A revisit to the gain and phase margins of linear quadratic regulators

In this paper, we revisit the well-known robustness properties of the linear quadratic regulator (LQR), namely, the guaranteed gain margin of -6 to +infinity dB and phase margin of -60 degrees to +60 degrees for single-input systems. We caution that these guaranteed margins need to be carefully interpreted. More specifically, we show via examples that an LQR may have a very small margin with respect to the variations of the gain and/or phase of the open-loop plant. Such a situation occurs in most practical systems, where the set of measurable state variables cannot be arbitrarily selected. Therefore the lack of robustness of the LQR can be very popular and deserves attention.