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Most existing works on optimal filtering of linear time-invariant (LTI)stochastic systems with arbitrary unknown inputs assume perfect knowledge ofthe covariances of the noises in the filter design. This is impractical andraises the question of whether and under what conditions one can identify theprocess and measurement noise covariances (denoted as $Q$ and $R$,respectively) of systems with unknown inputs. This paper considers theidentifiability of $Q$/$R$ using the correlation-based measurement differenceapproach. More specifically, we establish (i) necessary conditions under which$Q$ and $R$ can be uniquely jointly identified; (ii) necessary and sufficientconditions under which $Q$ can be uniquely identified, when $R$ is known; (iii)necessary conditions under which $R$ can be uniquely identified, when $Q$ isknown. It will also be shown that for achieving the results mentioned above,the measurement difference approach requires some decoupling conditions forconstructing a stationary time series, which are proved to be sufficient forthe well-known strong detectability requirements established by Hautus.

Identifiability Analysis of Noise Covariances for LTI Stochastic Systems with Unknown Inputs