Distance to Bifurcation in Multidimensional Parameter Space: Margin Sensitivity and Closest Bifurcations

The problem of operating or designing a system with robust stability with respect to many parameters can be viewed as a geometric problem in multidimensional parameter space of finding the position of nominal parameters λ0 relative to hypersurfaces at which stability is lost in a bifurcation. The position of λ0 relative to these hypersurfaces may be quantified by numerically computing the bifurcations in various directions in parameter space and the bifurcations closest to λ0. The sensitivity of the distances to these bifurcations yield hyperplane approximations to the hypersurfaces and optimal changes in parameters to improve the stability robustness. Methods for saddle-node, Hopf, transcritical, pitchfork, cusp, and isola bifurcation instabilities and constraints are outlined. These methods take full account of system nonlinearity and are practical in high dimensional parameter spaces. Applications to the design and operation of electric power systems, satellites, hydraulics and chemical process control are summarized.