In this paper, algorithms are proposed to design auxiliary signals for active fault detection based on a multi-model formulation of discrete-time systems. Two different scenarios are considered for this problem; the first one assumes there is no a priori information on initial conditions and no exogenous input signal, while the second allows for having a priori information and the possibility of having a known input in addition to the test signal. Approaches are proposed for solving these two types of problems which are capable of solving the problems efficiently. This is achieved by using a recursive approach based on the use of special Riccati equations. These algorithms can be used for systems of higher dimension and on longer time horizons than the existing methods.