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

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