Generic stabilisability for time-delayed feedback control
Philosophical Transactions A: Mathematical, Physical and Engineering Sciences
This is the author accepted manuscript. The final version is available from The Royal Society via the DOI in this record.
Reason for embargo
Time delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that there are no restrictions in principle. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilise this orbit with extended time-delayed feedback control. While the paper’s techniques are based on linear stability analysis, they exploit the specific properties of linearisations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.
Jan Sieber’s research has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement number 643073.
Philosophical Transactions A: Mathematical, Physical and Engineering Sciences, 2016, Volume 472, Issue 2189