The suggested methods for solving fault diagnosis and estimation problems are based on the use of the Jordan canonical form. The diagnostic observer, virtual sensor, interval, and sliding mode observer design problems are considered. Algorithms have been developed to solve these problems for both linear and nonlinear systems, considering the presence of external disturbances and measurement noise. It has been shown that the Jordan canonical form allows reducing the dimensions of interval observers and virtual sensors, thus simplifying the design process in comparison to the identification canonical form. The theoretical results are illustrated through examples.