Optimal expansions of multivariable ARX processes on Laguerre bases via the Newton–Raphson method

In this article, an optimal linear MIMO system approximation by using discrete-time MIMO autoregressive with exogenous input ARX model is proposed. Each polynomial function of the MIMO ARX model associated with the inputs and with the outputs is expanded on independent Laguerre orthonormal basis. The resulting model is entitled MIMO ARX-Laguerre model. The optimal approximation of which is ensured once the poles characterizing each Laguerre orthonormal basis are set to their optimal values. In this paper, a new method to estimate, from input/output measurements, the optimal Laguerre poles of the MIMO ARX-Laguerre model is proposed. The method consists in applying the Newton-Raphson's iterative technique in which the gradient and the Hessian are expressed analytically. The proposed algorithm is tested on a numerical example and on a benchmark system. Simulation results show the effectiveness of the proposed optimal modeling method. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  B. Wahlberg.  System identification using Laguerre models , 1991 .

[2]  B. Wahlberg.  System identification using Kautz models , 1994, IEEE Trans. Autom. Control..

[3]  G. Dumont,et al.  An optimum time scale for discrete Laguerre network , 1993, IEEE Trans. Autom. Control..

[4]  T. Oliveira e Silva,et al.  On the determination of the optimal pole position of Laguerre filters , 1995, IEEE Trans. Signal Process..

[5]  José Ragot,et al.  Online identification of the bilinear model expansion on Laguerre orthonormal bases , 2014, Int. J. Control.

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

[7]  Hiroto Hamane,et al.  Identification method for commercialized PI control using Laguerre function , 2011, SICE Annual Conference 2011.

[8]  Kais Bouzrara,et al.  Expansion of MIMO ARX model on Laguerre orthonormal bases , 2013, 2013 International Conference on Electrical Engineering and Software Applications.

[9]  Sachin C. Patwardhan,et al.  Closed-loop identification using direct approach and high order ARX/GOBF-ARX models , 2011 .

[10]  Pingkang Li,et al.  Closed-loop identification using Laguerre orthogonal functions for a virtual diesel engine , 2011, Int. J. Comput. Appl. Technol..

[11]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[12]  den Ac Bert Brinker,et al.  Model reduction by orthogonalized exponential sequences , 1996 .

[13]  B. Ninness,et al.  MIMO system identification using orthonormal basis functions , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[14]  J. Ragot,et al.  Dynamic SISO and MISO system approximations based on optimal Laguerre models , 1998, IEEE Trans. Autom. Control..

[15]  Jalel Ghabi,et al.  Robust Predictive Control using a GOBF Model for MISO Systems , 2007, Int. J. Comput. Commun. Control.

[16]  Noël Tanguy,et al.  Online optimization of the time scale in adaptive Laguerre-based filters , 2000, IEEE Trans. Signal Process..

[17]  José Ragot,et al.  Decomposition of an ARX model on Laguerre orthonormal bases. , 2012, ISA transactions.

[18]  Barbara D. Maccluer.  Elementary Functional Analysis , 2008 .

[19]  Didier Maquin,et al.  Optimality conditions for the truncated network of the generalized discrete orthonormal basis having real poles , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[20]  B. Ninness,et al.  A unifying construction of orthonormal bases for system identification , 1997, IEEE Trans. Autom. Control..

[21]  B. Wahlberg,et al.  Modelling and Identification with Rational Orthogonal Basis Functions , 2000 .

[22]  José Ragot,et al.  Online identification of the ARX model expansion on Laguerre orthonormal bases with filters on model input and output , 2013, Int. J. Control.