Assessment and Future Directions of Nonlinear Model Predictive ...

10 E.F. Camacho and C. Bordonscases and the second is that constraints, which are usually linear, are transformedinto a nonlinear set of constraints.MPC Based on Volterra Models: In some cases, the nlp shows a specialstructure that can be exploited to achieve an online feasible solution to thegeneral optimisation problem. If the process is described by a Volterra model,efficient solutions can be found, especially for second-order models. A controlstrategy can be devised that can solve the nonlinear problem by iteration of thelinear solution, based on the particular structure of Volterra models. This iterativeprocedure proposed by Doyle et al. [11] gives rise to an analytical solutionin the unconstrained case or a qp solution if constraints exist and allows an easysolution to the nonlinear problem. If a second-order model is used, the predictioncan be written as an extension of the linear process y = Gu + f + c(u), where fincludes the terms that depends on past and known values and the new term ctakes into account new terms that depend on crossed products between past andfuture control actions. The prediction depends on the unknowns (u) bothinalinear form (G u) and a quadratic form (c(u)) and cannot be solved analyticallyas in the linear unconstrained case. However, the iterative procedure proposed in[11] starts with an initial value of c and solves the problem. The new solution isused to recalculate c and the problem is solved again until the iterated solution isclose enough to the previous one. In the constrained case, u is computed solvinga qp. Due to the feasibility of its being implemented in real time, this methodhas been successfully applied to real plants, such as polymerisation processes[25] or biochemical reactors [11].In the simplified case that the process can be modelled by a Hammersteinmodel, the problem can be easily transformed into a linear one by inverting thenonlinear static part, g(.). The same idea can be applied to Wiener models, wherethe static nonlinearity goes after the linear dynamics. In [33] a pH neutralizationprocess is controlled in this way.Neural Networks: Artificial Neural Networks, apart from providing a modellingtool that enables accurate nonlinear model attainment from input-outputdata, can also be used for control. Since nns are universal approximators, theycan learn the behaviour of a nonlinear controller and calculate the control signalonline with few calculations, since the time-consuming part of the nn (training)is done beforehand. This has been applied to several processes in the processindustry [2], [4] and to systems with short sampling intervals (in the range ofmilliseconds) such as internal combustion engines [31]. An application of an nncontroller to a mobile robot is detailed at the end of the paper.Piecewise Affine Systems: In case the process cand be described by apwa state-space model the nmpc becomes a Mixed Integer Quadratic Problem(miqp), which can be solved as a series of qp problems. There are severalalgorithms to do that and one interesting approach is the one proposed in [34],that belongs to the class of Branch and Bound (B & B) methods. The procedureuses the concepts of reachable set combined to the specific B & B methods, in