Stabilization of unstable limit cycles in systems with limited controllability: Expanding the basin of convergence of OGY-type controllers
There are a lot of studies on the controllability of chaos, and a high percentage of them is covered by papers based on OGY method. Here the authors present the underlying geometric structure of the OGY control scheme. The authors reveal some of the main mechanisms that lead to failure of the control algorithm. It is shown that the requirement of ``closeness to the fixed point'' is not sufficient to insure controllability using a standard OGY controller, and that the limited controllability available in a system may lead to serious problems that will significantly enlarge the region of failure of the standard OGY controller. The authors consider limited controllability both due to the structure of the dynamic equations and due to limited power, energy or force available for control purposes, and reveal that for some systems in any vicinity of the fixed point, no matter how small, there will exist points where OGY scheme will fail. For these cases they develop a new control scheme that improves the basin of convergence, and prove that for some problems it may be more advantageous than the standard OGY method. Morover, it is shown that the minimal distance algorithm may easily fail for a broad category of situations. In order to solve these problems, the authors propose two new techniques: the penalized minimal distance method, and the multi-step OGY-type scheme, as a generalization for the standard OGY method and minimal distance algorithms. Finally, they give a necessary condition that estimates the region of controllability under the multi-step OGY-type control, and significantly improves the basin of convergence for the new multi-step OGY-type algorithm.
scientific article; zbMATH DE number 1192763