Asymptotically$H^2$-Optimal Tuning of Low Gain Robust Controllers for DPS

It is well known that a low-gain controller of the form C
_epsiv
(s)=sigma
_k=-n

ⁿ
epsivK
_k
/(s-iomega
_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
_epsiv
(s) as the positive scalar gain epsivrarr0
⁺
for exponentially stable plants with transfer function PisinH
^infin
. 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 epsivrarr0
⁺
the cost function decomposes into a sum of simpler cost functions, each depending only on K
_k
and P(iomega
_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(iomega
_k
) are needed, and it is shown that these can be measured from the open-loop plant with suitable input-output experiments