Nonlinear maneuvering with gradient optimization

The maneuvering problem involves two tasks. The first one, called the geometric task, is to force the system states to converge to a desired parametrized path. The second task, called the dynamic task, is to satisfy a desired dynamic behavior along the path. The desired geometric path /spl xi/ is viewed as a target set /spl Xi/ which is parametrized by a scalar variable /spl theta/. The proposed dynamic controller consists of a stabilization algorithm that drives the state x(t) to the point /spl xi/(/spl theta/(t)), and a smooth dynamic optimization algorithm that selects the point /spl xi/(/spl theta/) in the set /spl Xi/ that minimizes the weighted distance between x and /spl xi/. Choosing a gain /spl mu/ large in the optimization algorithm, induces a two-time scale behavior or a closed-loop plant. In the fast time-scale /spl theta/(t) rapidly converges to the minimizer, and in the slow time-scale x(t) converges to /spl Xi/. Two motivational examples illustrate the design and the achieved performance of the closed-loop.