Robust moving horizon estimation for nonlinear systems: From perfect to imperfect optimization

This paper examines the robust stability of moving-horizon estimators for nonlinear discrete-time systems that are detectable in the sense of incremental input/output-to-state stability and are affected by disturbances. The estimate of a moving-horizon estimator is derived from the on-line solution of a least-squares minimization problem at each time instant. The resulting stability guarantees depend on the optimization tolerance in solving these minimization problems. Specifically, two main contributions are established: (i) the robust stability of the estimation error, assuming the on-line minimization problem is solved exactly; (ii) the practical robust stability of the estimation error with state estimates obtained through imperfect minimization. Finally, the construction of such robust moving-horizon estimators and the performance resulting from the design based on the theoretical findings are showcased with two numerical examples.