Stabilization method based on oscillating full delayed feedback control

VERÓNICA E. PASTOR; GRACIELA A, GONZÁLEZ

Rio de Janeiro

Congreso; International Congress of Mathematicians (ICM2018); 2018

International Mathematical Union

Since Pyragas proposed the delayed feedback control (DFC) method in1992, there has been increasing interest in stabilizing unstable periodic orbitsvia this control design. The DFC has been re-formulated for the multi-delay,the observer-based and the predictor-based cases; and it has been extendedfor dealing with highly unstable orbits. A distinct feature of DFC and itsderivations, is that it does not require preliminary knowledge about the de-sired periodic orbit except its period.The DFC version for the stabilization of uncertain steady states uses thedifference between the present state x(t) and the delayed state x(t-\tau ), in-troducing a time delay \tau and a control gain K. Its stability analysis may berather intricate because it involves nonlinear delayed-differential equations.Some authors have worked on it but the analytical determination of the ade-quate values of K and \tau results a critical point. However, for a certain design method based on a full delayed feedback control, the stabilizing parameters may be obtained by solving some linear matrix inequalities.Moreover, if the linearization of the system in the equilibrium has either azero eigenvalue or an odd number of real positive eigenvalues, for no K and\tau stabilization is achieved; this restriction is known as the odd number limitation (ONL). Recently, we have proved that a periodically applied feedbackcontrol based on the difference between the past states x(t-2\tau) and x(t\tau)is able to stabilize in the general nonlinear scalar case; we refer to it as os-cillatory delayed feedback control (ODFC).Hence, we initiate a research on the implementation of the ODFC in then-dimensional case. Namely, conditions on system observability and contro-lability should be stated. To begin with, a scheme based on a full delayedfeedback control would result viable.