Representation of nonlinear systems: The NARMAX Model.
S. Chen · ePrints Soton (University of Southampton) · 1989
Representations of non-linear systemsand 4, using output-affine models to approximate the systems may require more terms and it is not obvious how the approximation of terms like y2(k -1) is achieved.Of course, if the response function of the system is a bounded polynomial, the system can then be modelled exactly by an output-affine model.The polynomial NARMAX model is more suitable as an approximation to the general system (3), and because power terms in both the inputs and outputs are allowed more parsimonious models can be obtained.When the system is operating close to an operating point where the linear approximation will be valid, it is desirable that the non-linear model degenerates to the linear model that is satisfied by the linearized system.A polynomial model can naturally be reduced to the linear model in such a situation.However, it is not clear how an output-affine model can achieve this.The linearized system around the origin for Example 1 is y(k + 1) = y(k) + u(k), and in this case the output-affine model fails completely to degenerate to this model when the system is operating close to the origin.For Example 1 it is also seen that the order of the (minimal) polynomial model is lower than that of the (minimal) output-affine model (n y = n = 1 compared with u r = 2).Evidently, a lower order representation is easier to implement in practice.To compute y(k + 1), we need to store y(k) and u(k) for Example 1 using the polynomial