Abstract
Stability is usually in the sense of Lyapunov's asymptotical stability, thus the solutions starting from points close to a stable equilibrium may have a very long transient. In the applications of time-delayed feedback controls, it is important not only to determine the stable regions in the gain plane or gain space, but also to find out the abscissa that can be used as an index of stability. Based on the D-subdivision method, this paper proposes a simple algorithm for finding and labeling the stable regions in feedback gain plane with abscissa.The labeled sub-regions with smaller abscissa are better in applications. The main results are presented for the controlled pendulum or inverted pendulum under a delayed feedback, and are illustrated with two case studies.
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