Reduced Conservatism Proof of the Balanced MIMO Gain and Phase Margins

For the determination of SISO gain and phase margins, the MIMO closed-loop system is broken at one singleloop channel by keeping the remaining loops closed, but these do not consider gain or phase uncertainty simultaneously in all channels. In this case, MIMO gain and phase margins are more suitable robustness measures. Three different MIMO gain and phase margins are known. It is stated in the literature without a formal proof that the MIMO gain and phase margins of the so-called balanced sensitivity transfer function are less conservative than the MIMO gain and phase margins of the complimentary sensitivity function and the sensitivity function. In this paper proofs for conservatism between the different MIMO gain and phase margins are provided.