State estimation for nonlinear systems using Finite Impulse Response (FIR) filters presents two key challenges: constructing effective filter structures and deriving optimal solutions. This paper introduces a novel nonlinear FIR filtering approach that addresses these challenges through the Koopman operator framework. The Koopman operator, a prominent data-driven method, provides intrinsic coordinates that enable global linearization of nonlinear dynamics. This transformation yields a linearized system that accurately represents the original non-linear behavior, offering new possibilities for nonlinear FIR filter design. To enhance robustness, we model the linearization error as bounded noise, enabling the filter to accommodate significant modeling mismatches within defined limits. Combined with the system’s Gaussian noise, this hybrid uncertainty description leverages the FIR structure’s inherent high degree of freedom. The optimal filter gain is then determined through multi-objective optimization. Simulation results demonstrate the proposed filter’s exceptional robustness against both modeling uncertainties and noise characterization errors, confirming its practical applicability.

In this paper, the set-membership state and parameter estimation problems are considered for linear time-varying systems. The noises in the system are unknown but bounded and the variation of parameters is considered as a bounded expansion. The entire estimator is an interactive estimation of parameters and states. In parameter estimation, the exact set of parameter estimates may be non-convex because of the uncertain states. A polytopic linear parameter varying (LPV) enclosure of the system regression expression is constructed as a convex relaxation. The result of parameter estimation is a set to include the true parameter of the model. Meanwhile, in the state estimation, a similar LPV enclosure of the state-space model is also used to bound the state matrix uncertainties. This LPV enclosure in the state estimation is narrowed by the parameter estimation, and then the set of state estimates is computed to include the true state. The constrained zonotope (CZ) is used to describe sets of interest and the necessary set operations are expanded to LPV mapping. Because of the highly adjustable accuracy of CZ, the proposed algorithm can make a better trade-off between computational accuracy and efficiency. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.