On consistency of least squares algorithm for ARMAX time-delay models

Abstract

Jointly identification of the sample delay and the parameters of time-delay discrete-time ARMAX models are addressed in this article. Like any identified problem, the proposed approach is based on the solution of an optimization problem. It consists then of two main steps: the first is the problem formulation and the second is the problem resolution. The problem formulation defines the criterion to be minimized according to an unknown parameter vector consisting of the sample delay and the parameters with their corresponding observation vector. Since the sample delay is of integer type, the obtained optimization problem cannot be efficiently solved using a real optimization algorithm. To overcome this difficulty, the rounding properties are used to transform the obtained problem into a real optimization problem. The solution is then done using the least square approach. Consistency of the estimates with their convergence rates is established under the persistent excitation condition. Finally, some numerical simulation and experimental results are presented to illustrate the performance of the proposed method.

On the combined estimation of the parameters and the states of fractional-order systems

“…Actually, most real-world industrial processes present nonlinear behavior. For decades, the study of nonlinear systems has attracted much research interest, including system identification, [1][2][3] system control, 4,5 state estimation, [6][7][8][9] and so on. In the identification area, 10 the works that are elaborated on the class of nonlinear systems concern some specific structure of models, such as black-box models, fuzzy and neural network models, and block-structured models [11][12][13][14] .…”