Density and flow: A different view on nonlinear control

Professor Anders Rantzer, Lund Institute of Technology, Sweden, Department of Automatic Control

Lyapunov's second theorem is a standard tool for stability analysis of nonlinar ordinary differential equations. In this lecture, we discuss a new category of results, which can be viewed as dual to Lyapunov's.

The new criteria are stated in terms of "denisty functions" and have implications in terms of "almost all trajectories" of the system. The implications are weaker than in Lyapunov theory, but this also makes them applicable to many systems where global stability in the classical sense does not hold. For example, we obtain new results on control of pendulums and phase locked loops.

Density functions also have remarkably nice properties for control synthesis. While the set of control Lyapunov functions for a given system may not even be connected, the corresponding set for density functions is always convex. We will demonstrate how this opens new possibilities for nonlinear control synthesis by convex optimization.