Yanjun Liu

It is known that fractional-order models can effectively represent complex high-order systems with fewer parameters. This paper focuses on the identification of a class of multiple-input single-output fractional Hammerstein models. When the commensurate order is assumed to be known, a greedy orthogonal least squares method is proposed to simultaneously identify the parameters and system orders, combined with a stopping rule based on the Bayesian information criterion. Subsequently, the commensurate order is determined by minimizing the normalized output error. The proposed method is validated by applying it to identify a CD-player arm system.

The multipath effect often reduce global positioning system (GPS) positioning accuracy. Traditional GPS positioning methods usually ignore the multipath effect errors, resulting in poor positioning estimation accuracy. This article proposes a variable projection algorithm based on the Aitken method, which takes into consideration of the multipath effect error with the aim at improving the estimation accuracy, and uses the Aitken acceleration method to increase the convergence rate. In addition, the proposed method is robust to the step-size. Theoretical analysis and simulation examples show the effectiveness of the proposed method.

This study proposes two iterative reweighted algorithms for systems whose data are contaminated by outliers. For the negative effect caused by the outliers, traditional least squares and gradient descent algorithms cannot obtain unbiased estimates, while the variational Bayesian and expectation-maximization algorithms have the assumption that the prior knowledge of the outlier is available. To deal with these dilemmas, two iterative reweighted algorithms are developed. By assigning suitable weights for each data set, unbiased parameter estimates can be obtained. In addition, the weights of the corrupted data sets become smaller and smaller with the increased number of iterations, and then the contaminated data can be picked out from the data sets. The proposed algorithms do not require the prior knowledge of the outliers. Convergence analysis and numerical experiments show effectiveness of the iterative reweighted algorithms.

This paper focuses on the identification of multiple inputs Wiener systems with piecewise-linear nonlinearity. The main objective is to jointly estimate the parameters and time-delays by using limited samples when the system orders are unknown. In order to identify the over-parameterized sparse system under the framework of greedy method, a Householder transformation-based greedy orthogonal least squares algorithm is proposed. In this scheme, the active columns of the information matrix are selected one by one according to a new greedy strategy, then the Householder transformation is employed to maintain an upper-triangular structure for the sub-information matrix, the back-substitution method is adopted to avoid the matrix inversion. Thus the problem of ill-conditioning frequently appears in nonlinear models can be eliminated. The Bayesian information criterion is used to choose the optimal sparsity level. Numerical experiments show that this new scheme can achieve almost optimal generalization performance while requiring less observations than the traditional schemes.

The expectation maximization (EM) algorithm has been extensively used to solve system identification problems with hidden variables. It needs to calculate a derivative equation and perform a matrix inversion in the EM-M step. The equations related to the EM algorithm may be unsolvable for some complex nonlinear systems, and the matrix inversion has heavy computational costs for large-scale systems. This paper provides two expectation based algorithms with the aim of constructing a comprehensive expectation framework concerning different kinds of time-delayed systems: (1) for a small-scale linear system, the classical EM algorithm can quickly obtain the parameter and time-delay estimates; (2) for a complex nonlinear system with low order, the proposed expectation gradient descent (EGD) algorithm can avoid derivative function calculation; (3) for a large-scale system, the proposed expectation multi-direction (EMD) algorithm does not require eigenvalue calculation and has less computational costs. These two algorithms are developed based on the gradient descent and multi-direction methods. Under such an expectation framework, different kinds of models are identified on a case-by-case basis. The convergence analysis and simulation examples show effectiveness of the algorithms.

In the traditional gradient descent (T-GD) algorithm, the convergence rate is strongly depend on the condition number of the information matrix: a larger condition number leads to a poor optimal convergence factor infimum $\mu _{\text{op}}$ , which sets a convergence rate ceiling. That is, once the information matrix is fixed, the convergence factor of the T-GD algorithm reaches at most the infimum $\mu _{\text{op}}$ . This article studies a varying infimum gradient descent algorithm, which can move down the infimum by using different order iterative preconditioning methods, as follows: first, for infinite iterative algorithm, the infimum becomes smaller and smaller with the increased iteration numbers; second, for finite iterative algorithm, the infimum is equal to zero, and the parameter estimates can be obtained in only one iteration; third, construct an adaptive interval between zero and $\mu _{\text{op}}$ , which can establish a link between the least squares and T-GD algorithms. Based on the varying infimum gradient descent algorithm, researchers can adaptively choose preconditioning matrices for different kinds of models on a case by case basis. The convergence analysis and simulation examples show effectiveness of the proposed algorithms.

A robust standard gradient descent (SGD) algorithm for ARX models using the Aitken acceleration method is developed. Considering that the SGD algorithm has slow convergence rates and is sensitive to the step size, a robust and accelerative SGD (RA-SGD) algorithm is derived. This algorithm is based on the Aitken acceleration method, and its convergence rate is improved from linear convergence to at least quadratic convergence in general. Furthermore, the RA-SGD algorithm is always convergent with no limitation of the step size. Both the convergence analysis and the simulation examples demonstrate that the presented algorithm is effective.