Homogeneity in control: Geometry and applications

This tutorial presentation surveys some history and the geometric foundations for the use of homogeneity in the analysis and design of control systems, from classical applications to active research. Reflecting on the success of linear systems theory in a nonlinear world, homogeneous systems may be considered the next step, providing a richer class of models but still amenable to explicit analysis and design. Linearity means additivity together with homogeneity. But much of the effectiveness and power persists when additivity is lost: even without a superposition principle, relying only on homogeneity, stability is still determined by the dynamics on a reduced space that is a nonlinear analogue of the union of eigenspaces. Homogeneity immediately ties global to local properties.