Convergence of the recursive identification algorithms for multivariate pseudo‐linear regressive systems

Abstract

The performance analysis of the recursive algorithms for the multivariate systems with an autoregressive moving average noise process is still open. This paper analyzes the convergence of two recursive identification algorithms, the multivariate recursive generalized extended least squares algorithm and the multivariate generalized extended stochastic gradient algorithm, for pseudo-linear multivariate systems and proves that the parameter estimation errors consistently converge to zero under persistent excitation conditions. The simulation results show that the proposed algorithms work well. 825 distillation column [29]. In the system identification area, Saadatzi et al. used two-input two-output transfer-function matrix models to describe the dynamics of the electric wheelchair on an arbitrary slope, and proposed a pre-compensator to decompose the plant's dynamics and to improve the identification accuracy [30]. Gu et al. combined the recursive least squares algorithm with the state filtering to identify the canonical state space systems [31].In practical industrial processes, the observed output data are always corrupted by various noise. Some methods assume that noise is independent identically distributed or white. However, it may be more reasonable to adopt the colored noise or the correlated noise because noise is unmeasurable [32,33]. The regressive systems involving the colored noise or correlated noise are called the pseudo-linear regressive systems. Cerutti et al. used the autoregressive moving average (ARMA) model to represent the dynamics of brain potentials [34]. Zheng et al. proposed a biased compensation based recursive least squares method for stochastic linear systems with correlated noise [35].On the convergence of the identification algorithms, early researchers always assumed that the observation noise has zero mean, finite second-order moment, and higher-order moment [36]. Some recent studies relaxed these assumptions. Liu and Ding studied the convergence properties of the recursive least squares algorithm for multivariable stochastic systems [37]; Ding and Gu studied the convergence of the auxiliary model based stochastic gradient algorithm for output-error state-delay systems [38]; Ding presented a coupled least squares algorithm for the multivariable stochastic systems and analyzed its convergence [39]. To the best of the authors' knowledge, the performance analysis for the multivariable ARMA systems has not been fully investigated. The main contributions of this paper lie in the following.By utilizing the measurable input-output data, this paper presents a multivariate generalized extended stochastic gradient (M-GESG) algorithm and a multivariate recursive generalized extended least squares (M-RGELS) algorithm for estimating the parameters of multivariate pseudo-linear regressive systems. By using the stochastic martingale theory, this paper analyzes the convergence of the M-GESG algorithm and the M-RGELS algorithm for multivariate systems with ARMA noise process.Briefly, the remainder is...

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