The filtered-x least mean square (FxLMS) algorithm has been proposed for an active noise control (ANC) system. However, due to the used mean square error (MSE) criterion, FxLMS suffers from performance degeneration for non-Gaussian noises, dramatically. To address this issue, a novel robust generalized hyperbolic tangent (GHT) criterion is first constructed in this paper. Then, the random Fourier features (RFF) method and the conjugate gradient (CG) method are used to address the nonlinearity existing in ANC and solve the quadratic optimization problem induced by the GHT criterion, respectively. Finally, a novel robust random Fourier conjugate gradient filtered-x generalized hyperbolic tangent (RFCGFxGHT) algorithm is proposed for ANC. The theoretical analyses regarding the convergence and computational complexity of RFCGFxGHT are also derived. Simulation experiments on nonlinear ANC systems corrupted by the synthetic logistic chaotic and$\alpha$-stable noises, as well as real-world functional magnetic resonance imaging (fMRI) and server room noises, are conducted to confirm the effectiveness, robustness, and desirable nonlinear learning ability of the proposed algorithm.

In this letter, a generalized measurement model (GMM) is established under the attack injecting additive and multiplicative false data, simultaneously. The GMM can be applied to any existing nonlinear Kalman filters (NKFs), such as extended Kalman filter (EKF), cubature Kalman filter (CKF), and geometric unscented Kalman filter (GUF). Based on the constructed GMM, a class of nonlinear Kalman filters with GMM (NKFs-GMM) is proposed to deal with additive false data, multiplicative false data, and simultaneous attacks on additive and multiplicative false data with no increase of computational complexity. Numerical simulations show that the filtering accuracy of NKFs-GMM is superior to that of corresponding NKFs.

Generally, the power system (PS) faces various anomalies such as non-Gaussian measurement noises and gross measurement errors, when the communication link is lost or the phasor synchronization is inaccurate. Therefore, the robust forecasting-aided state estimation (FASE) is crucial for PS stability. This letter develops a novel geometric unscented Kalman filter (GUF) with the generalized loss (GL) (GL-GUF) to estimate PS state for forecasting aid. In contrast to the minimum mean square error (MMSE) criterion in the original GUF framework, GL-GUF combines the strength of the GL in robust information learning against non-Gaussian disturbances and the advantages of GUF in handling strong model nonlinearity with high accuracy and good stability. More importantly, by establishing the linear regression model to obtain residual vectors and introducing the negative log-likelihood of GL, simultaneously, GL-GUF optimizes free parameters to avoid manual parameter tuning. Simulations on IEEE 14-bus and 30-bus test systems confirm the high precision and robustness of GL-GUF for non-Gaussian disturbances.

Quaternion adaptive filters have been widely used in processing three-dimensional and four-dimensional signals. To improve the performance of quaternion adaptive filtering algorithms, this letter first proposes a novel derivation rule named generalized HR (GHR)$q$-derivative based on the concept of$q$-derivative and quaternion GHR derivative. Then, the product rules of GHR$q$-derivative are deduced, and the results of GHR$q$-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR$q$-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the$q$-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed$q$-QLMS algorithm.