Guaranteed steady state bounds for uncertain (bio-)chemical processes using infeasibility certificates

Abstract

Analysis and safety considerations of chemical and biological processes require complete knowledge of the set of all feasible steady states. Nonlinearities, uncertain parameters, and discrete variables complicate the task of obtaining this set. In this paper, the problem of outer-approximating the region of feasible steady states, for processes described by uncertain nonlinear differential algebraic equations including discrete variables and discrete changes in the dynamics, is addressed. The calculation of the outer bounds is based on a relaxed version of the corresponding feasibility problem. It uses the Lagrange dual problem to obtain certificates for regions in state space not containing steady states. These infeasibility certificates can be computed efficiently by solving a semidefinite program, rendering the calculation of an outer bounding set computationally feasible. The derived method guarantees globally valid outer bounds for the feasible steady states. The method is exemplified by the analysis of a simple chemical reactor showing parametric uncertainties and large variability due to the appearance of bifurcations characterising the ignition and extinction of a reaction.