Frequency domain conditions for disturbance rejection

The solvability conditions for the problems of nulling the disturbance transfer function in a linear time-invariant multivariable system by: 1) linear state feedback and 2) linear state feedback and disturbance feedforward control are known in geometric terms. In this paper it is shown that alternative necessary conditions for solvability are that the order of each row of the open-loop control transfer function: 1) strictly dominates and 2) dominates the order of the corresponding row of open-loop disturbance transfer function. Moreover, it is shown that when there are at least as many control inputs as outputs, these necessary conditions for solvability are generically sufficient, and that in other cases both problems are generically unsolvable.