To improve the identification efficiency of the auxiliary model based gradient iterative algorithm, this note derives a couple of interval error correction auxiliary model based gradient iterative (IEC-AM-GI) algorithms for multi-rate ARX models. The unmeasurable outputs between two measured outputs are estimated by two interval error correction auxiliary models. Unlike the AM-GI algorithm, the IEC-AM-GI algorithms can estimate the missing outputs by the auxiliary model and further fine tune through the measurable outputs, which thus are more efficient and accurate than those by only using the auxiliary model. In addition, this note makes a comparison between these two interval error correction auxiliary model methods, and through which can choose the optimal interval error correction auxiliary model method for the unmeasurable outputs. The effectiveness of the proposed algorithms is demonstrated by the simulation results.

In this study, a multi-direction-based gradient iterative (GI) algorithm for Hammerstein systems with irregular sampling data is proposed. The algorithm updates the parameter estimates using several orthogonal directions at each iteration. The convergence rate is significantly improved with an increasing number of directions. The convergence property and two simulation examples are provided to demonstrate the effectiveness of the proposed algorithm. In addition, the multi-direction-based GI algorithm establishes a relationship between the traditional GI and least squares (LS) algorithms. Thus, our algorithm that combines the LS and GI algorithms constructs an identification framework for a significantly wider class of systems.

This paper develops a biased compensation recursive least squares-based threshold algorithm for a time-delay rational model. The time-delay rational model is transformed into an augmented model by using the redundant rule, and then, a recursive least squares algorithm is proposed to estimate the parameters of the augmented model. Since the output of the augmented model is correlated with the noise, a biased compensation method is derived to eliminate the bias of the parameter estimates. Furthermore, based on the structures of the augmented model parameter vector and the rational model parameter vector, the unknown time delay can be computed by using a threshold given in prior. A simulated example is used to illustrate the efficiency of the proposed algorithm.

A stochastic gradient (SG)-based particle filter (SG-PF) algorithm is developed for an ARX model with nonlinear communication output in this paper. This ARX model consists of two submodels, one is a linear ARX model and the other is a nonlinear output model. The process outputs (outputs of the linear submodel) transmitted over a communication channel are unmeasurable, while the communication outputs (outputs of the nonlinear submodel) are available, and both of the two-type outputs are contaminated by white noises. Based on the rich input data and the available communication output data, a SG-PF algorithm is proposed to estimate the unknown process outputs and parameters of the ARX model. Furthermore, a direct weight optimization method and the Epanechnikov kernel method are extended to modify the particle filter when the measurement noise is a Gaussian noise with unknown variance and the measurement noise distribution is unknown. The simulation results demonstrate that the SG-PF algorithm is effective.