In this work, we consider the state estimation problem for a class of non-autonomous Persidskii systems. This paper presents conditions on the existence and stability of a nonlinear observer based on the invariant manifold approach. The conditions are formulated using Linear Matrix Equalities (LME) and Inequalities (LMI). Two interesting applications of the result are presented: a reduced-order observer (e.g., an observer for unmeasured states) and regression, both in linear and nonlinear settings. An example to demonstrate the efficiency of results is provided.