Necessary conditions for transformation the nonlinear control system into the observer form via state and output coordinate changes

The paper gives more direct and simple necessary conditions for the existence of state and output coordinate transformations, allowing to transform the nonlinear single-input single-output control system into the observer form. Both the old and new conditions require that the certain n differential one-forms, associated with the nth order differential input-output equation (corresponding to the state equations), and depending on a unknown single-variable output function, are the total differentials of certain functions.