Modelling and control of Hammerstein system using B-spline approximation and the inverse of De Boor algorithm

Abstract

In this article a simple and effective controller design is introduced for the Hammerstein systems that are identified based on observational input/output data. The nonlinear static function in the Hammerstein system is modelled using a B-spline neural network. The controller is composed by computing the inverse of the B-spline approximated nonlinear static function, and a linear pole assignment controller. The contribution of this article is the inverse of De Boor algorithm that computes the inverse efficiently. Mathematical analysis is provided to prove the convergence of the proposed algorithm. Numerical examples are utilised to demonstrate the efficacy of the proposed approach. . For example, it is a suitable model for signal processing applications involving any nonlinear distortion followed by a linear filter, the modelling of the human heart in order to regulate the heart rate during treadmill exercises (Su 2007), and the modelling of hydraulic actuator friction dynamics (Kwak, Yagle, and Levitt 1998). The Hammerstein model has been widely researched (Billings and Fakhouri 1979;Stoica and So¨derstro¨m 1982;Greblicki and Pawlak 1986;Greblicki 1989Greblicki , 2002Verhaegen and Westwick 1996;Lang 1997;Bai and Fu 2002;Chen 2004;Chaoui, Giri, Rochdi, Haloua, and Naitali 2005). The model characterisation/representation of the unknown nonlinear static function is fundamental to the identification of Hammerstein model. Various approaches have been developed in order to capture the a priori unknown nonlinearity by use of both parametric (Verhaegen and

The system identification and control of Hammerstein system using non-uniform rational B-spline neural network and particle swarm optimization

“…With the number of knots and their location determined, conventional nonlinear optimization algorithms are applicable for determining the weights and the shaping parameters. Note that if l j ¼ 1=d (8j) the NURB network based Hammerstein systems becomes a nonrational B-spline based Hammerstein systems [26], for which the system identification can be carried out iteratively in practice. The number and locations of knots are predetermined to produce a model as small as possible that can still provide good modelling capability.…”

“…In this study, the controller consists of computing the inverse of the nonlinear static function approximated by NURB, followed by a linear pole assignment controller. The linearization of the closed loop system is achieved by inserting the inverse of the identified static nonlinearity via the inverse of De Boor algorithm [26] which was introduced for the control of B-spline based Hammerstein systems. It is shown that the inverse of De Boor algorithm [26] is also applicable to NURB based Hammerstein systems.…”

“…Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/neucom nonlinear dynamical systems [23][24][25], including the modelling and control of Hammerstein systems [26].…”

PSO Assisted NURB Neural Network Identification

“…Under the assumption that the inverse of De Boor algorithm [14]) can cancel the nonlinearity in the system which is modeled by the identified NURB model, the required controller polynomials are estimated. The reference signals r(t) are generated as a series of sinusoidal wave with its magnitude and frequency changing every 200 time steps.…”

“…In this study, the controller consists of computing the inverse of the nonlinear static function approximated by NURB, followed by a linear pole assignment controller. The linearization of the closed loop system is achieved by inserting the inverse of the identified static nonlinearity via the inverse of De Boor algorithm [14] which was introduced for the control of B-spline based Hammerstein systems.…”