Assessment and Future Directions of Nonlinear Model Predictive ...

4 E.F. Camacho and C. Bordonswhich corresponds to the widely used linear convolution model with the nonlinearityappearing as an extra term, that is, the nonlinearity is additive.Two special subclasses of the basic model are employed which reduce thecomplexity of the basic Volterra approach and have a reduced number of parameters.These are the Hammerstein and Wiener models. Hammerstein modelsbelong to the family of block-oriented nonlinear models, built from the combinationof linear dynamic models and static nonlinearities. They consist of a singlestatic nonlinearity g(.) connected in cascade to a single linear dynamic modeldefined by a transfer function H(z −1 ). Because of this, Hammerstein modelscan be considered diagonal Volterra models, since the off-diagonal coefficientsare all zero. Notice that this means that the behaviour that can be representedby this type of model is restricted. The Wiener model can be considered as thedual of the Hammerstein model, since it is composed of the same componentsconnected in reverse order. The input sequence is first transformed by the linearpart H(z −1 )toobtainΨ(t), which is transformed by the static nonlinearity g(.)to get the overall model output. The properties of Volterra and related modelsare extensively discussed in [11].Closely related to Volterra models are bilinear models. The main differencebetween this kind of model and the Volterra approach is that crossed productsbetween inputs and outputs appear in the model. Bilinear models have beensuccessfully used to model and control heat exchangers, distillation columns,chemical reactors, waste treatment plants, and pH neutralisation reactors [16]. Ithas been demonstrated that this type of model can be represented by a Volterraseries [23].Local Model NetworksAnother way of using input-output models to represent nonlinear behaviour is touse a local model network representation. The idea is to use a set of local modelsto accommodate local operating regimes [17], [44]. A global plant representationis formed using multiple models over the whole operating space of the nonlinearprocess. The plant model used for control provides an explicit, transparent plantrepresentation which can be considered an advantage over black-box approachessuch as neural networks (that will be described below).The basics of this operating regime approach are to decompose the spaceinto zones where linear models are adequate approximations to the dynamicalbehaviour within that regime, with a trade-off between the number of regimesand the complexity of the local model. The output of each submodel is passedthrough a local processing function that generates a window of validity of thatparticular submodel. The complete model output is then given byy(t +1)=F (Ψ(t),Φ(t)) =M∑f i (Ψ(t))ρ i (Φ(t))where the M local models f i (Ψ(t)) are linear arx functions of the measurementvector Ψ (inputs and outputs) and are multiplied by basis functions ρ i (Φ(t)) ofi=1