Two novel spline adaptive filtering (SAF) algorithms are proposed by combining different iterative gradient methods, i.e., Adagrad and RMSProp, named SAF-Adagrad and SAF-RMSProp, in this paper. Detailed convergence performance and computational complexity analyses are carried out also. Furthermore, compared with existing SAF algorithms, the influence of step-size and noise types on SAF algorithms are explored for nonlinear system identification under artificial datasets. Numerical results show that the SAF-Adagrad and SAF-RMSProp algorithms have better convergence performance than some existing SAF algorithms (i.e., SAF-SGD, SAF-ARC-MMSGD, and SAF-LHC-MNAG). The analysis results of various measured real datasets also verify this conclusion. Overall, the effectiveness of SAF-Adagrad and SAF-RMSProp are confirmed for the accurate identification of nonlinear systems.

To fully take advantage of LMS, LMAT, and SELMS, a novel adaptive estimator using the L1-norm and L0-norm of the estimated error is proposed in this paper. Then based on minimizing the mean-square deviation at the current time, the optimal step-size, parameters δ and θ of the proposed adaptive estimator are obtained. Besides, the stability and computational complexity of the mean estimation error is analyzed theoretically. Experimental results (both simulation and real mechanical system datasets) show that the proposed adaptive estimator is more robust to input signals and a variety of measurement noises (Gaussian and non-Gaussian noises). In addition, it is superior to LMS, LMAT, SELMS, the convex combination of LMS and LMAT algorithm, the convex combination of LMS and SELMS algorithm, and the convex combination of SELMS and LMAT algorithm. The theoretical analysis is consistent with the Monte-Carlo results. Both of them show that the adaptive estimator has an excellent performance in the estimation of unknown linear systems under various measurement noises.

Based on the superiority of adaptive filtering algorithms designed with hyperbolic function-like objective functions, this paper proposes generalized spline adaptive filtering (SAF) algorithms designed with hyperbolic function-like objective functions. Specifically, a series of generalized new SAF algorithms are proposed by introducing the q-deformed hyperbolic function as the cost function, named SAF-qDHSI, SAF-qDHCO, SAF-qDHTA & SAF-qDHSE algorithms, respectively. Then, the proposed algorithm is theoretically demonstrated with detailed mean convergence and computational complexity analysis; secondly, the effect of different q values on the performance of the new algorithm is verified through data simulation; the new algorithm still has better performance under the interference of Gaussian noise and non-Gaussian noise even when facing the system mutation; finally, the new algorithm is verified through the measured engineering data, and the results show that the new algorithm has better convergence and robustness compared with the existing algorithm. In conclusion, the generalized algorithm based on the new cost function proposed in this paper is more effective in nonlinear system identification.