Exact eigen value assignment of linear scalar systems with single delay using Lambert W function

Eigen value assignment problem of a linear scalar system with single discrete delay is analytically and completely solved. The existence condition on areal or complex desired eigenvalue is established when the current and delay states are included in the feedback. Design of the feedback controller is then followed by calculating the corresponding real feedback gains such that the rightmost eigenvalue of the closed loop system is exactly located at the desired value. The eigenvalue assignment for the input-delay system is also obtained as well. Numerical examples illustrate the procedure in assigning the desired eigenvalue.