Continuous pole placement method for time-delayed feedback controlled systems

Continuous pole placement method is adapted to time-periodic states of systems with time delay. The method is applied for finding an optimal control matrix in the problem of stabilization of unstable periodic orbits of dynamical systems via time-delayed feedback control algorithm. The optimal control matrix ensures the fastest approach of a perturbed system to the stabilized orbit. An application of the pole placement method to systems with time delay meets a fundamental problem, since the number of the Floquet exponents is infinity, while the number of control parameters is finite. Nevertheless, we show that several leading Floquet exponents can be efficiently controlled. The method is numerically demonstrated for the Lorenz system, which until recently has been considered as a system inaccessible for the standard time-delayed feedback control due to the odd-number limitation. The proposed optimization method is also adapted for an extended time-delayed feedback control algorithm and numerically demonstrated for the Rössler system.