This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming, for computing the test input. The developed methodology offers a competent continuous-time solution to the optimization-based computation of the test input via Hahn Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems.