This paper develops a systematic approach to fault diagnostic system design for sensor health monitoring in Lipschitz non-linear systems. The methodology applies to non-linear systems with three or more sensors in which the state is observable through any one of the sensor measurements. Two major issues are addressed in the paper-observer design for the non-linear system to ensure directional growth of residues for failure identification and use of linear matrix inequalities for explicit design of the observer gain. The use of the methodology is demonstrated through an illustrative example.