Normal vectors on critical manifolds for robust design of transient processes in the presence of fast disturbances

Information on steady-state bifurcations, most notably stability boundaries, is frequently used for the analysis and design of nonlinear systems. The bifurcation points separate regions with different dynamic behavior and thus give valuable information about nonlinear systems. They cannot, however, reflect the impact of fast disturbances on the transient behavior of nonlinear systems. The influence of fast disturbances can be addressed by bifurcation points that are defined as critical points during the transient behavior of a dynamic system in the presence of fast disturbances. Specifically, we consider two types of points-grazing points and end-points. At a grazing point the trajectory of a nonlinear system tangentially touches a hypersurface spanned by a state or output constraint. At an end-point the trajectory crosses the hypersurface at a specified final time. These critical points unfold to manifolds in the parameter space of the nonlinear system separating parts of the parameter space that admit trajectories that do not violate the constraint from those where the constraint is violated. The parametric distance between a candidate design of a nonlinear system and the critical manifold is used as a robustness measure. As the closest connection between the design and the critical manifold is along the normal direction of the critical manifold, normal vectors are used to formulate minimal-distance constraints for a nonlinear program. Thus it is possible to robustly take into account state and output constraints in the presence of fast disturbances for the design of a nonlinear system. Application of the approach to closed-loop systems allows for an integration of operating point and control design. Several case studies from chemical engineering are presented to illustrate the proposed method.