Convergence analysis of the alternating RGLS algorithm for the identification of the reduced complexity Volterra model.

In this paper we provide a convergence analysis of the alternating RGLS (Recursive Generalized Least Square) algorithm used for the identification of the reduced complexity Volterra model describing stochastic non-linear systems. The reduced Volterra model used is the 3rd order SVD-PARAFC-Volterra model provided using the Singular Value Decomposition (SVD) and the Parallel Factor (PARAFAC) tensor decomposition of the quadratic and the cubic kernels respectively of the classical Volterra model. The Alternating RGLS (ARGLS) algorithm consists on the execution of the classical RGLS algorithm in alternating way. The ARGLS convergence was proved using the Ordinary Differential Equation (ODE) method. It is noted that the algorithm convergence canno׳t be ensured when the disturbance acting on the system to be identified has specific features. The ARGLS algorithm is tested in simulations on a numerical example by satisfying the determined convergence conditions. To raise the elegies of the proposed algorithm, we proceed to its comparison with the classical Alternating Recursive Least Squares (ARLS) presented in the literature. The comparison has been built on a non-linear satellite channel and a benchmark system CSTR (Continuous Stirred Tank Reactor). Moreover the efficiency of the proposed identification approach is proved on an experimental Communicating Two Tank system (CTTS).

[1]  Gérard Favier,et al.  IDENTIFICATION OF SVD-PARAFAC BASED THIRD-ORDER VOLTERRA MODELS USING AN ARLS ALGORITHM , 2005 .

[2]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[3]  T. Lai,et al.  Extended least squares and their applications to adaptive control and prediction in linear systems , 1986 .

[4]  Bijan Afsari,et al.  Sensitivity Analysis for the Problem of Matrix Joint Diagonalization , 2008, SIAM J. Matrix Anal. Appl..

[5]  João Cesar M. Mota,et al.  Blind identification of multiuser nonlinear channels using tensor decomposition and precoding , 2009, Signal Process..

[6]  Maryam Dehghani,et al.  Identification of multivariable nonlinear systems in the presence of colored noises using iterative hierarchical least squares algorithm. , 2014, ISA transactions.

[7]  H. Messaoud,et al.  Online prediction model based on the SVD-KPCA method. , 2013, ISA transactions.

[8]  Walter Kellermann,et al.  Fast adaptation of frequency-domain volterra filters using inherent recursions of iterated coefficient updates , 2007, 2007 15th European Signal Processing Conference.

[9]  Thomas Bouilloc.  Applications de décompositions tensorielles à l'identification de modèles de Volterra et aux systèmes de communication MIMO-CDMA , 2011 .

[10]  Nicholas Kalouptsidis,et al.  Input-output identification of nonlinear channels using PSK, QAM and OFDM inputs , 2009, Signal Process..

[11]  David G. Messerschmitt,et al.  Nonlinear Echo Cancellation of Data Signals , 1982, IEEE Trans. Commun..

[12]  Lieven De Lathauwer,et al.  An enhanced line search scheme for complex-valued tensor decompositions. Application in DS-CDMA , 2008, Signal Process..

[13]  Lennart Ljung,et al.  On positive real transfer functions and the convergence of some recursive schemes , 1977 .

[14]  Lennart Ljung,et al.  Analysis of recursive stochastic algorithms , 1977 .

[15]  Hassani Messaoud,et al.  Identification of nonlinear stochastic systems described by a reduced complexity Volterra model using an ARGLS algorithm , 2012, 2012 5th International Symposium on Communications, Control and Signal Processing.

[16]  Pierre Comon,et al.  Enhanced Line Search: A Novel Method to Accelerate PARAFAC , 2008, SIAM J. Matrix Anal. Appl..

[17]  Rasmus Bro,et al.  A comparison of algorithms for fitting the PARAFAC model , 2006, Comput. Stat. Data Anal..

[18]  José Ragot,et al.  Nonlinear system modeling based on bilinear Laguerre orthonormal bases. , 2013, ISA transactions.

[19]  Junghsi Lee,et al.  A fast recursive least squares adaptive second order Volterra filter and its performance analysis , 1993, IEEE Trans. Signal Process..

[20]  Walter A. Frank.  An efficient approximation to the quadratic Volterra filter and its application in real-time loudspeaker linearization , 1995, Signal Process..

[21]  Gérard Favier,et al.  Blind equalization of nonlinear channels using a tensor decomposition with code/space/time diversities , 2009, Signal Process..

[22]  Yoshinobu Kajikawa.  Subband parallel cascade Volterra filter for linearization of loudspeaker systems , 2008, 2008 16th European Signal Processing Conference.