Checkable conditions for identifiability of linear systems operating in closed loop

Necessary and sufficient conditions are given for the identifiability of the open-loop transfer functions for linear discrete time systems operating in closed loop. It is shown that, provided certain structural constraints are satisfied, it is necessary and sufficient that the noise covariance matrix Σ associated with any spectral factorization of the joint input-output spectral density be block diagonal. A continuity result is also established showing that small off-diagonal elements in the noise covariance lead to small errors in the estimated open-loop transfer functions. This extends previously known results which required that the true noise covariance be block diagonal.