M

where ao is the coefficient of the constant basis function B\ and the sum is over all the basis functions Bm produced by the selection procedure.

Basis function selection is carried out in two steps. The first step is forward recursive partitioning which selects candidate basis functions. The second step is backward stepwise deletion which removes splines that duplicate similar information. Both steps are implemented by evaluating a lack-of-fit function [169]. A recent study reports the comparison of models developed by MARS and ANN with sigmoid functions [480].

Nonlinear Polynomial Models with Exponential and Trigonometric Terms (NPETM). If process behavior follows nonlinear functions such as trigonometric, exponential, or logarithmic functions, restricting model structure to polynomials would yield a model that has a large number of terms and acceptable accuracy over a limited range of predictor variables. Basically, several monomials are included in the model in order to describe approximately the functional behavior of that specific exponential or trigonometric relation. For example, l/i(t) = elVl(t - l)^*-1ยป + y\(t - 2) + 03 sin(042/i(t - 2)) . (4.181)

Consequently, inclusion of such functions in the pool of candidate terms would reduce the number of terms needed in the model and improve model accuracy. If the argument of such functions includes a parameter to be estimated (parameters 82 and O4 in Eq. 4.181), the model is not linear in the parameters and the parameter estimation problem becomes more challenging. If the nature of the functional relationship is not known a priori, the coupled problem of model structure determination and parameter estimation may not converge unless the initial guess is somewhat close to the correct values. Physical insight to the process or knowledge about model structure based on earlier modeling efforts provide vital information for the initial guess.

NPET models are feasible when changes in operating conditions necessitate a remodeling effort starting from an existing model. Some monomials and/or parameter values need to be changed, but the exponential or trigonometric type relations that are known remain the same for such cases. NPET models should be avoided when new models are being developed and information about the functional form of the nonlinearity is not known. The large number of nonlinear function types and the existence of unknown parameters in the function argument creates a search domain that is too large for most practical applications.

Cascade Systems

Cascade structures [210, 367] are composed of serially connected static nonlinear and dynamic linear transfer function blocks. This structure is appropriate when the process has static nonlinearities. The structure is called

Figure 4.20. General structure of Wiener-Hammerstein cascade model.

Figure 4.20. General structure of Wiener-Hammerstein cascade model.

a Hammerstein model if the static nonlinear element precedes the dynamic linear element, a Wiener model if the nonlinear element succeeds the linear element. In general, the models have polynomial terms that describe a continuous nonlinear steady state characteristic and/or a continuous function of a dynamic parameter.

Extensions of Linear Models for Describing Nonlinear Variations

Two other alternatives can be considered for developing linear models with better predictive capabilities than a traditional ARMAX model for nonlinear processes. If the nature of nonlinearity is known, a transformation of the variable can be utilized to improve the linear model. A typical example is the knowledge of the exponential relationship of temperature in reaction rate expressions. Hence, the log of temperature with the rate constant can be utilized instead of the actual temperature as a regressor. The second method is to build a recursive linear model. By updating model parameters frequently, mild nonlinearities can be accounted for. The rate of change of the process and the severity of the nonlinearities are critical factors for the success of this approach.

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