Controllability and Observability of Systems of Linear Delay Differential Equations via the Matrix Lambert W Function

Controllability and observability of linear time delayed systems have been studied, and various definitions and criteria have been presented since the 1960s. However, the lack of an analytical solution approach has limited the applicability of the existing theories. Recently, the solution to systems of linear delay differential equations has been derived using the matrix Lambert W function, in a form similar to the transition matrix in ordinary differential equations. The criteria for controllability and observability, and their Gramians, for systems of delay differential equations using the solution in terms of the matrix Lambert W function are presented for the first time and illustrated with examples.