− .l−1 u¯m

and steady-state regimes. Linear schemes have covered ARX, ARMAX, Box-Jenkins and state-space models.  The

Hence if the model has a term cluster of the form Ωyum , m = 1, 2, . . . , l, then the static function is rational, if not it is polynomial. The clusters coefficients are useful to write the models static functions and to implement the gray-box modelling techniques as shown in the Section IV.

The model structure of the NARMAX polynomials are au- tomatically chosen using the error reduction ratio (ERR) criterion [12, 13]. In the context of black-box modeling the parameters of such models are obtained by the extended least-squares (ELS) estimator [14, 15]. Other black-box models considered are the neural NARMAX model, a feed- forward multilayer perceptron with weights estimated us- ing the Leverberg–Marquardt algorithm available in Nor- gaards toolbox [16].

A representative simulation run is shown in Fig. 6, which when compared with the ensuing linear results   indicate

nonlinear method has included nonlinear autoregressive with moving average and exogenous variables (NARMAX) which uses free-run simulation. A comparison amongst the difference techniques has concluded that the NARMAX method generated models with better dynamic and static performance.

ACKNOWLEDGMENTS

The author would like to thank Mr. Mirza H Baig for carrying out the simulation studies. Also, he would like to thank the deanship for scientific research (DSR) at KFUPM for research support through project IN100018.

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