Frequency-shaped cost functionals: output or input weighting

The differences between using a frequency-dependent weight on the input (or control) and using a frequency-dependent weight on the output (or state) are considered. It is shown that with a frequency-dependent weight on the output and a constant weight (i.e. independent of frequency) on the input, the guaranteed robustness properties of a standard linear quadratic regulator are preserved, while if the weighting on the output is constant and the frequency-dependent weight is on the input, then the same guaranteed robustness properties cannot be obtained. An important conclusion is that to use frequency shaping effectively, both the control and state weights should be made frequency-dependent to get good high-frequency robustness and good performance at low frequencies