon Estimation~(MHE) is essentially an optimization-based approach designed to estimate the states of dynamic systems within a moving time horizon. Traditional MHE solutions become computationally prohibitive due to the \textit{curse of dimensionality} arising from increasing problem complexity and growing length of time horizon. To address this issue, we propose novel computationally efficient algorithms for solving nonlinear MHE problems. Specifically, we first introduce a distributed reformulation utilizing a time-splitting technique. Leveraging this reformulation, we develop the Efficient Gauss-Newton Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) to achieve computational efficiency. Additionally, to accommodate limited computational capabilities inherent in some sub-problem solvers, we propose the Efficient Sensitivity Assisted ALADIN, which enables sub-problems to be solved inexactly without hindering computational efficiency. Furthermore, recognizing scenarios where sub-problem solvers possess no computational power, we propose a Distributed Sequential Quadratic Programming (SQP) that relies solely on first- and second-order information of local objective functions. We demonstrate the performance and advantages of our proposed methods through numerical experiments on differential drive robots case, a practical nonlinear MHE problem. Our results demonstrate that the three proposed algorithms achieve computational efficiency while preserving high accuracy, thereby satisfying the real-time requirements of MHE.