This paper gives the optimal solutions to several robust fault detection problems such as H-/H-infinity, H-2/H-infinity and H-infinity/H-infinity and problems for linear time-varying systems in a time domain, which extends the previous results on linear time-invariant systems in a frequency domain. It is shown that all three problems have the same optimal detection filter and the filter is a simple observer obtained by solving a standard differential Riccati equation. Finally, an example is given to illustrate our results. (C) 2008 Elsevier Ltd. All rights reserved.