Control-Sharing and Merging Control Lyapunov Functions

Given two control Lyapunov functions (CLFs), a “merging” is a new CLF whose gradient is a positive combination of the gradients of the two parents CLFs. The merging function is an important trade-off since this new function may, for instance, approximate one of the two parents functions close to the origin, while being close to the other far away. For nonlinear control-affine systems, some equivalence properties are shown between the control-sharing property, i.e., the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given CLFs, and the existence of merging CLFs. It is shown that, even for linear time-invariant systems, the control-sharing property does not always hold, with the remarkable exception of planar systems. The class of linear differential inclusions is also discussed and similar equivalence results are presented. For this class of systems, linear matrix inequalities conditions are provided to guarantee the control-sharing property. Finally, a constructive procedure, based on the recently considered “R-functions,” is defined to merge two smooth positively homogeneous CLFs.