Linear Time-Delayed Systems

#Control #delay Consider the system \(\dot{x} (t) = A x(t)\) We know from control theory that this system is stable, if the matrix $A$ is Hurwitz, i.e. all of its eigenvalues are in the left half of the complex plane. Now, consider the system below: \(\dot{x} = A x(t) + A_d x(t - \tau)\) Studying this system is not as straightforward. This system has infinitely many eigenvalues, and its stability depends on $A$, $A_d$ and $\tau$. This is called a linear time-delayed system. There are several methods to handle these systems. One of the methods is using the Lambert W function.