Singular perturbation modeling of nonlinear processes with nonexplicit time-scale multiplicity

In this article, a modeling framework is proposed for two-time-scale chemical processes modeled by nonlinear ordinary differential equations (ODEs) with large parameters of the form 1/ε, to obtain a standard singularly perturbed representation where the slow and fast variables are explicitly separated. Initially, a result is derived that provides necessary and sufficient conditions for the existence and the explicit form of an ε-independent coordinate change that transforms the two-time-scale process into a standard singularly perturbed form. Whenever these conditions are not satisfied, it is established that an ε-dependent coordinate change, singular at ε = 0, has to be employed to obtain a standard singularly perturbed representation of the original two-time-scale process, and the construction of such a transformation is addressed. The application of the proposed framework in deriving standard singularly perturbed representations and its significance in the synthesis of well-conditioned controllers is demonstrated through chemical reactor applications.