Continuous pole placement method for time-delayed feedback controlled systems

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