Real-time robust and adaptive solutions to zero finding problems with uncertainty

This paper proposes extensions to a recent control Liapunov function (CLF) based method for designing dynamical systems with trajectories that converge to the zeros of a nonlinear vector function f . Specifically, the CLF design method is extended to the case when the Jacobian of the vector function can be decomposed into a known part and a partially known part, for which certain norm bounds are known. The extension of existing zero finding methods to this setting allows the consideration of functions which have singular Jacobians as well as the underdetermined case of a nonlinear system of equations in which there are more unknowns than equations. It is shown through simulation examples that methods that use dynamical systems that correspond to first order ordinary differential equations (ODEs) are less effective than those which use second order ODEs.