This paper proposes two parameter identification methods for a nonlinear membership function. An equation converted method is introduced to turn the nonlinear function into a concise model. Then a stochastic gradient algorithm and a gradient-based iterative algorithm are provided to estimate the unknown parameters of the nonlinear function. The numerical example shows that the proposed algorithms are effective.

Two M-decomposed based identification algorithms are proposed for large-scale systems in this study. Since the least squares algorithms involve matrix inversion calculation, they can be inefficient for large-scale systems whose information matrices are ill-conditioned. To overcome this difficulty, the Mdecomposed based least squares algorithm is developed, where the parameter vector is divided into M subvectors. Each sub-vector is estimated using the least squares algorithm, with the assumption that the other sub-vectors are known. The proposed algorithm has less computational efforts than those of the traditional least squares algorithm. To update the parameters with new arrived data, an M-decomposed based recursive least squares algorithm is also provided, this algorithm avoids matrix inversion calculation thus is more efficient. The simulation examples show the effectiveness of the proposed algorithms.

This paper proposes an improved redundant rule based lasso regression stochastic gradient (RR-LR-SG) algorithm for time-delayed models. The improved SG algorithm can update the parameter elements with different step-sizes and directions, thus it is more adaptive; while the lasso regression method can pick out the small weights from the redundant parameter vector, it therefore can obtain the time-delay easily. To show the effectiveness of the proposed algorithm, the convergence analysis is also given. The simulated numerical results are consistent with the analytically derived results of the proposed algorithm. INDEX TERMSParameter estimation, time-delayed model, lasso regression, redundant rule, stochastic gradient algorithm.

In this study, two modified gradient descent (GD) algorithms are proposed for time-delayed models. To estimate the parameters and time-delay simultaneously, a redundant rule method is introduced, which turns the time-delayed model into an augmented model. Then, two GD algorithms can be used to identify the time-delayed model. Compared with the traditional GD algorithms, these two modified GD algorithms have the following advantages: (1) avoid a high-order matrix eigenvalue calculation, thus, are more efficient for large-scale systems; (2) have faster convergence rates, therefore, are more practical in engineering practices. The convergence properties and simulation examples are presented to illustrate the efficiency of the two algorithms.

The self-organizing model and expectation-maximization method are two traditional identification methods for switching models. They interactively update the parameters and model identities based on offline algorithms. In this paper, we propose a flexible recursive least squares algorithm which constructs the cost function based on two kinds of errors: the neighboring two-parameter estimation errors and the output estimation errors. Such an algorithm has several advantages over the two traditional identification algorithms: it (1) can estimate the parameters of all the sub-models without prior knowledge of the model identities; (2) has less computational efforts; and (3) can update the parameters with newly arrived data. The convergence properties and simulation examples are provided to illustrate the efficiency of the algorithm.