Stabilization of Unstable Limit Cycles in Systems with Limited Controllability: Expanding the Basin of Convergence of OGY-Type Controllers

The underlying geometric structure of the standard OGY control scheme is analyzed. Some of the main mechanisms that under certain conditions lead to failure of the control algorithm are revealed. The limited controllability available in a system is investigated and it is shown that it may lead to serious problems that will significantly enlarge the state space region of failure of the standard OGY controller. A minimal distance algorithm is analyzed and shown to be, for some problems, more advantageous than the standard OGY technique. Nevertheless, for a broad category of problems, the minimal distance scheme is also shown to fail. As a solution for these problems, two new techniques are proposed: the penalized minimal distance and the multi-step OGY-type scheme. The standard OGY and minimal distance algorithms are particular cases of the new techniques proposed. Finally, we give a necessary condition that estimates the region of controllability under the multi-step OGY-type control. We demonstrate a significantly improved basin of convergence for the new multi-step OGY-type algorithm.