Reducing Complexity of Nonlinear Dynamic Systems

The complexity problem of nonlinear dynamic systems appears in a great number of scientific and engineering fields. The multi-model, also known as polytopic approach, constitutes an interesting tool for modeling dynamic nonlinear systems, in the framework of stability analysis and controller/observer design. A systematic procedure to transform a nonlinear system into a polytopic one will be briefly presented and illustrated by an academical example. This procedure gives the possibility of choosing between different polytopic structures, which is a degree of freedom used to ease the controllability, observability, stability analysis studies. In addition to that, the system transformation into polytopic form does not cause any information loss, contrarily to most existing studies in the field. In the second part of this chapter, a discussion about multiple time scale nonlinear systems, also known as singularly perturbed systems is proposed, by eliminating some structural constraints and by performing the identification and the separation of the time-scales. Robust observer synthesis with respect to internal/external perturbations, modeling parametrization errors and unknown inputs are presented for the estimation of different variables of interest, the state variables. The above-mentioned points will be applied to an activated sludge wastewater treatment plant (WWTP), which is a complex chemical and biological process. The variations in wastewater flow rate/composition and the time-varying bio-chemical reactions make this process nonlinear. Despite the process nonlinearity and complexity, there is a need to control the quality of the water rejected in the nature by the WWTPs in order to achieve the requirements of the European Union in terms of environmental protection. To this end, a Benchmark, proposed by the European program COST 624 to asses the control strategies of WWTPs, is used as an example in the present chapter.