Asymptotically H2-optimal tuning of low gain robust controllers for DPS

It is well known that a low-gain controller of the form C-epsilon(s) = Sigma(n)(k=-n) epsilon K-k/ (s - i omega(k)) is able to track and reject constants and finite linear combinations of sinusoidal reference and disturbance signals with known frequencies omega(k). In this note, we investigate the optimal tuning of the matrix gains K-k of the controller C-epsilon(s) as the positive scalar gain epsilon --> 0(+) for exponentially stable plants with transfer function P is an element of H-infinity. The cost function is taken to be the H-2-norm of the error between the reference signal and the measured output signal. It is shown that as epsilon --> 0(+) the cost function decomposes into a sum of simpler cost functions, each depending only on K-k and P(i omega(k)). Using this decomposition closed form solutions for the subproblems are found in certain special cases, and upper and lower bounds are given in the general case. No plant model is necessary since only the values P(i omega(k)) are needed, and it is shown that these can be measured from the open-loop plant with suitable input-output experiments.