On Multiloop Interaction and Relative and Bristol Gains

The relationship between multiloop design and Bristol gains of a multivariable plant is shown. The Bristol and relative gains are distinguished. The Bristol gains $B$ characterize the plant and the relative gains $Λ$ characterize the designed feedback system. It is $Λ$⁠, not $B$⁠, that determines the interaction in loop designs. $Λ$ is equal to $B$ in a loop only at frequencies where all other loop gains are much above $0dB$⁠, usually significantly below the respective loop gain crossover frequencies. $Λ=1$ in a loop at frequencies where all other loop gains are much below $0dB$⁠, usually significantly above the respective loop gain crossover frequencies. The transition of $Λ$ in a loop from $B$ to 1 in the important medium frequency range is one of the most interesting and under-researched aspects of loop design interaction.