Convergence Guarantees for Moving Horizon Estimation Based on the Real-Time Iteration Scheme

Abstract

Abstract-In this note, conditions are proven under which a realtime implementable moving horizon estimation (MHE) scheme is locally convergent. Specifically, the real-time iteration scheme of [17] is studied in which a single Gauss-Newton iteration is applied to approximate the solution to the respective MHE optimization problem at each timestep. Convergence is illustrated by a challenging small scale example, the Lorenz attractor with an unknown parameter. It is shown that the performance of the proposed real-time MHE algorithm is nearly identical to a fully converged MHE solution, while its fixed execution time per sample would allow one to solve 30 000 MHE problems per second on current hardware.

Real‐time nonlinear moving horizon observer with pre‐estimation for aircraft sensor fault detection and estimation

“…This results in a sparsely structured KKT system that is numerically better conditioned than the condensed form. 34 The multiple shooting method can be also applied to the MHO-PE (4) with the benefit of reducing computational complexity by avoiding explicit disturbance estimation compared with (2). However, the stability analysis of MHO-PE in the context of multiple shooting cannot easily follow what we have performed for the condensed MHO-PE in Section 4, which remains an open problem for future research.…”

“…Alternatively, all state variables can be kept as optimization variables, which is known as the multiple shooting method. Such multiple shooting were applied by Haverbeke and Wynn et al to solve and , respectively. This results in a sparsely structured KKT system that is numerically better conditioned than the condensed form .…”

“…Such multiple shooting were applied by Haverbeke and Wynn et al to solve and , respectively. This results in a sparsely structured KKT system that is numerically better conditioned than the condensed form . The multiple shooting method can be also applied to the MHO‐PE with the benefit of reducing computational complexity by avoiding explicit disturbance estimation compared with .…”

“…As a consequence, the established stability analysis for MHOs with fully solved optimizations in the works of Rao et al and Alessandri et al cannot be applied to such real‐time iteration schemes. For real‐time MHO using gradient, conjugate gradient, and Newton methods, different types of stability analyses were proposed by Alessandri and Gaggero and Wynn et al…”

“…As a consequence, the established stability analysis for MHOs with fully solved optimizations in the works of Rao et al 18 and Alessandri et al 33 cannot be applied to such real-time iteration schemes. For real-time MHO using gradient, conjugate gradient, and Newton methods, different types of stability analyses were proposed by Alessandri and Gaggero 22 and Wynn et al 34 Motivated by the idea of prestabilizing model predictive control, a new linear MHO was proposed in the works of Sui et al 35,36 by incorporating a pre-estimating observer for forward prediction. While model uncertainties in forward prediction are compensated for by disturbance estimates in a standard MHO formulation, 18 this is achieved by feedback injection from output measurements in an MHO with pre-estimation (MHO-PE).…”