The robust online identification problem of linear systems is considered in this article using a faster robust recursive expectation–maximization (RREM) framework. To improve the convergence rate, the outliers, which would deteriorate the identified models, are accommodated with a Laplace distribution instead of Student's $t$ -distribution. Then, the recursive transformation of the maximum likelihood function is realized with a recursive $Q$ -function. The extensively recognized autoregressive exogenous (ARX) models are used for the description of general linear systems. As a result, the unknown parameters, including the regression coefficient vector of the ARX models, the variance of the noise without outliers, and the scale parameter of the Laplace distribution, are determined in a recursive manner. The performance of the proposed approach is tested with a simulated continuous fermentation reactor system example and a coupled-tank experiment.

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.

An online variational Bayesian algorithm based on a skewed Student's t-distribution, referred to as OVB-skewt, is proposed for robust parameter estimation in linear AutoRegressive with eXogenous inputs (ARX) models. In practical applications, measurement data are often contaminated by non-Gaussian, heavy-tailed, and asymmetric noise, which poses significant challenges to conventional identification methods that typically assume Gaussian noise. To address this issue, the proposed method incorporates the generalized hyperbolic skewed Student's t-distribution into a variational Bayesian inference framework, allowing the noise characteristics to be adaptively learned during online estimation. By explicitly modeling both heavy-tailed behavior and skewness, the proposed algorithm improves robustness against outliers and asymmetric disturbances. The online formulation enables efficient processing of streaming data, making the method suitable for time-varying industrial environments. Numerical simulations and a continuous fermentation reactor (CFR) benchmark demonstrate that the proposed OVB-skewt approach achieves improved estimation accuracy compared with symmetric noise-based methods under skewed and contaminated noise conditions, while maintaining stable performance in online settings.