The frequency-domain approach to the analysis of closed-loop systems

Summary We compare two methods of estimating transfer functions from input/output records when the output contains an additive noise component of unknown structure. The two methods correspond to (a) a weighted least-squares estimate (  1 ), and (b) a simple least-squares estimate (  2 ). The estimate  1 is more attractive theoretically since it is equivalent to a maximum likelihood approach when the system is open loop and the noise process is Gaussian, but it is difficult to compute numerically. The alternative estimate,  2 , is much simpler to compute and is the one commonly used in practice, particularly as a first approximation in an iterative approach. Both  1 and  2 , however, will, in general, be biased estimates when the system is closed loop. We determine conditions under which the asymptotic form of A 2 will be identical with, or close to, the asymptotic form of A 1 in the case of a closed-loop system in which the feedback is linear with an additive noise component.