Frequency-domain methods and polynomial optimization for optimal periodic control of linear plants

We consider the problem of periodic trajectory design for single-output systems which may be subject to periodic external disturbances. We show how trajectories optimizing a possibly nonquadratic and nonconvex polynomial performance objective can be found by using the frequency-domain description of the plant by converting the problem to a polynomial optimization problem (POP) in the Fourier coefficients of the external input signals. The method is suited for distributed-parameter systems, since the system transfer functions are not required to be rational; the computational complexity of the method depends on the order of the polynomial nonlinearities in the performance objective as well as the number of required harmonics, but is independent of the underlying system dimension.