In identifying parameters of a continuous-time dynamical system, a difficulty arises when the observation noise covariance is unknown. The present paper solves this problem in the case of a linear time-invariant system with white noise affecting additively both the state and the observation. The problem is that the likelihood functional cannot be obtained when the observation noise covariance is unknown. A related procedure is suggested, however, and the estimates are obtained by finding roots of an appropriate functional. It is shown that the estimates obtained are consistent.
Bagchi, A. (1975). Continuous time systems identification with unknown noise covariance = Systèmes d'identification de temps continu avec co-variation de bruit (ou bruits) inconnus = Identifikation zeitkontinuierlicher systeme mit unbekannter rauschkovarianz. Automatica, 11(5), 533-536. https://doi.org/10.1016/0005-1098(75)90030-8