Local input-output modelling of non-linear systems using normal forms

The problem of structure determination in the modelling of non-linear systems is usually the most difficult in system identification. Traditional Volterra-Weiner methods are mathematically tractable but an enormously large number of data points are required to determine the parameters and moreover, the models obtained are still local in a sense. Some strategies adapt physical knowledge of the process to deduce the structure (from first principles) or perhaps simply consider linear approximations. Unfortunately, as is the case with many industrial processes, complete physical knowledge is sometimes either not available or too complicated to be of effective use and many such processes may be too highly non-linear to be modelled effectively by approximate linear schemes. However, for many such processes such as fed-batch processes, the feeding (inputs) and product (output) profiles generally have a known pattern, an optimal product trajectory having been established from years of experience. Thus even though the system typically goes through a wide range of operating conditions, any model obtained need only be local in the sense that there is only little deviation from the nominal trajectory. The principal reason for a model in this case is to predict output (or state) deviations for small input deviations, which can otherwise only be determined by costly laboratory analyses, thus enabling the implementation of optimal control schemes. In this paper, a local input-output approach based on the theory of bifurcations and normal forms is presented and then applied to the modelling of an industrial fed-batch process