When is a Matrix of Dimension Three Similar to a Metzler Matrix? Application to Interval Observer Design

A simple, necessary, and sufficient condition ensuring that a real matrix of dimension three is similar to a Metzler matrix is exhibited. When this condition is satisfied, a construction of the transfer matrix is given. This construction is used to design an interval observer for a family of continuous-time systems. An example is provided with interval observer design for the so-called love dynamics in the case of limit cycles