This paper addresses sensor fault detection and isolation problems for continuous- and discrete-time linear time-invariant systems. To that end, we employ a bank consisting of the same number of observers as there are sensors. Both the observer gain and the residual gain are considered. Unlike earlier work, the design conditions with $H_-/H_{infty}$ performance are derived in terms of linear, rather than nonlinear, matrix inequalities. An illustrative example is provided to show the effectiveness of the proposed methodology.