Assessment and Future Directions of Nonlinear Model Predictive ...

Nonlinear Model Predictive Control: An Introductory Review 5the current operating vector. These basis functions are chosen to give a valueclose to 1 in regimes where f i is a good approximation to the unknown F and avalue close to 0 in other cases.Notice that this technique allows the use of a linear predictive controller,avoiding the problems associated to computation time and optimality of thenonlinear solution. These strategies have been successfully tested on a pH neutralizsationplant (see [45] for details).Neural NetworksThe nonlinear dynamics of the process can also be captured by an artificial NeuralNetwork (nn). Neural networks are attractive tools to construct the modelof nonlinear processes since they have an inherent ability to approximate anynonlinear function to an arbitrary degree of accuracy [14]. This, together withthe availability of training techniques, has made them very successful in manypredictive control applications and commercial products. More details of the useof nn for control can be found in [32].Neural Networks are usually combined with linear models in practical applications,since they are not able to extrapolate beyond the range of their trainingdata set. Based on a model confidence index, the nn is gradually turned off whenits prediction looks unreliable, the predictions relying on the linear part.2.2 State Space ModelsThe linear state space model can naturally be extended to include nonlineardynamics. The following state space model can be used to describe a nonlinearplantx(t +1)=f(x(t),u(t)) y(t) =g(x(t)) (3)where x(t) is the state vector and f and g are generic nonlinear functions. Noticethat the same equation can be used for monovariable and multivariable processes.Notice also that this model can easily be derived from the differential equationsthat describe the model (if they are known) by converting them into a set offirst-order equations. Model attainment in this case is straightforward but theprocedure is very difficult to obtain from experimental data when no differentialequations are available.This kind of model is the most widely extended for nonlinear plants sinceit has given rise to a lot of theoretical results: the majority of results aboutstability and robustness have been developed inside this framework. It is alsoused in commercial tools such as nova nlc or nonlinear pfc.An important class of nonlinear state-space models are Piece Wise Affine(pwa) models, which can be used to approximate smooth nonlinear processesand are defined by:x k+1 = A i x k + B i u k + f i for: x k ∈X iwhere {X i } s i=1 is a polyhedral partition of the state space.